Solitons in air. Solitons

Doctor of Technical Sciences A. GOLUBEV.

A person, even without a special physical or technical education, is undoubtedly familiar with the words "electron, proton, neutron, photon". But the word "soliton", consonant with them, is probably the first time that many hear. This is not surprising: although what is designated by this word has been known for more than a century and a half, proper attention to solitons began to be paid only from the last third of the twentieth century. Soliton phenomena turned out to be universal and were found in mathematics, hydromechanics, acoustics, radiophysics, astrophysics, biology, oceanography, and optical technology. What is this - a soliton?

Painting by IK Aivazovsky "The Ninth Wave". Waves on water propagate like group solitons, in the middle of which, in the interval from the seventh to the tenth, the highest wave goes.

An ordinary linear wave has the shape of a regular sinusoid (a).

Science and Life // Illustrations

Science and Life // Illustrations

Science and Life // Illustrations

This is the behavior of a nonlinear wave on the water surface in the absence of dispersion.

This is what a group soliton looks like.

Shock wave in front of a ball flying six times faster than sound. By ear, it is perceived as a loud bang.

In all of the above areas, there is one common feature: in them or in their individual sections, wave processes are studied, or, more simply, waves. In the most general sense, a wave is the propagation of a disturbance of some physical quantity characterizing a substance or field. This spread usually occurs in some kind of environment - water, air, solids. And only electromagnetic waves can propagate in a vacuum. Everyone, undoubtedly, saw how spherical waves radiated from a stone thrown into the water, "disturbing" the calm surface of the water. This is an example of a "lonely" outrage spreading. Very often, disturbance is an oscillatory process (in particular, periodic) in various forms - swinging of a pendulum, vibrations of a string of a musical instrument, compression and expansion of a quartz plate under the action of alternating current, vibrations in atoms and molecules. Waves - propagating vibrations - can be of a different nature: waves on water, sound, electromagnetic (including light) waves. The difference in physical mechanisms that implement the wave process entails different ways of its mathematical description. But waves of different origins also have some general properties, for the description of which a universal mathematical apparatus is used. This means that you can study wave phenomena, distracting from their physical nature.

In wave theory, this is usually done by considering such properties of waves as interference, diffraction, dispersion, scattering, reflection and refraction. But at the same time, one important circumstance takes place: such a unified approach is legitimate provided that the studied wave processes of various natures are linear. We will talk about what is meant by this a little later, but now we will only note that only waves with too large an amplitude. If the amplitude of the wave is large, it becomes nonlinear, and this is directly related to the topic of our article - solitons.

Since we talk about waves all the time, it’s easy to guess that solitons are also something from the wave region. This is really so: a very unusual formation is called a soliton - a "solitary wave". The mechanism of its occurrence for a long time remained a mystery to researchers; it seemed that the nature of this phenomenon contradicted the well-known laws of the formation and propagation of waves. Clarity appeared relatively recently, and now they study solitons in crystals, magnetic materials, optical fibers, in the atmosphere of the Earth and other planets, in galaxies and even in living organisms. It turned out that tsunamis, nerve impulses, and dislocations in crystals (violations of the periodicity of their lattices) are all solitons! Soliton truly has many faces. By the way, this is the name of A. Filippov's excellent popular science book "The Many-Faced Soliton". We recommend it to the reader who is not afraid of rather a large number mathematical formulas.

In order to understand the basic ideas related to solitons, and at the same time to do practically without mathematics, it is necessary to talk first of all about the already mentioned nonlinearity and about dispersion - the phenomena that underlie the mechanism of formation of solitons. But first, let's talk about how and when the soliton was discovered. He first appeared to man in the "guise" of a solitary wave on the water.

This happened in 1834. John Scott Russell, a Scottish physicist and talented engineer-inventor, was asked to investigate the possibilities of navigating steam ships along the canal linking Edinburgh and Glasgow. At that time, transportation along the canal was carried out using small barges pulled by horses. To figure out how to convert barges when replacing horse traction with steam, Russell began observing barges of various shapes, moving at different speeds. And in the course of these experiments, he unexpectedly encountered a completely unusual phenomenon... This is how he described it in his "Report on the Waves":

“I was following the movement of the barge, which was rapidly being pulled along a narrow channel by a pair of horses when the barge suddenly stopped. speed and taking the form of a large single eminence - a rounded, smooth and well-defined mound of water. It continued its way along the channel, without changing its shape or decreasing speed. I followed him on horseback, and when I caught up with him, he continued to roll forward at about 8-9 miles per hour, retaining its original elevation profile of about thirty feet long and a foot to one and a half feet high. Its height gradually decreased, and after one or two miles of chase I lost it in the bends of the channel. "

Russell called the phenomenon he discovered "the solitary wave of transmission." However, his message was met with skepticism by recognized authorities in the field of hydrodynamics - George Airy and George Stokes, who believed that waves when moving on long distances cannot keep their shape. They had every reason for this: they proceeded from the equations of hydrodynamics generally accepted at that time. The recognition of the "solitary" wave (which was named a soliton much later - in 1965) happened during Russell's lifetime by the works of several mathematicians who showed that it could exist, and, in addition, Russell's experiments were repeated and confirmed. But the disputes around the soliton did not stop for a long time - the authority of Airy and Stokes was too great.

The Dutch scientist Diederik Johannes Korteweg and his student Gustav de Vries brought the final clarity to the problem. In 1895, thirteen years after Russell's death, they found the exact equation, the wave solutions of which completely describe the processes taking place. As a first approximation, this can be explained as follows. Korteweg-de Vries waves have a non-sinusoidal shape and become sinusoidal only when their amplitude is very small. With an increase in the wavelength, they take the form of humps far apart from each other, and at a very long wavelength, one hump remains, which corresponds to a "solitary" wave.

The Korteweg - de Vries equation (the so-called KdV equation) has played a very important role in our days, when physicists understood its universality and the possibility of applying it to waves of different nature. The great thing is that it describes nonlinear waves, and now we should dwell on this concept in more detail.

In wave theory, the wave equation is of fundamental importance. Without giving it here (this requires familiarity with higher mathematics), we only note that the sought-for function describing the wave and the quantities associated with it are contained in the first power. Such equations are called linear. The wave equation, like any other, has a solution, that is, a mathematical expression that, when substituted, turns into identity. A linear harmonic (sinusoidal) wave serves as a solution to the wave equation. Let us emphasize once again that the term "linear" is used here not in the geometric sense (sinusoid is not a straight line), but in the sense of using the first power of quantities in the wave equation.

Linear waves obey the principle of superposition (addition). This means that when multiple linear waves are superimposed, the resulting waveform is determined by simply adding the original waves. This happens because each wave propagates in the environment independently of the others, there is no energy exchange or other interaction between them, they freely pass one through the other. In other words, the principle of superposition means that the waves are independent, and that is why they can be added. Under normal conditions, this is true for sound, light and radio waves, as well as for waves that are considered in quantum theory. But for waves in a liquid, this is not always true: only waves of very small amplitude can be added. If we try to add the Korteweg - de Vries waves, then we will not get a wave that can exist at all: the equations of hydrodynamics are nonlinear.

It is important to emphasize here that the linearity property of acoustic and electromagnetic waves is observed, as already noted, under normal conditions, which mean, first of all, small wave amplitudes. But what does "small amplitudes" mean? The amplitude of sound waves determines the volume of sound, light waves determine the intensity of light, and radio waves determine the intensity of the electromagnetic field. Broadcasting, television, telephony, computers, lighting fixtures, and many other devices operate under the same "normal conditions", dealing with a variety of small amplitude waves. If the amplitude increases sharply, the waves lose their linearity and then new phenomena appear. In acoustics, shock waves have been known for a long time, propagating at a supersonic speed. Examples of shock waves are thunderclaps during a thunderstorm, the sounds of gunshots and explosions, and even the flapping of a whip: its tip moves faster than sound. Non-linear light waves are produced using high-power pulsed lasers. The passage of such waves through various media changes the properties of the media themselves; completely new phenomena are observed that constitute the subject of the study of nonlinear optics. For example, a light wave arises, the length of which is two times less, and the frequency, respectively, is twice that of the incoming light (the second harmonic is generated). If, say, a powerful laser beam with a wavelength of l 1 = 1.06 microns (infrared radiation invisible to the eye) is directed to a nonlinear crystal, then, in addition to infrared, green light with a wavelength of l 2 = 0.53 microns appears at the output of the crystal.

If nonlinear sound and light waves are formed only under special conditions, then hydrodynamics is nonlinear by its very nature. And since hydrodynamics shows nonlinearity even in the simplest phenomena, for almost a century it has been developing in complete isolation from "linear" physics. It simply never occurred to anyone to look for something similar to the "solitary" Russell wave in other wave phenomena. And only when new areas of physics were developed - nonlinear acoustics, radiophysics and optics - did the researchers remember the Russell soliton and asked the question: can such a phenomenon be observed only in water? To do this, it was necessary to understand the general mechanism of the formation of a soliton. The condition of nonlinearity turned out to be necessary, but insufficient: something else was required from the medium so that a "solitary" wave could be born in it. And as a result of research, it became clear that the missing condition was the presence of dispersion of the medium.

Let us briefly recall what it is. Dispersion is the dependence of the velocity of propagation of the wave phase (the so-called phase velocity) on the frequency or, which is the same, the wavelength (see "Science and Life" No.). According to the well-known Fourier theorem, a non-sinusoidal wave of any shape can be represented by a set of simple sinusoidal components with different frequencies (wavelengths), amplitudes and initial phases. These components, due to dispersion, propagate at different phase velocities, which leads to the "smearing" of the waveform during its propagation. But a soliton, which can also be represented as the sum of these components, as we already know, retains its shape during motion. Why? Recall that a soliton is a nonlinear wave. And here lies the key to the disclosure of his "secret". It turns out that a soliton arises when the nonlinearity effect, which makes the "hump" of the soliton steeper and tends to overturn it, is balanced by dispersion, which makes it flatter and tends to blur it. That is, a soliton appears "at the junction" of nonlinearity and dispersion, which cancel each other out.

Let us explain this with an example. Suppose that a hump has formed on the surface of the water, which begins to move. Let's see what happens if we don't take into account the variance. The speed of a nonlinear wave depends on its amplitude (linear waves have no such dependence). The top of the hump will move the fastest, and at some next moment its leading edge will become steeper. The steepness of the front increases, and over time, the wave will "overturn". We see a similar overturning of waves, watching the surf on the seashore. Now let's see what variance leads to. The initial hump can be represented by the sum of sinusoidal components with different wavelengths. The long-wave components travel at a higher speed than the short-wave ones, and, therefore, reduce the steepness of the leading edge, to a large extent equalizing it (see Science and Life, No. 8, 1992). At a certain shape and speed of the hump, a complete restoration of the original shape can occur, and then a soliton is formed.

One of the amazing properties of "solitary" waves is that they are much like particles. So, in a collision, two solitons do not pass through each other like ordinary linear waves, but rather repel each other like tennis balls.

Solitons of another type can also appear on the water, called group solitons, since their shape is very similar to groups of waves, which in reality are observed instead of an infinite sinusoidal wave and move with group velocity. The group soliton closely resembles amplitude-modulated electromagnetic waves; its envelope is non-sinusoidal, it is described by a more complex function - hyperbolic secant. The velocity of such a soliton does not depend on the amplitude, and this is how it differs from KdV solitons. There are usually no more than 14-20 waves under the envelope. The middle - the highest - the wave in the group is thus in the range from the seventh to the tenth; hence the well-known expression "ninth wave".

The scope of this article does not allow considering many other types of solitons, for example, solitons in crystalline solids - the so-called dislocations (they resemble "holes" in crystal lattice and are also able to move), related magnetic solitons in ferromagnets (for example, in iron), soliton-like nerve impulses in living organisms and many others. Let us restrict ourselves to considering optical solitons, which have recently attracted the attention of physicists by the possibility of their use in very promising optical communication lines.

