Geodesic
On two-dimensional hyperbolic equations with power-law non-linearity in the derivatives
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 1 (2015), pp. 12-19 Cet article a éte moissonné depuis la source Math-Net.Ru

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In recent years, extensive studies of nonlinear hyperbolic equations are carried out. Special attention is focused on equations of the Liouville type. However, of special interest is the study of nonlinear hyperbolic equations of a more general form, including those containing power-law nonlinearities in the derivatives. They are considered in this work. To study two-dimensional nonlinear hyperbolic equations containing power-law nonlinearities in the derivatives and a nonlinearity of an arbitrary type of an unknown function, the method of functional separation of variables is applied. For this class of equations, solutions of the traveling wave type and solutions depending on power and exponential functions of independent variables (in particular, self-similar solutions) were obtained, as well as solutions containing arbitrary functions of these variables. Solutions for regular and special values of parameters characterizing the nonlinearity have been obtained. The obtained solutions are valid for a wide class of two-dimensional hyperbolic equations with a power-law nonlinearity in derivative. The results can be generalized for multidimensional nonlinear hyperbolic equations with power-law nonlinearities.
Keywords: nonlinear hyperbolic equation, functional separation of variables, power-law non-linearity. 
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I. V. Rakhmelevich. On two-dimensional hyperbolic equations with power-law non-linearity in the derivatives. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 1 (2015), pp. 12-19. http://geodesic.mathdoc.fr/item/VTGU_2015_1_a1/

[1] Zhiber A. V., Sokolov V. V., “Tochno integriruemye giperbolicheskie uravneniya liuvillevskogo tipa”, Uspekhi matematicheskikh nauk, 56:1 (2001), 63–106 (in Russian) | DOI

[2] Kuznetsova M. N., “O nelineynykh giperbolicheskikh uravneniyakh, svyazannykh differentsial'nymi podstanovkami s uravneniem Kleyna–Gordona”, Ufimskiy matematicheskiy zhurnal, 4:3 (2012), 86–103 (in Russian)

[3] Polyanin A. D., Zaytsev V. F., Spravochnik po nelineynym uravneniyam matematicheskoy fiziki: tochnye resheniya, Fizmatlit Publ., M., 2002, 432 pp. (in Russian)

[4] Polyanin A. D., Zhurov A. I., “Obobshchennoe i funktsional'noe razdelenie peremennykh v matematicheskoy fizike i mekhanike”, Doklady RAN, 382:5 (2002), 606–611 (in Russian)

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

[6] Rakhmelevich I. V., “Ob uravneniyakh matematicheskoy fiziki, soderzhashchikh mul'tiodnorodnye funktsii ot proizvodnykh”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2014, no. 1(27), 42–50 (in Russian)

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

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

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