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 form-preserving 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.
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Acknowledgments
R.Ch. thanks the Organizing Committee of the 7th Workshop ‘Algebra, Geometry, and Mathematical Physics’ (Mulhouse, 24-26 October 2011) for the financial support.
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Cherniha, R., Davydovych, V. (2014). Reaction-Diffusion Systems with Constant Diffusivities: Conditional Symmetries and Form-Preserving Transformations. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds) Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55361-5_31
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