# Expansion Dynamics of a Two-Component Quasi-One-Dimensional Bose-Einstein Condensate: Phase Diagram, Self-Similar Solutions, and Dispersive Shock Waves

## TL;DR

This paper studies the expansion behavior of a two-component Bose-Einstein condensate in a quasi-one-dimensional trap, classifies initial states, derives self-similar solutions, and explores dispersive shock waves, providing both analytical and numerical insights.

## Contribution

It introduces a phase diagram for initial states, derives self-similar solutions, and analyzes dispersive shock waves in two-component condensates, advancing understanding of their expansion dynamics.

## Key findings

- Classified initial states with a phase diagram based on interaction constants.
- Derived self-similar solutions for specific cases of condensate expansion.
- Numerically validated analytical solutions and explored dispersive shock wave formation.

## Abstract

We investigate the expansion dynamics of a Bose-Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the non-uniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, what reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas-Fermi approximation. We obtained self-similar solutions of these differential equations for several important special cases and wrote out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms can be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves was also considered. We compare the numerical solutions of the Gross-Pitaevskii equations with their approximate analytical solutions and study numerically the situations when the analytical method admits no exact solutions.