Skip to main content
Log in

A new method for the construction of solutions of nonlinear wave equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We propose a simple new method for the construction of solutions of multidimensional nonlinear wave equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. L. V. Ovsiannikov,Group Analysis of Differential Equations, Academic Press, New York (1982).

    MATH  Google Scholar 

  2. J. Olver,Applications of Lie Groups to Differential Equations, Springer, New York (1986).

    MATH  Google Scholar 

  3. W. I. Fushchych, V. M. Stelen, and N. I. Serov,Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht (1993).

    Google Scholar 

  4. W. I. Fushchych and A. G. Nikitin,Symmetries of Maxwell's Equations, Reidel, Dordrecht (1987).

    Google Scholar 

  5. W. I. Fushchych and I. M. Tsyfra, “On a reduction and solutions of nonlinear wave equations with broken symmetry,”J. Phys. A: Math. Gen.,20, No. 2, L45-L48 (1987).

    Article  Google Scholar 

  6. D. Levi and P. Winternitz, “Non-classical symmetry reduction—example of the Boussinesq equation,”J. Phys. A,22, No. 2, 2915–2924 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. I. Fushchich and A. F. Barannik, “A new method for the construction of exact solutions of nonlinear wave equations,”Dopov. Akad. Nauk Ukr. No. 10, 48–51 (1996).

    MathSciNet  Google Scholar 

  8. A. F. Barannyk and I. I. Yuryk, “On some exact solutions of nonlinear wave equations,” in:Proceedings of the Second International Conference “Symmetry in Nonlinear Mathematical Physics,” Vol. 1 (1997), pp. 98–107.

  9. A. F. Barannyk and I. I. Yuryk, “On a new method for constructing exact solutions of nonlinear differential equations of mathematical physics,”J. Phys. A: Math. Gen.,31, 4899–4907 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  10. A. F. Barannik and I. I. Yurik, “Classification of maximal subalgebras of rankn of the conformal algebraAC(1,n),”Ukr. Mat. Zh.,50, No. 4, 459–470 (1998).

    MathSciNet  Google Scholar 

  11. A. M. Grunland, J. Harnad, and P. Winternitz, “Symmetry reduction of nonlinear relativistically invariant equations,”J. Math. Phys.,25, No. 4, 791–806 (1984).

    Article  MathSciNet  Google Scholar 

  12. V. I. Fushchich, L. F. Barannik, and A. F. Barannik,Subgroup Analysis of Galilei and Poincaré Groups and Reduction of Nonlinear Equations [in Russian], Naukova Dumka, Kiev (1991).

    MATH  Google Scholar 

  13. G. Cieciura and A. Grunland, “A certain class of solutions of the nonlinear wave equations,”J. Math. Phys.,25, No. 12, 3460–3469 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  14. V. I. Fushchich and R. Z. Zhdanov,Nonlinear Spinor Equations: Symmetry and Exact Solutions [in Russian], Naukova Dumka, Kiev (1992).

    MATH  Google Scholar 

  15. C. B. Collins, “Complex potential equations. I. A technique for solutions,”Math. Proc. Cambridge Philos. Soc.,80, Part 1, 165–187 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  16. V. I. Smirnov and S. L. Sobolev, “A new method for the solution of the two-dimensional problem of elastic vibrations,”Tr. Seismolog. Inst. Akad. Nauk SSSR, No. 20, 37–42 (1932).

    Google Scholar 

  17. V. I. Smirnov and S. L. Sobolev, “On the application of a new method to the investigation of elastic vibrations,”Tr. Seismolog. Inst. Akad. Nauk SSSR, No. 29, 43–51 (1933).

    Google Scholar 

  18. J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhauss, “Subgroups of the Poincaré group and their invariants,”J. Math. Phys.,17, No. 6, 977–985 (1976).

    Article  MATH  Google Scholar 

  19. V. I. Fushchich, A. F. Barannik, and Yu. D. Moskalenko, “On new exact solutions of the multidimensional nonlinear d'Alembert equation,”Dopov. Akad. Nauk Ukr., No. 2, 33–37 (1995).

    MathSciNet  Google Scholar 

  20. V. I. Fushchich, R. Z. Zhdanov, and I. V. Revenko, “General solutions of a nonlinear wave equation and the eikonal equation,”Ukr. Mat. Zh.,43, No. 11, 1471–1487 (1991).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp 583–593, May, 1999.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barannik, A.F., Yurik, I.I. A new method for the construction of solutions of nonlinear wave equations. Ukr Math J 51, 649–661 (1999). https://doi.org/10.1007/BF02591702

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02591702

Keywords