Periodic waves for the cubic-quintic non-linear Schr\"odinger equation: existence and orbital stability

TL;DR

This paper establishes the existence and orbital stability of explicit periodic standing waves with dnoidal profiles for the cubic-quintic nonlinear Schrödinger equation, using the implicit function theorem and spectral analysis.

Contribution

It introduces a method to construct explicit periodic waves and analyzes their stability, advancing understanding of wave behavior in nonlinear Schrödinger equations.

Findings

Existence of explicit dnoidal periodic waves.

Monotonicity of the period map with respect to energy levels.

Orbital stability of the constructed waves in the energy space.

Abstract

In this paper, we prove existence and orbital stability results of periodic standing waves for the cubic-quintic nonlinear Schr\"odinger equation. We use the implicit function theorem to construct a smooth curve of explicit periodic waves with \textit{dnoidal} profile and such construction can be used to prove that the associated period map is strictly increasing in terms of the energy levels. The monotonicity is also useful to obtain the behaviour of the non-positive spectrum for the associated linearized operator around the wave. Concerning the stability, we prove that the dnoidal waves are orbitally stable in the energy space restricted to the even functions.