Linear and Nonlinear Dispersive Waves
So far, in the previous chapters, we have considered nonlinear systems with finite degrees of freedom. The dynamics of these systems are governed by nonlinear ordinary differential equations (and maps). However, very often, the variation of a particular p
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So far, in the previous chapters, we have considered nonlinear systems with finite degrees of freedom. The dynamics of these systems are governed by nonlinear ordinary differential equations (and maps). However, very often, the variation of a particular physical property may depend in a continuous fashion on both space and time variables. Typical examples include the vibrations of a string, propagation of electromagnetic waves, waves on the surface of water and so on. Again such systems may be classified into linear and nonlinear ones. In these cases, the dynamics of the underlying systems are governed by partial differential equations and they are often considered to be infinite dimensional (in phase space) or as infinite degrees of freedom continuous systems. We are all familiar with many of the linear systems described by linear partial differential equations like the sound equation, heat equation, Maxwell equations, Laplace equation, Schrodinger equation and so on. Naturally one would expect many interesting and novel phenomena to occur in nonlinear continuous systems also in analogy with the case of finite dimensional systems. Among the nonlinear continuous systems, we will concentrate on the special case of the so-called nonlinear dispersive wave systems in the present and next few chapters while we will relegate the study of nonlinear diffusive systems to Chap. 15. The main reason is that a certain class of these nonlinear dispersive systems exhibits an exciting type of wave solution of finite energy, having remarkable stability properties, called soliton. Physically the solitons behave like stable particles, exhibiting in general elastic collision property on collision with other solitons in one spatial dimension. They occur in a wide variety of physical systems and lead to the identification of completely integrable, infinite dimensional, nonlinear dynamical systems. As the solitons are basically solitary waves, we will study such waves also in some detail in this chapter. As a prelude to such a study we will start our discussion with an introduction to the properties of linear dispersive waves.
11.1 Linear Waves We are all familiar with different kinds of waves even in our day to day life. Human beings are always fascinated by them. But what is a wave? One
M. Lakshmanan et al., Nonlinear Dynamics © Springer-Verlag Berlin Heidelberg 2003
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11. Linear and Nonlinear Dispersive Waves
can define it to be the result of some physical event that sends tremors or disturbances through a medium. Such tremors carry energy away from the point of disturbance [1].
Examples: (1) When a stone is thrown into a still pond, the waves radiate in a circular pattern: the energy given by the stone is transferred into the motion on the surface of the pond. This motion sets in waves that travel in the form of periodic undulations with crests and troughs but overall in the form of a wave packet. (2) When a girl plays (on) a veena, the notes we hear, like all sounds, are the result of vibrations pulsating through the air. (
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