An application of solvable structures to classical and nonclassical similarity solutions

摘要

Using exterior differential systems, we extend work by Harrison and Estabrook for deriving similarity solutions of hyperbolic and parabolic partial differential equations (PDEs). We use exterior calculus results to show that a symmetry (isovector) of the differential ideal corresponding to some hyperbolic or parabolic PDE can be used to generate a Cauchy characteristic vector field of a restricted exterior differential system defined on some four-dimensional regular submanifold of the first jet bundle. We then show that this restricted differential ideal has a Frobenius integrable annihilating space, which can be used to yield a similarity solution of the PDE by applying results from Lie and Cartan on integrating Frobenius integrable vector field distributions via symmetry. We also give an extension to conditional symmetries. (C) 2001 American Institute of Physics. [References: 30]