Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations

2006 ◽  
Vol 6 (2) ◽  
pp. 329-401 ◽  
Author(s):  
Peter A. Clarkson
Keyword(s):  
Painlevé Equations ◽  
Rational Solutions ◽  
Soliton Equations ◽  
Painleve Equations ◽  
Special Polynomials

RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION

2008 ◽  
Vol 06 (04) ◽  
pp. 349-369 ◽  
Author(s):  
PETER A. CLARKSON
Keyword(s):  
Infinite Number ◽  
Boussinesq Equation ◽  
Nonlinear Schrödinger Equations ◽  
Schrödinger Equations ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Symmetry Reductions ◽  
Nonlinear Schrodinger Equations ◽  
De Vries

Rational solutions of the Boussinesq equation are expressed in terms of special polynomials associated with rational solutions of the second and fourth Painlevé equations, which arise as symmetry reductions of the Boussinesq equation. Further generalized rational solutions of the Boussinesq equation, which involve an infinite number of arbitrary constants, are derived. The generalized rational solutions are analogs of such solutions for the Korteweg–de Vries and nonlinear Schrödinger equations.

scholarly journals Remarks on the Yablonskii-Vorob’ev polynomials

2000 ◽  
Vol 159 ◽  
pp. 87-111 ◽  
Author(s):  
Makoto Taneda
Keyword(s):  
Algebraic Approach ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Painleve Equation ◽  
Second Painlevé Equation

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

2011 ◽  
Vol 467 (2136) ◽  
pp. 3443-3468 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Exact Solutions ◽  
Linear Problem ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Group Symmetry ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Discrete Painlevé Equations ◽  
Painleve Equation ◽  
Second Painlevé Equation

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

scholarly journals Special polynomials and the Hirota Bilinear relations of the second and the fourth Painlevé equations

2000 ◽  
Vol 159 ◽  
pp. 179-200 ◽  
Author(s):  
Satoshi Fukutani ◽  
Kazuo Okamoto ◽  
Hiroshi Umemura
Keyword(s):  
Recurrence Relation ◽  
Rational Functions ◽  
Painlevé Equations ◽  
Algebraic Proof ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Bilinear Relations ◽  
Hirota Bilinear

We give a purely algebraic proof that the rational functions Pn(t), Qn(t) inductively defined by the recurrence relation (1), (2) respectively, are polynomials. The proof reveals the Hirota bilinear relations satisfied by the τ-functions.

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