Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1998-11-20
Nonlinear Sciences
Exactly Solvable and Integrable Systems
12 pages in LaTeX 2.09 (uses ioplppt.sty), to appear in Inverse Problems
Scientific paper
10.1088/0266-5611/15/1/019
The second Painlev\'e hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlev\'e equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlev\'e analysis to ordinary differential equations. We extend these techniques in order to derive auto-B\"acklund transformations for the second Painlev\'e hierarchy. We also derive a number of other B\"acklund transformations, including a B\"acklund transformation onto a hierarchy of P34 equations, and a little known B\"acklund transformation for P2 itself. We then use our results on B\"acklund transformations to obtain, for each member of the P2 hierarchy, a sequence of special integrals.
Clarkson Peter A.
Joshi Nalini
Pickering Andrew
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