Travelling solitons in the externally driven nonlinear Schr\"odinger equation

TL;DR

This paper investigates the existence, stability, and dynamics of travelling solitons in a driven nonlinear Schrödinger equation, revealing two stable families with distinct velocity-dependent stability properties and the formation of bound states.

Contribution

It introduces new localized solutions and characterizes their stability and interactions in the context of a periodically driven nonlinear Schrödinger equation.

Findings

Two families of stable solitons identified with different velocity regimes

Stable bound states of solitons can form in the low-velocity family

Existence of stable solitons depends on their traveling speed

Abstract

We consider the undamped nonlinear Schr\"odinger equation driven by a periodic external force. Classes of travelling solitons and multisoliton complexes are obtained by the numerical continuation in the parameter space. Two previously known stationary solitons and two newly found localised solutions are used as the starting points for the continuation. We show that there are two families of stable solitons: one family is stable for sufficiently low velocities while solitons from the second family stabilise when travelling faster than a certain critical speed. The stable solitons of the former family can also form stably travelling bound states.