Ukrainian Mathematical Journal

, Volume 46, Issue 10, pp 1480–1503 | Cite as

Separation of variables in two-dimensional wave equations with potential

  • R. Z. Zhdanov
  • I. V. Revenko
  • V. I. Fushchich


In the present paper, we solve the problem of separation of variables in the wave equationu tt u xx +V(x)u=0 and give a complete classification of potentialsV(x) forwhich equations of this type admit nontrivial separation of variables. Furthermore, we construct all coordinate systems in which this separation is possible.


Coordinate System Wave Equation Complete Classification 
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  1. 1.
    M. Bocher,Die Reihentwickelunger der Potentialtheorie, Teubner, Leipzig (1894).Google Scholar
  2. 2.
    G. Darboux,Lecons sur les Systémes Orthogonaux et les Coordonnées Curvilignes, Paris (1910).Google Scholar
  3. 3.
    L. P. Eisenhart, “Separable systems of Stäkel,”Ann. Math.,35, No. 2, 284–305 (1934).Google Scholar
  4. 4.
    V. V. Stepanov, “On the Laplace equation and some triorthogonal systems,”Mat. Sb.,11, 204–238 (1942).Google Scholar
  5. 5.
    M. N. Olevskii, “Triorthogonal systems in spaces with constant curvature where the equation Δ2u+λu=0 admits complete separation of variables,”Mat. Sb.,27, 379–426 (1950).Google Scholar
  6. 6.
    W. Miller,Symmetry and Separation of Variables, Addison-Wesley, Massachusetts (1977).Google Scholar
  7. 7.
    W. I. Fushchich and N. I. Serov, “The symmetry and some exact solutions of many-dimensional nonlinear d'Alembert, Liouville, and eikonal equations,”J. Phys. A: Math. Gen.,16, No. 15, 3645–3656 (1983).Google Scholar
  8. 8.
    W. I. Fushchich and R. Z. Zhdanov, “On some new exact solutions of the nonlinear d'Alembert-Hamiltonian system,”Phys. Lett. A,141, No. 3, 4, 113–115 (1989).Google Scholar
  9. 9.
    W. I. Fushchich, R. Z. Zhdanov, and I. A. Yegorchenko, “On the reduction of the nonlinear multi-dimensional wave equations and compatibility of the d'Alembert-Hamiltonian system,”J. Math. Anal. Appl.,161, No. 2, 352–360 (1991).Google Scholar
  10. 10.
    T. H. Koomwinder, “A precise definition of separation of variables,”Lect. Notes Math.,810, 240–263 (1980).Google Scholar
  11. 11.
    R. Z. Zhdanov, I. V. Revenko, and W. I. Fushchich, “On a new approach to the separation of variables in the wave equation with potential,”Dokl. Ukr. Akad. Nauk, Ser. A., No. 1, 9–11 (1993).Google Scholar
  12. 12.
    E. Kalnins and W. Miller, “Lie theory and separation of variables, II: The EPD equation,”J. Math. Phys.,17, No. 3, 369–377 (1976).Google Scholar
  13. 13.
    E. Kalnins and W. Miller, Lie theory and separation of variables, III: The equationf ttf ss2 fJ. Math. Phys.,15, No. 9, 1025–1032 (1974).Google Scholar
  14. 14.
    V. N. Shapovalov, “Separation of variables in a second-order linear differential equation,”Differents. Uravn.,16, No. 10, 1864–1874 (1980).Google Scholar
  15. 15.
    L. V. Ovsyannikov,Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).Google Scholar
  16. 16.
    P. Olver,Applications of Lie Groups to Differential Equations, Springer, New York (1986).Google Scholar
  17. 17.
    E. Kamke,Differentialgleichungen, Lösungsmethoden und Lösungen, Leipzig (1959).Google Scholar
  18. 18.
    A. Erdelyi et al.,Higher Transcendental Functions, Vol. 1, 2, McGraw & Hill, New York (1953).Google Scholar
  19. 19.
    G. Bluman and S. Kumei, “On invariance properties of the wave equation,”J. Math. Phys.,28, No. 2, 307–318 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • R. Z. Zhdanov
    • 1
  • I. V. Revenko
    • 1
  • V. I. Fushchich
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev

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