Asymptotics for a special solution to the second member of the Painleve I hierarchy

Abstract

We study the asymptotic behavior of a special smooth solution y(x,t) to the second member of the Painleve I hierarchy. This solution arises in random matrix theory and in the study of Hamiltonian perturbations of hyperbolic equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is known and relatively simple, but it turns out to be more subtle when x and t tend to infinity simultaneously. We distinguish a region of algebraic asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain rigorous asymptotics in both regions. We also discuss two critical transitional asymptotic regimes.

19 pages

References in corpus (4)

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• On Hamiltonian perturbations of hyperbolic systems of conservation laws
• Lectures on random matrix models. The Riemann-Hilbert approach
• Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation

Cited by in corpus (11)

• Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions
• On critical behaviour in generalized Kadomtsev--Petviashvili equations
• Pole-free solutions of the first Painlevé hierarchy and non-generic critical behavior for the KdV equation
• On an isomonodromy deformation equation without the Painlevé property
• "Quantization" of higher hamiltonian analogues of the Painleve I and Painleve II equations with two degrees of freedom
• Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation
• On the tritronquée solutions of P${}_I^2$
• On a class of compact perturbations of the special pole-free joint solution of KdV and $P_I^2.$
• Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis
• 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