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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.16507 |
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| _version_ | 1866909646179008512 |
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| author | Ficek, Filip Maliborski, Maciej |
| author_facet | Ficek, Filip Maliborski, Maciej |
| contents | We study time-periodic solutions for the cubic wave equation on an interval with Dirichlet boundary conditions. We begin by following the perturbative construction of Vernov and Khrustalev and provide a rigorous derivation of the fourth-order expansion in small amplitude, which we use to verify the Galerkin scheme. In the main part, we focus on exploring large solutions numerically. We find an intricate bifurcation structure of time-periodic solutions forming a fractal-like pattern and explore it for the first time. Our results suggest that time-periodic solutions exist for arbitrary frequencies, with appearance of fine bifurcation structure likely related to the Cantor set families of solutions described in previous rigorous works. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16507 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Periodic Solutions for the 1d Cubic Wave Equation with Dirichlet Boundary Conditions Ficek, Filip Maliborski, Maciej Analysis of PDEs Mathematical Physics 35B10, 35B32, 35L71, 65M70, 65P30 We study time-periodic solutions for the cubic wave equation on an interval with Dirichlet boundary conditions. We begin by following the perturbative construction of Vernov and Khrustalev and provide a rigorous derivation of the fourth-order expansion in small amplitude, which we use to verify the Galerkin scheme. In the main part, we focus on exploring large solutions numerically. We find an intricate bifurcation structure of time-periodic solutions forming a fractal-like pattern and explore it for the first time. Our results suggest that time-periodic solutions exist for arbitrary frequencies, with appearance of fine bifurcation structure likely related to the Cantor set families of solutions described in previous rigorous works. |
| title | Periodic Solutions for the 1d Cubic Wave Equation with Dirichlet Boundary Conditions |
| topic | Analysis of PDEs Mathematical Physics 35B10, 35B32, 35L71, 65M70, 65P30 |
| url | https://arxiv.org/abs/2407.16507 |