Uniformization and transcendence of solutions for the first and second Painlevé hierarchies

DOI10.1016/S0375-9601(97)00850-5zbMath0941.34077OpenAlexW2050380199MaRDI QIDQ1966520

Nikolay A. Kudryashov, Mikhail B. Soukharev

Publication date: 8 March 2000

Published in: Physics Letters. A (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/s0375-9601(97)00850-5

zbMATH Keywords

rational solutionsindicesfirst Painlevé hierarchyfirst and second Painlevé equations of fourth order

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) KdV equations (Korteweg-de Vries equations) (35Q53) Nonlinear ordinary differential equations and systems (34A34) Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies (34M55) Entire and meromorphic solutions to ordinary differential equations in the complex domain (34M05) Painlevé-type functions (33E17)

Related Items

Solution classes of the matrix second Painlevé hierarchy, Universality of a double scaling limit near singular edge points in random matrix models, Soliton, rational and special solutions of the Korteweg-de Vries hierarchy, Hierarchies of Difference Equations and Bäcklund Transformations, Paul Painlevé and his contribution to science, Higher Painlevé transcendents as special solutions of some nonlinear integrable hierarchies, Pole-free solutions of the first Painlevé hierarchy and non-generic critical behavior for the KdV equation, Nonlinear differential equations of the second, third and fourth order with exact solutions, A Lagrangian description of the higher-order Painlevé equations, Two hierarchies of ordinary differential equations and their properties, Instanton-type solutions for the second and the fourth Painlevé hierarchies with a large parameter, Asymptotics of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane, Polygons of differential equations for finding exact solutions, Power and non-power expansions of the solutions for the fourth-order analogue to the second Painlevé equation, Special polynomials and rational solutions of the hierarchy of the second Painlevé equation, A certain expression of the first Painlevé hierarchy, On an extended second Painlevé hierarchy, Transformations of ordinary differential equations via Darboux transformation technique, Lax pairs for one of hierarchies similar to the first Painlevé hierarchy, The generalized Yablonskii-Vorob'ev polynomials and their properties, Double Bäcklund transformations and special integrals for the \(K_{II}\) hierarchy, Twistor reductions of the KdV hierarchy, Lax pairs and rational solutions of similarity reductions for Kupershmidt and Sawada-Kotera hierarchies, Existence of a global solution of the Whitham equations, Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List, Nonisospectral scattering problems: A key to integrable hierarchies, Discrete equations corresponding to fourth-order differential equations of the P₂ and K₂ hierarchies

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Monodromy- and spectrum-preserving deformations. I
• Solitons in physics, mathematics, and nonlinear optics. Proceedings of two workshops which were an integral part of the 1988-89 IMA program on nonlinear waves
• A perturbative Painlevé approach to nonlinear differential equations
• Birational automorphism groups and differential equations
• Theta functions and non-linear equations
• The Painlevé property for partial differential equations
• A connection between nonlinear evolution equations and ordinary differential equations of P-type. II
• NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES
• From singular manifold equations to integrable evolution equations
• Rational Solutions of Painlevé Equations