# Solitons in inhomogeneous gauge potentials: integrable and nonintegrable dynamics

## TL;DR

This paper introduces an exactly integrable nonlinear model for spinor solitons in space-dependent gauge potentials, explores the effects of Zeeman splitting on soliton dynamics, and demonstrates a controllable transition between integrable and nonintegrable regimes.

## Contribution

The work presents a new integrable model for spinor solitons in inhomogeneous gauge potentials and analyzes how Zeeman splitting controls the transition between integrable and nonintegrable dynamics.

## Key findings

- At zero Zeeman splitting, solitons move without scattering in a disordered SOC landscape.
- Nonzero Zeeman splitting causes strong scattering and nonintegrable behavior.
- Large Zeeman splitting restores integrability, enabling control over soliton dynamics.

## Abstract

We introduce an exactly integrable nonlinear model describing the dynamics of spinor solitons in space-dependent matrix gauge potentials of rather general types. The model is shown to be gauge equivalent to the integrable system of vector nonlinear Schr\"odinger equations known as the Manakov model. As an example we consider a self-attractive Bose-Einstein condensate with random spin-orbit coupling (SOC). If Zeeman splitting is also included, the system becomes nonintegrable. We illustrate this by considering the random walk of a soliton in a disordered SOC landscape. While at zero Zeeman splitting the soliton moves without scattering along linear trajectories in the random SOC landscape, at nonzero splitting it exhibits strong scattering by the SOC inhomogeneities. For a large Zeeman splitting the integrability is recovered. In this sense the Zeeman splitting serves as a parameter controlling the crossover between two different integrable limits.