Orbital stability of periodic traveling wave solutions to the generalized zakharov equations

Abstract

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iut+uxx=uv+|u|2u,vtt-vxx=(|u|2)xx.First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].