An optical soliton is a typical group soliton. Its formation can be understood by the example of one of the nonlinear optical effects - the so-called self-induced transparency. This effect consists in the fact that a medium that absorbs light of low intensity, that is, opaque, suddenly becomes transparent when a powerful light pulse passes through it. To understand why this happens, let us remember what causes the absorption of light in matter.

A quantum of light, interacting with an atom, gives it energy and transfers it to a higher energy level, that is, to an excited state. In this case, the photon disappears - the medium absorbs light. After all the atoms of the medium are excited, the absorption of light energy stops - the medium becomes transparent. But such a state cannot last long: the photons flying after them force the atoms to return to their original state, emitting quanta of the same frequency. This is exactly what happens when a short light pulse of high power of the corresponding frequency is sent through such a medium. The leading edge of the pulse throws atoms to the upper level, being partially absorbed and becoming weaker. The maximum of the pulse is absorbed less, and the trailing edge of the pulse stimulates the reverse transition from the excited level to the main one. The atom emits a photon, its energy is returned to the impulse, which passes through the medium. In this case, the shape of the pulse turns out to correspond to the group soliton.

More recently, a publication appeared in one of the American scientific journals about the development of the transmission of signals over ultra-long distances through optical fibers using optical solitons by the well-known Bell Laboratories (USA, New Jersey). In conventional transmission over fiber-optic communication lines, the signal must be amplified every 80-100 kilometers (the fiber itself can serve as an amplifier when it is pumped with light of a certain wavelength). And every 500-600 kilometers it is necessary to install a repeater that converts an optical signal into an electrical one while maintaining all its parameters, and then back into an optical signal for further transmission. Without these measures, the signal at a distance of more than 500 kilometers is distorted beyond recognition. The cost of this equipment is very high: the transfer of one terabit (10 12 bits) of information from San Francisco to New York costs $ 200 million for each relay station.

The use of optical solitons, which retain their shape during propagation, makes it possible to carry out fully optical signal transmission over distances of up to 5-6 thousand kilometers. However, on the way to creating a "soliton line" there are significant difficulties that have been overcome only very recently.

The possibility of the existence of solitons in an optical fiber was predicted in 1972 by theoretical physicist Akira Hasegawa, a Bell employee. But at that time there were still no low-loss optical fibers in the wavelength regions where solitons can be observed.

Optical solitons can propagate only in a fiber with a small but finite dispersion value. However, an optical fiber that maintains the desired dispersion value over the entire spectral width of a multichannel transmitter simply does not exist. This makes "ordinary" solitons unsuitable for use in networks with long transmission lines.

A suitable soliton technology has been developed over a number of years under the leadership of Lynn Mollenauer, a leading specialist in the Optical Technology Department of the same Bell company. This technology is based on the development of optical fibers with controlled dispersion, which made it possible to create solitons, the shape of the pulses of which can be maintained indefinitely.

The control method is as follows. The amount of dispersion along the length of the optical fiber periodically changes between negative and positive values. In the first section of the fiber, the pulse expands and shifts in one direction. In the second section, which has a dispersion of the opposite sign, the pulse is compressed and shifted in the opposite direction, as a result of which its shape is restored. With further movement, the impulse expands again, then enters the next zone, which compensates for the action of the previous zone, and so on - a cyclic process of expansion and contraction occurs. The pulse undergoes ripple in width with a period equal to the distance between the optical amplifiers of a conventional fiber - from 80 to 100 kilometers. As a result, according to Mollenauer, a signal with an information volume of more than 1 terabit can pass without retransmission at least 5-6 thousand kilometers at a transmission rate of 10 gigabits per second per channel without any distortion. Such a technology for ultra-long-distance communication via optical lines is already close to the stage of implementation.

One of the most amazing and beautiful wave phenomena is the formation of solitary waves, or solitons, propagating in the form of impulses of constant shape and in many ways similar to particles. Soliton phenomena include, for example, tsunami waves, nerve impulses, etc.
In the new edition (1st ed. - 1985), the material of the book has been significantly revised taking into account the latest achievements.
For high school students, students, teachers.

Preface to the first edition 5
Preface to the second edition 6
Introduction 7

Part I. THE HISTORY OF SOLITON 16
Chapter 1.150 years ago 17
The beginning of the theory of waves (22). The Weber brothers study waves (24). On the benefits of wave theory (25). About the main events of the era (28). Science and Society (34).
Chapter 2. John Scott Russell's Great Solitary Wave 37
Until the fatal meeting (38). Meeting with a solitary wave (40). It can't be! (42). And yet it exists! (44). Solitary wave rehabilitation (46). Solitary wave isolation (49). Wave or particle? (50).
Chapter 3. Relatives of the soliton 54
Hermann Helmholtz and the nerve impulse (55). The further fate of the nerve impulse (58). Hermann Helmholtz and vortices (60). Kelvin's "vortex atoms" (68). Lord Ross and the vortices in space (69). Linearity and nonlinearity (71).

Part II. NONLINEAR OSCILLATIONS AND WAVES 76 Chapter 4. Portrait of a pendulum 77
Pendulum equation (77). Small oscillations of the pendulum (79). Galileo's pendulum (80). Similarity in dimensions (82). Conservation of energy (86). The language of phase diagrams (90). Phase portrait (97). Phase portrait of a pendulum (99). "Soliton" solution of the pendulum equation (103). Pendulum motions and "hand" soliton (104). Concluding remarks (107).
Waves in a chain of bound particles (114). Retreat into history. Bernoulli family and waves (123). D'Alembert waves and spores around them (125). Discrete and continuous (129). How the speed of sound was measured (132). Dispersion of waves in a chain of atoms 136 How to "hear" the Fourier expansion? (138). A few words about light dispersion (140). Dispersion of water waves 142 With what speed the flock of waves runs (146). How much energy is in the wave (150).

Part III. PRESENT AND FUTURE SOL ITONS 155
What is theoretical physics (155). The ideas of Ya.I. Frenkel '(158). Atomic model of a moving dislocation according to Frenkel and Kontorova (160). Interaction of dislocations 164 A "living" soliton atom 167. Dialogue between the reader and the author (168). Dislocations and pendulums (173). What have become sound waves(178). How to see the locations? (182). Table solitons (185). Other close relatives of dislocations in the mathematical line (186). Magnetic solitons 191
Can a person "be friends" with a computer (198). Many-sided chaos (202). Computer surprises Enrico Fermi (209) Return of Russell's soliton (215). Oceanic solitons: tsunami, "ninth wave" (227). Three solitons (232). Soliton telegraph (236). A nervous impulse is an "elementary particle" of thought (241). The ubiquitous whirlwinds (246). Josephson effect 255. Solitons in long Josephson junctions 260 Elementary particles and solitons 263 Unified theories and strings 267
Chapter 6. Frenkel's Solitons 155
Chapter 7. The second birth of a soliton 195
Applications
Short name index

