$\bar{\partial}$-problem for focusing nonlinear Schrödinger equation and soliton shielding
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929510485590016 |
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| author | Bertola, Marco Grava, Tamara Orsatti, Giuseppe |
| author_facet | Bertola, Marco Grava, Tamara Orsatti, Giuseppe |
| contents | We consider soliton gas solutions of the Focusing Nonlinear Schrödinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condensate in a bounded domain $\mathcal{D}$ in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a $\overline{\partial}$-problem on the domain. We prove the existence of the solution of this $\overline{\partial}$-problem by showing that the $τ$-function of the problem (a Fredholm determinant) does not vanish. We then represent the solution of the NLS equation via the $τ$ of the $\overline{\partial}$- problem. Finally we show that, when the domain $\mathcal{D}$ is an ellipse and the density of solitons is analytic, the initial datum of the Cauchy problem is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as $x \to - \infty$ while it vanishes exponentially fast as $x \to +\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14825 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\bar{\partial}$-problem for focusing nonlinear Schrödinger equation and soliton shielding Bertola, Marco Grava, Tamara Orsatti, Giuseppe Mathematical Physics Analysis of PDEs Pattern Formation and Solitons Exactly Solvable and Integrable Systems 35Q55 (Primary) 35Q15, 37K15, 35Q51 (Secondary) We consider soliton gas solutions of the Focusing Nonlinear Schrödinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condensate in a bounded domain $\mathcal{D}$ in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a $\overline{\partial}$-problem on the domain. We prove the existence of the solution of this $\overline{\partial}$-problem by showing that the $τ$-function of the problem (a Fredholm determinant) does not vanish. We then represent the solution of the NLS equation via the $τ$ of the $\overline{\partial}$- problem. Finally we show that, when the domain $\mathcal{D}$ is an ellipse and the density of solitons is analytic, the initial datum of the Cauchy problem is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as $x \to - \infty$ while it vanishes exponentially fast as $x \to +\infty$. |
| title | $\bar{\partial}$-problem for focusing nonlinear Schrödinger equation and soliton shielding |
| topic | Mathematical Physics Analysis of PDEs Pattern Formation and Solitons Exactly Solvable and Integrable Systems 35Q55 (Primary) 35Q15, 37K15, 35Q51 (Secondary) |
| url | https://arxiv.org/abs/2409.14825 |