The bidirectional NLS approximation for the one-dimensional Euler-Poisson system
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908527081029632 |
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| author | Liu, Huimin Lu, Yurui Pu, Xueke |
| author_facet | Liu, Huimin Lu, Yurui Pu, Xueke |
| contents | The nonlinear Schrödinger (NLS) equation is known as a universal equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet in various dispersive systems. In this paper, we prove that under a certain multiple scale transformation, solutions to the Euler-Poisson system can be approximated by the sums of two counter-propagating waves solving the NLS equations. It extends the earlier results [Liu and Pu, Comm. Math. Phys., 371(2), (2019)357-398], which justify the unidirectional NLS approximation to the Euler-Poisson system for the ion-coustic wave. We demonstrate that the solutions could be convergent to two counter-propagating wave packets, where each wave packet involves independently as a solution of the NLS equation. We rigorously prove the validity of the NLS approximation for the one-dimensional Euler-Poisson system by obtaining uniform error estimates in Sobolev spaces. The NLS dynamics can be observed at a physically relevant timespan of order $\mathcal{O}(ε^{-2})$. As far as we know, this result is the first construction and valid proof of the bidirectional NLS approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The bidirectional NLS approximation for the one-dimensional Euler-Poisson system Liu, Huimin Lu, Yurui Pu, Xueke Analysis of PDEs The nonlinear Schrödinger (NLS) equation is known as a universal equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet in various dispersive systems. In this paper, we prove that under a certain multiple scale transformation, solutions to the Euler-Poisson system can be approximated by the sums of two counter-propagating waves solving the NLS equations. It extends the earlier results [Liu and Pu, Comm. Math. Phys., 371(2), (2019)357-398], which justify the unidirectional NLS approximation to the Euler-Poisson system for the ion-coustic wave. We demonstrate that the solutions could be convergent to two counter-propagating wave packets, where each wave packet involves independently as a solution of the NLS equation. We rigorously prove the validity of the NLS approximation for the one-dimensional Euler-Poisson system by obtaining uniform error estimates in Sobolev spaces. The NLS dynamics can be observed at a physically relevant timespan of order $\mathcal{O}(ε^{-2})$. As far as we know, this result is the first construction and valid proof of the bidirectional NLS approximation. |
| title | The bidirectional NLS approximation for the one-dimensional Euler-Poisson system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.07371 |