Complete classification of rational solutions of \(A_{2n}\)-Painlevé systems. (English) Zbl 1472.37067
The authors provide a complete classification of the rational solutions of the Painlevé IV equation and its higher-order hierarchy, known as the \(A_{2n}\)-Painlevé or Noumi-Yamada system. First, they recall the equivalence between the Noumi-Yamada system and a cyclic dressing chain of Schrödinger operators. Then, they show by a careful investigation of the local expansions of the rational solutions around their poles, that the solutions have trivial monodromy, and therefore they must be expressible in terms of Wronskian determinants whose entries are Hermite polynomials.
Next, they use Maya diagrams to classify all the \( (2n + 1)\)-cyclic dressing chains and therefore achieve a complete classification. Finally, they connect their results with the geometric approach mastered by the Japanese school, showing a representation for the action of the symmetry group of Bäcklund transformations in terms of Maya cycles and oddly colored integer sequences. The natural extension of this work is to tackle the full classification of the rational solutions to the \(A_{2n+1}\)-Painlevé systems, which include the Painlevé V and its higher order extensions.
This is a very significative work in the study of the solutions of Painlevé IV equation.
Next, they use Maya diagrams to classify all the \( (2n + 1)\)-cyclic dressing chains and therefore achieve a complete classification. Finally, they connect their results with the geometric approach mastered by the Japanese school, showing a representation for the action of the symmetry group of Bäcklund transformations in terms of Maya cycles and oddly colored integer sequences. The natural extension of this work is to tackle the full classification of the rational solutions to the \(A_{2n+1}\)-Painlevé systems, which include the Painlevé V and its higher order extensions.
This is a very significative work in the study of the solutions of Painlevé IV equation.
Reviewer: Mengkun Zhu (Jinan)
MSC:
| 37J65 | Nonautonomous Hamiltonian dynamical systems (Painlevé equations, etc.) |
| 34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
Keywords:
Painlevé equations; rational solutions; Darboux dressing chains; Maya diagrams; Wronskian determinants; Hermite polynomialsReferences:
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