Stability of small solitary waves for the 1$d$ NLS with an attractive delta potential

TL;DR

This paper proves that small solitary waves in a 1D nonlinear Schrödinger equation with an attractive delta potential are asymptotically stable, with solutions decomposing into a solitary wave and decaying radiation.

Contribution

It establishes the asymptotic stability of small solitary waves for the 1D NLS with an attractive delta potential, a novel result in this setting.

Findings

Small initial data lead to solutions decomposing into solitary wave + radiation.

Radiation term decays and scatters as time goes to infinity.

Solitary waves are proven to be asymptotically stable.

Abstract

We consider the initial-value problem for the one-dimensional nonlinear Schr\"odinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes into a small solitary wave plus a radiation term that decays and scatters as $t\to\infty$. In particular, we establish the asymptotic stability of the family of small solitary waves.