The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves
| Autor: | Blackstone, Elliot, Gassot, Louise, Gérard, Patrick, Miller, Peter D. |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^\mathrm{ZD}(t,x;\epsilon)$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data $u_0$, such that the solution $u(t,x;\epsilon)$ of the Benjamin-Ono equation with dispersion parameter $\epsilon>0$ and initial data $u_0$ satisfies $u(t,x;\epsilon)-u^\mathrm{ZD}(t,x;\epsilon)\to 0$ in the locally uniform sense as $\epsilon\to 0$, provided a discriminant inequality holds implying that certain caustic curves in the $(t,x)$-plane are avoided. In some cases this convergence implies strong $L^2(\mathbb{R})$ convergence. The asymptotic profile $u^\mathrm{ZD}(t,x;\epsilon)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever. Comment: 63 pages, 18 figures |
| Databáze: | arXiv |
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