Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Schrödinger system with quintic nonlinearity: spectral stability of multiple sign-changing periodic waves

Guilherme de Loreno, Gabriel E. Bittencourt Moraes

TL;DR

This work analyzes spectral stability of periodic sign-changing standing waves in a coupled quintic NLS system, constructing cnoidal and snoidal (odd) periodic profiles and linearizing about synchronized waves. A diagonalization via a similarity transform reduces the problem to Hill operators, and the non-positive spectrum is examined through Floquet theory, the periodic Comparison Theorem, and Krein signature, yielding the Hamiltonian–Krein index that governs stability conclusions. The main findings show spectral instability for the cnoidal case in the full space in the case $B=1$, $\gamma=2\kappa$, while the snoidal profile admits detailed instability/stability results within the odd subspace, with several regimes remaining open in the full space. A corrected analysis of the VK-type criterion leads to stable orbital behavior for the snoidal wave in the subspace of odd functions, while the full-space problem remains inconclusive. Together, these results clarify how parity (odd vs full space) and parameter choices affect stability and highlight open problems in the quintic coupled NLS setting.

Abstract

This manuscript investigates the existence and spectral stability of multiple periodic standing wave solutions for a nonlinear Schrödinger system. By considering both cnoidal and snoidal profiles, we provide a comprehensive spectral analysis of the associated linearized operators, employing the Floquet theory and comparison theorems. Stability results are derived under periodic perturbations with the same period as the underlying standing waves. Furthermore, we apply the spectral stability theory via Krein signature to determine the spectral stability and instability results.

Schrödinger system with quintic nonlinearity: spectral stability of multiple sign-changing periodic waves

TL;DR

This work analyzes spectral stability of periodic sign-changing standing waves in a coupled quintic NLS system, constructing cnoidal and snoidal (odd) periodic profiles and linearizing about synchronized waves. A diagonalization via a similarity transform reduces the problem to Hill operators, and the non-positive spectrum is examined through Floquet theory, the periodic Comparison Theorem, and Krein signature, yielding the Hamiltonian–Krein index that governs stability conclusions. The main findings show spectral instability for the cnoidal case in the full space in the case , , while the snoidal profile admits detailed instability/stability results within the odd subspace, with several regimes remaining open in the full space. A corrected analysis of the VK-type criterion leads to stable orbital behavior for the snoidal wave in the subspace of odd functions, while the full-space problem remains inconclusive. Together, these results clarify how parity (odd vs full space) and parameter choices affect stability and highlight open problems in the quintic coupled NLS setting.

Abstract

This manuscript investigates the existence and spectral stability of multiple periodic standing wave solutions for a nonlinear Schrödinger system. By considering both cnoidal and snoidal profiles, we provide a comprehensive spectral analysis of the associated linearized operators, employing the Floquet theory and comparison theorems. Stability results are derived under periodic perturbations with the same period as the underlying standing waves. Furthermore, we apply the spectral stability theory via Krein signature to determine the spectral stability and instability results.
Paper Structure (8 sections, 11 theorems, 129 equations)

This paper contains 8 sections, 11 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Existence of a Smooth Curve of Periodic Waves
  3. Spectral analysis
  4. Spectral analysis for $\mathcal{L}$ in $\mathbb{H}^2_{\rm per}$ -- solution with cnoidal profile
  5. Spectral analysis for $\mathcal{L}$ in $H^2_{\rm per,odd}$ -- solution with snoidal profile
  6. Spectral stability
  7. Spectral Stability on subspace of odd functions
  8. Scalar NLS Equation: Orbital Stability on subspace of odd functions

Key Result

Theorem 1.2

The Cauchy problem associated with system NLS-system with initial data $U_0 = (u_0, v_0) \in H_{\text{per}}^1 \times H_{\text{per}}^1$ is locally well-posed. More precisely, for each $U_0 \in H_{\text{per}}^1 \times H_{\text{per}}^1$, there exists a time $T > 0$ and a unique solution $U = (u, v) \in is continuous.

Theorems & Definitions (24)

  • Remark 1.1
  • Theorem 1.2: Local Well-Posedness
  • Definition 1.3: Spectral Stability
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: Existence of Solutions with Cnoidal Profile
  • Proposition 2.2: Existence of Solutions with Snoidal Profile
  • Remark 2.3
  • Lemma 4.1
  • proof
  • ...and 14 more