Many have probably come across the word "soliton", which is consonant with words such as an electron or a proton. This book is dedicated to the scientific idea behind this easy-to-remember word, its history and creators.
It is designed for the widest circle of readers who have mastered the school course in physics and mathematics and are interested in science, its history and applications. Far from everything is told about solitons. But most of what remained after all the restrictions, I tried to set out in sufficient detail. At the same time, some well-known things (for example, about oscillations and waves) had to be presented a little differently than it was done in other popular science and quite scientific books and articles, which I, of course, widely used. It is absolutely impossible to list their authors and mention all the scientists whose conversations with whom have influenced the content of this book, and I apologize to them, along with deep gratitude.
I would especially like to thank S.P. Novikov for constructive criticism and support, L.G. Aslamazov and Ya.A. Smorodinsky for valuable advice, and also Yu.S. Gal'pern and S.R. Filonovich, who carefully read the manuscript and made many comments that contributed to its improvement.
This book was written in 1984, and while preparing a new edition, the author naturally wanted to talk about new interesting ideas that have been born recently. The main additions relate to optical and Josephson solitons, the observation and application of which have recently been the subject of very interesting work. The section on chaos has been expanded somewhat, and on the advice of the late Yakov Borisovich Zeldovich, more details are given about shock waves and detonation. At the end of the book, an essay on modern unified theories of particles and their interactions is added.It also made an attempt to give some idea of ​​relativistic strings - a new and rather mysterious physical object, with the study of which hopes are pinned on the creation of a unified theory of all interactions known to us. Added a small math appendix and a short index.
A lot of minor changes have also been made to the book - some have been thrown away, and some have been added. It is hardly worth describing this in detail. The author tried to greatly expand everything related to computers, but this idea had to be abandoned, it would be better to devote a separate book to this topic. I hope that the adventurous reader, armed with some kind of computer, will be able to come up with and carry out his own computer experiments based on the material of this book.
In conclusion, I am pleased to express my gratitude to all the readers of the first edition, who provided their comments and suggestions on the content and form of the book. To the best of my ability, I tried to take them into account.
Nowhere is the unity of nature and the universality of its laws manifested so clearly as in oscillatory and wave phenomena. Each student can easily answer the question: "What is common between a swing, a clock, a heart, an electric bell, a chandelier, a TV, a saxophone and an ocean liner?" - and will easily continue this list. The common thing, of course, is that oscillations exist or can be excited in all these systems.
Some of them we see with the naked eye, others we observe with the help of instruments. Some vibrations are very simple, such as swing vibrations, others are much more complicated - just look at the electrocardiogram or encephalogram, but we can always easily distinguish the oscillatory process by its characteristic repeatability, periodicity.
We know that oscillation is a periodic movement or change of state, and it does not matter what moves or changes state. The science of vibrations studies what is common in vibrations of the most varied nature.
In the same way, one can compare waves of completely different nature - ripples on the surface of a puddle, radio waves, a "green wave" of traffic lights on a highway - and many, many others. Wave science studies waves on their own, distracting from their physical nature. A wave is considered as a process of transfer of excitation (in particular, oscillatory motion) from one point of the medium to another. In this case, the nature of the environment and the specific nature of its excitations are insignificant. Therefore, it is natural that vibrational and sound waves and the connections between them are studied today by a single science - theory
vibrations and waves. The general nature of these connections is well known. The clock "ticks", the bell rings, the swing sways and creaks, emitting sound waves; a wave propagates through the blood vessels, which we observe by measuring the pulse; electromagnetic oscillations excited in the oscillatory circuit are amplified and carried away into space in the form of radio waves; "Vibrations" of electrons in atoms give rise to light, etc.
When a simple periodic wave of small amplitude propagates, the particles of the medium perform periodic movements... With a slight increase in the amplitude of the wave, the amplitude of these movements also increases proportionally. If, however, the amplitude of the wave becomes large enough, new phenomena may arise. For example, waves on water at high altitudes become steep, breakers form on them, and they eventually overturn. In this case, the nature of the motion of the wave particles completely changes. Particles of water in the crest of a wave begin to move completely randomly, that is, regular, oscillatory motion turns into irregular, chaotic. This is the most extreme degree of non-linearity of water waves. A weaker manifestation of nonlinearity is the dependence of the waveform on its amplitude.
To explain what nonlinearity is, one must first explain what linearity is. If the waves have a very small height (amplitude), then with an increase in their amplitude, say, two times, they remain exactly the same, their shape and propagation speed do not change. If one such wave runs over another, then the resulting more complex motion can be described by simply adding the heights of both waves at each point. This simple property of linear waves is the basis for the well-known explanation of the phenomenon of wave interference.
Waves with a sufficiently small amplitude are always linear. However, with an increase in amplitude, their shape and speed begin to depend on the amplitude, and they can no longer be simply added, the waves become nonlinear. At large amplitudes, nonlinearity generates breakers and leads to breaking waves.
Waveforms can be distorted not only due to nonlinearity. It is well known that waves of different lengths propagate, generally speaking, with different speeds. This phenomenon is called variance. Observing the waves scattering in circles from a stone thrown into the water, it is easy to see that long waves on the water travel faster than short ones. If a small elevation has formed on the surface of the water in a long and narrow groove (it is easy to make it with the help of partitions that can be quickly removed), then, due to dispersion, it will quickly disintegrate into separate waves of different lengths, scatter and disappear.
It is remarkable that some of these water mounds do not disappear, but live long enough, keeping their shape. It is not at all easy to see the birth of such unusual "solitary" waves, but nevertheless, 150 years ago, they were discovered and studied in experiments, the idea of ​​which was just described. The nature of this amazing phenomenon remained mysterious for a long time. It seemed to contradict the laws of wave formation and propagation, well established by science. Only many decades after the publication of reports on experiments with solitary waves, their riddle was partially solved. It turned out that they can be formed when the effects of nonlinearity, which make the mound more steep and tend to overturn it, and the effects of dispersion, which make it flatter and tend to blur it, are "balanced". Between the Scylla of nonlinearity and the Charybdis of dispersion, solitary waves are born, recently called solitons.
Already in our time, the most amazing properties of solitons were discovered, thanks to which they became the subject of fascinating scientific searches. They will be discussed in detail in this book. One of the great things about a solitary wave is that it looks like a particle. Two solitary waves can collide and scatter like billiard balls, and in some cases, you can imagine a soliton simply as a particle, the motion of which obeys Newton's laws. The most remarkable thing about a soliton is its versatility. Over the past 50 years, many solitary waves have been discovered and studied, similar to solitons on the surface of waves, but existing in completely different conditions.
Their common nature became clear relatively recently, in the last 20 - 25 years.
Now solitons are studied in crystals, magnetic materials, superconductors, in living organisms, in the atmosphere of the Earth and other planets, in galaxies. Apparently, solitons played an important role in the evolution of the Universe. Many physicists are now carried away by the idea that elementary particles (for example, a proton) can also be viewed as solitons. Modern theories of elementary particles predict various solitons that have not yet been observed, for example, solitons carrying a magnetic charge!
The use of solitons for storing and transmitting information is already beginning. The development of these ideas in the future can lead to revolutionary changes, for example, in communication technology. In general, if you have not heard about solitons yet, you will very soon. This book is one of the first attempts to tell about solitons in an accessible way. Of course, it is impossible to talk about all the solitons known today, and you shouldn't even try. Yes, this is not necessary.
Indeed, in order to understand what vibrations are, it is not at all necessary to get acquainted with the whole variety of vibrational phenomena found in nature and. technique. It is enough to understand the basic ideas of the science of vibrations using the simplest examples. For example, all small vibrations are similar to each other, and it is enough for us to understand how a weight on a spring or a pendulum in a wall clock vibrates. The simplicity of small fluctuations is associated with their linearity - the force that returns a weight or a pendulum to an equilibrium position is proportional to the deviation from this position. An important consequence of linearity is the independence of the oscillation frequency from their amplitude (swing).
If the linearity condition is violated, then the oscillations are much more varied. Nevertheless, some types of nonlinear oscillations can be distinguished, having studied which, one can understand the operation of a wide variety of systems - a clock, a heart, a saxophone, a generator of electromagnetic oscillations ...
The most important example of nonlinear oscillations is given by the movements of the same pendulum, if not limited to small amplitudes and arrange the pendulum so that it could not only swing, but also rotate. It is remarkable that, having understood well the pendulum, one can also understand the structure of the soliton! It is on this path that we, the reader, will try to understand what a soliton is.
Although this is the easiest road to the land where solitons live, many difficulties await us on it, and those who want to truly understand the soliton must be patient. First it is necessary to study the linear oscillations of the pendulum, then to understand the connection between these oscillations and linear waves, in particular to understand the nature of the dispersion of linear waves. It's not that hard. The relationship between nonlinear oscillations and nonlinear waves is much more complex and subtle. But still, we will try to describe it without complicated mathematics. We succeed in presenting only one type of solitons quite completely, while the rest will have to be dealt with by analogy.
Let the reader perceive this book as a journey to unfamiliar lands, in which he will get to know one city in detail, and stroll through the rest of the places, looking closely at everything new and trying to connect it with what has already been understood. You still need to get to know one city well enough, otherwise there is a risk of missing out on the most interesting things due to ignorance of the language, manners and customs of foreign lands.
So, on the Road, reader! Let this "collection of colorful chapters" be a guide to an even more colorful and diverse country, where vibrations, waves and solitons live. To make it easier to use this guide, you first need to say a few words about what it contains and what it does not.
Going to an unfamiliar country, it is natural to first get acquainted with its geography and history. In our case, this is almost the same thing, since the study of a given country is in fact just beginning, and we do not even know its exact boundaries.
The first part of the book presents the history of the solitary wave along with the basic concepts of it. Then it tells about things that, at first glance, are quite unlike a solitary wave on the surface of the water - about vortices and a nerve impulse. Their research also began in the last century, but the relationship with solitons was established quite recently.
The reader can truly understand this connection if he has the patience to get to the last chapter. By compensating for the expended efforts, he will be able to see a deep inner relationship of such dissimilar phenomena as tsunamis, forest fires, anticyclones, sunspots, hardening of metals during forging, magnetization of iron, etc.
But first, we will have to plunge into the past for a while, into the first half of XIX century, when ideas arose that were fully mastered only in our time. In this past, we will be primarily interested in the history of the theory of oscillations, waves and how, against this background, ideas arose, developed and were perceived, which later formed the foundation of the science of solitons. We will be interested in the fate of ideas, and not the fate of their creators. As Albert Einstein said, the history of physics is a drama, a drama of ideas. In this drama “... it is instructive to follow the changing destinies of scientific theories. They are more interesting than the changing fates of people, for each of them includes something immortal, at least a particle of eternal truth ”*).
*) These words belong to the Polish physicist Marian Smoluchowski, one of the founders of the theory Brownian motion... For the development of some basic physical ideas (such as wave, particle, field, relativity), the reader can follow the remarkable popular book by A. Einstein and T. Infeld "Evolution of Physics" (Moscow: GTTI, 1956).
Nevertheless, it would be wrong not to mention the creators of these ideas, and in this book a lot of attention is paid to people who first expressed certain valuable ideas, regardless of whether they became famous scientists or not. The author made special efforts to extract from oblivion the names of people who were insufficiently appreciated by their contemporaries and descendants, as well as to remind about some little-known works of quite famous scientists. (Here, for example, it tells about the life of several scientists, little known to a wide circle of readers and who expressed ideas that are in one way or another related to the soliton; about others, only brief data are given.)
This book is not a textbook, much less a textbook on the history of science. Perhaps not all of the historical information cited in it is presented absolutely accurately and objectively. The history of the theory of oscillations and waves, especially nonlinear ones, has not been sufficiently studied. The history of solitons has not yet been written at all. Perhaps the pieces of the mosaic of this story, collected by the author in different places, will be useful to someone for a more serious study. In the second part of the book, we will mainly focus on the physics and mathematics of nonlinear oscillations and waves in the form and volume in which it is necessary for a sufficiently deep acquaintance with the soliton.
There is comparatively a lot of mathematics in the second part. It is assumed that the reader has a fairly good understanding of what a derivative is and how the derivative is used to express velocity and acceleration. It is also necessary to remember some of the trigonometry formulas.
You can't do without mathematics, but in fact we will need a little more than Newton owned. Two hundred years ago, Jean Antoine Condorcet, a French philosopher, teacher and one of the reformers of school teaching, said: “At present, a young man, after leaving school, knows from mathematics more than that which Newton acquired through deep study or discovered with his genius; he knows how to master the tools of calculation with ease, then inaccessible. " We will add to what Condorcet envisioned for famous schoolchildren, a little of the achievements of Euler, the Bernoulli family, D'Alembert, Lagrange and Cauchy. This is quite enough for understanding modern physical concepts of a soliton. About modern mathematical theory solitons are not described - it is very complex.
We will nevertheless remind in this book of everything that is needed from mathematics, and, in addition, the reader who does not want or have no time to understand the formulas can simply skim through them, following only physical ideas. Things that are more difficult or take the reader away from the main road are highlighted in small print.
The second part to some extent gives an idea of ​​the doctrine of vibrations and waves, but many important and interesting ideas are not mentioned in it. On the contrary, what is needed to study solitons is described in detail. The reader who wants to get acquainted with the general theory of vibrations and waves should look in other books. Solitons are associated with so different
sciences that the author had in many cases to recommend other books for a more detailed acquaintance with some of the phenomena and ideas, which are said here too briefly. In particular, it is worth looking into other issues of the Kvant Library, which are often cited.
The third part describes in detail and consistently one type of solitons, which entered science 50 years ago, independently of the solitary wave on the vome and is associated with dislocations in crystals. The last chapter shows how in the end the fates of all solitons crossed and a general idea of ​​solitons and soliton-like objects was born. Computers played a special role in the birth of these general ideas. Computer calculations, which led to the second birth of a soliton, were the first example of a numerical experiment, when computers were used not just for calculations, but to discover new phenomena unknown to science. Numerical experiments on a computer undoubtedly have a great future, and they have been described in sufficient detail.
After that, we turn to a story about some of the modern concepts of solitons. Here the exposition gradually becomes more and more concise, and the last paragraphs of Ch. 7 give only a general idea of ​​the directions in which the science of solitons is developing. The purpose of this very short excursion is to give an idea of ​​the science of today and look a little into the future.
If the reader is able to grasp the inner logic and unity in the colorful picture presented to him, then the main goal set by the author will be achieved. The specific task of this book is to tell about the soliton and its history. The fate of this scientific idea seems unusual in many ways, but upon deeper reflection it turns out that many scientific ideas that today constitute our common wealth were born, developed and perceived with no less difficulty.
Hence, the broader task of this book arose - using the example of a soliton to try to show how science in general works, how it eventually, after many misunderstandings, delusions and errors, gets to the truth. The main goal of science is to obtain true and complete knowledge about the world, and it can benefit people only to the extent that it approaches this goal. The hardest part here is completeness. We ultimately establish the truth of a scientific theory by experimentation. However, no one can tell us how to come up with a new scientific idea, a new concept, with the help of which whole worlds of phenomena that were previously separated or even completely escaped our attention enter the sphere of harmonious scientific knowledge. One can imagine a world without solitons, but it will already be a different, poorer world. The soliton idea, like other great scientific ideas, is valuable not only because it is useful. It further enriches our perception of the world, revealing its inner beauty that eludes a superficial glance.
The author especially wanted to reveal to the reader this side of the scientist's work, which makes it akin to the work of a poet or composer, revealing to us the harmony and beauty of the world in areas more accessible to our senses. The work of a scientist requires not only knowledge, but also imagination, observation, courage and dedication. Maybe this book will help someone decide to follow the disinterested knights of science, whose ideas are told in it, or at least think and try to understand what made their thought work tirelessly, never satisfied with what has been achieved. The author would like to hope so, but, unfortunately, "we are not given to predict how our word will respond ..." What came out of the author's intention - to judge the reader.

