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Stability of Cnoidal waves for the damped nonlinear Schrödinger equation

Paolo Antonelli

https://orcid.org/0000-0001-8211-8296

Gran Sasso Science Institute, viale Francesco Crispi, 7, L’Aquila 67100, Italy

E-mail Address: paolo.antonelli@gssi.it

Corresponding author.

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and

Boris Shakarov

https://orcid.org/0009-0004-0877-9941

Institut de Mathématiques de Toulouse, 118, Route de Narbonne, Toulouse F-31062, France

E-mail Address: boris.shakarov@math.univ-toulouse.fr

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https://doi.org/10.1142/S0219891624500176Cited by:0 (Source: Crossref)

Abstract

In this paper, we consider the cubic nonlinear Schrödinger (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 the 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.

Communicated by F.Ancona

Keywords:

AMSC: 35Q55, 37L15

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