On stability of small solitons of the 1--D NLS with a trapping delta potential

TL;DR

This paper investigates the stability and asymptotic behavior of small solitons in a 1D nonlinear Schrödinger equation with a delta potential, extending previous results and analyzing dispersion in defocusing cases.

Contribution

It generalizes recent stability results for small solitons in the 1D NLS with delta potential and explores dispersion in defocusing scenarios.

Findings

Stability of small solitons under general nonlinearities

Asymptotic behavior of solutions in trapping delta potential

Dispersion results for defocusing equations with non-trapping delta potential

Abstract

We consider a Nonlinear Schr\"odinger Equation with a very general non linear term and with a trapping $\delta $ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. We give also a result of dispersion in the case of defocusing equations with a non--trapping delta potential.