Model equations for long waves in nonlinear dispersive systems

Abstract

Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation u$_{t}$ + u$_{x}$ + uu$_{x}$ - u$_{xxt}$ = 0, (a) whose solution u(x,t) is considered in a class of real nonperiodic functions defined for -$\infty $ < x < $\infty $, t $\geq $ 0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation u$_{t}$ + u$_{x}$ + uu$_{x}$ + u$_{xxx}$ = 0, (b) with which (a) is to be compared in various ways. It is contended that (a) is in important respects the preferable model, obviating certain problematical aspects of (b) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics. In section 2 the origins and immediate properties of equations (a) and (b) are discussed in general terms, and the comparative shortcomings of (b) are reviewed. In the remainder of the paper (section section 3, 4) - which can be read independently of the preceding discussion - an exact theory of (a) is developed. In section 3 3 the existence of classical solutions is proved; and following our main result, theorem 1, several extensions and sidelights are presented. In section 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of (a). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of (a) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of section 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in section 3 is established.