On the solutions of the non-linear stationary equations related to the generalised hierarchy of the second Painlevé equation

Authors

  • Valeri I. Gromak Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

Keywords:

hierarchies of Painlevé equations, meromorphic solutions, Bäcklund transformations

Abstract

This paper considers the analytical properties of the solutions of non-linear stationary equations of the generalised hierarchy of the second Painlevé equation and the hierarchies of the first Painlevé equation related with it, as well as the equation P34 from the Painlevé classification list. The local properties of the solutions are investigated, namely, expansion of the solutions in the neighbourhood of moving poles, construction of entire functions (tau functions) that provide a representation of meromorphic solutions. For the stationary hierarchies under consideration, Bäcklund transformations and auto-transformations are given, with the help of which transcendental and rational solutions are constructed. For the initial equations of the hierarchies under investigation, the first integrals are obtained, which are then used to prove the embeddability of the set of solutions of the hierarchy equation with a smaller number in the set of solutions of the hierarchy equation with a larger number. The relationships between the parameters of the equations with embeddable sets of solutions are given.

Author Biography

  • Valeri I. Gromak, Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

    doctor of science (physics and mathematics), full professor; professor at the department of differential equations and system analysis, faculty of mechanics and mathematics

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Published

2025-05-13

Issue

No. 1 (2025): Journal of the Belarusian State University. Mathematics and Informatics

Section

Differential Equations and Optimal Control

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How to Cite

[1]
Gromak, V.I. 2025. On the solutions of the non-linear stationary equations related to the generalised hierarchy of the second Painlevé equation. Journal of the Belarusian State University. Mathematics and Informatics. 1 (May 2025), 40–50.