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

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References

  1. I. G. Petrovsky, Lectures on Partial Differential Equations [in Russian], GITTL, Moscow—Leningrad (1950).

    Google Scholar 

  2. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, New York (1964).

  3. J. Hadamard, Lecture on Cauchy's Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven (1923).

    Google Scholar 

  4. M. Mathisson, “Le problème de Hadamard relatif à la diffusion des ondes,” Acta Math.,71, 249–282 (1939).

    Google Scholar 

  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. K. L. Stellmacher, “Ein Beispiel einer Huygensschen Differentialgleichung,” Nachr. Akad. Wiss. Göttingen Math.—Phys. Kl. IIa,10, 133–138 (1953).

    Google Scholar 

  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. 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. G. Darboux, “Sur la representations sphérique des surfaces,” Compt. Rend.,94, 1343–1345 (1882).

    Google Scholar 

  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. S. P. Novikov, Periodic problem for Korteweg—de Vries equation. I,” Funkts. Anal. Prilozhen.,8, No. 3, 54–63 (1974).

    Google Scholar 

  12. Yu. Yu. Berest, “Deformations preserving Huygens' principle,” to appear in J. Math. Phys. (1993).

  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. 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. M. A. Olshanetsky and A. M. Perelomov, “Quantum integrable systems related to Lie algebras,” Phys. Rep.,94, 313–404 (1983).

    Google Scholar 

  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. 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. 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. E. M. Opdam, “Some applications of hypergeometric shift operators,” Invent. Math.,98, 1–18 (1989).

    Google Scholar 

  20. C. F. Dunkl, “Differential-difference operators associated with reflection groups,” Trans. Amer. Math. Soc.,311, 167–183 (1989).

    Google Scholar 

  21. G. J. Heckman, “A remark on the Dunkl differential-difference operators,” Progr. Math.,101, 181–191 (1991).

    Google Scholar 

  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.

  23. V. N. Babich, “Hadamard's anzats: its analogs, generalizations, and applications,” Algebra Analiz,3, No. 5, 1–37 (1991).

    Google Scholar 

  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. J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” J. Math. Phys.,24, 522–526 (1983).

    Google Scholar 

  26. J. Weiss, “The sine-Gordon equation: complete and partial integrability,” J. Math. Phys.,25, 2226–2235 (1984).

    Google Scholar 

  27. K. L. Stellmacher, “Eine Klasse Huygensscher Differentialgleichungen und ihre Integration,” Math. Ann.,130, No. 3, 219–233 (1955).

    Google Scholar 

  28. I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1, Acad. Press, New York (1964).

    Google Scholar 

  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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Dedicated to I. M. Gel'fand on his 80th birthday

Moscow Physical Technical Institute. Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 1, pp. 3–15, January–March, 1994.

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Berest, Y.Y., Veselov, A.P. Hadamard's problem and coxeter groups: New examples of Huygens' equations. Funct Anal Its Appl 28, 3–12 (1994). https://doi.org/10.1007/BF01079005

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