Traveling-wave solutions for a higher-order Boussinesq system: existence and numerical analysis
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| Format: | Preprint |
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2025
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| _version_ | 1866911267220881408 |
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| author | Capistrano-Filho, Roberto de A. Muñoz, Juan Carlos Quintero, José R. |
| author_facet | Capistrano-Filho, Roberto de A. Muñoz, Juan Carlos Quintero, José R. |
| contents | We study the existence and numerical computation of traveling wave solutions for a family of nonlinear higher-order Boussinesq evolution systems with a Hamiltonian structure. This general Boussinesq evolution system includes a broad class of homogeneous and non-homogeneous nonlinearities. We establish the existence of traveling wave solutions using the variational structure of the system and the \textit{concentration-compactness} principle by P.-L. Lions, even though the nonlinearity could be non-homogeneous. For the homogeneous case, the traveling wave equations of the Boussinesq system are approximated using a spectral approach based on a Fourier basis, along with an iterative method that includes appropriate stabilizing factors to ensure convergence. In the non-homogeneous case, we apply a collocation Fourier method supplemented by Newton's iteration. Additionally, we present numerical experiments that explore cases in which the wave velocity falls outside the theoretical range of existence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_15106 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Traveling-wave solutions for a higher-order Boussinesq system: existence and numerical analysis Capistrano-Filho, Roberto de A. Muñoz, Juan Carlos Quintero, José R. Analysis of PDEs 76B15, 35A15, 37K40, 65M70, 65M06 We study the existence and numerical computation of traveling wave solutions for a family of nonlinear higher-order Boussinesq evolution systems with a Hamiltonian structure. This general Boussinesq evolution system includes a broad class of homogeneous and non-homogeneous nonlinearities. We establish the existence of traveling wave solutions using the variational structure of the system and the \textit{concentration-compactness} principle by P.-L. Lions, even though the nonlinearity could be non-homogeneous. For the homogeneous case, the traveling wave equations of the Boussinesq system are approximated using a spectral approach based on a Fourier basis, along with an iterative method that includes appropriate stabilizing factors to ensure convergence. In the non-homogeneous case, we apply a collocation Fourier method supplemented by Newton's iteration. Additionally, we present numerical experiments that explore cases in which the wave velocity falls outside the theoretical range of existence. |
| title | Traveling-wave solutions for a higher-order Boussinesq system: existence and numerical analysis |
| topic | Analysis of PDEs 76B15, 35A15, 37K40, 65M70, 65M06 |
| url | https://arxiv.org/abs/2502.15106 |