The Weakly Nonlinear Schrödinger Equation in Higher Dimensions with Quasi-periodic Initial Data
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918145975910400 |
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| author | Xu, Fei |
| author_facet | Xu, Fei |
| contents | In this paper, under the exponential/polynomial decay condition in Fourier space, we prove that the nonlinear solution to the quasi-periodic Cauchy problem for the weakly nonlinear Schrödinger equation in higher dimensions will asymptotically approach the associated linear solution within a specific time scale. The proof is based on a combinatorial analysis method present through diagrams. Our results and methods apply to {\em arbitrary} space dimensions and general power-law nonlinearities of the form $\pm|u|^{2p}u$, where $1\leq p\in\mathbb N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10006 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Weakly Nonlinear Schrödinger Equation in Higher Dimensions with Quasi-periodic Initial Data Xu, Fei Analysis of PDEs Mathematical Physics 35Q55, 35B15, 35C10, 35A01, 35A02, 35B40 In this paper, under the exponential/polynomial decay condition in Fourier space, we prove that the nonlinear solution to the quasi-periodic Cauchy problem for the weakly nonlinear Schrödinger equation in higher dimensions will asymptotically approach the associated linear solution within a specific time scale. The proof is based on a combinatorial analysis method present through diagrams. Our results and methods apply to {\em arbitrary} space dimensions and general power-law nonlinearities of the form $\pm|u|^{2p}u$, where $1\leq p\in\mathbb N$. |
| title | The Weakly Nonlinear Schrödinger Equation in Higher Dimensions with Quasi-periodic Initial Data |
| topic | Analysis of PDEs Mathematical Physics 35Q55, 35B15, 35C10, 35A01, 35A02, 35B40 |
| url | https://arxiv.org/abs/2409.10006 |