A convergent Fourier spectral Galerkin method for the fractional Camassa-Holm equation
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911161379717120 |
|---|---|
| author | Dwivedi, Mukul Rupp, Andreas |
| author_facet | Dwivedi, Mukul Rupp, Andreas |
| contents | We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent $α\in [1,2]$ with periodic boundary conditions. The semi-discrete scheme preserves both mass and energy invariants of the fCH equation. For the fractional Benjamin-Bona-Mahony reduction, we establish existence and uniqueness of semi-discrete solutions and prove strong convergence to the unique solution in $ C^1([0, T];H^α_{\mathrm{per}}(I))$ for given $T>0$. For the general fCH equation, we demonstrate spectral accuracy in spatial discretization with optimal error estimates $\mathcal{O}(N^{-r})$ for initial data $u_0 \in H^r(I)$ with $r \geq α+ 2$ and exponential convergence $\mathcal{O}(e^{-cN})$ for smooth solutions. Numerical experiments validate orbital stability of solitary waves achieving optimal convergence, confirming theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A convergent Fourier spectral Galerkin method for the fractional Camassa-Holm equation Dwivedi, Mukul Rupp, Andreas Numerical Analysis 65M18, 65M13, 35L17 We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent $α\in [1,2]$ with periodic boundary conditions. The semi-discrete scheme preserves both mass and energy invariants of the fCH equation. For the fractional Benjamin-Bona-Mahony reduction, we establish existence and uniqueness of semi-discrete solutions and prove strong convergence to the unique solution in $ C^1([0, T];H^α_{\mathrm{per}}(I))$ for given $T>0$. For the general fCH equation, we demonstrate spectral accuracy in spatial discretization with optimal error estimates $\mathcal{O}(N^{-r})$ for initial data $u_0 \in H^r(I)$ with $r \geq α+ 2$ and exponential convergence $\mathcal{O}(e^{-cN})$ for smooth solutions. Numerical experiments validate orbital stability of solitary waves achieving optimal convergence, confirming theoretical findings. |
| title | A convergent Fourier spectral Galerkin method for the fractional Camassa-Holm equation |
| topic | Numerical Analysis 65M18, 65M13, 35L17 |
| url | https://arxiv.org/abs/2508.13683 |