Solitary Waves, Bound Soliton States and Chaotic Soliton Trains in a Dissipative Boussinesq-Korteweg-de Vries Equation
A solitary wave that, eventually, becomes a soliton represents a unique and very attractive object in modern nonlinear science of spatially extended systems. The term “soliton” was coined by N.J. Zabusky and M.D. Kruskal [2.30 ] to characterize solitary w
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1 Introduction and Motivation A solitary wave that, eventually, becomes a soliton represents a unique and very attractive object in modern nonlinear science of spatially extended systems. The term “soliton” was coined by N.J. Zabusky and M.D. Kruskal [2.30] to characterize solitary waves in nonlinear systems whose properties (energy etc.) are mostly localized in a bounded region of space at any instant of time such that upon collision they act like particles (e.g. electron, proton, etc.) and, in the case they studied, elastically, crossing each other or interchanging places with no significant change of form. As such, solitons were for long time a concept in the realm of conservative (and integrable) systems and hence solutions of dissipationless nonlinear equations [2.2, 2.9, 2.10, 2.12, 2.21, 2.24]. Recently, this concept has been extended to driven-dissipative systems, where an appropriate input–output energy balance exists and helps sustain the particle-like traveling localized structure. The work of Zabusky and Kruskal referred to the Korteweg–de Vries (1895) equation, an equation earlier explicitly derived by Boussinesq (1872, 1877) (Fig. 2.1) [2.4–6,2.8,2.14]. Accordingly, in this book we shall refer to it as the Boussinesq–Korteweg– de Vries (BKdV) equation (Fig. 2.1). The localized structure and hence the solitary wave solution of the BKdV equation result from the appropriate (local) balance between nonlinearity (velocity depends on amplitude) and dispersion (velocity depends on color or wavelength), as Boussinesq and Lord Rayleigh [2.22] clearly understood (see, however, the work by Ursell [2.27]). Since the pioneering work by Zabusky and Kruskal, intensive and highly fruitful research has developed on solitons and soliton-bearing equations and related matters. In fact a new chapter in applied mathematics and a new research line with broad scope in physics and other sciences has appeared. Examples are the phenomena of self-focusing and self-induced transmittancy in optics, the description of developed turbulence in plasma, the interpretation of solitons as particles in field theory, the exploration of statistical properties of some models in solid-state physics and hydrodynamics, solitons in Josephson junctions, action potentials in neurodynamics, and so on. On the one hand, many numerical computations have been done on soliton-bearing equations. On the other hand, new analytical tools for solving nonlinear PDEs have been developed. As a result of these investigations the V. I. Nekorkin et al., Synergetic Phenomena in Active Lattices © Springer-Verlag Berlin Heidelberg 2002
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2. Solitary Waves, Bound Soliton States, and Chaotic Soliton Trains
fundamental role of solitons in nonlinear science has, indeed, been established. It has been found that they are stable to finite perturbations and capable of self-restoration after interaction with one another. Moreover a major finding is that they significantly define the character of evolution of wave perturbations in many nonlinear spatially extended systems. In oth
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