Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation

Abstract

This chapter examines the nonlinearly resonant surface waves and homoclinic bifurcation. The behavior of steady nonlinear water waves on the surface of an inviscid heavy fluid layer has received much attention, both from the mathematical and from the physical side. The basic equations for the interaction of traveling nonlinear surface waves with in-phase external pressure waves are derived. An inviscid fluid layer of mean depth is considered under gravity. On its free upper surface, where capillary forces may also act, it supports nonlinear surface waves of permanent form, traveling from right to left with constant speed. The upper bound could be estimated explicitly by quantitatively showing the validity of subsequently described, almost identical transformations and estimates on noncritical eigenvalues of the operator. The linear dispersion relation for cnoidal waves corresponds to the imaginary eigenvalues. It is found that the reduced phase space is four-dimensional and the existence of a homoclinic orbit is equivalent to the intersection of two curves. The existence of solitary waves on a curve in the parameter space is also elaborated. © 1988 Academic Press Inc.