The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation

TL;DR

This paper proves the existence of a real, pole-free solution to a fourth order analogue of the Painleve I equation, confirming a conjecture by Dubrovin and analyzing its asymptotic behavior.

Contribution

It establishes the existence of a pole-free real solution to a complex fourth order Painleve I analogue, using Riemann-Hilbert problem techniques.

Findings

Existence of a pole-free real solution y(x,T).

Asymptotic behavior of y(x,T) as x approaches infinity.

Confirmation of Dubrovin's conjecture.

Abstract

We establish the existence of a real solution y(x,T) with no poles on the real line of the following fourth order analogue of the Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x,T) as x\to\pm\infty.