Table des matières de ce fascicule | Article précédent | Article suivant
Plamen Iliev
On the heat kernel and the Korteweg--de Vries hierarchy
(Sur le noyau de la chaleur et la hiérarchie de Korteweg-de Vries)
Annales de l'institut Fourier, 55 no. 6 (2005), p. 2117-2127
Article: sur abonnement | Analyses MR 2187948 | Zbl 1078.35103
Class. Math.: 35Q53, 35K05, 37K10
Mots clés: Noyau de la chaleur, hiérarchie de KdV, fonctions tau
Nous donnons des formules explicites pour les coefficients d'Hadamard en termes de la
fonction tau de la hiérarchie de Korteweg-de Vries. A partir de cette formule nous
pouvons facilement démontrer les propriétés de ces coefficients.
[1] M. Adler & J. Moser, “On a class of polynomials connected with the Korteweg-de Vries equation”, Comm. Math. Phys. 61 (1978), p. 1-30
Article | MR 501106 | Zbl 0428.35067
[2] H. Airault, H. P. McKean & J. Moser, “Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem”, Comm. Pure Appl. Math. 30 (1977), p. 95-148 MR 649926 | Zbl 0338.35024
[3] G. Andrews, R. Askey & R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications 71, Cambridge University Press, 1990 Zbl 0920.33001
[4] I. Avramidi & R. Schimming, “A new explicit expression for the Korteweg-de Vries hierarchy”, Math. Nachr. 219 (2000), p. 45-64 MR 1791911 | Zbl 0984.37084
[5] N. Berline, E. Getzler & M. Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften 298, Springer-Verlag, 1992 MR 1215720 | Zbl 0744.58001
[6] M. Berger, Geometry of the Spectrum, Proc. Sympos. Pure Math. 27, Amer. Math. Soc., Providence, 1975 Zbl 0311.53055
[7] E. Date, M. Jimbo, M. Kashiwara & T. Miwa, Transformation groups for soliton equations, Proc. RIMS Symp. Nonlinear Integrable Systems - Classical and Quantum Theory (Kyoto 1981), World Scientific, Singapore, 1983, p. 39-119 Zbl 0571.35098
[8] L. A. Dickey, Soliton Equations and Hamiltonian Systems, 2nd Edition, Advanced Series in Mathematical Physics 26, World Scienti?c, 2003 MR 1964513 | Zbl 01843266
[9] J. J. Duistermaat & F. A. Grünbaum, “Differential equations in the spectral parameter”, Comm. Math. Phys. 103 (1986), p. 177-240
Article | MR 826863 | Zbl 0625.34007
[10] S. A. Fulling (ed.), Heat kernel techniques and quantum gravity (Winnipeg, MB, 1994), Discourses Math. Appl. 4, Texas A & M Univ., College Station, TX, 1995 MR 1424245 | Zbl 0845.00044
[11] P. Gilkey, Heat equation asymptotics, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) 54, Part 3, Amer. Math. Soc., 1993 MR 1216627 | Zbl 0791.58092
[12] F. A. Grünbaum & P. Iliev, “Heat kernel expansions on the integers”, Math. Phys. Anal. Geom. 5 (2002), p. 183-200 MR 1918052 | Zbl 0996.35077
[13] J. Hadamard, “Lectures on Cauchy's Problem”, New Haven, Yale Univ. Press (1923) JFM 49.0725.04
[14] L. Haine, “The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations”, to appear in Annales de l'Institut Fourier
Cedram
[15] R. Hirota, The direct method in soliton theory, Translated from the 1992 Japanese original and edited by Atsushi Nagai, Jon Nimmo and Claire Gilson (with a foreword by Jarmo Hietarinta and Nimmo), Cambridge University Press, 2004 Zbl 02117215
[16] P. Iliev, “Finite heat kernel expansions on the real line”, math-ph/0504046, http://arxiv.org/abs/math-ph/0504046
arXiv | Zbl 05135868
[17] M. Kac, “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966), p. 1-23 MR 201237 | Zbl 0139.05603
[18] H. P. McKean & I. Singer, “Curvature and the eigenvalues of the Laplacian”, J. Diff. Geom. 1 (1967), p. 43-69 MR 217739 | Zbl 0198.44301
[19] H. P. McKean & P. van Moerbeke, “The spectrum of Hill's equation”, Invent. Math. 30 (1975), p. 217-274
Article | MR 397076 | Zbl 0319.34024
[20] M. Sato & Y. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds”, Lect. Notes Num. Appl. Anal. 5 (1982), p. 259-271 MR 730247 | Zbl 0528.58020
[21] R. Schimming, “An explicit expression for the Korteweg-de Vries hierarchy”, Z. Anal. Anwendungen 7 (1988), p. 203-214 MR 951118 | Zbl 0659.35089
