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Ukrainian Mathematical Journal

, Volume 43, Issue 11, pp 1350–1364 | Cite as

Conditional symmetry of the equations of nonlinear mathematical physics

  • V. I. Fushchich
Article

Abstract

We present a survey of results regarding the investigation of the conditions of the symmetry of nonlinear equations of mathematical and theoretical physics: the wave equation, the Schrödinger, Boussinesq, Korteweg-de Vries, Maxwell, and Dirac equations. We construct families of exact solutions that cannot be obtained by the classical Lie approach.

Keywords

Exact Solution Mathematical Physic Wave Equation Theoretical Physic Nonlinear Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature cited

  1. 1.
    V. I. Fushchich, “On the symmetry and particular solutions of certain multidimensional equations of mathematical physics,” in: Algebra Theoretic Methods in Problems of Mathematical Physics [in Russian], Akad. Nauk Ukr. SSR, Kiev (1983), pp. 4–23.Google Scholar
  2. 2.
    V. I. Fushchich, “How can one extend the symmetry of differential equations?,” in: Symmetry and Solutions of Nonlinear Equations of Mathematical Physics [in Russian], Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1987), pp. 4–16.Google Scholar
  3. 3.
    V. I. Fushchich, “Symmetry and exact solutions of multidimensional nonlinear wave equations,” Ukr. Mat. Zh.,39, No. 1, 116–123 (1987).Google Scholar
  4. 4.
    W. I. Fushchich and I. M. Tsifra, “On a reduction and solutions of nonlinear wave equations with broken symmetry,” J. Phys. A,20, No. 2, L45-L48 (1987).Google Scholar
  5. 5.
    W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell's Equations, Reidel, Dordrecht (1987).Google Scholar
  6. 6.
    W. Fushchich and R. Zhdanov, “On some new exact solutions of nonlinear d'Alembert and Hamilton equations,” Preprint No. 468, Inst. for Math. and Its Appl., Univ. of Minnesota, Minneapolis (1988).Google Scholar
  7. 7.
    V. I. Fushchich, N. I. Serov, and V. I. Chopik, “Conditional invariance and nonlinear heat equations,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 9, 17–21 (1988).Google Scholar
  8. 8.
    V. I. Fushchich and N. I. Serov, “Conditional invariance and exact solutions of a nonlinear equation of acoustics,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 10, 27–31 (1988).Google Scholar
  9. 9.
    W. I. Fushchich and R. Z. Zhdanov, “Symmetry and exact solutions of nonlinear spinor equations,” Phys. Rep.,46, No. 2, 325–365 (1989).Google Scholar
  10. 10.
    V. I. Fushchich and N. I. Serov, “Conditional invariance and exact solutions of the Boussinesq equation,” in: Symmetry and Solutions of Equations of Mathematical Physics [in Russian], Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1989), pp. 95–102.Google Scholar
  11. 11.
    G. W. Bluman and J. D. Cole, “The general similarity solution of the heat equation,” J. Math. Mech.,18, No. 11, 1025–1042 (1969).Google Scholar
  12. 12.
    P. J. Olver and Ph. Rosenau, “The construction of special solutions to partial differential equations,” Phys. Lett. A,114, No. 3, 107–112 (1986).Google Scholar
  13. 13.
    P. A. Clarkson and M. D. Kruskal, “New similarity reductions of the Boussinesq equation,” J. Math. Phys.,30, No. 10, 2201–2213 (1989).Google Scholar
  14. 14.
    D. Levi and P. Winternitz, “Nonclassical symmetry reduction: example of the Boussinesq euation,” J. Phys. A.,22, No. 15, 2915–2924 (1989).Google Scholar
  15. 15.
    M. V. Shul'ga, “The symmetry and some particular solutions of the d'Alembert equation with a nonlinear condition,” in: Group-Theoretic Investigations of the Equations of Mathematical Physics [in Russian], Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1985).Google Scholar
  16. 16.
    V. I. Fushchich, R. Z. Zhdanov, and I. V. Revenko, “Compatibility and solutions of nonlinear d'Alembert and Hamilton equations,” Preprint No. 39, Akad. Nauk UkrSSR, Inst. Mat. (1990).Google Scholar
  17. 17.
    W. I. Fushchich and N. I. Serov, “On some exact solutions of the three-dimensional nonlinear Schrödinger equation,” J. Phys. A.,20, No. 15, L929-L933 (1987).Google Scholar
  18. 18.
    V. I. Fushchich and V. I. Chopik, “Conditional invariance of a nonlinear Schrödinger equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 4, 30–33 (1990).Google Scholar
  19. 19.
    V. I. Fushchich and N. I. Serov, “Conditional invariance and reduction of a nonlinear heat equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 24–28 (1990).Google Scholar
  20. 20.
    V. I. Fushchich, N. I. Serov, and T. K. Amerov, “Conditional invariance of the nonlinear heat equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 11, 16–21 (1990).Google Scholar
  21. 21.
    V. I. Fushchich, “On a generalization of S. Lie's method,” in: Algebra-Theoretic Analysis of Equations of Mathematical Physics [in Russian], Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1990), pp. 4–9.Google Scholar
  22. 22.
    V. I. Fushchich, V. M. Shtelen', and N. I. Serov, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
  23. 23.
    V. I. Fushchich, N. I. Serov, and T. K. Amerov, “On the conditional symmetry of the generalized Korteweg-de Vries equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 12, 28–30 (1991).Google Scholar
  24. 24.
    W. F. Ames, R. J. Lohner, and E. Adams, “Group properties of utt=[f(u)ux]x,” Int. J. Nonlinear Mech.,16, No. 5/6, 439–447 (1981).Google Scholar
  25. 25.
    V. I. Fushchich, N. I. Serov, and V. K. Repeta, “Conditional symmetry, reduction, and exact solutions of the nonlinear wave equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 5, 29–34 (1991).Google Scholar
  26. 26.
    V. I. Fushchich, V. I. Chopik, and P. P. Mironyuk, “Conditional invariance and exact solutions of three-dimensional nonlinear equations in acoustics,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 9, 25–28 (1990).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • V. I. Fushchich
    • 1
  1. 1.Institute of MathematicsAcademy of Sciences of the UkraineKiev

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