Direct Scattering of the Focusing Nonlinear Schrödinger Equation with Step-like Oscillatory Initial Data
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918367489687552 |
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| author | Grava, Tamara Jenkins, Robert Zhang, Xiaofan Zhang, Zechuan |
| author_facet | Grava, Tamara Jenkins, Robert Zhang, Xiaofan Zhang, Zechuan |
| contents | In this manuscript we set up the direct and inverse scattering problems for step-like traveling-wave solutions of the nonlinear Schrödinger equation. Specifically, we consider initial data $u(x,0)$ satisfying $u(x,0)\to u_0^\ell(x)$ as $x\to-\infty$ and $u(x,0)\to u_0^r(x)$ as $x\to+\infty$, where $u_0^\ell(x)$ and $u_0^r(x)$ are elliptic traveling waves. Under suitable assumptions on the initial data we formulate the direct scattering problem and establish analytic properties of the scattering data. We then formulate the inverse problem as a Riemann--Hilbert problem and prove its solvability. Finally, we observe that this Riemann--Hilbert formulation is a special case of the one arising for full soliton-gas initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02855 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Direct Scattering of the Focusing Nonlinear Schrödinger Equation with Step-like Oscillatory Initial Data Grava, Tamara Jenkins, Robert Zhang, Xiaofan Zhang, Zechuan Analysis of PDEs Mathematical Physics 35Q55, 37K15, 35P25 In this manuscript we set up the direct and inverse scattering problems for step-like traveling-wave solutions of the nonlinear Schrödinger equation. Specifically, we consider initial data $u(x,0)$ satisfying $u(x,0)\to u_0^\ell(x)$ as $x\to-\infty$ and $u(x,0)\to u_0^r(x)$ as $x\to+\infty$, where $u_0^\ell(x)$ and $u_0^r(x)$ are elliptic traveling waves. Under suitable assumptions on the initial data we formulate the direct scattering problem and establish analytic properties of the scattering data. We then formulate the inverse problem as a Riemann--Hilbert problem and prove its solvability. Finally, we observe that this Riemann--Hilbert formulation is a special case of the one arising for full soliton-gas initial data. |
| title | Direct Scattering of the Focusing Nonlinear Schrödinger Equation with Step-like Oscillatory Initial Data |
| topic | Analysis of PDEs Mathematical Physics 35Q55, 37K15, 35P25 |
| url | https://arxiv.org/abs/2603.02855 |