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We establish the existence of a real solution y(x,T) with no poles on thereal 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 theexistence part of a conjecture posed by Dubrovin. We obtain our result byproving the solvability of an associated Riemann-Hilbert problem through theapproach of a vanishing lemma. In addition, by applying the Deift/Zhousteepest-descent method to this Riemann-Hilbert problem, we obtain theasymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure

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