The conditional symmetry of a system of nonlinear reaction-diffusion equations is investigated. It is shown that the operators of conditional symmetry exist for the systems of nonlinear reaction-diffusion equations with an arbitrary number of independent variables. Moreover, these operators are found in the explicit form.
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Yu. A. Danilov, Group Analysis of the Turing Systems and of Its Analogues, Preprint IAE 3287/1, Kurchatov Institute of Atomic Energy.
R. Cherniha and I. R. King, “Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I,” J. Phys. A: Math. Gen., 33, 267–282 (2000).
R. Cherniha and I. R. King, “Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I. Addendum,” J. Phys. A: Math. Gen., 33, 7839–7841 (2000).
R. Cherniha and I. R. King, “Lie symmetries of nonlinear multidimensional reaction-diffusion systems: II,” J. Phys. A: Math. Gen., 36, 405–425 (2002).
A. G. Nikitin and R. Wiltshire, “Symmetries of systems of nonlinear reaction-diffusion equations,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 30, Part 1 (2000), pp. 47–59.
A. G. Nikitin and R. J.Wiltshire, “Systems of reaction-diffusion equations and their symmetry properties,” J. Math. Phys., 42, No. 4, 1667–1688 (2001).
A. G. Nikitin, “Group classification of systems of non-linear reaction-diffusion equations with general diffusion matrix. I. Generalized Ginsburg–Landau equations,” J. Math. Anal. Appl., 324, No. 1, 615–628 (2006).
A. G. Nikitin, “Group classification of systems of non-linear reaction-diffusion equations with general diffusion matrix. II. Generalized Turing systems,” J. Math. Anal. Appl., 332, No. 1, 666–690 (2007).
A. G. Nikitin, “Group classification of systems of non-linear reaction-diffusion equations with triangular diffusion matrix,” Ukr. Math. J., 59, No. 3, 395–441 (2007).
V. I. Fushchich and N. I. Serov, “Conditional invariance and reduction of the nonlinear heat-conduction equation,” Dokl. Akad. Nauk Ukr. SSR, No. 7, 24–28 (1990).
P. Clarkson and E. Mansfield, “Symmetry reductions and exact solutions of a class of nonlinear heat equations,” Physica D, 70, 250–288 (1993).
T. A. Barannyk, “Conditional symmetry and exact solutions of a multidimensional diffusion equation,” Ukr. Mat. Zh., 54, No. 10, 1416–1420 (2002); English translation: Ukr. Math. J., 54, No. 10, 1715–1721 (2002).
T. A. Barannyk and A. G. Nikitin, “Solitary wave solutions for heat equations,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 50, Part 1 (2004), pp. 34–39.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 11, pp. 1443–1449, November, 2015.
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Barannyk, T.A. Conditional Symmetry of a System of Nonlinear Reaction-Diffusion Equations. Ukr Math J 67, 1621–1628 (2016). https://doi.org/10.1007/s11253-016-1178-5
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DOI: https://doi.org/10.1007/s11253-016-1178-5