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Orbital stability of kinks in the NLS equation with competing nonlinearities

Justin Holmer, Panayotis G. Kevrekidis, Dmitry E. Pelinovsky

TL;DR

This work provides a rigorous orbital stability proof for kink solutions in the nonlinear Schrödinger equation with competing nonlinearities, beginning with the cubic–quintic model $i\psi_t = \psi_{xx} - \psi + 4|\psi|^2\psi - 3|\psi|^4\psi$ and its heteroclinic kink $\phi$. The authors develop a energy-expansion framework around the kink, establish spectral coercivity in a weighted space $\mathcal{H}_R$ with a crucial condition $\phi(R) > \sqrt{2}/\sqrt{3}$, and construct a two-parameter constrained decomposition to account for phase and translation symmetries. They prove a global-in-time bound showing that perturbations remain close to the kink orbit, up to phase and translation, and verify the key spectral criterion numerically. The results extend to general competing-power nonlinearities and provide a robust methodology for rigorous stability analyses of kinks in NLS-type models with nonzero background, with relevance to nonlinear optics and ultracold atomic gases.

Abstract

Kinks connecting zero and nonzero equilibria in the NLS equation with competing nonlinearities occur at the special values of the frequency parameter. Since they are minimizers of energy, they are expected to be orbitally stable in the time evolution of the NLS equation. However, the stability proof is complicated by the degeneracy of kinks near the nonzero equilibrium. The main purpose of this work is to give a rigorous proof of the orbital stability of kinks. We give details of analysis for the cubic--quintic NLS equation and show how the proof is extended to the general case.

Orbital stability of kinks in the NLS equation with competing nonlinearities

TL;DR

This work provides a rigorous orbital stability proof for kink solutions in the nonlinear Schrödinger equation with competing nonlinearities, beginning with the cubic–quintic model and its heteroclinic kink . The authors develop a energy-expansion framework around the kink, establish spectral coercivity in a weighted space with a crucial condition , and construct a two-parameter constrained decomposition to account for phase and translation symmetries. They prove a global-in-time bound showing that perturbations remain close to the kink orbit, up to phase and translation, and verify the key spectral criterion numerically. The results extend to general competing-power nonlinearities and provide a robust methodology for rigorous stability analyses of kinks in NLS-type models with nonzero background, with relevance to nonlinear optics and ultracold atomic gases.

Abstract

Kinks connecting zero and nonzero equilibria in the NLS equation with competing nonlinearities occur at the special values of the frequency parameter. Since they are minimizers of energy, they are expected to be orbitally stable in the time evolution of the NLS equation. However, the stability proof is complicated by the degeneracy of kinks near the nonzero equilibrium. The main purpose of this work is to give a rigorous proof of the orbital stability of kinks. We give details of analysis for the cubic--quintic NLS equation and show how the proof is extended to the general case.

Paper Structure

This paper contains 12 sections, 5 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Main result
  4. Methodology and organization of the paper
  5. Proof of the orbital stability of kinks
  6. Expansion of the energy
  7. Spectral estimates
  8. Decomposition near the kink orbit
  9. Proof of Theorem \ref{['th-main']}
  10. Numerical verification of the spectral criterion
  11. Generalization for competing power nonlinearity
  12. Proof of the local and global well-posedness in $\mathcal{E}$

Key Result

Theorem 1

Fix $R > 0$ sufficiently large such that $\phi(R) > \frac{\sqrt{2}}{\sqrt{3}}$. The kink with the profile $\phi$ is orbitally stable in the NLS equation (NLS) with respect to perturbations in $\mathcal{E} \cap \mathcal{H}_R$. To be precise, for every small $\epsilon > 0$ there is $\delta > 0$ such t

Figures (3)

  • Figure 1: Spectrum of the linearized operator (\ref{['operators']}) in the complex $\lambda$-plane, where $\lambda=\lambda_r + i \lambda_i$.
  • Figure 2: Left: plot of the lowest three numerically observed eigenvalues versus $R$, for the spectral problem \ref{['E:LR-trans']}. Right: plot of $(\lambda_1)_RF_R(0)$ versus $R$, showing that it remains positive and thus $F_R(0)<0$ for all $R$.
  • Figure 3: Plots of the first, second and third eigenfunctions obtained numerically for $R=-1$ (left) and $R=0.2$ (right). The horizontal axis is shown as $x$-values, although this computation was performed with the $z$ variable as described in the text.

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof