The development of the concept of Painlevé chains for the classification of integrable higher-order differential equations

The solvability of higher-order ordinary differential equations (ODE’s) by the inverse scattering transform appears to be related to the absence of movable critical points, i.e. the so-called Painlevé property. In this connection, the following three-step procedure can be used to test ODE’s for the Painlevé property: (1) determination of the dominant terms of the ODE and its balancing exponent p in a power series which characterizes the behavior of its solutions near the movable singularities; (2) solution of an indicial algebraic equation to determine the resonances r1,…,rn which indicate the terms where the integration constants can enter the above power series; (3) determination of the compatibility of the coefficients of the above power series with a pure Laurent series without any logarithmic terms entering at the resonances. The Painlevé test can fail at any of these three steps: (1) the balancing exponent p is not a negative integer; (2) the resonances are not integers or the indicial equation has a repeated root; or (3) the expressions for the coefficients of the power series terms at the resonances are incompatible with the identical zero values required for introduction of the integration constants. In the 1980s, I observed that the fundamental building blocks for ODE’s satisfying the first two steps of the above Painlevé test may be generated by strongly self-dominant differential equations of the type Dnu=Dm(u(m−n+p)/p) in which D is the derivative ∂/∂x, m and n the positive integers, and p a negative integer. Such differential equations having both a constant degree and a constant value of the difference n−m form a Painlevé chain. The three simple Painlevé chains which can satisfy the first two steps of the Painlevé test are generated by such successive differentiations of uxx=ku2 (KdV chain), uxx=ku3 (modified KdV chain), and ux=ku2 (Burgers chain). The length of the Painlevé chains obtained by this method is limited by the appearance of double roots in the indicial equation and/or a failure of the resulting ODE to pass the third step of the Painlevé test. Since the publication of my original papers more than 10 years ago, it has been shown that longer Painlevé chains can be generated by the substitution of more complicated operators (e.g. D+u+u′D−1 for the Burgers chain) for the simple derivative D.