Stablity of solitary waves in higher order Sobolev spaces

Abstract

The orbital stability of solitary waves has generally been established in Sobolev classes of relatively low order, such as $H^1$. It is shown here that at least for solitary-wave solutions of certain model equations, a sharp form of orbital stability is valid in $L^2$-based Sobolev classes of arbitrarily high order. Our theory includes the classical Korteweg-de Vries equation, the Benjamin- Ono equation and the cubic, nonlinear Schrödinger equation.