Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation

Abstract

We obtain conditions for the nonlinear stability (Theorem 5.1) of solitary waves fro two classes of nonlinear dispersive equations which arise in the mathematical description of long wave propagation:(u is in general complex-valued), where the linear operator L, typically nonlocal, and the nonlinearity, are of some general class.For the case of homogeneous nonlinearities, A(u), a new variational characterization of solitary waves for I and II is presented (Theorem 3.1) and is exploited in the dynamic stability analysis. The existence of solitary waves in higher dimensions is also considered (Theorem 7.1).