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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, col- lective effects arising from the interaction between particles in solid state physics, etc. But, in most of the cases, theoretical approaches were trying to avoid nonlin- earities, 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 en- gineer, John Scott Russell, the soliton initiated passion and debates. John Scott 1 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-85421-4 - Physics of Solitons Thierry Dauxois and Michel Peyrard Excerpt More information 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 his- tory 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 phe- nomenon 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 ex- hibits 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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-85421-4 - Physics of Solitons Thierry Dauxois and Michel Peyrard Excerpt More information 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. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-85421-4 - Physics of Solitons Thierry Dauxois and Michel Peyrard Excerpt More information 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.
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