Transformation methods for solving nonlinear field equations

Abstract

The collection of extended canonical transformations of first-order contact manifolds is studied. This collection is shown to form a group under target-source composition and to contain the group of all first prolongations of point transformation of the underlying graph space and all isogroups of completely integrable horizontal ideals. Extended canonical transformations are compared and contrasted with Bäcklund transformations. These results are used to construct an extended Hamilton-Jacobi method for systems of nonlinear PDE. The collection of all extended canonical transformations is also shown to contain infinitely many one-parameter families of transformations, but there is no Lie group structure that contains these one-parameter families, in general. Conditions are obtained under which a one-parameter family of extended canonical transformations will map a solution of the fundamental ideal that characterizes a given system of PDE into a one-parameter family of solutions. These results are applied to the Ω-Gordon equation ∂_x∂₁ φ = Ω(φ) and to the Navier-Stokes equations.