Existence and uniqueness of Tronquée solutions of the fourth-order Jimbo-Miwa second Painlevé equation

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 $\sigma$ of the complex plane. Given $\sigma$ we also prove that there exists a wider sector $\Sigma \supset \sigma$ 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.