Existence and Uniqueness of Tronquée Solutions of the Fourth-Order Jimbo-Miwa Second Painlevé Equation

Nalini Joshi and Tegan Morrison

Abstract

We consider the asymptotic limit as the independent variable approaches infinity, of the fourth-order second Painlevé equation obtained from a hierarchy based on the Jimbo-Miwa Lax pair. We prove that there exist two families of algebraic formal power series solutions and that there exist true solutions with these behaviours in sectors σ of the complex plane. Given σ we also prove that there exists a wider sector Σ ⊃ σ in which there exists a unique solution in each family. These provide the analogue of Boutroux's tri-tronquée solutions for the classical second Painlevé equation. Surprisingly, they also extend beyond the tri-tronquée solutions in the sense that we find penta-, hepta-, ennea-, and hendeca-tronquée solutions.

AMS Subject Classification: Primary 33E17; secondary 34M55.