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Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2015, Number 1(33), Pages 12–19 (Mi vtgu436)  

This article is cited in 7 scientific papers (total in 7 papers)

MATHEMATICS

On two-dimensional hyperbolic equations with power-law non-linearity in the derivatives

I. V. Rakhmelevich

Nizhny Novgorod State University, Nizhny Novgorod, Russian Federation

Abstract: 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.

DOI: https://doi.org/10.17223/19988621/33/2

Full text: PDF file (386 kB)
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UDC: 517.957
Received: 10.12.2014

Citation: 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
\Bibitem{Rak15}
\by I.~V.~Rakhmelevich
\paper On two-dimensional hyperbolic equations with power-law non-linearity in~the derivatives
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2015
\issue 1(33)
\pages 12--19
\mathnet{http://mi.mathnet.ru/vtgu436}
\crossref{https://doi.org/10.17223/19988621/33/2}
\elib{https://elibrary.ru/item.asp?id=23223189}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. V. Rakhmelevich, “O nekotorykh novykh resheniyakh mnogomernogo uravneniya v chastnykh proizvodnykh pervogo poryadka so stepennymi nelineinostyami”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2015, no. 3(35), 18–25  mathnet  crossref  elib
    2. I. V. Rakhmelevich, “O resheniyakh dvumernogo uravneniya Monzha–Ampera so stepennoi nelineinostyu po pervym proizvodnym”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2016, no. 4(42), 33–43  mathnet  crossref  elib
    3. I. V. Rakhmelevich, “O resheniyakh mnogomernogo differentsialnogo uravneniya proizvolnogo poryadka so smeshannoi starshei chastnoi proizvodnoi i stepennymi nelineinostyami”, Vladikavk. matem. zhurn., 18:4 (2016), 41–49  mathnet
    4. I. V. Rakhmelevich, “On multi-dimensional partial differential equations with power nonlinearities in first derivatives”, Ufa Math. J., 9:1 (2017), 98–108  mathnet  crossref  isi  elib
    5. I. V. Rakhmelevich, “Dvumernoe neavtonomnoe giperbolicheskoe uravnenie vtorogo poryadka so stepennymi nelineinostyami”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2017, no. 49, 52–60  mathnet  crossref  elib
    6. 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  crossref  mathscinet  zmath  isi  scopus
    7. I. V. Rakhmelevich, “O mnogomernykh determinantnykh differentsialno-operatornykh uravneniyakh”, Vladikavk. matem. zhurn., 22:2 (2020), 53–69  mathnet  crossref
  • Вестник Томского государственного университета. Математика и механика
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