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

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 Painlevé I equation: 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 → ±∞.