Painleve Chains for the Study of Integrable Higher Order Differential Equations

摘要

A previous paper defines the three fundamental homogeneous active Painleve chains which are generated from the equation u sub 2x = ku sub 2, u sub 2x = ku sub 3, and ux = ku sub 2. This paper shows how the concept of Painleve chains can be extended to Painleve chains generated from hybrid differential equations and the passive differential equation u sub 2 = ku 2 over x. In addition the Schwarzian derivative can be used to generate a Painleve chain with the interesting property of consecutive integral resonances. Finally the 2 x 2 eigenvalue problem of Zakharov and Shabat is shown to generate chains exhibiting some but not all of the features of Painleve chains.