THE HISTORY OF SOLITON

The science! you are a child of the Gray Times!
Changing everything with the attention of transparent eyes.
Why are you disturbing the poet's dream ...
Edgar Poe

The first officially recorded human encounter with a soliton took place 150 years ago, in August 1834, near Edinburgh. This meeting was, at first glance, accidental. A person did not prepare for it on purpose, and special qualities were required of him so that he could see the unusual in a phenomenon that others faced, but did not notice anything surprising in it. John Scott Russell (1808 - 1882) was fully endowed with just such qualities. He not only left us a scientifically accurate and vivid description of his meeting with a soliton *), not devoid of poetry *), but also devoted many years of his life to the study of this phenomenon that amazed his imagination.
*) He called it the translation wave or the great solitary wave. From the word solitary and the term "soliton" was later coined.
Russell's contemporaries did not share his enthusiasm, and the secluded wave did not become popular. From 1845 to 1965 no more than two dozen scientific works were published directly related to solitons. During this time, however, close relatives of the soliton were discovered and partially studied, however, the universality of soliton phenomena was not understood, and Russell's discovery was hardly remembered.
In the last twenty years began new life soliton, which turned out to be truly many-sided and omnipresent. Thousands of scientific papers on solitons in physics, mathematics, hydromechanics, astrophysics, meteorology, oceanography, and biology are published annually. Scientific conferences are being held specially devoted to solitons, books are being written about them, an increasing number of scientists are joining in the fascinating hunt for solitons. In short, the solitary wave has gone out of solitude into big life.
How and why this amazing turn in the fate of the soliton took place, which even Russell, who was in love with the soliton, could not have foreseen, the reader will know if he has the patience to read this book to the end. In the meantime, let's try to mentally travel back to 1834 in order to imagine the scientific atmosphere of that era. This will help us better understand the attitude of Russell's contemporaries to his ideas and the future fate of the soliton. Our excursion into the past will, of necessity, be very cursory, we will get acquainted mainly with those events and ideas that are directly or indirectly connected with the soliton.

Chapter 1
150 YEARS AGO

Nineteenth century, iron,
I have a cruel age ...
A. Block

Our poor age - how many attacks on him, what a monster he is considered! And all for the railways, for the steamers - these great victories of him, no longer over mothers only, but over space and time.
V. G. Belinsky

So, the first half of the last century, the time not only of Napoleonic wars, social shifts and revolutions, but also of scientific discoveries, the meaning of which was revealed gradually, decades later. Then few knew about these discoveries, and only a few could foresee their great role in the future of mankind. We now know about the fate of these discoveries and will not be able to fully assess the difficulties of their perception by contemporaries. But let's still try to strain our imagination and memory and try to break through the layers of time.
1834 ... There is still no telephone, radio, television, cars, airplanes, rockets, satellites, computers, nuclear power and much more. The first railway was built just five years ago, and the construction of steamships has just begun. The main type of energy used by people is the energy of heated steam.
However, ideas are already ripening that will ultimately lead to the creation of technical wonders of the 20th century. All this will take almost another hundred years. Meanwhile, science is still concentrated in universities. The time for narrow specialization has not yet come, and physics has not yet emerged as a separate science. Universities teach courses in "natural philosophy" (that is, natural science), the first physics institute will be created only in 1850. At that distant time, fundamental discoveries in physics can be made by very simple means, it is enough to have a brilliant imagination, observation and golden hands.
One of the most amazing discoveries of the last century was made with the help of a wire through which electricity, and a simple compass. This is not to say that this discovery was completely accidental. Russell's older contemporary, Hans Christian Oersted (1777 - 1851), was literally obsessed with the idea of ​​a connection between various phenomena nature, including between heat, sound, electricity, magnetism *). In 1820, during a lecture on the search for connections between magnetism, "galvanism" and electricity, Oersted noticed that when current was passed through a wire parallel to the compass needle, the needle was deflected. This observation aroused great interest in educated society, and in science gave rise to an avalanche of discoveries, begun by André Marie Ampere (1775 - 1836).
*) The close connection between electrical and magnetic phenomena was the first to notice at the end of the 18th century. Petersburg academician Franz Epinus.
In the famous series of works 1820 - 1825. Ampere laid the foundations for a unified theory of electricity and magnetism and called it electrodynamics. This was followed by the great discoveries of the brilliant self-taught Michael Faraday (1791 - 1867), made by him mainly in the 30s - 40s - from observation electromagnetic induction in 1831 before the formation of the concept of an electromagnetic field by 1852. Faraday also set up his experiments that amazed the imagination of his contemporaries, using the simplest means.
In 1853 Hermann Helmholtz, which will be discussed later, wrote: “I managed to get to know Faraday, indeed the first physicist of England and Europe ... He is simple, amiable and unassuming, like a child; I have never met such a disposed person ... He was always helpful, showed me everything that was worth seeing. But he had to inspect a little, since old pieces of wood, wire and iron serve him for his great discoveries. "
At this time, the electron is still unknown. Although Faraday suspected the existence of an elementary electric charge as early as 1834 in connection with the discovery of the laws of electrolysis, its existence became scientifically established only at the end of the century, and the term "electron" itself would be introduced only in 1891.
A complete mathematical theory of electromagnetism has not yet been developed. Its creator James Clark Maxwell was only three years old in 1834, and he grows up in the same city of Edinburgh, where the hero of our story lectures on natural philosophy. At this time, physics, which has not yet been divided into theoretical and experimental, is just beginning to mathematize. So, Faraday in his works did not even use elementary algebra. Although Maxwell would say later that he adheres not only to ideas, but also to the mathematical methods of Faraday, this statement can be understood only in the sense that Maxwell was able to translate Faraday's ideas into the language of contemporary mathematics. In A Treatise on Electricity and Magnetism, he wrote:
“Perhaps it was a happy circumstance for science that Faraday was not actually a mathematician, although he was perfectly familiar with the concepts of space, time and force. Therefore, he was not tempted to delve into interesting, but purely mathematical research, which his discoveries would require if they were presented in mathematical form ... Thus, he had the opportunity to go his own way and coordinate his ideas with the facts obtained, using natural, not technical language ... Having started to study the work of Faraday, I found that his method of understanding phenomena was also mathematical, although not represented in the form of ordinary mathematical symbols. I have also found that this method can be expressed in conventional mathematical form and thus be compared to the methods of professional mathematicians. "
If you ask me ... whether the present age will be called the Iron Age or the age of steam and electricity, I will answer without hesitation that our age will be called the age of the mechanical worldview ...
At the same time, the mechanics of systems of points and solids, as well as the mechanics of the motion of fluids (hydrodynamics), have already been substantially mathematized, that is, they have largely become mathematical sciences. The problems of the mechanics of systems of points were completely reduced to the theory of ordinary differential equations (Newton's equations - 1687, the more general Lagrange equations - 1788), and the problems of hydromechanics - to the theory of so-called partial differential equations (Euler's equations - 1755. , Navier equations - 1823). This does not mean that all tasks have been completed. On the contrary, in these sciences, deep and important discoveries, the flow of which does not dry up even today. It's just that mechanics and fluid mechanics have reached that level of maturity when the basic physical principles were clearly formulated and translated into the language of mathematics.
Naturally, these deeply developed sciences served as the basis for constructing theories of new physical phenomena. To understand a phenomenon for a scientist of the last century meant to explain it in the language of the laws of mechanics. Celestial mechanics was considered an example of the consistent construction of a scientific theory. The results of its development were summed up by Pierre Simon Laplace (1749 - 1827) in the monumental five-volume Treatise on Celestial Mechanics, published in the first quarter of a century. This work, which collected and summarized the achievements of the giants of the XVIII century. - Bernoulli, Euler, D'Alembert, Lagrange and Laplace himself had a profound influence on the formation of a "mechanical worldview" in the 19th century.
Note that in the same 1834 in a coherent picture classical mechanics Newton and Lagrange, a final stroke was added - the famous Irish mathematician William Rowan Hamilton (1805 - 1865) gave the equations of mechanics the so-called canonical form (according to S. I. Ozhegov's dictionary, "canonical" means "taken as a model, firmly established, corresponding to the canon") and discovered the analogy between optics and mechanics. The canonical equations of Hamilton were destined to play an outstanding role at the end of the century in the creation of statistical mechanics, and the optical-mechanical analogy, which established the connection between the propagation of waves and the motion of particles, was used in the 1920s by the creators of quantum theory. The ideas of Hamilton, who was the first to deeply analyze the concept of waves and particles and the relationship between them, played a significant role in the theory of solitons.
The development of mechanics and hydromechanics, as well as the theory of deformations of elastic bodies (the theory of elasticity), was spurred on by the needs of developing technology. J.C. Maxwell also worked extensively in the theory of elasticity, the theory of stability of motion with applications to the work of regulators, and structural mechanics. Moreover, while developing his electromagnetic theory, he constantly resorted to visual models: “... I retain the hope, while carefully studying the properties of elastic bodies and viscous liquids, to find a method that would make it possible to give a certain mechanical image for the electrical state as well ... ( Wed with work: William Thomson "On the mechanical representation of electrical, magnetic and galvanic forces", 1847) ".
Another famous Scottish physicist William Thomson (1824 - 1907), who later received the title of Lord Kelvin for his scientific merits, generally believed that all natural phenomena should be reduced to mechanical movements and explained in the language of the laws of mechanics. Thomson's views had a strong influence on Maxwell, especially in his younger years. It is surprising that Thomson, who knew and appreciated Maxwell, was one of the last to recognize his electromagnetic theory. This happened only after the famous experiments of Pyotr Nikolaevich Lebedev on measuring the light pressure (1899): "All my life I fought with Maxwell ... Lebedev forced me to surrender ..."

The beginning of wave theory
Although the basic equations describing the motion of a liquid, in the 30s of the XIX century. have already been obtained, the mathematical theory of water waves has just begun to be created. The simplest theory waves on the surface of the water was given by Newton in his " Mathematical principles natural philosophy ", first published in 1687 One hundred years later, the famous French mathematician Joseph Louis Lagrange (1736 - 1813) called this work" the greatest work of the human mind. " Unfortunately, this theory was based on the incorrect assumption that the water particles in a wave just oscillate up and down. Despite the fact that Newton did not give a correct description of water waves, he set the problem correctly, and his simple model gave rise to other studies. The correct approach to surface waves was first found by Lagrange. He understood how to construct a theory of waves on water in two simple cases - for waves with small amplitude ("shallow waves") and for waves in vessels, the depth of which is small compared to the wavelength ("shallow water"). Lagrange did not study detailed development of the theory of waves, since he was carried away by other, more general mathematical problems.
Are there many people who, while admiring the play of waves on the surface of a brook, think how to find equations that could be used to calculate the shape of any wave crest?
An exact and surprisingly simple solution of the equations describing
waves on the water. This is the first, and one of the few exact, solution to the equations of hydromechanics received in 1802 by a Czech scientist, professor of mathematics in
Prague Frantisek Joseph Gerstner (1756 - 1832) *).
*) Sometimes F. I. Gerstner is confused with his son, F. A. Gerstner, who lived in Russia for several years. Under his leadership in 1836 - 1837. the first railway in Russia was built (from St. Petersburg to Tsarskoe Selo).
In the Gerstner wave (Fig. 1.1), which can form only in "deep water", when the wavelength is much less than the depth of the vessel, liquid particles move in circles. The Gerstner wave is the first non-sinusoidal wave to be studied. From the fact that the particles of the LIQUID move in circles, we can conclude that the surface of the water has the shape of a cycloid. (from the Greek "kyklos" - a circle and "eidos" - a form), that is, a curve, which is described by some point of a wheel rolling on a flat road. Sometimes this curve is called a trochoid (from the Greek "trochos" - a wheel), and Gerstner's waves are called trochoidal *). Only for very small waves, when the height of the waves becomes much less than their length, the cycloid becomes similar to a sinusoid, and the Gerstner wave becomes sinusoidal. Although in this case the water particles deviate slightly from their equilibrium positions, they still move in circles, and do not swing up and down, as Newton believed. It should be noted that Newton was clearly aware of the erroneousness of this assumption, but considered it possible to use it for a rough approximate estimate of the wave propagation speed: in fact, it does not happen in a straight line, but rather in a circle, so I assert that time is given to these positions only approximately. " Here "time" is the period of oscillation T at each point; wave velocity v =% / T, where K is the wavelength. Newton showed that the speed of a wave on water is proportional to -y / K. In the future, we will see that this is the correct result, and we will find the coefficient of proportionality, which was known to Newton only approximately.
*) We will call curves described by points lying on the rim of the wheel cycloids, and curves described by points between the rim and the axle trochoids.
Gerstner's discovery did not go unnoticed. I must say that he himself continued to be interested in waves and used his theory for practical calculations of dams and dams. Soon, the beginning of the laboratory study of water waves was laid. This was done by the young brothers Weber.
The elder brother Erist Weber (1795 - 1878) later made important discoveries in anatomy and physiology, especially in the physiology of the nervous system. Wilhelm Weber (1804 - 1891) became a famous physicist and a long-term collaborator of K. Gauss's "control of mathematicians" in physics research. On the proposal and with the assistance of Gauss, he founded the world's first physics laboratory at the University of Göttingen (1831). Best known for his work on electricity and magnetism, as well as electromagnetic theory Weber, which was later supplanted by Maxwell's theory. He was one of the first (1846) to introduce the concept of individual particles of electrical matter - "electrical masses" and proposed the first model of the atom, in which the atom was likened to the planetary model Solar system... Weber also developed the basic idea of ​​Faraday's theory of elementary magnets in matter and invented several physical devices that were very perfect for their time.
Ernst, Wilhelm and their younger brother Eduard Weber became seriously interested in waves. They were real experimenters, and simple observations of the waves that can be seen "at every turn" could not satisfy them. Therefore, they made a simple device (Weber's tray), which, with various improvements, is still used today for experiments with water waves. Having built a long box with a glass side wall and simple devices for exciting waves, they made extensive observations of various waves, including Gerstner waves, whose theory they thus tested experimentally. They published the results of these observations in 1825 in a book entitled "The Teaching of Waves Based on Experiments." This was the first experimental study, in which waves of different shapes, the speed of their propagation, the relationship between wavelength and height, etc. were systematically studied. The observation methods were very simple, ingenious and quite effective. For example, to determine the shape of the wave surface, they dipped frosted glass into the bath
plate. When the wave reaches the middle of the plate, it is quickly pulled out; in this case, the front part of the wave is completely correctly imprinted on the plate. To observe the paths of the particles vibrating in the wave, they filled the tray with muddy water from the rivers. Zaale and observed the movements with the naked eye or with a weak microscope. In this way, they determined not only the shape, but also the dimensions of the particle trajectories. For example, they found that trajectories near the surface are close to circles, and when approaching the bottom, they flatten into ellipses; near the very bottom, the particles move horizontally. The Weber discovered many interesting properties of waves on water and other liquids.

