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Teoreticheskaya i Matematicheskaya Fizika, 2016, Volume 186, Number 3, Pages 371–385
DOI: https://doi.org/10.4213/tmf8889
(Mi tmf8889)
 

This article is cited in 6 scientific papers (total in 6 papers)

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
Full-text PDF (429 kB) Citations (6)
References:
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).
Received: 06.03.2015
Revised: 09.06.2015
English version:
Theoretical and Mathematical Physics, 2016, Volume 186, Issue 3, Pages 320–332
DOI: https://doi.org/10.1134/S0040577916030028
Bibliographic databases:
Document Type: Article
PACS: 03.65.Ge, 41.20.Jb, 02.30.Jr
Language: Russian
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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  • https://www.mathnet.ru/eng/tmf8889
  • https://doi.org/10.4213/tmf8889
  • https://www.mathnet.ru/eng/tmf/v186/i3/p371
  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
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    Abstract page:523
    Full-text PDF :155
    References:52
    First page:37
     
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