Reaction-Diffusion Systems with Constant Diffusivities: Conditional Symmetries and Form-Preserving Transformations
\(Q\) -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. Ch
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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.
1 Introduction The paper is devoted to the investigation of the two-component reaction-diffusion (RD) systems of the form ut = d1 uxx + F(u, v), (1) vt = d2 vxx + G(u, v). R. Cherniha (B) · V. Davydovych Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivs’ka Street 3, Kyiv 01601, Ukraine e-mail: [email protected] V. Davydovych e-mail: [email protected] R. Cherniha School of Mathematical Sciences, University of Nottingham, Nottingham 2RD NG7, UK A. Makhlouf et al. (eds.), Algebra, Geometry and Mathematical Physics, 533 Springer Proceedings in Mathematics & Statistics 85, DOI: 10.1007/978-3-642-55361-5_31, © Springer-Verlag Berlin Heidelberg 2014
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R. Cherniha and V. Davydovych
where u = u(t, x) and v = v(t, x) are two unknown functions representing the densities of populations (cells), the concentrations of chemicals, the pressures in thin films, etc. F and G are the given smooth functions describing interaction between them and environment, d1 and d2 are diffusivities assumed to be positive constants. The subscripts t and x denote differentiation with respect to these variables. The class of RD systems (1) generalizes many well-known nonlinear second-order models and is used to describe various processes in physics, biology, chemistry and ecology (see, e.g., the well-known books [1–5]). Nevertheless the search for Lie symmetries of the class of RD systems (1) was initiated about 30 years ago [6], this problem was completely solved only during the last decade because of its complexity. Now one can claim that all possible Lie symmetries of (1) were completely described in [7–9]. The time is therefore ripe for a complete description of non-Lie symmetries for the class of the RD systems (1). However, it seems to be extremely difficult task because, firstly, several definitions of non-Lie symmetries have been introduced (nonclassical symmetry [1, 10], conditional symmetry [11, 12], generalized conditional symmetry [13, 14] etc.), secondly, the complete description of non-Lie symmetries needs to solve the corresponding system of determining equations, which is non
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