Differential equations and Optimal control

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

V. I. Gromak

Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

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 $P_{34}$ 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, Bācklund 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.

Keywords: Hierarchies of Painlevé; equations; meromorphic solutions; Bācklund transformations

UDC: 517.925.7

Received: 29.01.2025
Revised: 20.02.2025
Accepted: 25.02.2025