ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF THEWHITHAM'S EQUATION

Abstract

In this thesis wok, the asymptotic behavior of the solutions of the initial value problem as t ! 1 is studied. Asymptotic formula, as t ! 1, uniform with respect to the spatial variables and with bounds for the re- mainder term, is found for the solutions of the Cauchy problem for dis- sipative equation of evolution, in particular, for the whitham equation. We used the method based on a detailed study of the nonlinear nonlocal equation in momentum representation and on incorporating of a special di- agram technique of the perturbation theory. The Whitham equation was proposed as an alternate model equation for the simpli ed description of unidirectional wave motion at the surface of an inviscid uid. An advan- tage of the Whitham equation over the Korteweg-de Vries(KdV) equation is that it provides a more faithful description of short waves of small am- plitude. The Whitham equation admits periodic traveling wave solutions. The other focus of this work is the stability of these solutions. The numer- ical results presented here suggest that all large-amplitude solutions are unstable, while small-amplitude solutions with large enough wavelength L are stable.