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Systematic construction of nonautonomous Hamiltonian equations of Painlevé‐type. II. Isomonodromic Lax representation
Author(s) -
Błaszak Maciej,
Domański Ziemowit,
Marciniak Krzysztof
Publication year - 2022
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12495
Subject(s) - integrable system , type (biology) , mathematics , lax pair , hamiltonian system , pure mathematics , construct (python library) , representation (politics) , algebra over a field , mathematical physics , computer science , ecology , politics , political science , law , biology , programming language
This is the second article in a suite of articles investigating relations between Stäckel‐type systems and Painlevé‐type systems. In this paper, we construct isomonodromic Lax representations for Painlevé‐type systems found in the previous paper by Frobenius integrable deformations of Stäckel‐type systems. We first construct isomonodromic Lax representations for Painlevé‐type systems in the so‐called magnetic representation and then, using a multitime‐dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé‐type systems in the nonmagnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painlevé‐type. We also present isomonodromic Lax representations for all one‐, two‐, and three‐dimensional Painlevé‐type systems originating in our scheme. Based on these results we propose complete hierarchies ofP I − P I V$P_{I}-P_{IV}$ that follow from our construction.