On travelling waves of the non linear Schr\"odinger equation escaping a potential well

TL;DR

This paper studies how nonlinear Schrödinger equation solitons escape potential wells, revealing different dynamical regimes depending on the potential's tail properties and constructing the associated nonlinear dynamics.

Contribution

It introduces a detailed analysis of soliton escape dynamics in NLS with potential, highlighting the influence of potential tail properties and constructing the corresponding nonlinear dynamical system.

Findings

Two regimes identified: fat tail potential dominates motion, weak tail leads to self-interaction.

Constructed nonlinear dynamical system for untrapped soliton motions.

Demonstrated the fundamental role of potential/soliton interaction in escape dynamics.

Abstract

In this paper we consider the NLS equation with focusing nonlinearities in the presence of a potential. We investigate the compact soliton motions that correspond to a free soliton escaping the well created by the potential. We exhibit the dynamical system driving the exiting trajectory and construct associated nonlinear dynamics for untrapped motions. We show that the nature of the potential/soliton is fundamental, and two regimes may exist: one where the tail of the potential is fat and dictates the motion, one where the tail is weak and the soliton self interacts with the potential defects, hence leading to different motions.