The formation of a soliton gas condensate for the focusing Nonlinear Schrödinger equation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915759261745152 |
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| author | Gkogkou, Aikaterini Mazzuca, Guido McLaughlin, Kenneth D. T-R |
| author_facet | Gkogkou, Aikaterini Mazzuca, Guido McLaughlin, Kenneth D. T-R |
| contents | In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schrödinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which exhibit the formation (as $N \to \infty$) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov - Shabat operator for the NLS equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from $0$, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of $(x,t)$. We then consider more complex solutions with an extra soliton component, and provide rigorous justification of the predictions of the kinetic theory of solitons in this deterministic setting. This is to be distinguished from previous analyses of soliton gasses where the norming constants were tending to zero with $N$, and the asymptotic description only included elliptic waves in the long-time asymptotics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The formation of a soliton gas condensate for the focusing Nonlinear Schrödinger equation Gkogkou, Aikaterini Mazzuca, Guido McLaughlin, Kenneth D. T-R Analysis of PDEs Mathematical Physics Pattern Formation and Solitons Exactly Solvable and Integrable Systems In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schrödinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which exhibit the formation (as $N \to \infty$) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov - Shabat operator for the NLS equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from $0$, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of $(x,t)$. We then consider more complex solutions with an extra soliton component, and provide rigorous justification of the predictions of the kinetic theory of solitons in this deterministic setting. This is to be distinguished from previous analyses of soliton gasses where the norming constants were tending to zero with $N$, and the asymptotic description only included elliptic waves in the long-time asymptotics. |
| title | The formation of a soliton gas condensate for the focusing Nonlinear Schrödinger equation |
| topic | Analysis of PDEs Mathematical Physics Pattern Formation and Solitons Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2502.14749 |