The benefits of wave theory
No one is looking for his own, but each is the benefit of the other.
Apostle Paul
Independently of this, the development of Lagrange's ideas took place, associated mainly with the names of the French mathematicians Augustin Louis Cauchy (1789 - 1857) and Simon Denis Poisson (1781 - 1840). Our compatriot Mikhail Vasilyevich Ostrogradsky (1801 - 1862) also took part in this work. These famous scientists did a lot for science, their names bear numerous equations, theorems and formulas. Less well known are their works on the mathematical theory of small-amplitude waves on the surface of water. The theory of such waves can be applied to some storm waves at sea, to the movement of ships, to waves on shallows and near breakwaters, etc. The value of the mathematical theory of such waves for engineering practice is obvious. But at the same time, the mathematical methods developed for solving these practical problems were later applied to the solution of completely different problems, far from hydromechanics. We will come across similar examples of the "omnivorousness" of mathematics and the practical benefits of solving mathematical problems that at first glance are related to "pure" ("useless") mathematics.
Here it is difficult for the author to resist a small digression on one episode associated with the emergence of a single
the development of Ostrogradsky's work on the theory of will. This mathematical work not only brought distant benefits to science and technology, but also had a direct and important impact on the fate of its author, which does not happen very often. This is how the outstanding Russian shipbuilder, mathematician and engineer, academician Alexei Nikolaevich Krylov (1863 - 1945) describes this episode. “In 1815, the Paris Academy of Spider made the theory of will the theme for the Grand Prize in Mathematics. Cauchy and Poisson took part in the competition. An extensive (about 300 pages) memoir by Cauchy was awarded, Poisson's memoir earned an honorable mention ... At the same time (1822) M.V. im imprisoned in Clichy (a debt prison in Paris). Here he wrote "The Theory of the Will in a Cylindrical Vessel" and sent his memoir to Cauchy, who not only approved this work and presented it to the Paris Academy spider for printing in her works, but also, being rich, bought Ostrogradsky from a debt prison and recommended him for the position of a teacher of mathematics in one of the lyceums in Paris. A number of mathematical works by Ostrogradskiy drew the attention of the St. Petersburg Academy of Sciences, and in 1828 he was elected to its associate, and then to ordinary academicians, having only a certificate of a student of Kharkov University, who had been dismissed, and had not finished the course ”.
We add to this that Ostrogradsky was born into a poor family of Ukrainian nobles, at the age of 16 he entered the physics and mathematics faculty of Kharkov University at the behest of his father, against his own wishes (he wanted to become a military man), but very soon his outstanding abilities in mathematics showed up. In 1820, he passed the examinations for a candidate with honors, but the Minister of Public Education and Spiritual Affairs, Kiyaz A. N. Golitsyn, did not only refuse to award him a candidate's degree, but also deprived him of the previously issued university diploma. The basis was the accusation of "atheism and freethinking", that he "did not visit not only
lectures on philosophy, on the knowledge of God and Christian doctrine. " As a result, Ostrogradsky left for Paris, where he diligently attended lectures by Laplace, Cauchy, Poisson, Fourier, Ampere and other prominent scientists. Subsequently, Ostrogradsky became a correspondent member of the Paris Academy of Sciences, a member of the Turin,
Roman and American Academies, etc. In 1828 Ostrogradsky returned to Russia, to St. Petersburg, where, on the personal order of Nicholas I, he was taken under secret police surveillance *). This circumstance did not prevent, however, the career of Ostrogradsky, who gradually took a very high position.
The work on waves, mentioned by A. N. Krylov, was published in the proceedings of the Paris Academy of Sciences in 1826. It is devoted to waves of small amplitude, that is, the problem on which Cauchy and Poissoy worked. Ostrogradsky never returned to studying waves. In addition to purely mathematical works, his research on Hamiltonian mechanics is known, one of the first works on the study of the influence of the nonlinear force of the treium on the movement of projectiles in the air (this problem was posed by
*) Emperor Nicholas I generally regarded scientists with distrust, considering all of them, not without reason, to be free-thinkers.
Euler). Ostrogradskiy was one of the first to recognize the need to study nonlinear oscillations and found an ingenious way to approximately account for small nonlinearities in the oscillations of a pendulum (Poisson's problem). Unfortunately, he did not complete many of his scientific endeavors - he had to devote too much effort to pedagogical work, paving the way for new generations of scientists. For this alone, we should be grateful to him, as well as to other Russian scientists of the beginning of the last century, who worked hard to create the foundation for the future development of science in our country.
Let's return, however, to our conversation about the benefits of waves. A remarkable example of the application of the ideas of the theory of waves to a completely different range of phenomena can be cited. We are talking about Faraday's hypothesis about the wave nature of the process of propagation of electrical and magnetic interactions.
Faraday became a famous scientist during his lifetime; many studies and popular books have been written about him and his works. However, few people even today know that Faraday was seriously interested in water waves. Not possessing the mathematical methods known to Cauchy, Poisson and Ostrogradsky, he very clearly and deeply understood the basic ideas of the theory of water waves. Thinking about the propagation of electric and magnetic fields in space, he tried to imagine this process by analogy with the propagation of waves on water. This analogy, apparently, led him to the hypothesis about the finiteness of the speed of propagation of electrical and magnetic interactions and about the wave nature of this process. On March 12, 1832, he recorded these thoughts in a special letter: "New views currently to be kept in a sealed envelope in the archives of the Royal Society." The thoughts expressed in the letter were far ahead of their time; in fact, the idea of ​​electromagnetic waves was formulated here for the first time. This letter was buried in the archives of the Royal Society, it was discovered only in 1938. Evidently, and Faraday himself forgot about it (he gradually developed a serious illness associated with memory loss). He outlined the main ideas of the letter later in the work of 1846.
Of course, today it is impossible to accurately reconstruct the train of thought of Faraday. But his reflections and experiments on water waves shortly before composing this remarkable letter are reflected in the work he published in 1831. It is devoted to the study of small ripples on the surface of the water, ie, the so-called "capillary" waves *) (more about them will be discussed in Chapter 5). To study them, he came up with an ingenious and, as always, very simple device. Subsequently, the Faraday method was used by Russell, who observed other subtle, but beautiful and interesting phenomena with capillary waves. The experiments of Faraday and Russell are described in § 354 - 356 of Rayleigh's book (John William Stratt, 1842 - 1919) "The Theory of Sound", which was first published in 1877, but is still not outdated and can give great pleasure to the reader (there is a Russian translation). Rayleigh not only did a lot for the theory of oscillations and waves, but was one of the first to recognize and appreciate the solitary wave.

