Abstract
We consider the zero-dispersion limit for the Benjamin–Ono equation on the torus. We prove that when the initial data is bell shaped, the zero-dispersion limit exists in the weak sense and is uniform on every compact time interval. Moreover, the limit is equal to the signed sum of branches for the multivalued solution of the inviscid Burgers equation obtained by the method of characteristics. This result is similar to the one obtained by Miller and Xu for the Benjamin–Ono equation on the real line for decaying and positive initial data. We also establish some precise asymptotics of the spectral data with initial data \(u_0(x)=-\beta \cos (x)\), \(\beta >0\), justifying our approximation method, which is analogous to the work of Miller and Wetzel concerning a family of rational potentials for the Benjamin–Ono equation on the real line.
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Acknowledgements
The author is grateful to Patrick Gérard for relevant advice on this problem, in particular, for providing notes on Proposition 21 about the cosine function. The author would also like to warmly thank the reviewer for precise and constructive remarks about this work. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Hamiltonian Methods in Dispersive and Wave Evolution Equations” program.
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Communicated by A. Ionescu.
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Gassot, L. Zero-Dispersion Limit for the Benjamin–Ono Equation on the Torus with Bell Shaped Initial Data. Commun. Math. Phys. 401, 2793–2843 (2023). https://doi.org/10.1007/s00220-023-04701-0
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DOI: https://doi.org/10.1007/s00220-023-04701-0