On meromorphic solutions of the equations related to the non-stationary hierarchy of the second Painlevé equation - Gromak, Elena V.; Gromak, Valeri I. (Journal of the Belarusian State University. Mathematics and Informatics, 2023)
The non-stationary hierarchy of the second Painlevé equation is herein considered. It is a sequence of polynomial ordinary differential equations of even order with a single differential-algebraic structure determined by the operator LN. The first member of this hierarchy for N = 1 is the second Painlevé equation, and the subsequent equations of 2N order contain arbitrary parameters. They are also named generalised higher analogues of the second Painlevé equation of 2N order. The hierarchies of the first Painlevé equation and the equation P34 from the classification list of canonical Painlevé equations are also associated with this hierarchy. In this paper, we also consider a second order linear equation the coefficients of which are determined by solutions of the hierarchy of the second Painlevé equation and the equation P34. Using the Frobenius method, we obtain sufficient conditions for the meromorphicity of the general solution of second-order linear equations with the coefficients defined by the solutions of the first three equations of the non-stationary hierarchy of the second Painlevé equation and the equation P34. We also find sufficient conditions for the rationality of the general solution of second-order linear equations with coefficients determined by rational solutions of the equations of the non-stationary hierarchy of the second Painlevé equation and the equation P34.
Keywords: Painlevé equations; the hierarchy of the second Painlevé equation; meromorphic solutions