scholarly journals Quantizing the Bäcklund transformations of Painlevé equations and the quantum discrete Painlevé VI equation

Author(s):  
Koji Hasegawa
2018 ◽  
Vol 32 (17) ◽  
pp. 1850181 ◽  
Author(s):  
Haifeng Wang ◽  
Chuanzhong Li

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.


2004 ◽  
Vol 22 (5) ◽  
pp. 1103-1115 ◽  
Author(s):  
P.R. Gordoa ◽  
U. Muğan ◽  
A. Pickering ◽  
A. Sakka

2002 ◽  
Vol 54 (3) ◽  
pp. 648-670 ◽  
Author(s):  
Yuan Wenjun ◽  
Li Yezhou

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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