The Riemann Hilbert dressing method and wave breaking for two (2 + 1)-dimensional integrable equations

Author:

Lu Huanhuan1,Ren Xinan1,Zhang Yufeng2,Zhang Hongyi1

Affiliation:

1. School of Mathematics , 12392 China University of Mining and Technology , Xuzhou 221116 , P.R. China

2. School of Mathematics and Information Sciences , Weifang University , Weifang 261061 , P.R. China

Abstract

Abstract In this article, we present a method for generating (2 + 1)-dimensional integrable equations, resulting in the generalized Pavlov equation and dispersionless Kadomtsev–Petviashvili (dKP) equation, which can further be reduced to the standard Pavlov equation and dKP equation. Inspired by the inverse spectral transform presented in existing literature, we introduce the Riemann–Hilbert (RH) dressing method to construct the formal solutions of the Cauchy problems for the generalized Pavlov equation and dKP equation, providing a spectral representation of the solutions. Subsequently, we also extensively investigate the longtime behavior of solutions to these two equations in specific space regions. In particular, for the generalized dKP equation, we conduct a dedicated study on its implicit solutions expressed by arbitrary differential function through linearizing their RH problems. In the final section, we elaborate in detail on the analytic aspects of the wave breaking of a localized two-dimensional wave evolving according to the Hopf equation. With the assistance of a transformation, the longtime breaking of solutions to the generalized dKP equation can then be further characterized.

Funder

National Natural Science Foundation of China

SuQian Sci&Tech Program

Graduate Innovation Program of China University of Mining and Technology

Postgraduate Research & Practice Innovation Program of Jiangsu Province

Publisher

Walter de Gruyter GmbH

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