Painlevé equations from Darboux chains Part 1 : P III −
نویسنده
چکیده
We show that the Painlevé equations PIII −PV I can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrödinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three different branches, each described by an infinite chain of equations. The Painlevé equations are obtained by closing the chain periodically at the lowest nontrivial level(s). The chains provide “symmetric forms” for the Painlevé equations, from which Hirota bilinear forms and Lax pairs are derived. In this paper (Part 1) we analyze in detail the cases PIII − PV , while PV I will be studied in Part 2.
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