A class of nonlinear symmetry operators has been constructed for the many-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation quadratic in independent variables and derivatives. The construction of each symmetry operator includes an interwining operator for the auxiliary linear equations and additional nonlinear algebraic conditions. Symmetry operators for the one-dimensional equation with a constant influence function have been constructed in explicit form and used to obtain a countable set of exact solutions.
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References
L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
P. J. Olver, Application of Lie Groups to Differential Equations, Springer, N. Y. (1986).
G. W. Bluman, A. F. Cheviakov, and S. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, N. Y. (2010).
R. A. Fisher, Annu. Eugenics, 7, 255–369 (1937).
A. N. Kolmogorov, N. G. Petrovskii, and N. S. Piskunov, Bull. MGU, Matem. Mekh., 1, No. 6, 1–16 (1937).
E. A. Levchenko, A. V. Shapovalov, and A. Yu. Trifonov, J. Math. An. Appl., 395, 716–726 (2012).
M. A. Fuentes, M. N. Kuperman, and V. M. Kenkre, Phys. Rev. Lett., 91, 158104–158108 (2003).
J. A. R. Da Cunha, A. L. A. Penna, M. H. Vainstein, et al., Phys. Lett. A, 373, 661–667 (2009).
V. M. Kenkre, Physica A, 342, 242–248 (2004).
H. Berestycki et al., Nonlinearity, 22, 2813–2844 (2009).
V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow, (1973).
M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization [in Russian], Nauka, Moscow (1991).
A. Yu. Trifonov and A. V. Shapovalov, Russ. Phys. J., 52, No. 9, 899–911 (2009).
V. G. Budanov, Teor. Mat. Fiz., 61, No. 3, 347–363 (1984).
H. Bateman and A. Erdelyi, Higher Transcendental Functions: Bessel Functions, Parabolic Cylinder Functions, and Orthogonal Polynomials [Russian translation], Nauka, Moscow (1966).
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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 86–95, December, 2013.
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Levchenko, E.A., Trifonov, A.Y. & Shapovalov, A.V. Symmetry Operators of the Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Equation with a Quadratic Operator. Russ Phys J 56, 1415–1426 (2014). https://doi.org/10.1007/s11182-014-0194-x
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DOI: https://doi.org/10.1007/s11182-014-0194-x