Nonlinear stability of multi-solitons for the Hirota equation

Abstract

In this paper, we study the nonlinear stability of solitons and multi-solitons for the following Hirota equationiut+uxx+2|u|2u+iα(uxxx+6|u|2ux)=0,(t,x)∈R2,α∈R. Firstly, by employing the variational approach with suitable conservation laws, we establish the orbital stability of the Hasimoto solitons in Sobolev space H1(R). Secondly, by choosing a suitable Lyapunov functional and investigating the spectrum of second variation operator around the solitons, we derive that the Hasimoto solitons are also orbitally stable in Hm(R) with m∈N for any α∈R. Finally, pure N-solitons of the Hirota equation are shown to be dynamically stable in HN(R). By constructing a suitable Lyapunov functional, it is found that the multi-solitons are non-isolated constrained minimizers satisfying a 2N order elliptic equation, and the dynamical stability issue is reduced to study of the spectrum of explicit linearized systems. Our approach in the spectral analysis consists in an invariant for the multi-solitons and new operator identities motivated by recursion operators of the Hirota equation. As a direct consequence, we recover a new proof of the stability of N-solitons for the nonlinear Schrödinger equation [24].