The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves
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| Format: | Preprint |
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2024
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| _version_ | 1866909359762571264 |
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| author | Blackstone, Elliot Gassot, Louise Gérard, Patrick Miller, Peter D. |
| author_facet | Blackstone, Elliot Gassot, Louise Gérard, Patrick Miller, Peter D. |
| contents | 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 $u^\mathrm{ZD}(t,x;ε)$ 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 $u(t,x;ε)$ of the Benjamin-Ono equation with dispersion parameter $ε>0$ and initial data $u_0$ satisfies $u(t,x;ε)-u^\mathrm{ZD}(t,x;ε)\to 0$ in the locally uniform sense as $ε\to 0$, 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 $u^\mathrm{ZD}(t,x;ε)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_17405 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves Blackstone, Elliot Gassot, Louise Gérard, Patrick Miller, Peter D. Analysis of PDEs 35C20, 35Q51, 41A60 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 $u^\mathrm{ZD}(t,x;ε)$ 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 $u(t,x;ε)$ of the Benjamin-Ono equation with dispersion parameter $ε>0$ and initial data $u_0$ satisfies $u(t,x;ε)-u^\mathrm{ZD}(t,x;ε)\to 0$ in the locally uniform sense as $ε\to 0$, 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 $u^\mathrm{ZD}(t,x;ε)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever. |
| title | The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves |
| topic | Analysis of PDEs 35C20, 35Q51, 41A60 |
| url | https://arxiv.org/abs/2410.17405 |