Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911785896902656 |
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| author | Cadiot, Matthieu Lessard, Jean-Philippe Nave, Jean-Christophe |
| author_facet | Cadiot, Matthieu Lessard, Jean-Philippe Nave, Jean-Christophe |
| contents | In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in $H^{l}$ as well as bounded linear operators from $L^{2}$ to $H^{l}$. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in $H^{4}(\mathbb{R})$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_12877 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods Cadiot, Matthieu Lessard, Jean-Philippe Nave, Jean-Christophe Analysis of PDEs Dynamical Systems In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in $H^{l}$ as well as bounded linear operators from $L^{2}$ to $H^{l}$. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in $H^{4}(\mathbb{R})$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave. |
| title | Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2302.12877 |