Linear limit continuation: Theory and an application to two-dimensional Bose-Einstein condensates

TL;DR

This paper introduces a systematic theoretical framework for constructing nonlinear solitary waves from linear limits, applied to two-dimensional Bose-Einstein condensates, revealing a wide variety of waveforms and potential for future extensions.

Contribution

The authors develop a novel method combining classification of linear degenerate states with numerical continuation to generate exact nonlinear solitary waves in BECs.

Findings

Classified linear degenerate sets in harmonic potentials.

Generated a diverse set of nonlinear waveforms in 2D BECs.

Method is extendable to 3D and multi-component condensates.

Abstract

We present a coherent and effective theoretical framework to systematically construct numerically exact nonlinear solitary waves from their respective linear limits. First, all possible linear degenerate sets are classified for a harmonic potential using lattice planes. For a generic linear degenerate set, distinct wave patterns are identified in the near-linear regime using a random searching algorithm by suitably mixing the linear degenerate states, followed by a numerical continuation in the chemical potential extending the waves into the Thomas-Fermi regime. The method is applied to the two-dimensional, one-component Bose-Einstein condensates, yielding a spectacular set of waveforms. Our method opens a remarkably large program, and many more solitary waves are expected. Finally, the method can be readily generalized to three dimensions, and also multi-component condensates, providing a highly powerful technique for investigating solitary waves in future works.