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

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, x = Ty − ( 1 6 y + 1 24 (y x + 2yyxx) + 1 240 yxxxx ) . 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 → ±∞.