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Functional Analysis and Its Applications

, Volume 28, Issue 1, pp 3–12 | Cite as

Hadamard's problem and coxeter groups: New examples of Huygens' equations

  • Yu. Yu. Berest
  • A. P. Veselov
Article

Keywords

Functional Analysis Coxeter Group 
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References

  1. 1.
    I. G. Petrovsky, Lectures on Partial Differential Equations [in Russian], GITTL, Moscow—Leningrad (1950).Google Scholar
  2. 2.
    R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, New York (1964).Google Scholar
  3. 3.
    J. Hadamard, Lecture on Cauchy's Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven (1923).Google Scholar
  4. 4.
    M. Mathisson, “Le problème de Hadamard relatif à la diffusion des ondes,” Acta Math.,71, 249–282 (1939).Google Scholar
  5. 5.
    L. Asgeirsson, “Some hints on Huygens' principle and Hadamard's conjecture,” Comm. Pure Appl. Math.,9, No. 3, 307–327 (1956).Google Scholar
  6. 6.
    K. L. Stellmacher, “Ein Beispiel einer Huygensschen Differentialgleichung,” Nachr. Akad. Wiss. Göttingen Math.—Phys. Kl. IIa,10, 133–138 (1953).Google Scholar
  7. 7.
    J. E. Lagnese and K. L. Stellmacher, “A method of generating classes of Huygens' operators,” J. Math. Mech.,17, No. 5, 461–472 (1967).Google Scholar
  8. 8.
    J. E. Lagnese, “A solution of Hadamard's problem for a restricted class of operators,” Proc. Amer. Math. Soc.,19, 981–988 (1968).Google Scholar
  9. 9.
    G. Darboux, “Sur la representations sphérique des surfaces,” Compt. Rend.,94, 1343–1345 (1882).Google Scholar
  10. 10.
    M. Adler and J. Moser, “On a class of polynomials connected with the Korteweg—de Vries equation,” Commun. Math. Phys.,61, No. 1, 1–30 (1978).Google Scholar
  11. 11.
    S. P. Novikov, Periodic problem for Korteweg—de Vries equation. I,” Funkts. Anal. Prilozhen.,8, No. 3, 54–63 (1974).Google Scholar
  12. 12.
    Yu. Yu. Berest, “Deformations preserving Huygens' principle,” to appear in J. Math. Phys. (1993).Google Scholar
  13. 13.
    Yu. Yu. Berest and A. P. Veselov, “Huygens' principle and Coxeter groups,” Usp. Mat. Nauk,48, No. 3, 181–182 (1993).Google Scholar
  14. 14.
    M. A. Olshanetsky and A. M. Perelomov, “Quantum completely integrable systems connected with semisimple Lie algebras,” Lett. Math. Phys.,2, 7–13 (1977).Google Scholar
  15. 15.
    M. A. Olshanetsky and A. M. Perelomov, “Quantum integrable systems related to Lie algebras,” Phys. Rep.,94, 313–404 (1983).Google Scholar
  16. 16.
    F. Calogero, “Solution of the one-dimensionaln-body problem with quadratic and/or inversely quadratic pair potential,” J. Math. Phys.,12, 419–436 (1971).Google Scholar
  17. 17.
    O. A. Chalykh and A. P. Veselov, “Commutative rings of partial differential operators and Lie algebras,” Preprint of FIM, ETH, Zürich (1988); Commun. Math. Phys.,126, 597–611 (1990).Google Scholar
  18. 18.
    A. P. Veselov, K. L. Styrkas, and O. A. Chalykh, “Algebraic integrability for the Schrödinger equation and the groups generated by reflections,” Teor. Mat. Fiz.,94, No 2, 253–275 (1993).Google Scholar
  19. 19.
    E. M. Opdam, “Some applications of hypergeometric shift operators,” Invent. Math.,98, 1–18 (1989).Google Scholar
  20. 20.
    C. F. Dunkl, “Differential-difference operators associated with reflection groups,” Trans. Amer. Math. Soc.,311, 167–183 (1989).Google Scholar
  21. 21.
    G. J. Heckman, “A remark on the Dunkl differential-difference operators,” Progr. Math.,101, 181–191 (1991).Google Scholar
  22. 22.
    G. Felder and A. P. Veselov, “Shift operator for Calogero—Sutherland quantum problems via the Knizhnik—Zamolodchikov equation,” Preprint FIM, ETH, Zürich (1993); to appear in Commun. Math. Phys.Google Scholar
  23. 23.
    V. N. Babich, “Hadamard's anzats: its analogs, generalizations, and applications,” Algebra Analiz,3, No. 5, 1–37 (1991).Google Scholar
  24. 24.
    N. H. Ibragimov and A. O. Oganesyan, “Hierarchy of Huygens' equations in spaces with a nontrivial conformal group,” Usp. Mat. Nauk,46, No. 3, 111–146 (1991).Google Scholar
  25. 25.
    J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” J. Math. Phys.,24, 522–526 (1983).Google Scholar
  26. 26.
    J. Weiss, “The sine-Gordon equation: complete and partial integrability,” J. Math. Phys.,25, 2226–2235 (1984).Google Scholar
  27. 27.
    K. L. Stellmacher, “Eine Klasse Huygensscher Differentialgleichungen und ihre Integration,” Math. Ann.,130, No. 3, 219–233 (1955).Google Scholar
  28. 28.
    I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1, Acad. Press, New York (1964).Google Scholar
  29. 29.
    I. N. Bernshtein, I. M. Gel'fand, and S. I. Gel'fand, “Shubert cells and the cohomology ofG/P,” Usp. Mat. Nauk,28, No. 3, 3–26 (1973).Google Scholar

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© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Yu. Yu. Berest
  • A. P. Veselov

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