Introduction

Cambridge University Press 978-0-521-85421-4 - Physics of Solitons Thierry Dauxois and Michel Peyrard Excerpt More information

Introduction

The nineteenth century and the ﬁrst half of the twentieth century can be viewed as the triumph of linear physics, which started with Maxwell’s equations and culmi- nated with quantum mechanics, based on a linear formalism emphasising a super- position principle. The familiar mathematical tools of physics such as the Fourier transform, the linear response theory and perturbative expansions, were themselves intrinsically linear. Of course physicists had noticed the importance of nonlinear phenomena which appeared in the Navier–Stokes equation of hydrodynamics, gravitational theory, collective effects arising from the interaction between particles in solid state physics, etc. But, in most of the cases, theoretical approaches were trying to avoid nonlinearities, or to treat them as perturbations of linear theories. The picture dramatically changed in the last 40 years. The importance of an intrinsic analysis of nonlinear phenomena has been gradually understood, and led to two concepts that revolutionalised previous ideas, the strange attractor and the soliton. Both are related to astonishing properties of nonlinear systems, and they seem to contradict each other. The strange attractor is linked to the idea of chaos in a system which is described by deterministic equations. It shows up in systems with a small number of degrees of freedom, which could have been viewed as ‘simple’, while solitons appear in systems with a very large number of degrees of freedom. Although it seems that adding degrees of freedom should make the behaviour of such systems even more complex, this is not necessarily the case. Collective effects can lead to spatially coherent structures, which result in a self-organisation. Understanding the coexistence between coherent structures and chaos in nonlinear systems is still an open question. A soliton is a solitary wave, i.e. a spatially localised wave, with spectacular stability properties. Since its ﬁrst observation [162] in 1834 by a hydrodynamic engineer, John Scott Russell, the soliton initiated passion and debates. John Scott

2 Introduction Russell himself had been so fascinated by his unexpected observation that he devoted ten years of his life to study this phenomenon while theories, based on linearised approaches, were showing . . . that solitons could not exist. The early history of solitons has been marked by long eclipses. While the first observation had been made in 1834, one had to wait until 1895 for a theory [104] that could describe solitons, thanks to an equation derived by Korteweg and de Vries. Then this phenomenon was forgotten until a numerical experiment, carried out by Fermi, Pasta and Ulam, in 1953, with one of the first computers in Los Alamos, exhibited a result that appeared to contradict thermodynamics. A one-dimensional lattice of particles coupled to each other by an anharmonic potential was not necessarily reaching thermal equilibrium. The energy, initially injected in one particular mode, was first transferred to the other modes as one would have expected, but then it was coming back, almost perfectly, to the mode that had been originally excited. It is only ten years later that an explanation could be provided by Zabusky and Kruskal [190]. As we shall see in this book, it involves solitons and it is their work that introduced the word soliton. This name, which sounds like the name of a particle, was chosen on purpose. A soliton is a wave, but it is also a local maximum in the energy density, which preserves its shape and velocity when it moves, exactly as a particle does. It corresponds to a solution of a classical field equation which simultaneously exhibits wave and quasi-particle properties. These are features that one would expect from a quantum system and not from a classical one. The quantum analogy goes so far that soliton tunnelling has been found [137]. The study of Zabusky and Kruskal is a landmark in the history of solitons. Since then solitons have stayed in the front of the scene and have been the object of a huge number of investigations in mathematics and physics. Equations having soliton solutions, in the exact mathematical sense, provide remarkable examples of completely integrable systems with an infinite number of degrees of freedom. This is why they have interested mathematicians so much, and this also explains why most of the books on solitons are mainly devoted to their mathematical properties. However solitons are also of major interest to physicists. They are essential to describe phenomena such as the propagation of some hydrodynamic waves, localised waves in astrophysical plasmas, the propagation of signals in optical fibres or, at the microscopic level, charge transport in conducting polymers, localised modes in magnetic crystals and the dynamics of biological molecules such as DNA and proteins, for instance. All these systems are only approximately described by the equations of the mathematical theory of solitons. One should rather speak of ‘quasi- solitons’. But the remarkable feature of solitons is their exceptional stability against perturbations so that these ‘quasi-solitons’ exhibit most of the spectacular properties of actual solitons. Moreover they can emerge spontaneously in a physical system

