Nonlocal models for nonlinear, dispersive waves

Abstract

Considered herein are model equations for the unidirectional propagation of small-amplitude, nonlinear, dispersive, long waves such as those governed by the classical Korteweg-de Vries equation. Of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial, so that the operator modelling dispersive effect is nonlocal. Particular cases in view here are the Benjamin-Ono equation and the intermediate long-wave equation which arise in internal-wave theory, and Smith's equation which governs certain types of continental-shelf waves. The initial-value problem for these equations is shown to be globally well posed in the classical sense, including continuous dependence upon the initial data and, in certain cases upon the modelling of nonlinear and dispersive effects. Whilst the results are stated for the specific equations listed above, the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear, dispersive evolution equations. Particularly worthy of note is our theorem implying that solutions of the intermediate long-wave equation converge strongly to solutions of the Korteweg-de Vries equation, or to solutions of the Benjamin-Ono equation, in appropriate asymptotic limits.