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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
M. F. Prokhorova, Factorization of Partial Differential Equations: Structure of the Parabolic Equations Category, Preprint No. 10/2005 (St. Petersburg Dep. Steklov Inst. Math., Russ. Acad. Sci., 2005) [in Russian].
M. F. Prokhorova, Modeling of the Heat Equation and Stefan Problem (IMM UrO RAN, Yekaterinburg, 2000), Available from VINITI, No. 347-V00 [in Russian].
M. F. Prokhorova, “The structure of the category of parabolic equations,” http://arxiv.org/pdf/math/0512094v5.pdf, Submitted May 29, 2009.
P. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1986).
A. M. Vinogradov, “Geometry of nonlinear differential equations,” in Results in Science and Technology (VINITI, Moscow, 1980), Ser. Geometry Problems, Vol. 11, pp. 89–134 [in Russian].
A. M. Vinogradov, “Category of nonlinear differential equations,” in Global Analysis. Studies and Applications I, Ed. by Yu. G. Borisovich and Yu. E. Gliklikh (Springer, Berlin, 1984), Ser. Lecture Notes in Mathematics, Vol. 1108, pp. 77–102.
A. V. Bocharov, A. M. Verbovetskii, A. M. Vinogradov, et al., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (Amer. Math. Soc., Providence, RI, 1999).
M. Hirsch, Differential Topology (Springer, New York, 1976).
F. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer, New York, 1971).
Original Russian Text © M.F. Prokhorova, 2013, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 4.
About this article
Cite this article
Prokhorova, M.F. Factorization of the reaction-diffusion equation, the wave equation, and other equations. Proc. Steklov Inst. Math. 287, 156–166 (2014). https://doi.org/10.1134/S0081543814090156
- category of partial differential equations
- reaction-diffusion equation
- heat equation
- wave equation.