[22] P. van Moerbeke, Integrable foundations of string theory, Singapore: World Scientific, 1994, p. 163-267 Zbl 0850.81049
Plamen Iliev
On the heat kernel and the Korteweg--de Vries hierarchy
(Sur le noyau de la chaleur et la hiérarchie de Korteweg-de Vries)
Annales de l'institut Fourier, 55 no. 6 (2005), p. 2117-2127
Article: sur abonnement | Analyses MR 2187948 | Zbl 1078.35103
Class. Math.: 35Q53, 35K05, 37K10
Mots clés: Noyau de la chaleur, hiérarchie de KdV, fonctions tau
Résumé - Abstract
Bibliographie
Article | MR 501106 | Zbl 0428.35067
[2] H. Airault, H. P. McKean & J. Moser, “Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem”, Comm. Pure Appl. Math. 30 (1977), p. 95-148 MR 649926 | Zbl 0338.35024
[3] G. Andrews, R. Askey & R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications 71, Cambridge University Press, 1990 Zbl 0920.33001
[4] I. Avramidi & R. Schimming, “A new explicit expression for the Korteweg-de Vries hierarchy”, Math. Nachr. 219 (2000), p. 45-64 MR 1791911 | Zbl 0984.37084
[5] N. Berline, E. Getzler & M. Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften 298, Springer-Verlag, 1992 MR 1215720 | Zbl 0744.58001
[6] M. Berger, Geometry of the Spectrum, Proc. Sympos. Pure Math. 27, Amer. Math. Soc., Providence, 1975 Zbl 0311.53055
[7] E. Date, M. Jimbo, M. Kashiwara & T. Miwa, Transformation groups for soliton equations, Proc. RIMS Symp. Nonlinear Integrable Systems - Classical and Quantum Theory (Kyoto 1981), World Scientific, Singapore, 1983, p. 39-119 Zbl 0571.35098
[8] L. A. Dickey, Soliton Equations and Hamiltonian Systems, 2nd Edition, Advanced Series in Mathematical Physics 26, World Scienti?c, 2003 MR 1964513 | Zbl 01843266
[9] J. J. Duistermaat & F. A. Grünbaum, “Differential equations in the spectral parameter”, Comm. Math. Phys. 103 (1986), p. 177-240
Article | MR 826863 | Zbl 0625.34007
[10] S. A. Fulling (ed.), Heat kernel techniques and quantum gravity (Winnipeg, MB, 1994), Discourses Math. Appl. 4, Texas A & M Univ., College Station, TX, 1995 MR 1424245 | Zbl 0845.00044
[11] P. Gilkey, Heat equation asymptotics, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) 54, Part 3, Amer. Math. Soc., 1993 MR 1216627 | Zbl 0791.58092
[12] F. A. Grünbaum & P. Iliev, “Heat kernel expansions on the integers”, Math. Phys. Anal. Geom. 5 (2002), p. 183-200 MR 1918052 | Zbl 0996.35077
[13] J. Hadamard, “Lectures on Cauchy's Problem”, New Haven, Yale Univ. Press (1923) JFM 49.0725.04
[14] L. Haine, “The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations”, to appear in Annales de l'Institut Fourier
Cedram
[15] R. Hirota, The direct method in soliton theory, Translated from the 1992 Japanese original and edited by Atsushi Nagai, Jon Nimmo and Claire Gilson (with a foreword by Jarmo Hietarinta and Nimmo), Cambridge University Press, 2004 Zbl 02117215
[16] P. Iliev, “Finite heat kernel expansions on the real line”, math-ph/0504046, http://arxiv.org/abs/math-ph/0504046
arXiv | Zbl 05135868
[17] M. Kac, “Can one hear the shape of a drum?”, Amer. Math. Monthly 73 (1966), p. 1-23 MR 201237 | Zbl 0139.05603
[18] H. P. McKean & I. Singer, “Curvature and the eigenvalues of the Laplacian”, J. Diff. Geom. 1 (1967), p. 43-69 MR 217739 | Zbl 0198.44301
[19] H. P. McKean & P. van Moerbeke, “The spectrum of Hill's equation”, Invent. Math. 30 (1975), p. 217-274
Article | MR 397076 | Zbl 0319.34024
[20] M. Sato & Y. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds”, Lect. Notes Num. Appl. Anal. 5 (1982), p. 259-271 MR 730247 | Zbl 0528.58020
[21] R. Schimming, “An explicit expression for the Korteweg-de Vries hierarchy”, Z. Anal. Anwendungen 7 (1988), p. 203-214 MR 951118 | Zbl 0659.35089
[22] P. van Moerbeke, Integrable foundations of string theory, Singapore: World Scientific, 1994, p. 163-267 Zbl 0850.81049
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310