On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation

TL;DR

This paper analyzes the well-posedness and stability of black solitons in a one-dimensional Gross-Pitaevskii equation with a Dirac potential, revealing conditions for stability and bifurcation of stationary solutions.

Contribution

It provides a detailed analysis of the Cauchy problem for a singularly perturbed Gross-Pitaevskii equation and characterizes the stability of stationary solutions under Dirac perturbations.

Findings

Persistence of black solitons under perturbation

Existence of a new branch of stationary waves

Stability or instability depending on perturbation type

Abstract

We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we show the persistence of the stationary black soliton of the unperturbed problem as a solution. We also prove the existence of another branch of non-trivial stationary waves. Depending on the attractive or repulsive nature of the Dirac perturbation and of the type of stationary solutions, we prove orbital stability via a variational approach, or linear instability via a bifurcation argument.