Continuous binary Darboux transformation as an abstract framework for KdV soliton gases
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909960319795200 |
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| author | Rybkin, Alexei |
| author_facet | Rybkin, Alexei |
| contents | We present a unified operator-theoretic framework for constructing deterministic KdV soliton gases and step-type KdV solutions. Starting from Dyson's determinantal formula, we obtain a broad class of reflectionless solutions and describe their basic spectral and analytic properties, including their interpretation as deterministic soliton gases. We then introduce a continuous binary Darboux transformation that acts directly on the scattering data and generates general step-type solutions, with particular emphasis on reflectionless hydraulic-jump-type profiles modelling a soliton condensate on the left and vacuum on the right. The paper is methodological in nature: our goal is not to develop a full kinetic or probabilistic theory, but to show how classical tools from spectral and scattering theory can be combined into a conceptually simple framework that accommodates both reflectionless and non-reflectionless soliton gas configurations, including step-like backgrounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous binary Darboux transformation as an abstract framework for KdV soliton gases Rybkin, Alexei Mathematical Physics We present a unified operator-theoretic framework for constructing deterministic KdV soliton gases and step-type KdV solutions. Starting from Dyson's determinantal formula, we obtain a broad class of reflectionless solutions and describe their basic spectral and analytic properties, including their interpretation as deterministic soliton gases. We then introduce a continuous binary Darboux transformation that acts directly on the scattering data and generates general step-type solutions, with particular emphasis on reflectionless hydraulic-jump-type profiles modelling a soliton condensate on the left and vacuum on the right. The paper is methodological in nature: our goal is not to develop a full kinetic or probabilistic theory, but to show how classical tools from spectral and scattering theory can be combined into a conceptually simple framework that accommodates both reflectionless and non-reflectionless soliton gas configurations, including step-like backgrounds. |
| title | Continuous binary Darboux transformation as an abstract framework for KdV soliton gases |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2512.12495 |