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Stability of Cnoidal Waves for the Damped Nonlinear Schrödinger Equation

Paolo Antonelli, Boris Shakarov

TL;DR

The paper analyzes the stability of the cubic NLS on the 1D torus under a small linear damping $ε$, showing that the family of cnoidal waves remains orbitally stable when the mass decays as $m(t)=m_0 e^{-2ε t}$. It introduces a modulation framework around evolving cnoidal waves and, because exact dissipative solitons do not exist, constructs a first-order corrected profile $Q_{m,ε}$ to achieve uniform coercivity and improved $H^1$ control via a modified Lyapunov functional. A key technical ingredient is the uniform coercivity of the linearized operator on the even half-anti-periodic subspace, which, together with the mass-damping dynamics, yields robust stability bounds for the perturbations. The work thereby extends orbital stability concepts to dissipative, periodic settings and provides a rigorous mechanism for dissipative soliton dynamics through the introduction of an approximate dissipative profile and a corresponding Lyapunov framework.

Abstract

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 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.

Stability of Cnoidal Waves for the Damped Nonlinear Schrödinger Equation

TL;DR

The paper analyzes the stability of the cubic NLS on the 1D torus under a small linear damping , showing that the family of cnoidal waves remains orbitally stable when the mass decays as . It introduces a modulation framework around evolving cnoidal waves and, because exact dissipative solitons do not exist, constructs a first-order corrected profile to achieve uniform coercivity and improved control via a modified Lyapunov functional. A key technical ingredient is the uniform coercivity of the linearized operator on the even half-anti-periodic subspace, which, together with the mass-damping dynamics, yields robust stability bounds for the perturbations. The work thereby extends orbital stability concepts to dissipative, periodic settings and provides a rigorous mechanism for dissipative soliton dynamics through the introduction of an approximate dissipative profile and a corresponding Lyapunov framework.

Abstract

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 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 -norm of the perturbation around the approximated solitons.
Paper Structure (8 sections, 14 theorems, 304 equations)

This paper contains 8 sections, 14 theorems, 304 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Periodic spaces
  4. Jacobi elliptic functions
  5. Linearized Operator
  6. Lyapunov control around the modulated solitary wave
  7. The approximated solitary wave and improved error bounds
  8. Appendix

Key Result

Theorem 1.1

Let $\psi_0 \in H^1(\mathbb T) \cap \mathcal{A}^+$, $\psi \in C([0,\infty), H^1(\mathbb T) \cap \mathcal{A}^+)$ be the corresponding solution and let $m$ be defined by eq:mass2, where $m_0=M[\psi_0]$. There exists $\varepsilon^* = \varepsilon^*(m_0)> 0$ such that if then there exist $\gamma\in C^1(\mathbb R;\mathbb R)$ and $C = C(m_0) >0$ such that, for any $t >0$, we have

Theorems & Definitions (30)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 20 more