Estabrook-Wahlquist Prolongations and Infinite-Dimensional Algebras

I. Estabrook-Wahlquist Prolongations and Zero-Curvature Requirements Motivated by a desire to find new solutions of physically-interesting partial differential equations, we think of a k-th order pde as a variety, Y , of a finite jet bundle, J (M, N), with M the independentand N the dependent-variables for the pde. From this geometric approach, we can look for point symmetries, contact symmetries, generalized symmetries, or even non-local symmetries, where the system is prolonged further, to a fiber space over J∞, with fibers W , where vertical flows map solution spaces of one pde into another, satisfied by the additional dependent variables, w, that coordinatize the fibers. The compatibility conditions for such flows to exist are “zero-curvature conditions.” Solutions of these conditions may be found using the tangent structure or the co-tangent structure, over J∞×W . We describe both, but follow the approach via differential forms, following Cartan, , Estabrook and Wahlquist, and Pirani, believing that it provides better guides for the intuition, for complicated (systems of) pde’s.