New conditional symmetries and exact solutions of reaction–diffusion systems with power diffusivities

Abstract

A wide range of new Q-conditional symmetries for reaction-diffusion systems with power diffusivities is constructed. The relevant non-Lie ansätze to reduce the reactiondiffusion systems to ODE systems and examples of exact solutions are obtained. Relation of the solutions obtained with the development of spatially inhomogeneous structures is discussed. In 1952 A.C. Turing published the remarkable paper [1], in which a revolutionary idea about mechanism of morphogenesis (the development of structures in an organism during the life) has been proposed. Roughly speaking, his idea says that the diffusion process can interact with the chemical reaction process in such a way that this can stimulate the development and growth of different forms and structures in an organism. Moreover this differentiation is often impossible without diffusion, i.e., only one may destabilize the spatially homogeneous structures. This effect is known as the Turing instability and mathematically leads to systems of reaction-diffusion equations with nonlinearities of the special form [2,3]. Since 1952 the reaction-diffusion systems have been extensively studied by means of different mathematical methods, including group-theoretical methods. The main attention was paid to investigation of the two-component RD systems of the formwhere U = U(t, x) and V = V (t, x) are two unknown functions representing the densities of cells and chemicals, F (U, V ) and G(U, V ) are two given functions describing interaction between them and environment, the functions D 1 (U) and D 2 (V ) are the relevant diffusivities (usually they are assumed to be constants or power functions), and the subscripts t and x denote differentiation with respect to these variables. At the present time one can claim that all possible Lie symmetries of (1) with the constant diffusivities are completely described in [4,5,6], while in [7,8,9] it has been done for the non-constant diffusivities.The problem of construction of conditional symmetries for (1) is still not solved even in the case of Q-conditional symmetries (non-classical symmetries). Moreover, to our best knowledge, 1 to appear in J.Phys.A: Math.Theor.

Reaction-Diffusion Systems with Constant Diffusivities: Conditional Symmetries and Form-Preserving Transformations

“…To the best of our knowledge, there are not many paper devoted to search of Q-conditional symmetries for the systems of PDEs [20][21][22][23][24]. One may easily check that Definition 2 was only used in all these papers.…”

Nonclassical Symmetries for a Class of Reaction-Diffusion Equations: the Method of Heir-Equations

“…Since then the nonclassical symmetry method has been applied to various equations and systems in hundreds of published papers, e.g. [27], [37], [16], [28], [17], [54], [13], [11], [12], [15], [51], [4], the latest b being [14], [31], [55], [10].…”

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.…”