Factorization of the reaction-diffusion equation, the wave equation, and other equations

Abstract

We investigate equations of the form D _t u = Δu + ξ∇ _u for an unknown function u(t, x), t ∈ ℝ, x ∈ X, where D _t u = a ₀(u, t) + Σ ^r _k=1 a _k (t, u)∂ ^k _t u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ∇ _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ∇)f*v = f*(Δ + Ξ∇)v holds for any smooth function v: Y → ℝ. In this case, the quotient equation is D _t v = Δv + Ξ∇v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.