Conditional Discretization of a
Generalized Reaction–Diffusion Equation
Decio Levi, Miguel A. Rodríguez, and Zora Thomova
Abstract A PDE modeling a reaction–diffusion physical system is discretized
using its conditional symmetries. Discretization is carried out using two specific
conditional symmetries. Explicit solutions of the difference equation are constructed
when the symmetry is projective.
Keywords Symmetry · Integrable systems · Difference equations · Conditional
symmetry · Invariant discretization
1 Introduction
Partial differential equations (PDE) modeling interesting models in Physics are
generically hard to solve and exact solutions are rare and difficult to obtain. The Lie
symmetries of these equations provide, in many cases, explicit solutions, or at least
hints on how to find those solutions. This is a well-known topic, see, for example,
the following monographs dedicated to it [1, 3, 15, 16].
A particularly useful application of this method is the reduction of the order
of the equation or of the number of variables. The invariants of the vector field
corresponding to the infinitesimal symmetry provide a new set of coordinates and
the PDE, written in this new system, can be usually solved and solutions of the
original equation can be obtained. The well-known example is the construction of
the fundamental solution of the heat equation [2].
D. Levi
INFN, Sezione Roma Tre, Roma, Italy
e-mail: decio.levi@roma3.infn.it
M. A. Rodríguez
Depto. de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain
e-mail: rodrigue@ucm.es
Z. Thomova ()
Department of Mathematics and Physics, SUNY Polytechnic Institute, Utica, NY, USA
e-mail: zora.thomova@sunypoly.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_14
149
Generalized Reaction–Diffusion Equation
Decio Levi, Miguel A. Rodríguez, and Zora Thomova
Abstract A PDE modeling a reaction–diffusion physical system is discretized
using its conditional symmetries. Discretization is carried out using two specific
conditional symmetries. Explicit solutions of the difference equation are constructed
when the symmetry is projective.
Keywords Symmetry · Integrable systems · Difference equations · Conditional
symmetry · Invariant discretization
1 Introduction
Partial differential equations (PDE) modeling interesting models in Physics are
generically hard to solve and exact solutions are rare and difficult to obtain. The Lie
symmetries of these equations provide, in many cases, explicit solutions, or at least
hints on how to find those solutions. This is a well-known topic, see, for example,
the following monographs dedicated to it [1, 3, 15, 16].
A particularly useful application of this method is the reduction of the order
of the equation or of the number of variables. The invariants of the vector field
corresponding to the infinitesimal symmetry provide a new set of coordinates and
the PDE, written in this new system, can be usually solved and solutions of the
original equation can be obtained. The well-known example is the construction of
the fundamental solution of the heat equation [2].
D. Levi
INFN, Sezione Roma Tre, Roma, Italy
e-mail: decio.levi@roma3.infn.it
M. A. Rodríguez
Depto. de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain
e-mail: rodrigue@ucm.es
Z. Thomova ()
Department of Mathematics and Physics, SUNY Polytechnic Institute, Utica, NY, USA
e-mail: zora.thomova@sunypoly.edu
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_14
149