Nonlinear PDE Phenomena

Abstract

In Section 2.6 the reader encountered the Korteweg-deVries (KdV) equation which has been successfully used to describe the propagation of solitons in various physical contexts, the most historically famous being for shallow water waves in a rectangular canal. Using subscripts to denote partial derivatives with respect to the distance x travelled and the time t elapsed, the KdV equation for the (normalized) transverse displacement ψ of the water waves is $$ {\psi _t} + \alpha \psi {\psi _x} + {\psi _{xxx}} = 0 $$ ((9.1)) Soliton solutions of such a nonlinear PDE are stable (stable against collisions) solitary waves. Recall that a solitary wave is a localized pulse which travels at constant speed without change of shape despite the “competition” between the nonlinearity and other (e.g., dispersive) terms. Not all nonlinear PDEs support solitary waves. How does one go about finding solitary wave solutions to a given nonlinear PDE or search for other possible physically important solutions? In this chapter, a few of the basic analytic methods for studying nonlinear PDE phenomena will be outlined. The survey will be far from complete, our goal being to study some of the more important phenomena. For example, the Lie symmetry method, which provides a sophisticated mathematical approach to studying PDEs and for which Maple has a liesymm package, will not be covered. For simplicity, we shall further confine our attention to PDEs which involve only one spatial dimension and time.KeywordsSolitary WaveTravel Wave SolutionSolitary Wave SolutionPhase Plane AnalysisNonlinear SuperpositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.