Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, Number 1(33), Pages 12–19
This article is cited in 7 scientific papers (total in 7 papers)
On two-dimensional hyperbolic equations with power-law non-linearity in the derivatives
I. V. Rakhmelevich
Nizhny Novgorod State University, Nizhny Novgorod, Russian Federation
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.
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”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, no. 1(33), 12–19
Citation in format AMSBIB
\paper On two-dimensional hyperbolic equations with power-law non-linearity in~the derivatives
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
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I. V. Rakhmelevich, “A multidimensional nonautonomous equation containing a product of powers of partial derivatives”, Vestn. St Petersb. Univ.-Math., 51:1 (2018), 87–94
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