Stability of Cnoidal Waves for the Damped Nonlinear Schr\"odinger Equation

TL;DR

This paper proves the orbital stability of cnoidal wave solutions in the damped cubic nonlinear Schrödinger equation on a torus, extending stability concepts to a non-Hamiltonian, dissipative setting.

Contribution

It establishes the orbital stability of cnoidal waves under damping by constructing an approximate solution and employing a Lyapunov functional to control perturbations.

Findings

Cnoidal waves are orbitally stable in the damped NLS.

A first order approximation accounts for dissipative effects.

Lyapunov functional effectively controls perturbations.

Abstract

We consider the cubic nonlinear Schr\"odinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation is well known to admit a family of periodic waves given by Jacobi elliptic functions of cnoidal type. We show that the family of cnoidal waves is orbitally stable. More precisely, by considering a sufficiently small perturbation of a given cnoidal wave at initial time, the evolution will always remain close (up to symmetries of the equation) to the cnoidal wave whose mass is modulated according to the dissipative dynamics. This result extends the concept of orbital stability to this non-Hamiltonian evolution. Since cnoidal waves are not exact solutions to the damped NLS, the perturbation is forced away from the family of solitary wave profiles. In order to control this secular growth of the error, we find a first order approximation of the solitary wave that takes into account the dissipative term. Then we use a suitable, exponentially decreasing Lyapunov functional that controls the $H^1$-norm of the perturbation around the approximated solitons.