Nonlinear stability of strong rarefaction waves for the generalized KdV–Burgers–Kuramoto equation with large initial perturbation

Abstract

In was shown in Ruan et al. (2008) [3] that rarefaction waves for the generalized KdV–Burgers–Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small. The main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized KdV–Burgers–Kuramoto equation with large initial perturbation. In our results, we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in H 1 ( R ) , while for a general smooth nonlinear flux function, we need to ask for the L 2 -norm of the initial perturbation to be small but the L 2 -norm of the first derivative of the initial perturbation can be large and, consequently, the H 1 ( R ) -norm of the initial perturbation can also be large.