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Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $(\mathrm{D}_7)$ Equation
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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It is well known that the Painlevé 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 Painlevé-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.
Keywords: isomonodromy method, Weierstraß equation.
Mots-clés : Painlevé-III (D$_7$) equation, algebraic solutions 
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Robert J. Buckingham; Peter D. Miller. Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $(\mathrm{D}_7)$ Equation. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a7/

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