Nonlinear reaction–diffusion systems with a non-constant diffusivity: Conditional symmetries in no-go case

Abstract

a b s t r a c t Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. The work is a natural continuation of our paper "Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities" (Cherniha and Davydovych, 2012) [1] in order to extend the results on so-called no-go case. Using the notion of Q-conditional symmetries of the first type, an exhaustive list of reaction-diffusion systems admitting such symmetry is derived. The results obtained are compared with those derived earlier. The symmetries for reducing reaction-diffusion systems to two-dimensional dynamical systems (ODE systems) and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for nonlinear reaction-diffusion systems with an arbitrary power-law diffusivity are constructed and their properties for possible applicability are established.

New Conditional Symmetries and Exact Solutions of the Diffusive Two-Component Lotka–Volterra System

“…The problem of finding Q-conditional symmetries of the first type for some reactiondiffusion systems (in particular, two-and three-component DLV systems) are considered in monograph [31] (see Chapters 3 and 4). In paper [29], such symmetries were constructed in the no-go case for a wide class of reaction-diffusion systems with nonconstant diffusivities.…”

“…Notably, following Fushchych's proposal dating back to the 1980s [24,25], we use the terminology 'Qconditional symmetry' instead of 'nonclassical symmetry' (see also a discussion concerning terminology in Chapter 3 of [22]). Although this method was suggested 50 years ago, its successful applications for solving nonlinear systems of PDEs were accomplished only in the 2000s, and the majority of such papers were published during the last 10 years (see [17,18,[26][27][28][29][30]). This occurred because application of the nonclassical method (such a terminology was used in [23] instead of nonclassical symmetries) leads to very complicated nonlinear equations to-be-solved.…”

A Novel Solution for the General Diffusion

“…It may be that any such nonlinear diffusive behavior could simply be captured in a modified diffusion constant, rather, experimentalists may find unique diffusion constants for various media, and, never think it due to some reactive medium response [3,4,5]. In fact, any exotic, complex diffusivity should be focused upon the diffusion constant, per se, and not extricated to describe the medium response itself.…”

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

“…A complete description of Lie symmetries of the system is presented in [16]. The conditional symmetries for (9) are studied in [43][44][45][46]. The second-order CLBS (DC) admitted by the system (9) is discussed in [21].…”

“…Once the symmetries of the considered system (9) have been identified, one can algorithmically implement the reduction procedure and thereby determine all solutions that are invariant under the resulting symmetries. In [16,21,[43][44][45][46], a wide range of exact solutions has been established due to various symmetry reductions therein.…”