The Kadomtsev-Petviashvili equation in conformal variables for waves over topography
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918439360135168 |
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| author | Andrade, David Flamarion, Marcelo V. |
| author_facet | Andrade, David Flamarion, Marcelo V. |
| contents | The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by means of asymptotic expansions, obtain a Kadomtsev-Petviashvili type equation formulated in conformal variables as a model for weakly transversal surface waves propagating over topography. A key advantage of this formulation is that the topography, defined in the physical domain, does not need to be a smooth function, or even a function in the classical sense because, our asymptotic analysis relies on the effective depth, which comes through the Jacobian of the conformal map which is assumed to be a slowly varying function. The resulting equation provides a consistent extension of several well known weakly nonlinear dispersive wave models previously reported in the literature. Numerical simulations are performed to illustrate the newly derived equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09646 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Kadomtsev-Petviashvili equation in conformal variables for waves over topography Andrade, David Flamarion, Marcelo V. Analysis of PDEs The conformal mapping approach is a well established technique for solving the Euler equations for potential flows with one spatial dimension. In this work, we extend this framework to problems with a weakly transversal dependence and, by means of asymptotic expansions, obtain a Kadomtsev-Petviashvili type equation formulated in conformal variables as a model for weakly transversal surface waves propagating over topography. A key advantage of this formulation is that the topography, defined in the physical domain, does not need to be a smooth function, or even a function in the classical sense because, our asymptotic analysis relies on the effective depth, which comes through the Jacobian of the conformal map which is assumed to be a slowly varying function. The resulting equation provides a consistent extension of several well known weakly nonlinear dispersive wave models previously reported in the literature. Numerical simulations are performed to illustrate the newly derived equation. |
| title | The Kadomtsev-Petviashvili equation in conformal variables for waves over topography |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.09646 |