An explicitly solvable NLS model with discontinuous standing waves

TL;DR

This paper analyzes a solvable nonlinear Schrödinger equation model with a point interaction, revealing explicit stationary states and their dependence on parameters, including existence, uniqueness, and bifurcation phenomena.

Contribution

It introduces an explicitly solvable NLS model with a combined delta and dipole point interaction, providing exact stationary solutions and detailed bifurcation analysis.

Findings

Existence and uniqueness of ground states at any mass for subcritical nonlinearity

Existence of excited states beyond a mass threshold

Explicit stationary states and their dependence on interaction strength

Abstract

We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction.