| Autor: |
Buckingham, Robert J., Miller, Peter D. |
| Rok vydání: |
2023 |
| Předmět: |
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| Zdroj: |
SIGMA 20 (2024), 008, 27 pages |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.3842/SIGMA.2024.008 |
| Popis: |
It is well known that the Painlev\'e equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlev\'e-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstrass equation. |
| Databáze: |
arXiv |
| Externí odkaz: |
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