Journal of Mathematical Sciences

, Volume 138, Issue 2, pp 5603–5612 | Cite as

Time-dependent deformations of powers of the wave operator

  • S. P. Khekalo


Conditions under which the linear differential operators of the second order are equivalent to operators not containing “friction” (first partial derivatives) are investigated. One can construct iso-Huygens deformations for powers of the wave operator with time-dependent coefficients. The fundamental solutions of these deformations and conditions under which the Huygens principle holds are found. Bibliography: 17 titles.


Huygens Fundamental Solution Scalar Curvature Wave Operator Adjoint Action 
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  1. 1.
    M. Riesz, “L’integrale de Riemann-Liouville et le probleme de Cauchy, Acta Math., 81, 1–223 (1949).MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Y. Berest and Y. Molchanov, “Fundamental solution for partial differential equations with reflection group invariance,” J. Math. Phys., 36(8), 4324–4339 (1995).MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Y. Berest and A. P. Veselov, “The Huygens principle and integrability,” Usp. Mat. Nauk, 49,6(300), 8–78 (1994).MathSciNetGoogle Scholar
  4. 4.
    N. Kh. Ibragimov and A. O. Oganesyan, “A hierarchy of Huygens equations in spaces with nontrivial conformal group,” Usp. Mat. Nauk, 46,3(278), 111–146 (1991).MATHMathSciNetGoogle Scholar
  5. 5.
    S. P. Khekalo, “Iso-Huygens deformations of the ultrahyperbolic operator,” Zap. Nauchn. Semin. POMI, 285, 207–223 (2002).MATHGoogle Scholar
  6. 6.
    V. M. Babich, “The Hadamard ansatz, its analogs, generalizations, applications,” Algebra Analiz, 3, 1–37 (1991).MathSciNetGoogle Scholar
  7. 7.
    Y. Berest, “Hierarchies of Huygens’ Operators and Hadamard’s Conjecture,” Acta Appl. Math., 53, 125–185 (1998).MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    V. I. Semyanistyi, “Some problems of integral geometry in pseudo-Euclidean and non-Euclidean spaces, ” MGU, Trudy Semin. Vekt. Tenz. Anal., 13, 244–302 (1963).Google Scholar
  9. 9.
    B. Rubin, “Zeta integrals and integral geometry in the space of rectangular matrices,” Preprint, The Hebrew University, Jerusalem (2004).Google Scholar
  10. 10.
    I. M. Gel’fand, S. G. Gindikin, and M. I. Graev, Selected Problems in Integral Geometry [in Russian], Dobrosvet, Moscow (2000).Google Scholar
  11. 11.
    N. Kh. Ibragimov, Groups of Transformations in Mathematical Physics [in Russian], Fizmatgiz, Moscow (1983).Google Scholar
  12. 12.
    E. Cotton, “Sur les invariants differentieles de quelques equations lineaires aux derivées partielles du second ordre,” Ann. Sci. Ecole Norm. Sup., 17, 211–244 (1900).MATHMathSciNetGoogle Scholar
  13. 13.
    V. A. Zorich, Mathematical Analysis [in Russian], Vol. 2, Nauka, Moscow (1984).Google Scholar
  14. 14.
    I. M. Gel’fand and G. E. Shilov, Generalized Functions [in Russian], Vol. 1, Fizmatgiz, Moscow (1985).Google Scholar
  15. 15.
    J. F. Treves, Lectures on Linear Equations in Partial Derivatives with Constant Coefficients [Russian translation], Mir, Moscow (1965).Google Scholar
  16. 16.
    S. P. Khekalo, “Gauge related deformations of ordinary linear differential equations with constant coefficients,” Zap. Nauchn. Semin. POMI, 308, 235–251 (2004).Google Scholar
  17. 17.
    J. E. Lagnese and K. L. Stellmacher, “A method of generating classes of Huygens’ operators,” J. Math. Mech., 17, No. 5, 461–472 (1967).MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. P. Khekalo
    • 1
  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

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