The Riemann Hilbert dressing method and wave breaking for two (2 + 1)-dimensional integrable equations
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Huanhuan Lu
and
Hongyi Zhang
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.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: No. 12371256
Funding source: SuQian Sci&Tech Program
Award Identifier / Grant number: No. K202225
Funding source: Graduate Innovation Program of China University of Mining and Technology
Award Identifier / Grant number: No.2024WLKXJ114
Funding source: Postgraduate Research & Practice Innovation Program of Jiangsu Province
Award Identifier / Grant number: No.KYCX24 2673
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Research ethics: Not applicable.
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Author contributions: Huanhuan Lu: writing – original draft, methodology, formal analysis, funding acquisition; Xinan Ren: formal analysis; Yufeng Zhang: formal analysis, funding acquisition. Hongyi Zhang: software.
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Competing interests: The authors state no conflict of interest.
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Research funding: This work was supported by the National Natural Science Foundation of China (grant No. 12371256); the SuQian Sci&Tech Program (grant No. K202225); the Graduate Innovation Program of China University of Mining and Technology (No. 2024WLKXJ114); the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX24 2673).
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Data availability: Not applicable.
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Articles in the same Issue
- Frontmatter
- Research Articles
- Analytical and numerical studies of a cancer invasion model with nonlocal diffusion
- Exploration of different multi-peak solitons and vibrant breather type waves’ solutions of nonlinear Schrödinger equations with advanced dispersion and cubic–quintic nonlinearity, unveiling their applications
- Leader-following consensus tracking control for fractional-order multi-motor systems via disturbance-observer
- A numerical approach for solving nonlinear fractional Klein–Gordon equation with applications in quantum mechanics
- On the construction of various soliton solutions of two space-time fractional nonlinear models
- RANS study of surface roughness effects on ship resistance
- Complexiton and interaction solutions to a specific form of extended Calogero–Bogoyavlenskii–Schiff equation via its bilinear form
- An investigation on explicit exact non-traveling wave solutions of the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation
- The Riemann Hilbert dressing method and wave breaking for two (2 + 1)-dimensional integrable equations
