Long-time asymptotics of the coupled nonlinear Schödinger equation in a weighted Sobolev space
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| Format: | Preprint |
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2025
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| author | Huang, Yubin Ling, Liming Zhang, Xiaoen |
| author_facet | Huang, Yubin Ling, Liming Zhang, Xiaoen |
| contents | We study the Cauchy problem for the focusing coupled nonlinear Schrödinger (CNLS) equation with initial data $\mathbf{q}_0$ lying in the weighted Sobolev space and the scattering data having $n$ simple zeros. Based on the corresponding $3\times3$ matrix spectral problem, we deduce the Riemann-Hilbert problem (RHP) for CNLS equation through inverse scattering transform. We remove discrete spectrum of initial RHP using Darboux transformations. By applying the nonlinear steepest-descent method for RHP introduced by Deift and Zhou, we compute the long-time asymptotic expansion of the solution $\mathbf{q}(x,t)$ to an (optimal) residual error of order $\mathcal{O}\left(t^{-3 / 4+1/(2p)}\right)$ where $2\le p<\infty$. The leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions. Our work strengthens and extends the earlier work regarding long-time asymptotics for solutions of the nonlinear Schrödinger equation with a delta potential and even initial data by Deift and Park. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21315 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-time asymptotics of the coupled nonlinear Schödinger equation in a weighted Sobolev space Huang, Yubin Ling, Liming Zhang, Xiaoen Exactly Solvable and Integrable Systems We study the Cauchy problem for the focusing coupled nonlinear Schrödinger (CNLS) equation with initial data $\mathbf{q}_0$ lying in the weighted Sobolev space and the scattering data having $n$ simple zeros. Based on the corresponding $3\times3$ matrix spectral problem, we deduce the Riemann-Hilbert problem (RHP) for CNLS equation through inverse scattering transform. We remove discrete spectrum of initial RHP using Darboux transformations. By applying the nonlinear steepest-descent method for RHP introduced by Deift and Zhou, we compute the long-time asymptotic expansion of the solution $\mathbf{q}(x,t)$ to an (optimal) residual error of order $\mathcal{O}\left(t^{-3 / 4+1/(2p)}\right)$ where $2\le p<\infty$. The leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions. Our work strengthens and extends the earlier work regarding long-time asymptotics for solutions of the nonlinear Schrödinger equation with a delta potential and even initial data by Deift and Park. |
| title | Long-time asymptotics of the coupled nonlinear Schödinger equation in a weighted Sobolev space |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2504.21315 |