Lie transformations, nonlinear evolution equations, and Painlevé forms

Abstract

We present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one-parameter continuous (Lie) group of transformations. It is shown that, in general, the corresponding invariant variables (the subclass of which is the usual similarity variables) lead to ordinary differential equations of Painlevé type in the case of inverse scattering transform solvable equations, as conjectured by Ablowitz, Ramani, and Segur. This is found to be also true for certain higher spatial dimensional versions such as the Kadomtsev–Petviashivilli, two dimensional sine–Gordon, and Ernst equations. For the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points.