The existence of a real pole-free solution of the fourth order analogue of the Painleve 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 Painleve I equation: x = T y - (1/6y(3) + 1/24(y(x)(2) +2yy(xx)) + 1/240y(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 -> +/-infinity.

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