Huygens' principle in Minkowski spaces and soliton solutions of the Korteweg-de Vries equation

Communications in Mathematical Physics

Volumn 190, Issue 1, 1997, Pages 113-132

  Berest, Yuri Yu ^a   Loutsenko, Igor M ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031329338     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050235     Document Type: Article

References (42)

1. Ablowitz, M.J. and Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press, 1991
2. Adler, M. and Moser, J. On a class of polynomials connected with the Korteweg-de Vries Equation. Commun. Math. Phys. 61, 1-30 (1978)
3. Airault, H., McKean, H. and Moser, J.: Rational and elliptic solutions of the KdV-equation and related many body problems. Commun. Pure Appl. Math. 30, 95-148 (1977)
4. Berest, Yu.Yu.: Hierarchies of Huygens' operators and Hadamard's conjecture. Preprint CRM-2296, to be published in Acta Math Appl. (1997)
5. Berest, Yu.Yu.: Lacunae of hyperbolic Riesz kernels and commutative rings of partial differential operators. Lett. Math. Phys. 41, 227-235 (1997)
6. Berest, Yu.Yu.: Solution of a restricted Hadamard's problem in Minkowski spaces. Commun. Pure Appl. Math. 50, 1021-1054 (1997)
7. Berest, Yu.Yu. and Molchanov, Yu.A.: Fundamental solutions for partial differential equations with reflection group invariance. J. Math. Phys. 36 (8), 4324-4339 (1995)
8. Berest, Yu.Yu. and Veselov, A.P.: Huygens' principle and integrability. Uspekhi Mat. Nauk, 49 (6), 5-77 (1994)
9. Berest, Yu.Yu. and Veselov, A.P.: Huygens' principle and Coxeter groups. Uspekhi Mat. Nauk, 48 (2), 181-182 (1993)
10. Berest, Yu.Yu. and Veselov, A.P.: Hadamard's problem and Coxeter groups: new examples of Huygens' equations. Funct. Anal. Appl. 28 (1), 3-15 (1994)
11. Burchnall, J.L. and Chaundy, T.W.: A set of differential equations which can be solved by polynomials. Proc. London Math. Soc. 30, 401-414 (1929-1930)
12. Carminati, J. and McLenaghan, R.G.: On the validity of Huygens' principle for the wave equation on curved space-time. In: Huygens' principle 1690-1990: Theory and Applications. (H. Blok, H.A. Ferwerda, H.K. Kuiken, Eds.), New York: Elsevier Publishers B.V.,1992, pp. 435-442
13. Chalykh, O.A. and Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras. Commun. Math. Phys. 126, 597-611 (1990)
14. Chalykh, O.A. and Veselov, A.P.: Integrability in the theory of the Schrödinger operator and harmonic analysis. Commun. Math. Phys. 152, 29-40 (1993)
15. Courant, R. and Hilbert, D.: Methods of Mathematical Physics II, New York: Interscience Publ., 1962
16. Crum, M.: Associated Sturm-Liouville systems. Quart. J. Math. 6 (2), 121-127 (1955)
17. Darboux, G.: Sur la representation sphérique des surfaces. Compt. Rend. 94, 1343-1345 (1882)
18. Deift, P.A.: Application of a commutation formula. Duke Math. Journal, 45 (2), 267-310 (1978)
19. Duistermaat, J.J.: Huygens' principle for linear partial differential equations. In Huygens' principle 1690-1990: Theory and Applications, (H. Blok, H.A. Ferwerda, H.K. Kuiken, Eds.), New York: Elsevier Publishers B.V. 1992, pp. 273-297
20. Duistermaat, J.J. and Grünbaum, F.A.: Differential equations in the spectral parameter. Commun. Math. Phys. 103, 177-240 (1986)
21. Friedlander, F.G.: The Wave Equation on a Curved Space-time, Cambridge: Cambridge University Press, 1975
22. Gabrielov, A.M. and Palamodov, V.P.: Huygens' principle and its generalizations. In: I. G. Petrovskii, Collected papers, v. 1, Nauka, Moscow, 1986 (in Russian) pp. 449-456
23. Gel'fand, I.M. and Shilov, G.E.: Generalized Functions, V. 1, New York: Acad. Press, 1964
24. P. Günther, P.: Huygens' Principle and Hyperbolic Equations, Boston: Acad. Press, 1988
25. Günther, P.: Huygens' principle and Hadamard's conjecture. The Mathematical Intelligencer, 13(2), 56-63 (1991)
26. Hadamard, J.: Lectures on Cauchy's Problem in Linear Partial Differential Equations. New Haven: Yale University Press, 1923
27. Helgason, S.: Wave equations on homogeneous spaces. Lect. Notes in Math. 1077, Berlin: Springer-Verlag, (1984) pp.252-287
28. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Dordrecht: Reidel, 1985
29. Ince, E.L.: Ordinary Differential Equations. New-York: Dover, 1956
30. Infeld, L. and Hull, T.E.: The factorization method. Rev. Mod. Phys. 23 (1), 21-70 (1951)
31. Lagnese, J.E. and Stellmacher, K.L.: A method of generating classes of Huygens' operators. J. Math. Mech. 17 (5), 461-472 (1967)
32. Lagnese, J.E.: A solution of Hadamard's problem for a restricted class of operators. Proc. Amer. Math. Soc. 19, 981-988 (1968)
33. Mathisson, M.: Le problème de M. Hadamard relatif à la diffusion des ondes. Acta Math., 71, 249-282 (1939)
34. Matveev, V.B. and Salle, M.A.: Darboux Transformations and Solitons. Berlin-Heidelberg-New York: Springer-Verlag, 1991
35. McLenaghan, R.G.: Huygens' principle. Ann. Inst. Henri Poincaré, 37 (3), 211-236 (1982)
36. Olshanetsky, M.A. and Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Reps. 94, 313-404 (1983)
37. Riesz, M.: L'intégrale de Riemann-Liouville et le problème de Cauchy. Acta Math. 81, 1-223 (1949)
38. Schimming, R.: Korteweg-de Vries-Hierarchie und Huygenssches Prinzip. Dresdener Seminar für Theoretische Physik, Sitzungsberichte, N 26 (1986)
39. Schimming, R.: Laplace-like linear differential operators with a logarithmic-free elementary solution. Math. Nachr. 148, 145-174 (1990)
40. Stellmacher, K.: Ein Beispiel einer Huygensschen Differentialgleichung. Nachr. Akad. Wiss. Göttingen, Math.-Phys. K1, IIa, Bd. 10, 133-138 (1953)
41. Veselov, A.P., Styrkas, K.L. and Chalykh, O.A.: Algebraic integrability for the Schrödinger operator and reflection groups. Theor. Math. Phys. 94, 253-275 (1993)
42. Veselov, A.P., Feigin, M.V. and Chalykh, O.A.: New integrable deformations of quantum Calogero-Moser problem. Usp. Mat. Nauk. 51 (3), 185-186 (1996)