Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate

TL;DR

This paper classifies flow patterns in two-component Bose-Einstein condensates from initial discontinuities, deriving solutions for polarization waves, including novel contact dispersive shock waves, and confirms findings with numerical simulations.

Contribution

It provides a comprehensive classification and analytical solutions for polarization wave dynamics in two-component BECs, introducing contact dispersive shock waves absent in traditional models.

Findings

Derived full set of one-phase periodic solutions.

Connected Whitham modulation equations with Riemann invariants.

Identified and characterized contact dispersive shock waves.

Abstract

We provide a classification of the possible flow of two-component Bose-Einstein condensates evolving from initially discontinuous profiles. We consider the situation where the dynamics can be reduced to the consideration of a single polarization mode (also denoted as "magnetic excitation") obeying a system of equations equivalent to the Landau-Lifshitz equation for an easy-plane ferro-magnet. We present the full set of one-phase periodic solutions. The corresponding Whitham modulation equations are obtained together with formulas connecting their solutions with the Riemann invariants of the modulation equations. The problem is not genuinely nonlinear, and this results in a non-single-valued mapping of the solutions of the Whitham equations with physical wave patterns as well as to the appearance of new elements --- contact dispersive shock waves --- that are absent in more standard, genuinely nonlinear situations. Our analytic results are confirmed by numerical simulations.