Estabrook-Wahlquist Prolongations and Infinite-Dimensional Algebras

Abstract

Detailed mappings between zero-curvture equations for prolongation structures of nonlinear pde's and Estabrook-Wahlquist algorithms for same are given. The differences are exemplified by studies of the sine-Gordon equation. An example where the prolongation structure must be infinite-dimensional is given by the Robinson-Trautman equation, where the minimal algebra is $K_2$. In general these algorithms require integration of vector-field valued pde's; solutions of simultaneous flow equations are given. Applications to coupled systems of flow equations are given, where the result describes Lie algebras of vector fields vertical over fibers of pseudopotentials over a jet bundle appropriate for a given system of pde's; algebras invariant under sl(2,C) are of special interest.