Orbital stability of kinks in the NLS equation with competing nonlinearities

TL;DR

This paper rigorously proves the orbital stability of kink solutions in the nonlinear Schrödinger equation with competing nonlinearities, addressing degeneracy issues near nonzero equilibria and extending results from the cubic-quintic case to general scenarios.

Contribution

It provides a rigorous proof of orbital stability for kinks in NLS equations with competing nonlinearities, including detailed analysis for the cubic-quintic case and its extension.

Findings

Established orbital stability of kinks in the cubic-quintic NLS.

Extended stability proof to general NLS with competing nonlinearities.

Addressed degeneracy issues near nonzero equilibria.

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.