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

T. Claeys·M. Vanlessen
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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 Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{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\to\pm\infty.

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T. Claeys, M. Vanlessen. 2006-04-20. The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation. https://doi.org/10.1088/0951-7715%2F20%2F5%2F006

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