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

Abstract

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207), an exhaustive list of reaction-diffusion systems admitting such symmetry is derived. The formpreserving transformations for this class of systems are constructed and it is shown that this list contains only non-equivalent systems. The obtained symmetries permit to reduce the reaction-diffusion systems under study to two-dimensional systems of ordinary differential equations and to find exact solutions. As a non-trivial example, multiparameter families of exact solutions are explicitly constructed for two nonlinear reaction-diffusion systems. A possible interpretation to a biologically motivated model is presented.

Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model

“…The proof of Theorem 1 is quite similar to that for the class reaction-diffusion systems (1) with D k = constant (k = 1, 2) presented in [6].…”

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

“…It turns out that 4h u + 1 = 0 ⇔ d 1 = δ 1 (u + α 2 ) −4 (otherwise restrictions (16) leading to Lie symmetry are automatically fulfilled), therefore two different cases arise…”

“…The case a) for system (3) was examined in our previous papers [1,16,17]. Here we restrict ourselves on the case b), i.e.…”

“…It is well-known that constructing Q-conditional symmetries for evolution equations and evolution systems leads to the requirement to analyze two essentially different cases a) ξ 0 = 0; b) ξ 0 = 0, ξ 1 = 0 (see operator (5)). The case a) for system (3) was examined in our previous papers [1,16,17]. Here we restrict ourselves on the case b), i.e.…”

“…take place. Thus, we take into account only such solutions of ( 8)-( 14), which do not satisfy at least one of the equations from (16). Moreover, since Q-conditional symmetry of the first type is automatically one of the second type, we should also check the same for coefficients of the operator obtained by multiplying (5) on any smooth functions.…”