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Exact Analytical Solutions for Nonautonomic Nonlinear Klein-Fock-Gordon Equation

  • Eron L. Aero
  • A. N. BulyginEmail author
  • Yu. V. Pavlov
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 87)

Abstract

We develop methods of construction of functionally invariant solutions U(xyzt) for the nonlinear nonautonomic Klein-Fock-Gordon equation. The solutions U(xyzt) are found in the form of an arbitrary function, which depends on one, \(\tau (x,y,z,t)\), or two, \(\alpha (x,y,z,t)\), \(\beta (x,y,z,t)\) specially constructed functions. The functions are called ansatzes. The ansatzes \((\tau , \alpha , \beta )\) are defined as solutions of the special equations (algebraic or mixed type—algebraic and partial differential equations). The equations for defining of the ansatzes contain arbitrary functions, depending on \((\tau , \alpha , \beta )\). The offered methods allow to find the solution U(xyzt) for particular, but wide class of the nonautonomic nonlinear Klein-Fock-Gordon equations. The methods are illustrated by examples of finding exact analytical solutions of the nonautonomic Liouville equation.

Notes

Acknowledgements

This work was supported by Russian Foundation for Basic Research, grants 16-01-00068-a and 17-01-00230-a.

References

  1. 1.
    Jacobi, C.G.J.: Über eine particuläre Lösung der partiellen Differentialgleichung \(\frac{\partial ^2 V}{\partial x^2} + \frac{\partial ^2 V}{\partial y^2} + \frac{\partial ^2 V}{\partial z^2} =0\). Crelle Journal für die reine und angewandte Mathematik 36, 113–134 (1848)CrossRefGoogle Scholar
  2. 2.
    Forsyth, A.R.: New solutions of some of the partial differential equations of mathematical physics. Messenger Math. 27, 99–118 (1898)Google Scholar
  3. 3.
    Bateman, H.: The Mathematical Analysis of Electrical and Optical Wave-Motion on the Basis of Maxwell’s Equations. Cambridge University Press, Cambridge (1915)zbMATHGoogle Scholar
  4. 4.
    Smirnoff, V., Soboleff, S.: Sur une méthode nouvelle dans le problème plan des vibrations élastiques. Trudy Seismologicheskogo Inst. 20, 1–37 (1932)Google Scholar
  5. 5.
    Smirnoff, V., Soboleff, S.: Sur le problème plan des vibrations élastiques. C. R. Acad. Sci. Paris 194, 1437–1439 (1932)Google Scholar
  6. 6.
    Sobolev, S.L.: Functionally invariant solutions of wave equation. Trudy Fiz.-Mat. Inst. V.A. Steklov 5, 259–264 (1934)Google Scholar
  7. 7.
    Erugin, N.P.: On functionally invariant solutions. Uchenye Zap. Leningarad University 15(96), 101–134 (1948)Google Scholar
  8. 8.
    Aero, E.L., Bulygin, A.N., Pavlov, Yu.V.: Solutions of the three-dimensional sine-Gordon equation. Theor. Math. Phys. 158, 313–319 (2009).  https://doi.org/10.1007/s11232-009-0025-3
  9. 9.
    Aero, E.L., Bulygin, A.N., Pavlov, Yu.V.: Solutions of generalized (3+1) sine-Gordon equation [In Russian]. Nelineinii Mir 7, 513–517 (2009)Google Scholar
  10. 10.
    Aero, E.L., Bulygin, A.N., Pavlov, Yu.V.: New approach to the solution of the classical sine-Gordon equation and its generalizations. Differ. Equ. 47, 1442–1452 (2011).  https://doi.org/10.1134/S0012266111100077
  11. 11.
    Aero, E.L., Bulygin, A.N., Pavlov, Yu.V.: Functionally invariant solutions of nonlinear Klein-Fock-Gordon equation. Appl. Math. Comput. 223, 160–166 (2013).  https://doi.org/10.1016/j.amc.2013.07.088
  12. 12.
    Aero, E.L., Bulygin, A.N., Pavlov, Yu.V.: Solutions of the sine-Gordon equation with a variable amplitude. Theor. Math. Phys. 184, 961–972 (2015).  https://doi.org/10.1007/s11232-015-0309-8
  13. 13.
    Porubov, A.V., Fradkov, A.L., Andrievsky, B.R.: Feedback control for some solutions of the sine-Gordon equation. Appl. Math. Comput. 269, 17–22 (2015)MathSciNetGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering of Russian Academy of SciencesSaint PetersburgRussia

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