Computer-Assisted Proofs of Gap Solitons in Bose-Einstein Condensates
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915812538843136 |
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| author | Ayala, Miguel García-Azpeitia, Carlos Lessard, Jean-Philippe |
| author_facet | Ayala, Miguel García-Azpeitia, Carlos Lessard, Jean-Philippe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_04701 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computer-Assisted Proofs of Gap Solitons in Bose-Einstein Condensates Ayala, Miguel García-Azpeitia, Carlos Lessard, Jean-Philippe Dynamical Systems Analysis of PDEs 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. |
| title | Computer-Assisted Proofs of Gap Solitons in Bose-Einstein Condensates |
| topic | Dynamical Systems Analysis of PDEs |
| url | https://arxiv.org/abs/2503.04701 |