Stability of standing waves for NLS-log equation with $\delta$-interaction

TL;DR

This paper analyzes the stability of standing wave solutions with peak-Gausson profiles in a nonlinear logarithmic Schrödinger equation featuring delta interactions, addressing spectral challenges with advanced mathematical techniques.

Contribution

It provides a novel analytical approach to determine the orbital stability of standing waves in the NLS-log equation with delta interactions, including both attractive and repulsive cases.

Findings

Established conditions for stability and instability of standing waves.

Computed the number of negative eigenvalues of the linearized operator.

Applied perturbation and extension theories to spectral analysis.

Abstract

We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schr\"odinger equation with $\delta$-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.