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Existence and spectral stability of small-amplitude periodic waves for the 2D nonlinear focusing Schrodinger equation

Fabio Natali

TL;DR

This work proves the existence of small-amplitude double-periodic standing waves for the 2D focusing nonlinear Schrödinger equation on a bi-torus and establishes their spectral stability with respect to perturbations of the same period. Existence is obtained through local and global bifurcation theory, with complementary variational minimization arguments that yield periodic minimizers. For both $p=1$ and general $p\ge 2$, explicit small-amplitude expansions are derived, and global continuation is shown for $c>2/p$. Spectral stability is then established using the Kapitula–Kevrekidis–Sandstede Krein-index framework, demonstrating stability for the small-amplitude waves in the appropriate periodic function spaces, while also highlighting the limitations on orbital stability depending on kernel structure and symmetry.

Abstract

The purpose of this paper is to establish the existence and spectral stability, with respect to perturbations of the same period, of double-periodic standing waves for the nonlinear focusing Schrödinger equation posed on the bi-dimensional torus. We first show that such double-periodic solutions can be constructed via local and global bifurcation theory, under the assumption that the kernel of the linearized operator around the equilibrium solution is one-dimensional. In addition, we prove that these local and global solutions minimize an appropriate variational problem, which enables us to derive spectral properties of the linearized operator about the periodic wave. Finally, we establish the spectral stability of small-amplitude periodic waves by applying the techniques developed in \cite{KapitulaKevrekidisSandstedeII} and \cite{KapitulaKevrekidisSandstedeI}.

Existence and spectral stability of small-amplitude periodic waves for the 2D nonlinear focusing Schrodinger equation

TL;DR

This work proves the existence of small-amplitude double-periodic standing waves for the 2D focusing nonlinear Schrödinger equation on a bi-torus and establishes their spectral stability with respect to perturbations of the same period. Existence is obtained through local and global bifurcation theory, with complementary variational minimization arguments that yield periodic minimizers. For both and general , explicit small-amplitude expansions are derived, and global continuation is shown for . Spectral stability is then established using the Kapitula–Kevrekidis–Sandstede Krein-index framework, demonstrating stability for the small-amplitude waves in the appropriate periodic function spaces, while also highlighting the limitations on orbital stability depending on kernel structure and symmetry.

Abstract

The purpose of this paper is to establish the existence and spectral stability, with respect to perturbations of the same period, of double-periodic standing waves for the nonlinear focusing Schrödinger equation posed on the bi-dimensional torus. We first show that such double-periodic solutions can be constructed via local and global bifurcation theory, under the assumption that the kernel of the linearized operator around the equilibrium solution is one-dimensional. In addition, we prove that these local and global solutions minimize an appropriate variational problem, which enables us to derive spectral properties of the linearized operator about the periodic wave. Finally, we establish the spectral stability of small-amplitude periodic waves by applying the techniques developed in \cite{KapitulaKevrekidisSandstedeII} and \cite{KapitulaKevrekidisSandstedeI}.

Paper Structure

This paper contains 10 sections, 14 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Existence of periodic waves and spectral analysis
  3. Existence of periodic solutions via local and global bifurcation for the case $p=1$
  4. Existence of periodic minimizers - Case $p=1$
  5. Existence of periodic solutions - Case $p\geq2$
  6. Spectral stability
  7. Remarks on the existence and stability of small-amplitude periodic waves
  8. Existence of periodic waves - Case $p=1$.
  9. Existence of periodic waves - Case $p\geq2$
  10. Spectral stability of periodic waves.

Key Result

Theorem 1.3

Let $p\geq1$ be a fixed integer. Consider the small-amplitude periodic wave solution $\varphi\in H^1_{\rm per,s}$ of the equation ode which solves the minimization problem in $(infB2)$. The wave $\Psi = (\varphi, 0)$ is spectrally stable for all $c\sim\frac{2}{p}^{+}$ in the sense of Definition def-

Theorems & Definitions (41)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 31 more