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The inverse problem for the Gross-Pitaevskii equation
Boris A Malomed1, Yury A Stepanyants
1Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel.
Chaos (Woodbury, N.Y.)
|April 8, 2010
Summary
Two novel methods generate exact solutions for the stationary Gross-Pitaevskii equation (GPE). These approaches yield diverse localized solutions, including solitary vortices, for 1D and 2D systems with varying nonlinearities.
Area of Science:
- Quantum mechanics
- Mathematical physics
Background:
- The Gross-Pitaevskii equation (GPE) describes Bose-Einstein condensates.
- Finding exact solutions is crucial for understanding GPE behavior.
Purpose of the Study:
- To develop new methods for generating exact solutions to the stationary GPE.
- To explore these solutions in one and two dimensions.
Main Methods:
- Method 1: Exploits the similarity between the GPE and the integrable Gardner equation for 1D systems.
- Method 2: Employs an inverse problem approach to construct potentials yielding desired GPE solutions.
Main Results:
- Both methods successfully generate wide classes of exact GPE solutions.
- Localized solutions, including solitary vortices, are demonstrated for attractive and repulsive nonlinearities.
- Stability analysis of 1D solutions was performed using direct simulations.
Conclusions:
- The proposed methods provide powerful tools for solving the stationary GPE.
- These techniques facilitate the study of diverse physical phenomena in Bose-Einstein condensates.
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