About the main events of the era
The improvement of science should be expected not from the ability or agility of any individual person, but from the consistent activity of many generations replacing each other.
F. Bacon
In the meantime, it is time for us to end a somewhat protracted historical excursion, although the picture of science at that time turned out, perhaps, too one-sided. In order to somehow correct this, let us very briefly recall the events of those years that historians of science rightly consider the most important. As already mentioned, all the basic laws and equations of mechanics were formulated in 1834 in the same form in which we use them today. By the middle of the century, the basic equations describing the motion of fluids and elastic bodies (hydrodynamics and the theory of elasticity) were written and began to be studied in detail. As we have seen, waves in liquids and in elastic bodies were of interest to many scientists. Physicists, however, were much more attracted by light waves at this time.
*) These waves are associated with the forces of surface tension of water. The same forces cause the rise of water in the thinnest, hair-thick tubes (the Latin word capillus means hair).
In the first quarter of a century, mainly thanks to the talent and energy of Thomas Jung (1773 - 1829), Augustin Jean Fresnel (1788 - 1827) and Dominique François Arago (1786 - 1853), the wave theory of light won out. The victory was not easy, for among the numerous opponents of the wave theory were such prominent scientists as Laplace and Poisson. The critical experiment that finally confirmed the wave theory was made by Arago at a meeting of the commission of the Paris Academy of Sciences, which discussed Fresnel's work on light diffraction submitted to the competition. In the report of the commission, this is described as follows: “One of the members of our commission, Monsieur Poisson, deduced from the integrals reported by the author that an amazing result that the center of the shadow from a large opaque screen should be the same illuminated as if the screen were not existed ... This consequence was verified by direct experience and observation fully confirmed these calculations. "
This happened in 1819, and the following year, the already mentioned discovery of Oersted caused a sensation. Orsted's publication of his work "Experiments Relating to the Action of an Electric Conflict on a Magnetic Needle" gave rise to an avalanche of experiments on electromagnetism. It is generally recognized that the greatest contribution to this work was made by Ampere. Oersted's work was published in Copenhagen at the end of July, in early September Arago announces this discovery in Paris, and in October the well-known Bio-Savard-Laplace law appears. Since the end of September, Ampere has been speaking almost weekly (!) With reports of new results. The results of this pre-Faraday era in electromagnetism are summed up in Ampere's book "A theory of electrodynamic phenomena derived exclusively from experience."
Note how quickly news of events that aroused general interest spread at that time, although the means of communication were less perfect than today (the idea of ​​telegraph communication was expressed by Ampere in 1829, and only in 1844 in North America the first commercial telegraph line). The results of Faraday's experiments quickly became widely known. This, however, cannot be said about the spread of Faraday's theoretical ideas that explained his experiments (the concept of lines of force, an electrotonic state, that is, of an electromagnetic field)
The first to appreciate the depth of Faraday's ideas was Maxwell, who was able to find a suitable mathematical language for them.
But this happened already in the middle of the century. The reader may ask why the ideas of Faraday and Ampere were so differently perceived. The point, apparently, is that the electrodynamics of Ampere had already matured, "was floating in the air." Without belittling the great merits of Ampere, who was the first to give these ideas an exact mathematical form, it must nevertheless be emphasized that Faraday's ideas were much deeper and more revolutionary. Oii not "were in the air", but were born by the creative power of thought and imagination of their author. Their perception was complicated by the fact that they were not clothed in mathematical clothes. If Maxwell had not appeared, Faraday's ideas might have been forgotten for a long time.
The third most important direction in physics in the first half of the last century was the beginning of the development of the theory of heat. The first steps in the theory of thermal phenomena, naturally, were associated with the operation of steam engines, and general theoretical ideas were formed with difficulty and penetrated into science slowly. The remarkable work of Sadi Carnot (1796 - 1832) "Reflections on the driving force of fire and on machines capable of developing this force", published in 1824, went completely unnoticed. It was remembered only thanks to the work of Clapeyron, which appeared in 1834, but the creation of a modern theory of heat (thermodynamics) is already a matter of the second half of the century.
Two works are closely related to the questions of interest to us. One of them is the famous book of the outstanding mathematician, physicist and Egyptologist *) Jean Baptiste Joseph Fourier (1768 - 1830) "The Analytical Theory of Heat" (1822), devoted to solving the problem of heat propagation; in it, the method of decomposition of functions into sinusoidal components (Fourier decomposition) was developed in detail and applied to the solution of physical problems. The birth of mathematical physics as an independent science is usually counted from this work. Its significance for the theory of oscillatory and wave processes is enormous - for more than a century, the main method of studying wave processes has been the decomposition of complex waves into simple sinusoidal
*) After the Napoleonic campaign in Egypt, he compiled a "Description of Egypt" and collected a small but valuable collection of Egyptian antiquities. Fourier guided the first steps of the young Jaia-Fraisois Champolioia, the ingenious decoder of hieroglyphic writing, the founder of Egyptology. Thomas Jung was also fond of deciphering hieroglyphs, not without success. After studying physics, this was perhaps his main hobby.
(harmonic) waves, or "harmonics" (from "harmony" in music).
Another work is the report of the twenty-six-year-old I Elmholtz "On the conservation of strength", made in 1847 at a meeting of the Physics Society founded by him in Berlin. Hermann Ludwig Ferdinand Helmholtz (1821 - 1894) is rightfully considered one of the greatest naturalists, and some historians of science put this work on a par with the most outstanding works of scientists who laid the foundations natural sciences... It deals with the most general formulation of the principle of conservation of energy (then it was called "force") for mechanical, thermal, electrical ("galvanic") and magnetic phenomena, including processes in an "organized being". It is especially interesting for us that here Helmholtz was the first to note the oscillatory nature of the discharge of the Leyden jar and wrote an equation, from which W. Thomson soon derived a formula for the period of electromagnetic oscillations in an oscillatory circuit.
In this little work, one can discern hints of future remarkable research by Helmholtz. Even a simple listing of his achievements in physics, hydromechanics, mathematics, anatomy, physiology and psychophysiology would lead us very far away from the main topic of our story. We will only mention the theory of vortices in a liquid, the theory of the origin of sea waves and the first definition of the speed of propagation of an impulse in a nerve. All these theories, as we will see shortly, are most directly related to modern research solitons. Among his other ideas, it is necessary to mention, for the first time, expressed by him in a lecture on the physical views of Faraday (1881), the idea of ​​the existence of an elementary ("smallest possible") electric charge ("electrical atoms"). Experimentally, the electron was discovered only sixteen years later.
Both described works were theoretical, they formed the foundation of mathematical and theoretical physics. The final formation of these sciences is undoubtedly associated with the works of Maxwell, and in the first half of the century, a purely theoretical approach to physical phenomena was, in general, alien to the majority
puppies. Physics was considered a purely "experimental" science and the main words even in the titles of works were "experiment", "based on experiments", "derived from experiments." It is interesting that Helmholtz's work, which even today can be considered a model of depth and clarity of presentation, was not accepted by the physics journal as theoretical and too large in volume and was later published as a separate brochure. Shortly before his death, Helmholtz talked about the history of the creation of his most famous work:
“Young people are most willing to take on the most profound tasks at once, and I was also interested in the question of the mysterious creature of life force ... I found that ... the theory of life force ... attributes to every living body the properties of a 'perpetual motion machine' ... Looking through the works of Daniel Bernoulli, D'Alembert and other mathematicians of the last century ... I came across the question: “what relations should exist between various forces of nature, if we accept that a“ perpetual motion machine ”is generally impossible and whether all these relations are actually fulfilled. .. "I only intended to give a critical assessment and systematics of the facts in the interests of physiologists. It would not have come as a surprise to me if, in the end, knowledgeable people told me: “Yes, all this is well known. What does this young physician want when he goes on about these things in such detail? " To my surprise, those authorities in physics with whom I had to come into contact looked at the matter in a completely different way. They tended to reject the justice of the law; in the midst of the zealous struggle they waged with Hegel's natural philosophy, and my work was considered fantastic speculation. Only the mathematician Jacobi recognized the connection between my reasoning and the thoughts of mathematicians of the last century, became interested in my experience and protected me from misunderstandings. "
These words clearly characterize the mindset and interests of many scientists of that era. There is, of course, a regularity and even a necessity in such resistance of the scientific society to new ideas. So let's not rush to condemn Laplace, who did not understand Fresnel, Weber, who did not recognize the ideas of Faraday, or Kelvin, who opposed the recognition of Maxwell's theory, but rather ask ourselves whether it is easy for us to assimilate new ideas that are unlike everything we have gotten used to. ... We recognize that some conservatism is embedded in our human nature, and therefore in the science that people do. They say that a certain "healthy conservatism" is even necessary for the development of science, since it prevents the spread of empty fantasies. However, this is by no means consoling when one remembers the fate of geniuses who looked into the future, but were not understood and not recognized by their era.

Your age, wondering at you, did not comprehend the prophecies
And with flattery he mixed insane reproaches.
V. Bryusov
Perhaps the most striking examples of such a conflict with the era in the time of interest to us (about 1830) we see in the development of mathematics. The face of this science was then determined, probably, by Gauss and Cauchy, who, together with others, completed the construction of the great edifice of mathematical analysis, without which modern science is simply unthinkable. But we cannot forget that at the same time, not appreciated by contemporaries, young Abel (1802 - 1829) and Galois (1811 - 1832) died, that from 1826 to 1840. published their works on non-Euclidean geometry Lobachevsky (1792 - 1856) and Boyai (1802 - I860), who did not live to see the recognition of their ideas. The reasons for this tragic misunderstanding are deep and varied. We cannot delve into them, but we will give only one more example, which is important for our story.
As we will see later, the fate of our hero, the soliton, is closely related to computers. Moreover, history presents us with a striking coincidence. In August 1834, while Russell was observing the solitary wave, the English mathematician, economist and engineer-inventor Charles Bab-badge (1792 - 1871) completed the development of the basic principles of his "analytical" machine, which later formed the basis of modern digital computing machines. Babbage's ideas were far ahead of their time. It took over a hundred years to realize his dream of building and using such machines. It is difficult to blame Babbage's contemporaries for this. Many understood the need for computers, but technology, science and society were not yet ripe for its implementation. bold projects... The Prime Minister of England, Sir Robert Peel, who had to decide the fate of financing the project presented by Babbage to the government, was not an ignoramus (he graduated from Oxford first in mathematics and classics). He formally held a thorough discussion of the project, but as a result came to the conclusion that the creation of a universal computing machine was not a priority for the British government. It was not until 1944 that the first automatic digital machines appeared, and an article titled "Babbage's Dream Come True" appeared in the English journal Nature.

Science and society
The squad of scientists and writers ... is always ahead in all the races of enlightenment, in all attacks of education. They should not be faint-heartedly indignant at the fact that they are always determined to endure the first shots and all the hardships, all the dangers.
A. S. Pushkin
Of course, both the successes of science and its failures are associated with the historical conditions of the development of society, on which we cannot keep the reader's attention. It is no coincidence that at that time such a pressure of new ideas arose that science and society did not have time to master them.
Development of science in different countries went on uneven paths.
In France scientific life was united and organized by the Academy to such an extent that the work, which was not noticed and supported by the Academy or even by well-known academicians, had little chance of interest in scientists. But the works that came to the attention of the Academy were supported and developed. This sometimes provoked protests and indignation on the part of young scientists. In an article dedicated to the memory of Abel, his friend Segi wrote: “Even in the case of Abel and Jacobi, the favor of the Academy did not mean recognition of the undoubted merits of these young scientists, but rather a desire to encourage the study of certain problems concerning a strictly defined range of issues, beyond which, in the opinion Academy, there can be no progress in science and no valuable discoveries can be made ... We will say something completely different: young scientists, do not listen to anyone except your own inner voice. Read the works of geniuses and reflect on them, but never turn into disciples deprived of their own
military opinion ... Freedom of views and objectivity of judgments - this should be your motto. " (Perhaps “not listening to anyone” is a polemical exaggeration, the “inner voice” is not always right.)
In many small states located on the territory of the future German Empire (only by 1834 customs offices between most of these states were closed), scientific life was concentrated in numerous universities, in most of which research work was also carried out. It was there at this time that schools of scientists began to take shape and a large number of scientific journals were published, which gradually became the main means of communication between scientists, beyond the control of space and time. Modern scientific journals also follow their pattern.
In the British Isles, there was neither a French-style academy that promoted the achievements it recognized, nor such scientific schools as in Germany. Most of the British scientists worked alone *). These loners managed to blaze completely new paths in science, but their work often remained completely unknown, especially when they were not sent to the journal, but were only reported at meetings of the Royal Society. The life and discoveries of an eccentric nobleman and a brilliant scientist, Lord Henry Cavendish (1731 - 1810), who worked all alone in his own laboratory and published only two works (the rest, containing discoveries rediscovered by others only tens of years later, were found and published by Maxwell), especially vividly illustrate these features of science in England at the turn of the XVIII - XIX centuries. Such tendencies in scientific work persisted in England for quite a long time. For example, the already mentioned Lord Rayleigh also worked as an amateur; he performed most of his experiments on his estate. This "amateur", in addition to the book on the theory of sound, was written
*) Don't take it too literally. Any scientist needs constant communication with other scientists. In England, the center of such communication was the Royal Society, which also had considerable funds to finance scientific research.
more than four hundred works! For several years Maxwell also worked alone in his ancestral nest.
As a result, as the English historian of science wrote about this time, “the largest number of works perfect in form and content that have become classical ... belongs, probably, to France; the largest amount of scientific work was carried out, probably in Germany; but of the new ideas that have fertilized science throughout the century, England probably has the largest share. " The last statement can hardly be attributed to mathematics. If we talk about physics, then this judgment does not seem too far from the truth. Let's not forget that Russell's contemporary *) was the great Charles Darwin, who was born a year later and died the same year as him.
What is the reason for the success of single researchers, why were they able to come up with such unexpected ideas that many other equally gifted scientists thought they were not just wrong, but even almost insane? If we compare Faraday and Darwin, two great natural scientists of the first half of the last century, then their extraordinary independence from the teachings that prevailed at that time, their trust in their own eyesight and reason, great ingenuity in posing questions and the desire to fully understand the unusual that they managed to observe. It is also important that an educated society is not indifferent to scientific research. If there is no understanding, then there is interest, and a circle of fans and sympathizers usually gathers around the pioneers and innovators. Even the misunderstood Babbage, who by the end of his life became a misanthrope, had people who loved and appreciated him. He was understood and highly appreciated by Darwin, his close employee and the first programmer of his analytical machine was an outstanding mathematician, Byron's daughter, lady
*) Most of the contemporaries we mentioned were probably familiar with each other. Of course, the members of the Royal Society met in meetings, but they also maintained personal contacts. For example, it is known that Charles Darwin visited Charles Babbage, who was friends with John Herschel from his student years, who knew John Russell intimately, etc.
Ada Augusta Lovelace. Babbage was also appreciated by Faraday and other prominent people of his time.
The social significance of scientific research has already become clear to many educated people, and this sometimes helped scientists to obtain the necessary funds, despite the lack of centralized funding for science. By the end of the first half of the 18th century. The Royal Society and the leading universities had more funds than any leading scientific institutions on the continent. "... A galaxy of outstanding physicists like Maxwell, Rayleigh, Thomson ... could not have arisen if ... in England at that time there would have been no cultural scientific community that correctly evaluates and supports the activities of scientists" (P . L. Kapitsa).

