Lower estimates of the growth order of solutions to the second Painlevé hierarchy

Abstract

This article concerns with the second Painlevé hierarchy, i.e., the 2nth order analogues of the second Painlevé equation. Though several higher order analogues of the second Painlevé equation are proposed by several authors, we investigate one derived from the KdV hierarchy by similarity reduction, and give the lower estimate of the growth order of transcendental meromorphic solutions to the second Painlevé hierarchy with an integer parameter. Moreover, the first lemma in Appendix supplements an incompleteness of Li and He [“On analytic properties of higher analogs of the second Painlevé equation,” J. Math. Phys. 43, 1106–1115 (2002)]10.1063/1.1420396 and Li [“Some properties of the solutions of higher analogue of the Painlevé equation,” Acta Math. Appl. Sin. 22, 59–64 (2006)]10.1007/s10255-005-0285-0.

Weierstrass’ Elliptic Function Solutions to the Autonomous Limit of the String Equation

“…. The author proved a theorem similar to Theorem A for the second Painlevé hierarchy [15], and, in its proof, auxiliary differential polynomials play important roles. So, for the first Painlevé hierarchy as well, the auxiliary differential polynomials should exist.…”