Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations

TL;DR

This paper proves the stability and asymptotic stability in the energy space of a sum of N solitons for subcritical gKdV equations, extending known results even for the classical KdV case.

Contribution

It establishes stability and asymptotic stability of multi-soliton solutions in H^1 for subcritical gKdV equations, including the classical KdV, using energy and monotonicity methods.

Findings

Stability of N solitons in H^1 for subcritical gKdV.

Asymptotic stability derived from a rigidity theorem.

Results are new even for the classical KdV (p=2).

Abstract

We prove in this paper the stability and asymptotic stability in H^1 of a decoupled sum of N solitons for the subcritical generalized KdV equations $u_t+(u_{xx}+u^p)_x=0$ (1<p<5). The proof of the stability result is based on energy arguments and monotonicity of local L^2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [15].