An Explicit Formula for the Benjamin-Ono Hierarchy with Applications to Traveling Waves and Zero-Dispersion Limits
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| Format: | Preprint |
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2026
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| _version_ | 1866914498810478592 |
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| author | Gérard, Patrick He, Jiao |
| author_facet | Gérard, Patrick He, Jiao |
| contents | In this paper, we first extend the explicit formula \cite{gerard2023explicit} for the classical Benjamin-Ono equation to each flow of the Benjamin-Ono hierarchy on line. We then use this representation to derive two main applications. First, we obtain a complete classification of traveling wave solutions for all higher-order flows in the hierarchy. Second, we analyze the zero-dispersion limit for the corresponding small-dispersion flows. For every fixed time $t\in\mathbb R$, we prove that, at any time, the solution converges weakly in $L^2(\mathbb R)$ as the dispersion parameter tends to $0$, and we provide a geometric characterization of the limit in terms of an alternating sum, which yields the higher-order analogue of the formula obtained in \cite{miller2011zero}, \cite{Gerard2025small} for the Benjamin-Ono equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_20464 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Explicit Formula for the Benjamin-Ono Hierarchy with Applications to Traveling Waves and Zero-Dispersion Limits Gérard, Patrick He, Jiao Analysis of PDEs 37K10, 35Q53, 35Q51 In this paper, we first extend the explicit formula \cite{gerard2023explicit} for the classical Benjamin-Ono equation to each flow of the Benjamin-Ono hierarchy on line. We then use this representation to derive two main applications. First, we obtain a complete classification of traveling wave solutions for all higher-order flows in the hierarchy. Second, we analyze the zero-dispersion limit for the corresponding small-dispersion flows. For every fixed time $t\in\mathbb R$, we prove that, at any time, the solution converges weakly in $L^2(\mathbb R)$ as the dispersion parameter tends to $0$, and we provide a geometric characterization of the limit in terms of an alternating sum, which yields the higher-order analogue of the formula obtained in \cite{miller2011zero}, \cite{Gerard2025small} for the Benjamin-Ono equation. |
| title | An Explicit Formula for the Benjamin-Ono Hierarchy with Applications to Traveling Waves and Zero-Dispersion Limits |
| topic | Analysis of PDEs 37K10, 35Q53, 35Q51 |
| url | https://arxiv.org/abs/2604.20464 |