EXISTENCE AND UNIQUENESS OF TRONQUEE SOLUTIONS OF THE FOURTH-ORDER JIMBO-MIWA SECOND PAINLEVE EQUATION

We consider the asymptotic limit as the independent variable approaches infinity, of the fourth-order second Painleve 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 sigma of the complex plane. Given sigma we also prove that there exists a wider sector Sigma superset of sigma in which there exists a unique solution in each family. These provide the analogue of Boutroux's tri-tronquee solutions for the classical second Painleve equation. Surprisingly, they also extend beyond the tri-tronquee solutions in the sense that we. find penta-, hepta-, ennea-, and hendeca-tronquee solutions.