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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
× corresponding author
# (joint) last author

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