Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866913395122372608 |
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| author | Cheng, Qiaoyuan Yang, Yiling Fan, Engui |
| author_facet | Cheng, Qiaoyuan Yang, Yiling Fan, Engui |
| contents | We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schrödinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_00919 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data Cheng, Qiaoyuan Yang, Yiling Fan, Engui Analysis of PDEs Exactly Solvable and Integrable Systems We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schrödinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation |
| title | Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2011.00919 |