Title: Rational solutions to the Painlevé II equation from particular polynomials

Authors: Pierre Gaillard

Addresses: Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche Comté, 9 avenue Alain Savary BP 47870, 21078 Dijon Cedex, France

Abstract: The Painlevé equations were derived by Painlevé and Gambier in 1895-1910. Given a rational function R in w, w′ and analytic in z, they searched what were the second order ordinary differential equations of the form w″ = R(z, w, w′) with the properties that the singularities other than poles of any solution or this equation depend on the equation only and not of the constants of integration. They proved that there are 50 equations of this type, and the Painlevé II is one of these. Here, we construct solutions to the Painlevé II equation (PII) from particular polynomials. We obtain rational solutions written as a derivative with respect to the variable x of a logarithm of a quotient of a determinant of order n + 1 by a determinant of order n. We obtain an infinite hierarchy of rational solutions to the PII equation. We give explicitly the expressions of these solutions solution for the first orders.

Keywords: Painlevé equation II; PII; rational solutions; determinants.

DOI: 10.1504/IJANS.2022.125305

International Journal of Applied Nonlinear Science, 2022 Vol.3 No.3, pp.189 - 196

Received: 25 Aug 2021
Accepted: 07 Apr 2022
Published online: 06 Sep 2022 *