Solitons in Inhomogeneous Gauge Potentials: Integrable and Nonintegrable Dynamics

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 Schrödinger 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 restored. In this sense, the Zeeman splitting serves as a parameter controlling the crossover between two different integrable limits.

Figure 1

Evolution of the average density of the dominant component $⟨|{\psi }_2{|}^2⟩$ for $\alpha =0.3$, $v=0.3$ in the initial condition ${\mathit{\psi }}_0$, and different strengths of the Zeeman field. Distributions are shown up to $t={10}^3$. Dotted lines indicate the initial wave-packet position (vertical line) and center trajectory in the integrable case $\mathrm{\Omega }=0$ (oblique line).

Figure 2

(a) Dynamics of the averaged soliton center for $\mathrm{\Omega }=0$ (curve 1), 0.06 (curve 2), 0.5 (curve 3), and 1 (the lowest curve 4). (b) Averaged displacement at $t={10}^3$ vs Zeeman splitting $\mathrm{\Omega }$ (red dots $\alpha =0.3$, white dots $\alpha =0.42$). (c) The mean squared displacement for $\Omega =0$ (curve 1), 0.06 (curve 2), 0.3 (curve 3), and 1 (curve 4). (d) The mean squared displacement at $t={10}^3$ vs Zeeman splitting $\mathrm{\Omega }$. In all cases $v=0.3$, $\alpha =0.3$.

Figure 3

Averaged displacement of the soliton center (a), and the MSD vs SOC strength $\alpha$ (b), at $t={10}^3$, for $v=0.3$ and $\mathrm{\Omega }=0.3$.

Figure 4

(a) Evolution of averaged pseudospin components at $\mathrm{\Omega }=0.15$. (b) Final averaged pseudospin components at $t={10}^3$ vs Zeeman splitting. In all cases $v=0.3$ and $\mathrm{\Omega }=0.3$.