Author:
Rosenhaus V.,Shankar Ravi,Squellati Cody
Abstract
AbstractIn this paper, we look at the method of separation of variables of a PDE from its symmetry transformation point of view. Specifically, we discuss the relation between the existence of additively and multiplicatively separated variables of a PDE, and the form of its symmetry operators. We show that solutions in the form of separated variables are in fact, invariant solutions, i.e. solutions invariant under some subalgebra of the symmetry operators of the equation. For the case of two independent variables, we obtain the form of Lie point symmetry operators corresponding to additively and multiplicatively separated solutions, and generalize our results for the case when separated variables are any functions of independent variables. We also discuss the role of contact symmetry transformations and differential invariants for the existence of separated solutions, and outline the role of variational symmetries, as well as conditional (non-classical) symmetry operators. We demonstrate that the symmetry approach is a valuable tool for obtaining information regarding existence of solutions with separated variables.
Publisher
Springer Science and Business Media LLC
Reference30 articles.
1. Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18(11), 1025–1042 (1969)
2. Broadbridge, P., Chanu, C.M., Miller, W., Jr.: Solutions of Helmholtz and Schrödinger equations with side condition and nonregular separation of variables. SIGMA Symmetry Integrability Geom. Methods Appl. 8, 089 (2012)
3. Buchnev, A.A.: Dinamika Sploshnoi Sredy 7, 212–214 (1971). ((in Russian))
4. Chou, K.-S., Qu, C.: Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations. J. Phys. A Math. Gen. 32(35), 6271–6286 (1999)
5. Doyle, P.W., Vassiliou, P.J.: Separation of variables for the 1-dimensional non-liner diffusion equation. Int. J. Non-Linear Mech. 33(2), 315–326 (1997)