Solution of a restricted Hadamard problem on Minkowski spaces

Communications on Pure and Applied Mathematics

Volumn 50, Issue 10, 1997, Pages 1019-1052

  Berest, Yuri Yu ^a  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031498011     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199710)50:10<1019::AID-CPA3>3.0.CO;2-F     Document Type: Article

References (34)

1. Adler, M., and Moser, J., On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61, 1978, pp. 1-30.
2. Airault, H., McKean, H. P., and Moser, J., Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30, 1977, pp. 95-148.
3. Ásgeirsson, L., Some hints on Huygens' principle and Hadamard's conjecture, Comm. Pure Appl. Math. 9, 1956, pp. 307-326.
4. Babich, V. M., Hadamard's ansatz, its analogues, generalizations and applications, St. Petersburg Math. J. 3(5), 1992, pp. 937-972.
5. Berest, Yu. Yu., Hierarchies of Huygens operators and Hadamard's conjecture, Acta Math. Appl., 1997, to appear.
6. Berest, Yu. Yu., and Lutsenko, I. M., Huygens' principle in Minkowski spaces and soliton solutions of the Korteweg-de Vries equation, Comm. Math. Phys., 1997, to appear.
7. Berest, Yu. Yu., and Veselov, A. P., Huygens' principle and Coxeter groups, Russian Math. Surveys 48(3), 1993, pp. 183-184.
8. Berest, Yu. Yu., and Veselov, A. P., The Hadamard problem and Coxeter groups: New examples of Huygens equations, Funct. Anal. Appl. 28(1), 1994, pp. 3-12.
9. Berest, Yu. Yu., and Veselov, A. P., The Huygens principle and integrability (Russian), Uspekhi Mat. Nauk 49, no. 6(300), 1994, pp. 7-78; translation in Russian Math. Surveys, 49(6), 1994, pp. 3-77.
10. Berest, Yu. Yu., and Veselov, A. P., The Huygens principle and integrability (Russian), Uspekhi Mat. Nauk 49, no. 6(300), 1994, pp. 7-78; translation in Russian Math. Surveys, 49(6), 1994, pp. 3-77.
11. Burchnall, J. L., and Chaundy, T. W., A set of differential equations which can be solved by polynomials, Proc. London Math. Soc. 30, 1929-30, pp. 401-414.
12. Chalykh, A. O., and Veselov, A. P., Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126, 1990, pp. 597-611.
13. Chalykh, A. O., and Veselov, A. P., Integrabilitv in the theory of Schrödinger operator and harmonic analysis, Comm. Math. Phys. 152, 1993, pp. 29-40.
14. Courant, R., and Hilbert, D., Methods of Mathematical Physics II, Interscience Publishers, New York, 1962.
15. Crum, M. M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6, 1955, pp. 121-127.
16. Darboux, G., Sur la representation sphérique des surfaces, C. R. Acad. Sci. Paris Sér. I Math. 94, 1882, pp. 1343-1345.
17. Dubrovin, B. A., Matveev, V. B., and Novikov, S. P., Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties (Russian), Uspehi Mat. Nauk 31, no. 1(187), 1976, pp. 55-136; translation in Russian Math. Surveys 31(1), 1976, pp. 55-136.
18. Dubrovin, B. A., Matveev, V. B., and Novikov, S. P., Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties (Russian), Uspehi Mat. Nauk 31, no. 1(187), 1976, pp. 55-136; translation in Russian Math. Surveys 31(1), 1976, pp. 55-136.
19. Friedlander, F. G., The Wave Equation on a Curved Space-Time, Cambridge University Press, Cambridge-New York, 1975.
20. Günther, P., Huygens' Principle and Hyperbolic Equations. Academic Press, Boston, 1988.
21. Hadamard, J., Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven-London, 1923.
22. Hadamard, J., The problem of diffusion of waves, Ann. of Math. 43(3), 1942, pp. 510-522.
23. Helgason, S., Wave equations on homogeneous spaces, pp. 252-287 in: Lie Group Representations, III (College Park, Md., 1982/1983), Lecture Notes in Mathematics No. 1077, Springer, Berlin-New York, 1984.
24. Ibragimov, N. K., Transformation Groups Applied to Mathematical Physics, D. Reidel, Dordrecht-Boston-Hingham, Mass., 1985.
25. Lagnese, J. E., A solution of Hadamard's problem for a restricted class of operators, Proc. Amer. Math. Soc. 19, 1968, pp. 981-988.
26. Lagnese, J. E., and Stellmacher, K. L., A method of generating classes of Huygens' operators, J. Math. Mech. 17(5), 1967, pp. 461-472.
27. Mathisson, M., Le problème de M. Hadamard relatif à la diffusion des ondes, Acta Math. 71, 1939, pp. 249-282.
28. McLenaghan, R. G., Huygens' principle, Ann. Inst. H. Poincaré Sect. A (N.S.) 37(3), 1983, pp. 211-236.
29. Riesz, M., L'intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81, 1949, pp. 1-223.
30. Schimming, R., An explicit expression for the Korteweg-de Vries hierarchy, Z. Anal. Anwendungen 7, 1988, pp. 203-214.
31. Schimming, R., Laplace-like linear differential operators with a logarithm-free elementary solution, Math. Nachr. 148, 1990, pp. 145-174.
32. Stellmacher, K. I., Ein Beispiel einer Huygensschen Differentialgleichung, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Chem. Abt. 10, 1953, pp. 133-138.
33. van der Waerden, B. L., Modern Algebra I, Ungar, New York, 1953.
34. Veselov, A. P., Styrkas, K. L., and Chalykh, O. A., Algebraic integrability for the Schrödinger equation, and groups generated by reflections, Theoret, and Math. Phys. 94(2), 1993, pp. 182-197.