Introduction 3 in which some energy is fed in, for instance as thermal energy or by an excitation with an electromagnetic wave or a mechanical stress, even if the excitation does not match exactly the soliton solution. This feature explains the interest of solitons in physics because, if a system possesses the necessary properties to allow the existence of solitons, it is highly likely that any large excitation will indeed lead to their formation. Moreover, as we shall show later, many physical systems meet the necessary criteria to sustain solitons, at least for some range of excitations. Very often solitons provide a fruitful approach to describe the physics of a nonlinear system. Rather than making a linear approximation and then attempting to take into account nonlinearities as perturbations, it may be much more efﬁcient to approximately describe the physics of the system by the most appropriate soliton equation and then to consider the possible perturbations of the exact soliton solution to improve the theory. The goal of this book is to explain the physics of solitons by showing how this concept enters in many areas of physics. It proposes a three-step journey in the world of solitons:

r The ﬁrst part introduces the main classes of soliton equations from examples chosen in macroscopic physics. For each case we start from a simple situation where a direct observation of solitons is easy, and we show how the basic laws of physics lead to nonlinear ﬁeld equations having soliton solutions. This part introduces the main properties of solitons and explains the basic features that a physical system must have in order to allow their existence. The last chapter of Part I discusses in detail the process leading to modelling the physical properties of a system in terms of solitons. We consider the example of plasma physics and show in particular how one given system can be described by several equations depending on the situation of interest. r The second part introduces some mathematical methods for the study of solitons. The mathematical aspects are not the primary aim of the book but the methods that we introduce in this part are relevant for physics because a real system is never exactly described by a soliton equation. It is therefore necessary to study and evaluate the role of the features which were neglected when a soliton equation was derived for the system. In this part we introduce some methods which are seldom discussed in the mathematical theory of solitons precisely because they are relevant for systems which are not exactly described by a soliton equation. Another topic of interest for a physicist is the time evolution of a given initial condition. An answer can be provided by a very elegant mathematical theory, the inverse scattering method, which succeeds in reducing the derivation of the solution of a nonlinear equation to a series of linear steps. We give an introduction to this method.

4 Introduction r The last step, presented in Parts III and IV of the book, is devoted to the physics of microscopic systems, such as atomic, solid state or biomolecular physics, where solitons have been used to study various problems. At such a scale, one cannot ‘see’ the solitons. They must be detected indirectly by their role in the properties of the system. Therefore, beyond the derivation of the soliton equations, as it was done in Part I, it is also necessary to discuss how solitons can be detected. Moreover, at the microscopic scale, thermal fluctuations can no longer be neglected. They can interact with the solitons, which must be studied in a background which is not at rest but coupled to a thermal bath. Besides their interactions with the thermal fluctuations, solitons themselves can play a role in the thermodynamic properties of a system, which can be significantly affected by their existence. Using the examples of ferroelectric materials and DNA we show that the concept of solitons can be a powerful tool to theoretically investigate the thermodynamic properties of some systems. The beauty of nonlinear science is in the links that it exhibits between very different systems which share common mathematical properties. The generality of the theory is similar to that of thermodynamics which can put an extreme variety of systems in a common framework. This book shows how a few fundamental equations can be applied to a wide range of physical situations, from the macroscopic to the microscopic scale, unifying topics which are often considered as completely different such as hydrodynamics and the dynamics of biological molecules. Of course there are still plenty of open questions. The concepts that we introduce are beginning to be applied in domains as different as biology, sociology, economy, epidemiology, ecology etc. We hope that this book will stimulate the reader to explore these open questions, keeping in mind that the astonishing ability of many systems to create extremely stable spatially coherent structures can have a profound influence on their properties.

Part I Different classes of solitons

1 Nontopological solitons: the Korteweg– de Vries equation

This chapter introduces the concept of solitons by discussing their ﬁrst experimental observation. Then it studies their main features, and particularly the conditions that are required for their existence. It also shows how the soliton solution can be derived with an elementary method which can be used to predict the possible existence of solitary waves in various systems. Then we consider a ﬁrst physical example, easy to build experimentally, which allows us to explain in detail the process that leads to the introduction of a contin- uous soliton equation to describe a discrete system, i.e. a system made of separate elements connected into a network. This is a very common situation, for instance in solid state physics, as discussed in Part III. A second example, the case of blood pressure waves, shows that solitons can exist even in situations where they would not be expected!

1.1 The discovery 1.1.1 John Scott Russell’s observations The very ﬁrst observation of a soliton was made in 1834 by the hydrodynamic engineer John Scott Russell while he was riding his horse along a canal near Edinburgh. When a barge abruptly stopped he was struck by the sight of what he called ‘the great solitary wave’, that he followed for a few miles before losing it in the meanders of the canal. The description that he gave shows the enthusiasm of a scientist who then devoted about ten years of his life to study this phenomenon. As we shall see later it contains all the basic ingredients which are required to derive and solve an equation that allows the analysis of the observations:

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent

8 The Korteweg–de Vries equation

Figure 1.1. Schematic picture of the time evolution of a perturbation of the water surface in a reservoir, driven by a piston moving downward or upward.

agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel, apparently without change of form or diminution of speed. I followed it on horseback and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished and after a chase of one or two miles I lost it in the winding of the channel. Such in the month of August 1834 was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.

