Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915942544441344 |
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| author | Rozmej, Piotr Karczewska, Anna |
| author_facet | Rozmej, Piotr Karczewska, Anna |
| contents | This article concludes the study of (2+1)-dimensional nonlinear wave equations that can be derived in a model of an ideal fluid with irrotational motion. In the considered case of identical scaling of the $x,y$ variables, obtaining a (2+1)-dimensional wave equation analogous to the KdV equation is impossible. Instead, from a system of two first-order Boussinesq equations, a non-linear wave equation for the auxiliary function $f(x,y,z)$ defining the velocity potential can be obtained, and only from its solutions can the surface wave form $η(x,y,t)$ be obtained. We demonstrate the existence of families of (2+1)-dimensional traveling wave solutions, including solitary and periodic solutions, of both cnoidal and superposition types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model Rozmej, Piotr Karczewska, Anna Pattern Formation and Solitons Mathematical Physics This article concludes the study of (2+1)-dimensional nonlinear wave equations that can be derived in a model of an ideal fluid with irrotational motion. In the considered case of identical scaling of the $x,y$ variables, obtaining a (2+1)-dimensional wave equation analogous to the KdV equation is impossible. Instead, from a system of two first-order Boussinesq equations, a non-linear wave equation for the auxiliary function $f(x,y,z)$ defining the velocity potential can be obtained, and only from its solutions can the surface wave form $η(x,y,t)$ be obtained. We demonstrate the existence of families of (2+1)-dimensional traveling wave solutions, including solitary and periodic solutions, of both cnoidal and superposition types. |
| title | Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model |
| topic | Pattern Formation and Solitons Mathematical Physics |
| url | https://arxiv.org/abs/2506.08941 |