Geodesic
On solutions of the Monge–Ampere equation with power-law non-linearity with respect to first derivatives
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2016), pp. 33-43 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this work, the two-dimensional Monge–Ampere equation the right-hand side of which includes arbitrary non-linearity with respect to the unknown function and power-law nonlinearity with respect to its first derivatives is considered. To solve this equation, the method of functional separation of variables is used. We study the case when one of the unknown functions used in the method of separation of variables is linear and also the case when all these functions are arbitrary. The exact solutions in the implicit form of the considered equation are received in the presence of arbitrary nonlinearity with respect to the unknown function. For the case when the equation does not contain unknown function explicitly, its solutions are found in an explicit form. The solutions have been analyzed for different values of parameters characterizing the nonlinearity.
Keywords: Monge–Ampere equation, functional separation of variables, power-law non-linearity. 
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I. V. Rakhmelevich. On solutions of the Monge–Ampere equation with power-law non-linearity with respect to first derivatives. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 4 (2016), pp. 33-43. http://geodesic.mathdoc.fr/item/VTGU_2016_4_a3/

[1] Polyanin A. D., Zaitsev V. F., Handbook of Nonlinear Partial Differential Equations, Chapman Hall/CRC Press, Boca Raton, 2004 | MR | Zbl

[2] Khabirov S. V., “Nonisentropic one-dimensional gas motions constructed by means of the contact group of the nonhomogeneous Monge–Ampere equation”, Mathematics of the USSR-Sbornik, 71:2 (1992), 447–462 | DOI | MR | Zbl | Zbl

[3] Shablovsky O. N., “Parametric solutions for the Monge–Ampere equation and gas flows with variable entropy”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika - Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 1(33), 105–118 | DOI | MR

[4] Kushner A. G., “Transformation of hyperbolic Monge–Ampere equations into linear equations with constant coefficients”, Doklady Mathematics, 78:3 (2008), 907–909 | DOI | MR | Zbl

[5] Polyanin A. D., Zaitsev V. F., Zhurov A. I., Methods of solving nonlinear equations of mathematical physics and mechanics, Fizmatlit Publ., M., 2005

[6] Polyanin A. D., Zhurov A. I., “The generalized and functional separation of variables in mathematical physics and mechanics”, Doklady Mathematics, 65:1 (2002), 129–134 | MR | Zbl

[7] Rakhmelevich I. V., “On application of the variable separation method to mathematical physics equations containing homogeneous functions of derivatives”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika - Tomsk State University Journal of Mathematics and Mechanics, 2013, no. 3(23), 37–44

[8] Rakhmelevich I. V., “On equations of mathematical physics containing multi-homogeneous functions of derivatives”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika - Tomsk State University Journal of Mathematics and Mechanics, 2014, no. 1, 42–50

[9] Rakhmelevich I. V., “On the Solutions of Multi-dimensional Clairaut Equation with Multi-homogeneous Function of the Derivatives”, Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya Matematika, mekhanika, informatika, 14:4–1 (2014), 374–381 | Zbl

[10] Rakhmelevich I. V., “On two-dimensional hyperbolic equations with power-law nonlinearity in the derivatives”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika - Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 1(33), 12–19 | DOI

[11] Rakhmelevich I. V., “On some new solutions of the multi-dimensional first order partial differential equation with power-law nonlinearities”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika - Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 3(35), 18–25 | DOI | MR

[12] 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 | MR | Zbl

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