An Exact Perturbative Existence and Uniqueness Theorem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912097413103616 |
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| author | Nikolaev, Nikita |
| author_facet | Nikolaev, Nikita |
| contents | We investigate singularly perturbed nonlinear complex differential systems of the form $\hbar \partial_x f = F (x, \hbar, f)$ where $\hbar$ is a small complex perturbation parameter. Under a geometric assumption on the eigenvalues of the Jacobian matrix of $F$, we prove an Existence and Uniqueness Theorem for exact perturbative solutions; i.e., holomorphic solutions with prescribed perturbative expansions in $\hbar$. In fact, these solutions are the Borel resummation of the formal perturbative solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_04526 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An Exact Perturbative Existence and Uniqueness Theorem Nikolaev, Nikita Classical Analysis and ODEs Mathematical Physics Analysis of PDEs 34M6 (primary), 34M04, 34M30, 34E15, 34E20, 34A12, 34A34 We investigate singularly perturbed nonlinear complex differential systems of the form $\hbar \partial_x f = F (x, \hbar, f)$ where $\hbar$ is a small complex perturbation parameter. Under a geometric assumption on the eigenvalues of the Jacobian matrix of $F$, we prove an Existence and Uniqueness Theorem for exact perturbative solutions; i.e., holomorphic solutions with prescribed perturbative expansions in $\hbar$. In fact, these solutions are the Borel resummation of the formal perturbative solutions. |
| title | An Exact Perturbative Existence and Uniqueness Theorem |
| topic | Classical Analysis and ODEs Mathematical Physics Analysis of PDEs 34M6 (primary), 34M04, 34M30, 34E15, 34E20, 34A12, 34A34 |
| url | https://arxiv.org/abs/2201.04526 |