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TMF, 2016, Volume 186, Number 3, Pages 371–385 (Mi tmf8889)  

This article is cited in 1 scientific paper (total in 1 paper)

Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations

V. M. Zhuravlev

Technological Research Institute of Ulyanovsk State University, Ulyanovsk, Russia

Abstract: We discuss an extension of the theory of multidimensional second-order equations of the elliptic and hyperbolic types related to multidimensional quasilinear autonomous first-order partial differential equations. Calculating the general integrals of these equations allows constructing exact solutions in the form of implicit functions. We establish a connection with hydrodynamic equations. We calculate the number of free functional parameters of the constructed solutions. We especially construct and analyze implicit solutions of the Laplace and d'Alembert equations in a coordinate space of arbitrary finite dimension. In particular, we construct generalized Penrose–Rindler solutions of the d'Alembert equation in $3{+}1$ dimensions.

Keywords: exact solution of multidimensional nonlinear hyperbolic equations, exact solution of multidimensional nonlinear elliptic equations, multivalued solution, system of nonlinear equations of hydrodynamic type, electromagnetic wave equation, Laplace equation, d'Alembert equation

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 14.Z50.31.0015
This research was supported by the Ministry of Education and Science of the Russia Federation (State Mission and Project No. 14.Z50.31.0015).


DOI: https://doi.org/10.4213/tmf8889

Full text: PDF file (429 kB)
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English version:
Theoretical and Mathematical Physics, 2016, 186:3, 320–332

Bibliographic databases:

PACS: 03.65.Ge, 41.20.Jb, 02.30.Jr
MSC: 35F50,35J62, 35L72, 76L05, 78A40
Received: 06.03.2015
Revised: 09.06.2015

Citation: V. M. Zhuravlev, “Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations”, TMF, 186:3 (2016), 371–385; Theoret. and Math. Phys., 186:3 (2016), 320–332

Citation in format AMSBIB
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\by V.~M.~Zhuravlev
\paper Multidimensional quasilinear first-order equations and multivalued solutions of the~elliptic and hyperbolic equations
\jour TMF
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\vol 186
\issue 3
\pages 371--385
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\transl
\jour Theoret. and Math. Phys.
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\vol 186
\issue 3
\pages 320--332
\crossref{https://doi.org/10.1134/S0040577916030028}
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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. V. M. Zhuravlev, “Multidimensional nonlinear Klein–Gordon equations and rivertons”, Theoret. and Math. Phys., 197:3 (2018), 1701–1713  mathnet  crossref  crossref  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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