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TMF, 2013, Volume 174, Number 2, Pages 272–284 (Mi tmf8391)  

This article is cited in 4 scientific papers (total in 4 papers)

Multidimensional nonlinear wave equations with multivalued solutions

V. M. Zhuravlev

Ulyanovsk State University, Ulyanovsk, Russia

Abstract: We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d'Alembert equations, nonlinear Klein–Gordon equations, and nonlinear telegraph equations.

Keywords: exact solution of multidimensional hyperbolic equations, breaking wave, multivalued solution, electromagnetic waves in a medium with nonlinear polarization

DOI: https://doi.org/10.4213/tmf8391

Full text: PDF file (640 kB)
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English version:
Theoretical and Mathematical Physics, 2013, 174:2, 236–246

Bibliographic databases:

PACS: 03.65.Ge, 41.20.Jb, 02.30.Jr
Received: 11.07.2012
Revised: 08.08.2012

Citation: V. M. Zhuravlev, “Multidimensional nonlinear wave equations with multivalued solutions”, TMF, 174:2 (2013), 272–284; Theoret. and Math. Phys., 174:2 (2013), 236–246

Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf8391
  • http://mi.mathnet.ru/eng/tmf/v174/i2/p272

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. M. Zhuravlev, “Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations”, Theoret. and Math. Phys., 186:3 (2016), 320–332  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. E. V. Trifonov, “Families of exact solutions for linear and nonlinear wave equations with a variable speed of sound and their use in solving initial boundary value problems”, Theoret. and Math. Phys., 192:1 (2017), 974–981  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. X.-Ya. Tang, X.-Zh. Hao, Z.-F. Liang, “Interacting waves of Davey-Stewartson III system”, Comput. Math. Appl., 74:6 (2017), 1311–1320  crossref  mathscinet  isi
    4. 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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