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arXiv:2309.01605

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

Wenlong Wang

Published Sep 4, 2023• 0 citations
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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.

DOI
10.48550/arxiv.2309.01605

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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.
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Abstract We systematically construct vector solitary waves in harmonically trapped one-dimensional two-component Bose–Einstein condensates with unequal dispersion coefficients by a numerical continuation in chemical potentials from the respective analytic low-density linear limits to the high-density nonlinear Thomas-Fermi regime. The main feature of the linear states herein is that the component with the larger quantum number has instead a smaller linear eigenenergy, enabled by suitable unequal dispersion coefficients, leading to new series of solutions compared with the states similarly obtained in the equal dispersion setting. Particularly, the lowest-lying series gives the well-known dark-anti-dark waves, and the second series yields the dark-multi-dark states, and the following series become progressively more complex in their wave structures. The Bogoliubov-de Gennes spectra analysis shows that most of these states bear unstable modes, but they can be long-lived and remarkably all of them can be fully stabilized in suitable parameter regimes.
In the present work, we revisit two-component Bose-Einstein condensates in their fully three-dimensional (3D) form. Motivated by earlier studies of dark-bright solitons in the 1D case, we explore the stability of these structures in their fully 3D form in two variants. In one the dark soliton is planar and trapping a planar bright (disk) soliton. In the other case, a dark spherical shell soliton creates an effective potential in which a bright spherical shell of atoms is trapped in the second component. We identify these solutions as numerically exact states (up to a prescribed accuracy) and perform a Bogolyubov-de Gennes linearization analysis that illustrates that both structures can be dynamically stable in suitable intervals of sufficiently low chemical potentials. We corroborate this finding theoretically by analyzing the stability via degenerate perturbation theory near the linear limit of the system. When the solitary waves are found to be unstable, we explore their dynamical evolution via direct numerical simulations which, in turn, reveal wave forms that are more robust. Finally, using the SO(2) symmetry of the model, we produce multi-dark-bright planar or shell solitons involved in pairwise oscillatory motion.
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The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose–Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.