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TMF, 2015, Volume 184, Number 1, Pages 79–91 (Mi tmf8821)  

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

Solutions of the sine-Gordon equation with a variable amplitude

E. L. Aero, A. N. Bulygin, Yu. V. Pavlov

Institute of Problems in Mechanical Engineering, RAS, St.~Petersburg, Russia

Abstract: We propose methods for constructing functionally invariant solutions $u(x,y,z,t)$ of the sine-Gordon equation with a variable amplitude in $3{+}1$ dimensions. We find solutions $u(x,y,z,t)$ in the form of arbitrary functions depending on either one $(\alpha(x,y,z,t))$ or two $(\alpha(x,y,z,t),\beta(x,y,z,t))$ specially constructed functions. Solutions $f(\alpha)$ and $f(\alpha,\beta)$ relate to the class of functionally invariant solutions, and the functions $\alpha(x,y,z,t)$ and $\beta(x,y,z,t)$ are called the ansatzes. The ansatzes $(\alpha,\beta)$ are defined as the roots of either algebraic or mixed (algebraic and first-order partial differential) equations. The equations defining the ansatzes also contain arbitrary functions depending on $(\alpha,\beta)$. The proposed methods allow finding $u(x,y,z,t)$ for a particular, but wide, class of both regular and singular amplitudes and can be easily generalized to the case of a space with any number of dimensions.

Keywords: sine-Gordon equation, wave equation, eikonal equation, functionally invariant solution, ansatz

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00224_a

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/tmf8821

Full text: PDF file (1625 kB)
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English version:
Theoretical and Mathematical Physics, 2015, 184:1, 961–972

Bibliographic databases:

PACS: 02.30.Jr, 05.45.-a
MSC: 39A14
Received: 20.11.2014
Revised: 24.02.2015

Citation: E. L. Aero, A. N. Bulygin, Yu. V. Pavlov, “Solutions of the sine-Gordon equation with a variable amplitude”, TMF, 184:1 (2015), 79–91; Theoret. and Math. Phys., 184:1 (2015), 961–972

Citation in format AMSBIB
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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. E. L. Aero, A. N. Bulygin, Yu. V. Pavlov, “Mathematical methods for solution of nonlinear model of deformation of crystal media with complex lattice”, Proceedings of the International Conference Days on Diffraction 2015, IEEE, 2015, 8–13  isi
    2. E. L. Aero, A. N. Bulygin, Yu. V. Pavlov, “Methods of construction of exact analytical solutions for nonautonomic nonlinear Klein-Fock-Gordon equation”, Proceedings of the International Conference on Days on Diffraction 2016 (DD), eds. O. Motygin, A. Kiselev, P. Kapitanova, L. Goray, A. Kazakov, A. Kirpichnikova, IEEE, 2016, 9–14  crossref  isi
    3. L. T. Stepien, “On certain exact solutions for some equations in field theory”, New trends in analysis and interdisciplinary applications, Trends in Mathematics, eds. P. Dang, M. Ku, T. Qian, L. Rodino, Birkhauser Boston, 2017, 327–335  crossref  mathscinet  zmath  isi
    4. M. Kamranian, M. Dehghan, M. Tatari, “Study of the two-dimensional sine-Gordon equation arising in Josephson junctions using meshless finite point method”, Int. J. Numer. Model.-Electron. Netw. Device Fields, 30:6 (2017), e2210  crossref  isi  scopus
    5. E. L. Aero, A. N. Bulygin, Yu. V. Pavlov, “The solutions of nonlinear equations of plane deformation of the crystal media allowing martensitic transformations: complex representation for macrofield equations”, Mater. Phys. Mech., 35:1 (2018), 1–9  crossref  isi  scopus
    6. E. L. Aero, A. N. Bulygin, Yu. V. Pavlov, “Exact analytical solutions for nonautonomic nonlinear Klein-Fock-Gordon equation”, Advances in Mechanics of Microstructured Media and Structures, Advanced Structured Materials, 87, eds. F. DellIsola, V. Eremeyev, A. Porubov, Springer, 2018, 21–33  crossref  mathscinet  isi  scopus
    7. A. N. Bulygin, Yu. V. Pavlov, “Complex representation of general solution of equations for nonlinear model of plane deformation of crystal media with a complex lattice”, 2018 Days on Diffraction (DD), eds. O. Motygin, A. Kiselev, L. Goray, A. Kazakov, A. Kirpichnikova, M. Perel, IEEE, 2018, 49–53  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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