Systematics of Strongly Self-Dominant Higher Order Differential Equations Based on the Painleve Analysis of Their Singularities

摘要

This paper presents a simple way of classifying higher order differential equations based on the requirements of the Painleve' property, i.e., the presence of no movable critical points. The fundamental building blocks for such equations may be generated by strongly self-dominant differential equations. Such differential equations having both a constant degree d and a constant value of the difference n-m form a Painleve' chain; however, only three of the many possible Painleve' chains can have the Painleve' property. Among the three Painleve' chains which can have the Painleve' property, one contains the Burgers' equation; another contains the dominant terms of the first Painleve' transcendent, the isospectral Korteweg-de Vries equation, and the isospectral Boussinesq equation; and the third contains the dominant terms of the second Painleve' transecendent and the isospectral modified (cubic) Korteweg-de Vries equation. Differential equations of the same order and having the same value of the quotient (n-m)/(d-1) can be mixed to generate a new hybrid differential equation. In such cases a hybrid can have the Painleve' property even if only one of its components has the Painleve' property. Such hybridization processes can be used to generate the various fifth-order evolution equations of interest, namely the Caudrey-Dodd-Gibbon, Kuperschmidt, and Morris equations.