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arXiv · 2503.04701

Computer-Assisted Proofs of Gap Solitons in Bose-Einstein Condensates

Miguel Ayala·Carlos García-Azpeitia·Jean-Philippe Lessard
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Abstract

We provide a framework for turning a numerical simulation of a gap soliton in the one-dimensional Gross-Pitaevskii equation into a rigorous mathematical proof of its existence. These nonlinear localized solutions play a central role in the study of Bose-Einstein condensates (BECs). We reformulate the problem of proving their existence as the search for homoclinic orbits in a dynamical system. We then apply computer-assisted proof techniques to obtain verifiable conditions under which a numerically approximated trajectory corresponds to a true homoclinic orbit. This work also presents the first examples of computer-assisted proofs of gap solitons in the Gross-Pitaevskii equation on non-perturbative parameter regimes.

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Miguel Ayala, Carlos García-Azpeitia, Jean-Philippe Lessard. 2025-03-06. Computer-Assisted Proofs of Gap Solitons in Bose-Einstein Condensates. https://doi.org/10.1007/s00332-026-10242-2

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