COUNT OF CHAPTER AND FRAGMEHTA BOOK

annotation... The report is devoted to the possibilities of the soliton approach in supramolecular biology, primarily for modeling a wide class of natural wavelike and oscillatory movements in living organisms. The author has identified many examples of the existence of soliton-like supramolecular processes ("biosolitons") in locomotor, metabolic and other phenomena of dynamic biomorphology at various lines and levels of biological evolution. By biosolitons, we mean, first of all, characteristic one-humped (unipolar) local deformations moving along the biobody while maintaining their shape and velocity.

Solitons, sometimes called "wave atoms", are endowed with properties unusual from the classical (linear) point of view. They are capable of acts of self-organization and self-development: self-localization; capturing energy; reproduction and death; the formation of ensembles with dynamics of a pulsating and other nature. Solitons were known in plasma, liquid and solid crystals, classical liquids, nonlinear lattices, magnetic and other polydomain media, etc. The discovery of biosolitons indicates that, due to its mechanochemistry, living matter is a soliton medium with various physiological uses of soliton mechanisms. Research hunting in biology for new types of solitons is possible - breathers, wobblers, pulsons, etc., deduced by mathematicians at the "tip of the pen" and only then discovered by physicists in nature. The report is based on the monographs: S.V. Petukhov “Biosolitons. Fundamentals of Soliton Biology ", 1999; S. V. Petukhov "Biperiodic table genetic code and the number of protons ", 2001.

Solitons are an important subject of modern physics. The intensive development of their theory and applications began after the publication in 1955 by Fermi, Pasta and Ulam on the computer calculation of oscillations in a simple nonlinear system of a chain of weights connected by nonlinear springs. The necessary mathematical methods were soon developed to solve soliton equations, which are nonlinear partial differential equations. Solitons, sometimes called "wave atoms", possess the properties of waves and particles at the same time, but they are not in the full sense either one or the other, but constitute a new object of mathematical natural science. They are endowed with properties unusual from the classical (linear) point of view. Solitons are capable of self-organization and self-development acts: self-localization; capturing energy coming from outside into a "soliton" environment; reproduction and death; the formation of ensembles with non-trivial morphology and dynamics of a pulsating and other nature; self-complication of these ensembles when additional energy enters the medium; overcoming the tendency towards disorder in soliton media containing them; and so on. They can be interpreted as a specific form of organization of physical energy in matter, and, accordingly, we can talk about "soliton energy" by analogy with the well-known expressions "wave energy" or "vibrational energy". Solitons are realized as states of special nonlinear media (systems) and have fundamental differences from ordinary waves. In particular, solitons are often stable self-trapped bunches of energy with a characteristic shape of a one-humped wave moving with the same shape and speed without dissipating its energy. Solitons are capable of non-destructive collisions, i.e. when they meet, they are able to pass through each other without breaking their shape. They have numerous technical uses.

A soliton is usually understood as a solitary wave-like object (a localized solution of a nonlinear partial differential equation belonging to a certain class of so-called soliton equations), which is able to exist without dissipating its energy and, when interacting with other local perturbations, always restores its original form, i.e. ... capable of non-destructive collisions. As is known, soliton equations “arise in the most natural way in the study of weakly nonlinear dispersion systems of various types on various spatial and temporal scales. The universality of these equations turns out to be so striking that many were inclined to see this as something magical ... But this is not so: dispersive weakly damped or persistent nonlinear systems behave the same, regardless of whether they occur when describing plasmas, classical liquids, lasers or nonlinear gratings ". Accordingly, solitons are known in plasmas, liquid and solid crystals, classical liquids, nonlinear lattices, magnetic and other multidomain media, etc. (The motion of solitons in real media is often not absolutely nondissipative, accompanied by small energy losses, which theorists take into account by adding small dissipative terms into soliton equations).

Note that living matter is permeated by many nonlinear lattices: from molecular polymer networks to supramolecular cytoskeletons and organic matrix. Rearrangements of these lattices have an important biological significance and may well behave in a soliton-like manner. In addition, solitons are known as forms of motion of the fronts of phase rearrangements, for example, in liquid crystals (see, for example,). Since many systems of living organisms (including liquid crystal ones) exist on the verge of phase transitions, it is natural to assume that the fronts of their phase rearrangements in organisms will also often move in a soliton form.

Even the pioneer of solitons Scott Russell in the last century experimentally showed that a soliton acts as a concentrator, trap and transporter of energy and matter, capable of non-destructive collisions with other solitons and local disturbances. It is obvious that these features of solitons can be beneficial for living organisms, and therefore biosoliton mechanisms can be specially cultivated in living nature by the mechanisms of natural selection. Here are some of these benefits:

• - 1) spontaneous capture of energy, matter, etc., as well as their spontaneous local concentration (self-trapping) and gentle, lossless transportation in a dosage form inside the body;
• - 2) ease of control over the flows of energy, matter, etc. (when they are organized in a soliton form) due to the possible local switching of the characteristics of the nonlinearity of the biological medium from soliton to nonsoliton type of nonlinearity and vice versa;
• - 3) decoupling for many of those simultaneously and in one place occurring in the body, i.e. overlapping processes (locomotor, blood supply, metabolic, growth, morphogenetic, etc.), which need the relative independence of their course. This decoupling can be provided precisely by the ability of solitons to undergo non-destructive collisions.

For the first time, our study of supramolecular cooperative processes in living organisms from a soliton point of view revealed the presence in them of many macroscopic soliton-like processes. The subject of study was, first of all, directly observed locomotor and other biological movements, the high energy efficiency of which has long been assumed by biologists. At the first stage of the study, we found that in many living organisms, biological macro-movements often have a soliton-like form of a characteristic one-humped wave of local deformation moving along a living body while maintaining its shape and speed and sometimes demonstrating the ability to non-destructive collisions. These "biosolitons" are realized at various branches and levels of biological evolution in organisms that differ in size by several orders of magnitude.

The report contains numerous examples of such biosolitons. In particular, an example of the crawling of a Helix snail, which occurs due to a one-humped wave-like deformation running through its body, while maintaining its shape and speed, is considered. Detailed recordings of this type of biological movement are taken from the book. In one variant of crawling (with one "gait"), local stretching deformations occur in the cochlea, going along the supporting surface of its body from front to back. With another, slower version of crawling on the same body surface, local compression deformations occur, going in the opposite direction from the tail to the head. Both the named types of soliton deformations - direct and retrograde - can be realized in the cochlea simultaneously with counter-collisions between them. We emphasize that their collision is non-destructive, typical of solitons. In other words, after a collision, they retain their shape and speed, that is, their individuality: “the presence of large retrograde waves does not affect the propagation of normal and much shorter direct waves; both types of waves propagated without any sign of mutual interference. " This biological fact has been known since the beginning of the century, although researchers have never connected with solitons before us.

As Gray and other classics of the study of locomotion (spatial movement in organisms) have emphasized, the latter are highly energy-efficient processes. This is essential for the vital provision of the body with the ability to move without fatigue over long distances in search of food, escape from danger, etc. (organisms in general are extremely careful with energy, which is not at all easy for them to store). So, in a snail, a soliton local deformation of the body, due to which the movement of its body in space is carried out, occurs only in the zone of separation of the body from the support surface. And the entire part of the body in contact with the support is undeformed and rests relative to the support. Accordingly, during the entire time the soliton-like deformation flows through the snail's body, such wave-like locomotion (or the process of mass transfer) does not require energy expenditures to overcome the forces of friction of the snail against the support, being the most economical in this regard. Of course, it can be assumed that part of the energy during locomotion is nevertheless dissipated into the mutual friction of tissues inside the snail's body. But if this locomotor wave is soliton-like, then it also minimizes friction losses inside the body. (As far as we know, the issue of energy losses due to intra-body friction during locomotion has not been sufficiently studied experimentally, however, the body is unlikely to have passed by the opportunity to minimize them). With the considered organization of locomotion, all (or almost all) energy consumption for it is reduced to the cost of the initial creation of each such soliton-like local deformation. It is the physics of solitons that provides extremely energy-efficient possibilities for handling energy. And its use by living organisms looks natural, especially since the world saturated with soliton media and solitons.

It should be noted that, at least since the beginning of the century, researchers have presented undulating locomotion as a kind of relay process. At that time of "pre-soliton physics", the natural physical analogy for such a relay process was the combustion process, in which local body deformation was transmitted from point to point like ignition. This idea of ​​relay-race dissipative processes such as combustion, now called autowave, was the best possible at that time, and it has long become familiar to many. However, physics itself did not stand still. In recent decades, it has developed the concept of solitons as a new type of non-dissipative relay processes of the highest energy efficiency with previously unthinkable paradoxical properties, which provides the basis for a new class of nonlinear models of relay processes.

One of the important advantages of the soliton approach over the traditional autowave approach when simulating processes in a living organism is determined by the ability of solitons to non-destructive collisions. Indeed, autowaves (describing, for example, the movement of the combustion zone along a burning cord) are characterized by the fact that a zone of non-excitability (a burnt cord) remains behind them, and therefore two autowaves, when they collide with each other, cease to exist, unable to move along the already “burned out site ". But in the areas of a living organism, many biomechanical processes simultaneously occur - locomotor, blood supply, metabolic, growth, morphogenetic, etc., and therefore, modeling them with autowaves, the theorist is faced with the following problem of mutual destruction of autowaves. One autowave process, moving along the considered area of ​​the body due to the continuous burning out of energy reserves on it, makes this environment non-excitable for other autowaves for some time until the energy reserves for their existence are restored in this area. In living matter, this problem is especially urgent also because the types of energy-chemical reserves in it are highly unified (organisms have a universal energy currency - ATP). Therefore, it is difficult to believe that the fact of the simultaneous existence of many processes in one area in the body is ensured by the fact that each autowave process in the body moves by burning out its specific type of energy, without burning out energy for others. For soliton models, this problem of mutual annihilation of biomechanical processes colliding in one place does not exist in principle, since solitons, due to their ability to non-destructive collisions, calmly pass through each other and in one section at the same time their number can be as large as you like. According to our data, the sine-Gordon soliton equation and its generalizations are of particular importance for modeling biosoliton phenomena of living matter.

As is known, in multidomain media (magnets, ferroelectrics, superconductors, etc.), solitons act as interdomain walls. In living matter, the phenomenon of polydomain plays an important role in morphogenetic processes. As in other multidomain environments, in multidomain biological environments it is associated with the classical Landau-Lifshitz principle of minimizing energy in a medium. In these cases, the soliton interdomain walls turn out to be places of increased energy concentration, in which biochemical reactions often occur especially actively.

The ability of solitons to play the role of locomotives transporting portions of matter to the desired place within the soliton medium (organism) according to the laws of nonlinear dynamics also deserves every attention in connection with bioevolutionary and physiological problems. We add that the biosoliton physical energy is able to coexist harmoniously in a living organism with the known chemical types of its energy. The development of the concept of biosolitons allows, in particular, to open a research "hunt" in biology for analogues of different types of solitons - breathers, wobblers, pulsons, etc., deduced by mathematicians "at the tip of the pen" when analyzing soliton equations and then discovered by physicists in nature. Many oscillatory and wave physiological processes can eventually receive for their description meaningful soliton models associated with the nonlinear, soliton nature of biopolymer living matter.

