The Method of Linear Determining Equations to Evolution System and Application for Reaction-Diffusion System with Power Diffusivities

Lina Ji

doi:10.3390/sym8120157

Abstract

The method of linear determining equations is constructed to study conditional Lie–Bäcklund symmetry and the differential constraint of a two-component second-order evolution system, which generalize the determining equations used in the search for classical Lie symmetry. As an application of the approach, the two-component reaction-diffusion system with power diffusivities is considered. The conditional Lie–Bäcklund symmetries and differential constraints admitted by the reaction-diffusion system are identified. Consequently, the reductions of the resulting system are established due to the compatibility of the corresponding invariant surface conditions and the original system.

Highlights

The method of differential constraint (DC) is pretty old, dating back at least to the time of Lagrange.Lagrange used DC to find the total integral of a first-order nonlinear equation
The survey of this method was presented by Sidorvo, Shapeev and Yanenko in [2], where the method of DC was successfully introduced into practice on gas dynamics
From the symmetry point of view, conditional Lie–Bäcklund symmetry (CLBS) related to sign-invariants [29,30,31,32,33], separation of variables [34] and invariant subspaces [35,36,37] are proved to be very effective to study the classifications and reductions of second-order nonlinear diffusion equations

Summary

Introduction

The method of differential constraint (DC) is pretty old, dating back at least to the time of Lagrange. From the symmetry point of view, CLBSs related to sign-invariants [29,30,31,32,33], separation of variables [34] and invariant subspaces [35,36,37] are proved to be very effective to study the classifications and reductions of second-order nonlinear diffusion equations. We will present the linear determining equations to identify DC (3) and CLBS (5) in the general form of second-order evolution system (4), which is exactly the extension of the results for the scalar evolution equation in [40,41,42]. The method of linear determining equations is proposed for finding the general form of DC η = un + g(t, x, u, u1 , u2 , · · · , un−1 ) = 0. The last section is devoted to the final discussions and conclusions

Preliminaries

Conclusions