A full theoretical understanding of John Scott Russell’s observation had to wait until 1895 with the studies of Korteweg and de Vries who derived the equation which nowadays bears their names (abbreviated as the KdV equation). This equation was however in an implicit form [131] in the earlier studies of Joseph Valentin de Boussinesq (1842–1929) published in 1872 [33]. The KdV equation is one of the prototype equations of soliton theory because it has remarkable mathematical properties. Its study leads to the understanding of the fundamental ideas that lie behind the soliton concept, but its derivation from the basic equations of hydrodynamics is tedious. It is given in Appendix A and is only valid when the depth of the ﬂuid and the height of the wave are small with respect to its spatial extent along the direction in which it propagates. One can notice that the second condition matches the observation of John Scott Russell who describes a wave which is thirty feet long and one and a half feet high. Figure 1.1 shows a schematic picture of the device used by John Scott Russell to experimentally investigate ‘the great solitary wave’. Waves are generated by the motion of a piston at the end of a canal. John Scott Russell observed the following features:

1.1 The discovery 9 r Depending on its amplitude, the initial perturbation can create one, two or several solitary waves. r √ Nonlinear waves have a speed higher than the speed c0 = gh of long- wavelength linear waves (where g is the strength of the gravitational ﬁeld and h the water depth in the canal). The deviation from c0 is proportional to the height η of the wave, so that the speed of nonlinear waves evolves according to the law v = c0(1 + Aη), where A is a positive constant. r There are no solitary waves with a negative amplitude, i.e. that would move as localised pits.

The studies of John Scott Russell triggered a lot of controversy in the scientiﬁc community of his time, which assumed that nonlinear effects were of secondary importance. Many debates were raised, which highlights how surprising the properties of solitons are. Hydrodynamic solitons are dynamic structures. They move with a constant speed and shape, but they cannot exist at rest. On both sides of the soliton the state of the medium is the same. They are called nontopological solitons in contrast to another class of solitons, introduced in Chapter 2, which interpolates between two different states of a medium, and can exist at rest.

Following the observation of a solitary wave on a canal near Edinburgh, the Scottish engineer JOHN SCOTT RUSSELL (1808–82) devoted many years of his life to investigate the soliton phenomenon. To him this discovery was a real rev- elation. Unfortunately, at the time people never shared his enthusiasm, and one had to wait more than 130 years until scientists really understood how important his discovery was. As a son of a clergyman he was expected to per- petuate this tradition [62] but his passion for sci- ences turned him otherwise! (1850 photograph) John Scott Russell graduated at the age of 16 from Glasgow University, after studying in St Andrews and Edinburgh Universities. Although he had a real talent for scientiﬁc studies, he decided to work for two years in industry. After this short period he attempted to come back to Edinburgh University as a teacher. Although he had well acknowledged teaching abilities, and in spite of a very laudatory recommendation letter from John Hamilton, his application was not successful and the position was given to James David Forbes, known for contributions to the theory of heat transfer and glaciology.

10 The Korteweg–de Vries equation

In contrast his career as an engineer was very bright. He invented an improved steam-driven road carriage in 1833, which quickly gave him a good reputation as an inventor. This is why the ‘Union Canal Society’ of Edinburgh asked him to set up a navigation system with steam boats on the canals of Edinburgh and Glasgow, in order to replace the boats drawn by horses. He was hired to design new barges thanks to experiments performed in a part of the canal devoted to his studies. It was during these investigations, six miles from the centre of Edinburgh, and very close to the present campus of the Heriot–Watt University, that he observed a soliton for the first time in August 1834. Following this discovery he performed several experiments on canals, rivers and lakes, but also in a specially designed, 10 m long tank, in his backyard. He gave a first report in 1838 before publishing the results of several experiments in 1844 [162]. This paper was however very badly received by two scientists who ruined all his expectations. First the well known astronomer Sir G. B. Airy (1801–92) strongly criticised his work in a paper on waves and tides which appeared in 1845. The main argument of Airy was that the formula derived by John Scott Russell from his experiments did not agree with his own theory of shallow water waves! Although he had studied the work of Russell more carefully, G. G. Stokes, one of the found- ing fathers of fluid mechanics, concluded that a solitary wave could not exist in a nonviscous fluid. This stopped all the research of John Scott Russell. He turned his attention to other fields and performed one of the first experimental measurements of the frequency shift of a moving source, which the Austrian physicist Doppler (1803–53) had described earlier, and which is now called the Doppler effect. He also contributed to the design of a gigantic boat, the ‘Great Eastern’, 207 m long and 25 m wide, which installed the first transatlantic cable, connecting England to the United States. Nobody knew at that time that another class of solitons than the one discovered by John Scott Russell, the optical solitons, would then become the best candidates for transatlantic telecommunications in the twenty-first century! However John Scott Russell did not die without a well deserved recognition of his achievements when the French scientist Joseph Valentine de Boussinesq (1842–1929) proposed a new theory of shallow water waves, which had solutions which agreed with his observations. These results were confirmed in 1876 by some investigations by Lord Rayleigh (1842–1919), who, ironically, was Stokes’ former student. Then, in 1885, Adhémar Jean-Claude Barré de Saint Venant (1797–1886) established a correct mathematical theory for these phenomena, which explained Airy and Stokes’ mistakes. During a meeting devoted to solitons and their applications, which took place at the Heriot–Watt University of Edinburgh in 1982, scientists tried to recreate the solitary wave in the famous canal in which John Scott Russell had seen it for the first