Geodesic
On multi-dimensional partial differential equations with power nonlinearities in first derivatives
Ufa mathematical journal, Tome 9 (2017) no. 1, pp. 98-108 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a class of multi-dimensional partial differential equations involving a linear differential operator of arbitrary order and a power nonlinearity in the first derivatives. Under some additional assumptions for this operator, we study the solutions of multi-dimensional travelling waves that depend on some linear combinations of the original variables. The original equation is transformed to a reduced one, which can be solved by the separation of variables. Solutions of the reduced equation are found for the cases of additive, multiplicative and combined separation of variables.
Keywords: partial differential equation, reduced equation, method of separation of variables, power nonlinearity. 
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I. V. Rakhmelevich. On multi-dimensional partial differential equations with power nonlinearities in first derivatives. Ufa mathematical journal, Tome 9 (2017) no. 1, pp. 98-108. http://geodesic.mathdoc.fr/item/UFA_2017_9_1_a8/

[1] Polyanin A. D., Zaitsev V. F., Spravochnik po nelineinym uravneniyam matematicheskoi fiziki: tochnye resheniya, Fizmatlit, M., 2002

[2] Polyanin A. D., Zaitsev V. F., Zhurov A. I., Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki, Fizmatlit, M., 2005

[3] Zaitsev V. F., Polyanin A. D., Spravochnik po differentsialnym uravneniyam s chastnymi proizvodnymi pervogo poryadka, Fizmatlit, M., 2003

[4] Polyanin A. D., Zhurov A. I., “Obobschennoe i funktsionalnoe razdelenie peremennykh v matematicheskoi fizike i mekhanike”, Doklady RAN, 382:5 (2002), 606–611 | MR | Zbl

[5] Rakhmelevich I. V., “O dvumernykh giperbolicheskikh uravneniyakh so stepennoi neline inostyu po proizvodnym”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2015, no. 1(33), 12–19 | DOI

[6] Rakhmelevich I. V., “O nekotorykh novykh resheniyakh mnogomernogo uravneniya v chastnykh proizvodnykh pervogo poryadka so stepennymi nelineinostyami”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2015, no. 3(35), 18–25 | DOI

[7] Rakhmelevich I. V., “O primenenii metoda razdeleniya peremennykh k uravneniyam matematicheskoi fiziki, soderzhaschim odnorodnye funktsii ot proizvodnykh”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2013, no. 3(23), 37–44

[8] Rakhmelevich I. V., “Ob uravneniyakh matematicheskoi fiziki, soderzhaschikh multiodnorodnye funktsii ot proizvodnykh”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2014, no. 1(27), 42–50

[9] Rakhmelevich I. V., “O resheniyakh mnogomernogo uravneniya Klero s multiodnorodnoi funktsiei ot proizvodnykh”, Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya Matematika, mekhanika, informatika, 14:4-1 (2014), 374–381 | Zbl

[10] J. Miller (Jr.), Rubel L. A., “Functional separation of variables for Laplace equations in two dimensions”, Journal of Physics A, 26 (1993), 1901–1913 | DOI | MR | Zbl

[11] R. Z. Zhdanov, “Separation of variables in the non-linear wave equation”, Journal of Physics A, 27 (1994), L291–L297 | DOI | MR | Zbl

[12] A. M. Grundland, E. Infeld, “A family of non-linear Klein–Gordon equations and their solutions”, Journal of Mathematical Physics, 33:7 (1992), 2498–2503 | DOI | MR | Zbl