Title: The existence of a real pole-free solution of the fourth order analogue of the Painlevé I equation Authors: Claeys, Tom ×Vanlessen, Maarten # Issue Date: May-2007 Publisher: IOP Pub. Series Title: Nonlinearity vol:20 issue:5 pages:1163-1184 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: $\begin{equation*}x=Ty-\left(\case 1 6 y^3+\case{1}{24} (y_x^2+2yy_{xx}) +\case {1}{240} y_{xxxx}\right).\end{equation*}$ 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 → ±∞. ISSN: 0951-7715 Publication status: published KU Leuven publication type: IT Appears in Collections: Analysis Section