Solitons and soliton interactions in repulsive spinor Bose-Einstein condensates with non-zero background

TL;DR

This paper analyzes soliton solutions and their interactions in one-dimensional repulsive spinor Bose-Einstein condensates with non-zero background, using inverse scattering transform and symmetry reductions to classify different soliton types.

Contribution

It provides a complete characterization of multi-soliton solutions, introduces canonical forms, and classifies interactions for ferromagnetic and polar states in spinor BECs.

Findings

Derived compact multi-soliton solutions using IST.

Showed solutions can be reduced to canonical form via unitary transformations.

Classified two-soliton interaction scenarios based on soliton types.

Abstract

We characterize the soliton solutions and their interactions for a system of coupled evolution equations of nonlinear Schr\"odinger (NLS) type that models the dynamics in one-dimensional repulsive Bose-Einstein condensates with spin one, taking advantage of the representation of such model as a special reduction of a 2 x 2 matrix NLS system. Specifically, we study in detail the case in which solutions tend to a non-zero background at space infinities. First we derive a compact representation for the multi-soliton solutions in the system using the Inverse Scattering Transform (IST). We introduce the notion of canonical form of a solution, corresponding to the case when the background is proportional to the identity. We show that solutions for which the asymptotic behavior at infinity is not proportional to the identity, referred to as being in non-canonical form, can be reduced to canonical form by unitary transformations that preserve the symmetric nature of the solution (physically corresponding to complex rotations of the quantization axes). Then we give a complete characterization of the two families of one-soliton solutions arising in this problem, corresponding to ferromagnetic and to polar states of the system, and we discuss how the physical parameters of the solitons for each family are related to the spectral data in the IST. We also show that any ferromagnetic one-soliton solution in canonical form can be reduced to a single dark soliton of the scalar NLS equation, and any polar one-soliton solution in canonical form is unitarily equivalent to a pair of oppositely polarized displaced scalar dark solitons up to a rotation of the quantization axes. Finally, we discuss two-soliton interactions and we present a complete classification of the possible scenarios that can arise depending on whether either soliton is of ferromagnetic or polar type.