Pole-free solutions of the first Painlevé hierarchy and non-generic critical behavior for the KdV equation

Abstract

We establish the existence of real pole-free solutions to all even members of the Painlevé I hierarchy. We also obtain asymptotics for those solutions and describe their relevance in the description of critical asymptotic behavior of solutions to the KdV equation in the small dispersion limit. This was understood in the case of a generic critical point, and we generalize it here to the case of non-generic critical points.

29 pages

References in corpus (5)

• Universality of a double scaling limit near singular edge points in random matrix models
• On Hamiltonian perturbations of hyperbolic systems of conservation laws
• Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation
• Asymptotics for a special solution to the second member of the Painleve I hierarchy
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Cited by in corpus (5)

• From gap probabilities in random matrix theory to eigenvalue expansions
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• On a class of compact perturbations of the special pole-free joint solution of KdV and $P_I^2.$
• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics
• Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type