The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation
(2007) Nonlinearity — Vol. 20, n° 5, p. 1163-1184 (2007)
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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 = 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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Claeys, T., & Vanlessen, M. (2007). The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation. Nonlinearity, 20(5), 1163-1184. https://doi.org/10.1088/0951-7715/20/5/006 (Original work published 2007)