# Expansion dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate: Phase diagram, self-similar solutions, and dispersive shock waves

## 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 nonuniformity 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, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.

- Authors:

- Russian Academy of Sciences, Institute of Spectroscopy (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22756526

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Experimental and Theoretical Physics

- Additional Journal Information:
- Journal Volume: 124; Journal Issue: 4; Other Information: Copyright (c) 2017 Pleiades Publishing, Inc.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1063-7761

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANALYTICAL SOLUTION; ASYMPTOTIC SOLUTIONS; BIOMEDICAL RADIOGRAPHY; DIFFERENTIAL EQUATIONS; EXACT SOLUTIONS; MATHEMATICAL EVOLUTION; NONLINEAR PROBLEMS; NUMERICAL ANALYSIS; NUMERICAL SOLUTION; PHASE DIAGRAMS; SHOCK WAVES; THOMAS-FERMI MODEL

### Citation Formats

```
Ivanov, S. K., E-mail: ivanoff.iks@gmail.com, and Kamchatnov, A. M., E-mail: kamchatnov@gmail.com.
```*Expansion dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate: Phase diagram, self-similar solutions, and dispersive shock waves*. United States: N. p., 2017.
Web. doi:10.1134/S1063776117040136.

```
Ivanov, S. K., E-mail: ivanoff.iks@gmail.com, & Kamchatnov, A. M., E-mail: kamchatnov@gmail.com.
```*Expansion dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate: Phase diagram, self-similar solutions, and dispersive shock waves*. United States. https://doi.org/10.1134/S1063776117040136

```
Ivanov, S. K., E-mail: ivanoff.iks@gmail.com, and Kamchatnov, A. M., E-mail: kamchatnov@gmail.com. Sat .
"Expansion dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate: Phase diagram, self-similar solutions, and dispersive shock waves". United States. https://doi.org/10.1134/S1063776117040136.
```

```
@article{osti_22756526,
```

title = {Expansion dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate: Phase diagram, self-similar solutions, and dispersive shock waves},

author = {Ivanov, S. K., E-mail: ivanoff.iks@gmail.com and Kamchatnov, A. M., E-mail: kamchatnov@gmail.com},

abstractNote = {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 nonuniformity 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, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.},

doi = {10.1134/S1063776117040136},

url = {https://www.osti.gov/biblio/22756526},
journal = {Journal of Experimental and Theoretical Physics},

issn = {1063-7761},

number = 4,

volume = 124,

place = {United States},

year = {2017},

month = {4}

}