Discontinuous Ground States for NLSE on $\mathbb{R}$ with a F\"{u}l\"{o}p-Tsutsui $\delta$ interaction

TL;DR

This paper investigates the existence and stability of ground states in a one-dimensional nonlinear Schrödinger equation with a F"{u}l"{o}p-Tsutsui delta defect, using variational methods and stability theory.

Contribution

It introduces a novel analysis of ground states with discontinuous delta defects in NLSE, combining variational and stability techniques.

Findings

Existence of ground states proved via variational methods.

Stability established using Grillakis-Shatah-Strauss theory.

Ground states can be discontinuous at the defect location.

Abstract

We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schr\"{o}dinger equation with a focusing power nonlinearity and a defect located at the origin. In this paper a ground state is defined as a global minimizer of the action functional on the Nehari manifold and the defect considered is a F\"{u}l\"{o}p-Tsutsui $\delta$ type, namely a $\delta$ condition that allows discontinuities. The existence of ground states is proved by variational techniques, while the stability results from the Grillakis-Shatah-Strauss theory.