On discrete Painlevé equations as Bäcklund transformations

2001 ◽  
pp. 129-139
Author(s):  
P. Clarkson ◽  
E. Mansfield ◽  
H. Webster
Keyword(s):  
Painlevé Equations ◽  
Bäcklund Transformations ◽  
Painleve Equations ◽  
Backlund Transformations ◽  
Discrete Painlevé Equations

Bäcklund transformation of Frobenius Painlevé equations

2018 ◽  
Vol 32 (17) ◽  
pp. 1850181 ◽  
Author(s):  
Haifeng Wang ◽  
Chuanzhong Li
Keyword(s):  
Lax Pair ◽  
Painlevé Equations ◽  
Bäcklund Transformations ◽  
Kp Hierarchy ◽  
Painleve Equations ◽  
Virasoro Constraint ◽  
Backlund Transformations ◽  
Hirota Bilinear Equation ◽  
Hirota Bilinear ◽  
Bilinear Equation

In this paper, in order to generalize the Painlevé equations, we give a [Formula: see text]-Painlevé IV equation which can apply Bäcklund transformations to explore. And these Bäcklund transformations can generate new solutions from seed solutions. Similarly, we also introduce a Frobenius Painlevé I equation and Frobenius Painlevé III equation. Then, we find the connection between the Frobenius KP hierarchy and Frobenius Painlevé I equation by the Virasoro constraint. Further, in order to seek different aspects of Painlevé equations, we introduce the Lax pair, Hirota bilinear equation and [Formula: see text] functions. Moreover, some Frobenius Okamoto-like equations and Frobenius Toda-like equations can also help us to explore these equations.

scholarly journals Bäcklund transformations for higher order Painlevé equations

2004 ◽  
Vol 22 (5) ◽  
pp. 1103-1115 ◽  
Author(s):  
P.R. Gordoa ◽  
U. Muğan ◽  
A. Pickering ◽  
A. Sakka
Keyword(s):  
Higher Order ◽  
Painlevé Equations ◽  
Bäcklund Transformations ◽  
Painleve Equations ◽  
Backlund Transformations

Rational Solutions of Painlevé Equations

2002 ◽  
Vol 54 (3) ◽  
pp. 648-670 ◽  
Author(s):  
Yuan Wenjun ◽  
Li Yezhou
Keyword(s):  
Transformation Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Painlevé Equations ◽  
Bäcklund Transformations ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Backlund Transformations ◽  
Special Relations ◽  
Necessary And Sufficient

AbstractConsider the sixth Painlevé equation (P6) below where α, β, γ and δ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation (P6) in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generated by means of transformation group found by Okamoto [7] and Bäcklund transformations found by Fokas and Yortsos [4]. A list of rational solutions is included in the appendix. For the sake of completeness, we collected all the corresponding results of other five Painlevé equations (P1)−(P5) below, which have been investigated by many authors [1]–[7].

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