A $\mathcal C^\infty$-structure-based approach to traveling wave solutions of the gKdV equation
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909797811486720 |
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| author | Pan-Collantes, Antonio J. |
| author_facet | Pan-Collantes, Antonio J. |
| contents | A novel geometric method is applied to the problem of describing traveling wave solutions of the generalized Korteweg--de Vries (gKdV) equation in the form $$ u_t + u_{xxx} + a(u)u_x = 0, $$ where $a(u)$ is a smooth function characterizing the nonlinearity. Using the traveling wave ansatz, the gKdV equation reduces to an ordinary differential equation (ODE), which we analyze via the $\mathcal{C}^\infty$-structure-based method, a geometric framework involving sequences of involutive distributions and Pfaffian equations. Starting with the symmetry $\partial_z$, we construct a $\mathcal{C}^\infty$-structure for the ODE and apply the stepwise integration algorithm to obtain an implicit general solution. Then we derive explicit solutions for specific forms of $a(u)$, including the modified KdV and Schamel--KdV equations, as well as power-law nonlinearities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07112 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A $\mathcal C^\infty$-structure-based approach to traveling wave solutions of the gKdV equation Pan-Collantes, Antonio J. Analysis of PDEs Exactly Solvable and Integrable Systems A novel geometric method is applied to the problem of describing traveling wave solutions of the generalized Korteweg--de Vries (gKdV) equation in the form $$ u_t + u_{xxx} + a(u)u_x = 0, $$ where $a(u)$ is a smooth function characterizing the nonlinearity. Using the traveling wave ansatz, the gKdV equation reduces to an ordinary differential equation (ODE), which we analyze via the $\mathcal{C}^\infty$-structure-based method, a geometric framework involving sequences of involutive distributions and Pfaffian equations. Starting with the symmetry $\partial_z$, we construct a $\mathcal{C}^\infty$-structure for the ODE and apply the stepwise integration algorithm to obtain an implicit general solution. Then we derive explicit solutions for specific forms of $a(u)$, including the modified KdV and Schamel--KdV equations, as well as power-law nonlinearities. |
| title | A $\mathcal C^\infty$-structure-based approach to traveling wave solutions of the gKdV equation |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2507.07112 |