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arXiv:2603.17996

Systematic solitary waves by linear limit continuation from two anisotropic traps in two-dimensional Bose-Einstein condensates

Wenlong Wang

Published Mar 18, 2026• 0 citations
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Abstract

Linear limit continuation was recently developed as a systematic and effective method for constructing numerically exact solitary waves from their respective linear limits. In this work, we apply the technique to two typical anisotropic harmonic traps in two-dimensional Bose-Einstein condensates to further establish the method and also to find more solitary waves. Many wave patterns are identified in the near-linear regime and they are subsequently continued into the Thomas-Fermi regime, and then they are further continued into the isotropic trap if possible. Finally, the parametric connectivity of the pertinent solitary waves is also discussed.

DOI
10.48550/arxiv.2603.17996

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Linear limit continuation was recently developed as a systematic and effective method for constructing numerically exact solitary waves from their respective linear limits. In this work, we apply the technique to two typical anisotropic harmonic traps in two-dimensional Bose-Einstein condensates to further establish the method and also to find more solitary waves. Many wave patterns are identified in the near-linear regime and they are subsequently continued into the Thomas-Fermi regime, and then they are further continued into the isotropic trap if possible. Finally, the parametric connectivity of the pertinent solitary waves is also discussed.
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.
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.
We systematically construct a series of vector solitary waves in harmonically trapped one-dimensional three-, four-, and five-component Bose-Einstein condensates. These stationary states are continued in chemical potentials from the analytically tractable low-density linear limit of respective states, as independent linear quantum harmonic oscillator states, to the high-density nonlinear Thomas-Fermi regime. A systematic interpolation procedure is proposed to achieve this sequential continuation via a trajectory in the multidimensional space of the chemical potentials. The Bogoliubov-de Gennes spectral analysis shows that all of the states considered herein can be fully stabilized in suitable chemical potential intervals in the Thomas-Fermi regime. Finally, we present some typical SU(n)-rotation-induced and driving-induced dynamics. This method can be extended to higher dimensions and shows significant promise for finding a wide range of solitary waves ahead.