On stability properties of the Cubic-Quintic Schrodinger equation with delta-point interaction

TL;DR

This paper investigates the stability of peak-standing-wave solutions in the cubic-quintic nonlinear Schrödinger equation with delta-point interaction, combining analytical and numerical methods to analyze spectral and orbital stability.

Contribution

It introduces a novel approach using extension theory of symmetric operators to determine the Morse index for stability analysis in this context.

Findings

Identified conditions for orbital stability and instability.

Determined the Morse index for specific self-adjoint operators.

Established spectral instability implications.

Abstract

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrodinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the case of an attractive-attractive case and an focusing interaction we give an approach based in the extension theory of symmetric operators for determining the Morse index.