Bâcklund transformations of Painleve equations and discrete equations of Painleve type
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With the help of the Schlesinger transformations, we obtain the B¨acklund transformations of the classical continuous Painlev´e equations (PII-PVI). Then using these B¨acklund transformations we derived the corresponding discrete equations. The main idea in obtaining these equations is to eliminate y 0 from the B¨acklund transformations, Ti and Tj , in such a way that Ti ◦ Tj = Tj ◦ Ti = I. Then we obtain an algebraic relation between yi , y and yj . This algebraic relation can be considered as as a discrete equation of the Painlev´e type.
Discrete Painlev´e Equations
QC20.7.D5 T67 2004