Zero-dispersion limit for the Benjamin-Ono equation
Elliot Blackstone1 ; Louise Gassot2 ; Patrick Gérard3 ; Peter D. Miller1
1 Department of Mathematics, University of Michigan, Ann Arbor, MI
2 CNRS and Department of Mathematics, University of Rennes, Rennes, France
3 Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France
Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 6, 16 p.

We consider the Benjamin-Ono equation on the line with a small dispersion parameter going to zero. After the shock time for the underlying inviscid Burgers equation, a dispersive shock wave appears in the solution when the parameter is small enough. We show that the solution is asymptotic to the multi-phase solution of Dobrokhotov and Krichever (generalizing periodic traveling waves) for the Benjamin-Ono equation, modulated by slow-varying parameters that depend only on the branches of the Burgers equations obtained by the method of characteristics. The proof relies on a solution formula of the Benjamin-Ono equation established by Gérard [28] and that we simplify for rational initial data in [7]. A paper on the zero-dispersion asymptotics will appear soon in [8].

Publié le :
DOI : 10.5802/slsedp.169
Elliot Blackstone 1 ; Louise Gassot 2 ; Patrick Gérard 3 ; Peter D. Miller 1

1 Department of Mathematics, University of Michigan, Ann Arbor, MI
2 CNRS and Department of Mathematics, University of Rennes, Rennes, France
3 Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France 
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Elliot Blackstone; Louise Gassot; Patrick Gérard; Peter D. Miller. Zero-dispersion limit for the Benjamin-Ono equation. Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 6, 16 p. doi : 10.5802/slsedp.169. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.169/

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