Variational Approach for the Singular Perturbation Domain Wall System
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916448459292672 |
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| author | Monreal, Javier Kowalczyk, Michał |
| author_facet | Monreal, Javier Kowalczyk, Michał |
| contents | In this article we study a coupled system of differential equations with Allen-Cahn type non-linearity. Motivated by physical phenomena one of the unknowns in the system is accompanied by a singular perturbation parameter $ε^2$ . By employing variational techniques, we establish the existence of solutions for all values of $ε$ and get results on their qualitative properties, including regularity. Additionally, we analyse the behaviour of solutions as $ε {\to} 0$, demonstrating their pointwise convergence to the solution of the problem for $ε = 0$. We establish the uniqueness of this solution modulo translations. Additionally, in the final section, through an appropriate change of scale, we relate this problem and the second Painlevé equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variational Approach for the Singular Perturbation Domain Wall System Monreal, Javier Kowalczyk, Michał Analysis of PDEs In this article we study a coupled system of differential equations with Allen-Cahn type non-linearity. Motivated by physical phenomena one of the unknowns in the system is accompanied by a singular perturbation parameter $ε^2$ . By employing variational techniques, we establish the existence of solutions for all values of $ε$ and get results on their qualitative properties, including regularity. Additionally, we analyse the behaviour of solutions as $ε {\to} 0$, demonstrating their pointwise convergence to the solution of the problem for $ε = 0$. We establish the uniqueness of this solution modulo translations. Additionally, in the final section, through an appropriate change of scale, we relate this problem and the second Painlevé equation. |
| title | Variational Approach for the Singular Perturbation Domain Wall System |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.14413 |