The Yablonskii - Vorob'ev polynomials for the second Painlev'e hierarchy

Abstract

Special polynomials associated with rational solutions of the second Painlev'e equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of the polynomials is established. Formulaes for their coefficients are found. The degree of every polynomial is obtained. The main achievement of the method lies in the fact that it enables one to construct the family of polynomials corresponding to any member of the second Painlev'e hierarchy. Our approach can be applied for deriving the polynomials related to rational or algebraic solutions of other nonlinear differential equations.

23 pages, 2 figures

References in corpus (2)

• Power expansions for solution of the fourth-order analog to the first Painlevé equation
• Explicit form of the Yablonskii - Vorob'ev polynomials

Cited by in corpus (6)

• Invariant algebraic curves for Liénard dynamical systems revisited
• The method of Puiseux series and invariant algebraic curves
• Relations for zeros of special polynomials associated to the Painleve equations
• Special polynomials associated with the K2 hierarchy
• Remarks on rational solutions for the Korteveg - de Vries hierarchy
• On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation