Asymptotic stability of solitary waves for the $1d$ NLS with an attractive delta potential

TL;DR

This paper proves the asymptotic stability of solitary waves in a 1D nonlinear Schrödinger equation with an attractive delta potential, extending previous results to larger waves and broader spectral conditions.

Contribution

It establishes asymptotic stability for a family of solitary waves in the 1D NLS with delta potential, beyond small waves and orbital stability.

Findings

Asymptotic stability holds under specific spectral conditions.

Extended stability results to larger solitary waves.

Generalized previous orbital stability results to asymptotic stability.

Abstract

We consider the one-dimensional nonlinear Schr\"odinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result beyond the regime of small solitary waves and extend the results of Fukuizumi-Ohta-Ozawa and Kaminaga-Ohta from orbital to asymptotic stability for a suitable family of solitary waves.