Factorization of nonlinear heat equation posed on Riemann manifold

Abstract

There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE systems, and its concrete investigation is developing for the heat equation case. There is considered the category of second order parabolic equations posed on arbitrary manifolds. In this category, for the given nonlinear heat equation we could find morphisms from it to other parabolic equations with the same or a smaller number of independent variables. This allows to receive some classes of solutions of original equation from the class of all solutions of such a reduced equation. Classification of morphisms (with the selection from every equivalence class of the simplest "canonical" representatives) is carried out. There are derived the necessary and sufficient conditions for canonical morphisms of heat equation to the parabolic equation on the other manifold. These conditions are formulated in the differential geometry language. The comparison with invariant solutions classes, obtained by the Lie group methods, is carried out. It is proved that discovered solution classes are richer than invariant solution classes, even if we find any (including discontinuous) symmetry groups of original equation.