For example, this refers to the basic physiological movements of a living biopolymer substance such as heartbeats, etc. Recall that in a human embryo at the age of three weeks, when it has a growth of only four millimeters, the heart is the first to move in motion. The onset of cardiac activity is due to some internal energy mechanisms, since at this time the heart still does not have any nerve connections to control these contractions and it begins to contract when there is still no blood to pump. At this moment, the embryo itself is essentially a piece of polymer mucus, in which the internal energy self-organizes into energy-efficient pulsations. The same can be said about the occurrence of heartbeats in the eggs and eggs of animals, where the supply of energy from the outside is minimized by the existence of the shell and other insulating covers. Such forms of energy self-organization and self-localization are known in polymeric media, including those of a non-biological type, and, according to modern concepts, have a soliton nature, since solitons are the most energy-efficient (nondissipative or low-dissipative) self-organizing structures of a pulsating and other nature. Solitons are realized in a variety of natural environments surrounding living organisms: solid and liquid crystals, classical liquids, magnets, lattice structures, plasma, etc. The evolution of living matter with its mechanisms of natural selection did not pass by unique properties solitons and their ensembles.

Do these materials have anything to do with synergy? Yes, absolutely. As defined in Hagen's monograph / 6, p. 4 /, “within the framework of synergetics, such a joint action of individual parts of a disordered system is studied, as a result of which self-organization occurs - macroscopic spatial, temporal or space-time structures arise, and are considered as deterministic and stochastic processes ”. There are many types of non-linear processes and systems that are studied in the framework of synergetics. Kurdyumov and Knyazeva / 7, p. 15 /, listing a number of these types, specially note that among them one of the most important and intensively studied are solitons. In recent years, the international journal Chaos, Solitons & Fractals has been published. Solitons observed in a variety of natural environments are a vivid example of nonlinear cooperative behavior of many elements of a system, leading to the formation of specific spatial, temporal, and spatio-temporal structures. The most famous, although far from the only, type of such soliton structures is the self-localizing, shape-stable, single-hump local deformation of the medium described above, running at a constant speed. Solitons are actively used and studied in modern physics... Since 1973, starting with the work of Davydov / 8 /, solitons are also used in biology to model molecular biological processes. At present, all over the world there are many publications on the use of such "molecular solitons" in molecular biology, in particular, for understanding the processes in proteins and DNA. Our works / 3, 9 / were the first publications in the world literature on the topic of "supramolecular solitons" in biological phenomena of the supramolecular level. Let us emphasize that the existence of molecular biosolitons (which, in the opinion of many authors, has yet to be proved) does not in any way follow the existence of solitons in cooperative biological supramolecular processes that unite myriads of molecules.

LITERATURE:

1. Dodd R. et al. Solitons and Nonlinear Wave Equations. M., 1988, 694 p.
2. Kamensky V.G. ZhETF, 1984, v. 87, no. 4 (10), p. 1262-1277.
3. S.V. Petukhov Biosolitons. Fundamentals of Soliton Biology. - M., 1999, 288 p.
4. Gray J. Animal locomotion. London, 1968.
5. S.V. Petukhov Biperiodic table of the genetic code and the number of protons. - M., 2001, 258 p.
6. Hagen G. Synergetics. - M., Mir, 1980, 404 p.
7. Knyazeva E.N., Kurdyumov S.P. The laws of evolution and self-organization of complex systems. - M., Nauka, 1994, 220 p.
8. Davydov A.S. Solitons in biology. - Kiev, Naukova Dumka, 1979.
9. S.V. Petukhov Solitons in biomechanics. Deposited at VINITI RAS on February 12, 1999, No. 471-B99. (Index of VINITI "Deposited scientific works", No. 4 for 1999)

Summary ... The report discusses the opportunities opened up by a solitonic approach to supramolecular biology, first of all, for modeling a wide class of natural wave movements in living organisms. The results of author's research demonstrate the existence of soliton-like supramolecular processes in locomotor, metabolic and other manifestations of dynamic biomorphology on a wide variety of branches and levels of biological evolution.

Solitons, named sometimes "wave atoms", have unusual properties from the classical (linear) viewpoint. They have ability for self-organizing: auto-localizations; catching of energy; formation of ensembles with dynamics of pulsing and other character. Solitons were known in plasma, liquid and firm crystals, classical liquids, nonlinear lattices, magnetic and others poly-domain matters, etc. The reveal of biosolitons points out that biological mechano-chemistry makes living matter as solitonic environment with opportunities of various physiological use of solitonic mechanisms. The report is based on the books: S.V. Petoukhov “Biosolitons. Bases of solitonic biology ”, Moscow, 1999 (in Russian).

Petukhov S.V., Solitons in cooperative biological processes of the supramolecular level // "Academy of Trinitarianism", M., El No. 77-6567, publ. 13240, 21.04.2006

Scientists have proven that words can revive dead cells! During the research, scientists were amazed at how powerful the word is. And also the unthinkable experiment of scientists on the influence of creative thought on cruelty and violence.
How did they manage to achieve this?

Let's start in order. Back in 1949, researchers Enrico Fermi, Ulam and Pasta studied nonlinear systems - oscillatory systems, the properties of which depend on the processes taking place in them. These systems behaved unusually under certain conditions.

Studies have shown that the systems memorized the conditions of action on them, and this information was stored in them for quite a long time. A typical example is a DNA molecule that stores the informational memory of an organism. Even in those days, scientists asked themselves the question, how is it possible that an unreasonable molecule that does not have brain structures or nervous system, can have memory that is superior to any modern computer in accuracy. Later, scientists discovered mysterious solitons.

Solitons

Soliton is a structurally stable wave found in nonlinear systems. There was no limit to the surprise of scientists. After all, these waves behave like intelligent beings. And only after 40 years have scientists been able to advance in these studies. The essence of the experiment was as follows - with the help of specific devices, scientists were able to trace the path of these waves in the DNA chain. Passing the chain, the wave completely read the information. This can be compared to a person reading an open book, only hundreds of times more accurate. During the study, all experimenters had the same question - why do solitons behave like this, and who gives them such a command?

Scientists continued their research at the Mathematical Institute of the Russian Academy of Sciences. They tried to influence solitons with human speech recorded on an information medium. What the scientists saw surpassed all expectations - under the influence of words, the solitons came to life. The researchers went further - they sent these waves to wheat grains, which had previously been irradiated with such a dose of radioactive radiation at which DNA strands are broken, and they become unviable. After exposure, wheat seeds germinated. The restoration of DNA destroyed by radiation was observed under a microscope.

It turns out that human words were able to revive a dead cell, i.e. under the influence of words, solitons began to possess life-giving power. These results have been repeatedly confirmed by researchers from other countries - Great Britain, France, America. Scientists have developed a special program in which human speech was transformed into vibrations and superimposed on soliton waves, and then influenced the DNA of plants. As a result, the growth and quality of plants was significantly accelerated. Experiments were carried out with animals, after exposure to them, an improvement in blood pressure was observed, the pulse was leveled, and somatic indicators improved.

Scientists' research did not stop there either.

Together with colleagues from scientific institutions The USA, India conducted experiments on the impact of human thought on the state of the planet. The experiments were carried out more than once; 60 and 100 thousand people took part in the latter. This is truly a huge number of people. The main and necessary rule for performing the experiment was the presence of creative thought in people. For this, people voluntarily gathered in groups and sent their positive thoughts to a certain point on our planet. At that time, this point was the capital of Iraq - Baghdad, where bloody battles were going on at that time.

During the experiment, the battles abruptly stopped and did not resume for several days, and also on the days of the experiment, the crime rates in the city sharply decreased! The process of influence of creative thought was recorded by scientific instruments, which registered a powerful flow of positive energy.

Scientists are confident that these experiments have proven the materiality of human thoughts and feelings, and their incredible ability to withstand evil, death and violence. Once again, thanks to their pure thoughts and aspirations, learned minds scientifically confirm the ancient common truths - human thoughts can both create and destroy.

The choice remains with the person, because it depends on the direction of his attention whether a person will create or negatively influence others and himself. Human life is a constant choice and you can learn to make it correctly and consciously.

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After thirty years of searching, nonlinear differential equations with three-dimensional soliton solutions were found. The key idea was the "complexification" of time, which can find further applications in theoretical physics.

When studying any physical system, the stage of "initial accumulation" of experimental data and their interpretation first goes. Then the baton is passed on to theoretical physics. The task of a theoretical physicist is to derive and solve mathematical equations for this system based on the accumulated data. And if the first step, as a rule, does not pose a particular problem, then the second - precise solving the obtained equations is often an incomparably more difficult task.

It just so happens that the evolution in time of many interesting physical systems are described non-linear differential equations : such equations for which the principle of superposition does not work. This immediately deprives theoreticians of the opportunity to use many standard techniques (for example, combine solutions, expand them in a series), and as a result, for each such equation, they have to invent a completely new solution method. But in those rare cases when such an integrable equation and a method for its solution is found, not only the original problem is solved, but also a whole series of related mathematical problems. That is why theoretical physicists sometimes, abandoning the "natural logic" of science, first look for such integrable equations, and only then try to find applications for them in different areas theoretical physicists.

One of the most remarkable properties of such equations is the solutions in the form solitons- limited in space "pieces of the field" that move over time and collide with each other without distortion. Being limited in space and indivisible "bunches", solitons can give a simple and convenient mathematical model of many physical objects. (For more details about solitons, see the popular article by N. A. Kudryashov, Nonlinear Waves and Solitons, SOZh, 1997, No. 2, pp. 85-91 and A. T. Filippov's book The Many-Faced Soliton.)

Unfortunately, different species very few solitons are known (see the Portrait Gallery of Solitons), and all of them are not very suitable for describing objects in three-dimensional space.

For example, ordinary solitons (which occur in the Korteweg – de Vries equation) are localized in just one dimension. If such a soliton is "launched" in a three-dimensional world, then it will look like an infinite flat membrane flying forward. In nature, however, such infinite membranes are not observed, which means that the original equation for describing three-dimensional objects is not suitable.

Not so long ago, soliton-like solutions (for example, dromions) of more complex equations were found, which are already localized in two dimensions. But in three-dimensional form they are also infinitely long cylinders, that is, they are also not very physical. The real three-dimensional until now it has not been possible to find solitons for the simple reason that the equations that could produce them were unknown.

Recently, the situation has changed dramatically. The Cambridge mathematician A. Focas, the author of the recent publication A. S. Focas, Physical Review Letters 96, 190201 (May 19, 2006), managed to make a significant step forward in this area of ​​mathematical physics. His short three-page article contains two discoveries at once. First, he found a new way to derive integrable equations for multidimensional space, and secondly, he proved that these equations have multidimensional soliton-like solutions.

Both of these achievements were made possible by a bold step taken by the author. He took the already known integrable equations in two-dimensional space and tried to consider time and coordinates as complex, not real numbers. In this case, a new equation was automatically obtained for four-dimensional space and two-dimensional time... The next step, he imposed non-trivial conditions on the dependence of solutions on coordinates and "times", and the equations began to describe three-dimensional a situation that depends on a single time.

It is interesting that such a "blasphemous" operation as the transition to two-dimensional time and the allocation of a new time in it O axis, did not greatly spoil the properties of the equation. They are still integrable, and the author was able to prove that among their solutions there are the much-desired three-dimensional solitons. Now it remains for scientists to write these solitons in the form of explicit formulas and study their properties.

The author is confident that the benefits of the method of "complexification" of time developed by him are not at all limited to the equations that he has already analyzed. He lists a number of situations in mathematical physics in which his approach can yield new results, and encourages colleagues to try to apply it in the most diverse areas of modern theoretical physics.