Nonlinear waves in two-component Bose-Einstein condensates: Manakov system and Kowalevski equations

TL;DR

This paper investigates nonlinear traveling waves in two-component Bose-Einstein condensates described by the Manakov system, revealing connections to Kowalevski equations and classifying wave types based on their phase relations.

Contribution

It establishes a link between Bose-Einstein condensate wave solutions and Kowalevski equations, providing explicit solutions in terms of elliptic functions for the first time.

Findings

Solutions reduce to Kowalevski top equations.

Both density and polarization waves are characterized.

Explicit elliptic function solutions are derived.

Abstract

Traveling waves in two-component Bose-Einstein condensates whose dynamics is described by the Manakov limit of the Gross-Pitaevskii equations are considered in general situation with relative motion of the components when their chemical potentials are not equal to each other. It is shown that in this case the solution is reduced to the form known in the theory of motion of S.~Kowalevski top. Typical situations are illustrated by the particular cases when the general solution can be represented in terms of elliptic functions and their limits. Depending on the parameters of the wave, both density waves (with in-phase motions of the components) and polarization waves (with counter-phase their motions) are considered.