On the heat kernel and the Korteweg--de Vries hierarchy
[Sur le noyau de la chaleur et la hiérarchie de Korteweg-de Vries]
Iliev, Plamen
Annales de l'Institut Fourier, Tome 55 (2005), p. 2117-2127 / Harvested from Numdam
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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.

We give explicit formulas for Hadamard's coefficients in terms of the tau-function of the Korteweg-de Vries hierarchy. We show that some of the basic properties of these coefficients can be easily derived from these formulas.

Publié le : 2005-01-01
DOI : https://doi.org/10.5802/aif.2154
Classification:  35Q53,  35K05,  37K10
Mots clés: Noyau de la chaleur, hiérarchie de KdV, fonctions tau 
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     title = {On the heat kernel and the Korteweg--de Vries hierarchy},
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     year = {2005},
     pages = {2117-2127},
     doi = {10.5802/aif.2154},
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Iliev, Plamen. On the heat kernel and the Korteweg--de Vries hierarchy. Annales de l'Institut Fourier, Tome 55 (2005) pp. 2117-2127. doi : 10.5802/aif.2154. http://gdmltest.u-ga.fr/item/AIF_2005__55_6_2117_0/

[1] M. Adler; J. Moser On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys., Tome 61 (1978), pp. 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., Tome 30 (1977), pp. 95-148 | Article | MR 649926 | Zbl 0338.35024

[3] G. Andrews; R. Askey; R. Roy Special Functions, Cambridge University Press, Encyclopedia of Mathematics and Its Applications, Tome 71 (1990) | Zbl 0920.33001

[4] I. Avramidi; R. Schimming A new explicit expression for the Korteweg-de Vries hierarchy, Math. Nachr., Tome 219 (2000), pp. 45-64 | Article | MR 1791911 | Zbl 0984.37084

[5] N. Berline; E. Getzler; M. Vergne Heat kernels and Dirac operators, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome 298 (1992) | MR 1215720 | Zbl 0744.58001

[6] M. Berger Geometry of the Spectrum, Amer. Math. Soc., Providence, Proc. Sympos. Pure Math., Tome 27 (1975) | Zbl 0311.53055

[7] E. Date; M. Jimbo; M. Kashiwara; T. Miwa; M. Jimbo And T. Miwa Transformation groups for soliton equations, World Scientific, Singapore (Proc. RIMS Symp. Nonlinear Integrable Systems - Classical and Quantum Theory (Kyoto 1981)) (1983), pp. 39-119 | Zbl 0571.35098

[8] L. A. Dickey Soliton Equations and Hamiltonian Systems, 2nd Edition, World Scienti?c, Advanced Series in Mathematical Physics, Tome 26 (2003) | MR 1964513 | Zbl 01843266

[9] J. J. Duistermaat; F. A. Grünbaum Differential equations in the spectral parameter, Comm. Math. Phys., Tome 103 (1986), pp. 177-240 | Article | MR 826863 | Zbl 0625.34007

[10] S. A. Fulling (Ed.) Heat kernel techniques and quantum gravity (Winnipeg, MB, 1994), Texas A & M Univ., College Station, TX, Discourses Math. Appl., Tome 4 (1995) | MR 1424245 | Zbl 0845.00044

[11] P. Gilkey Heat equation asymptotics, Amer. Math. Soc., Providence, RI, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), Tome 54, Part 3 (1993) | MR 1216627 | Zbl 0791.58092

[12] F. A. Grünbaum; P. Iliev Heat kernel expansions on the integers, Math. Phys. Anal. Geom., Tome 5 (2002), pp. 183-200 | Article | 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) | Numdam

[15] R. Hirota; Cambridge Tracts In Mathematics The direct method in soliton theory, Cambridge University Press, Cambridge (Translated from the 1992 Japanese original and edited by Atsushi Nagai, Jon Nimmo and Claire Gilson (with a foreword by Jarmo Hietarinta and Nimmo)) Tome 155 (2004) | Zbl 02117215

[16] P. Iliev Finite heat kernel expansions on the real line (math-ph/0504046, http://arxiv.org/abs/math-ph/0504046) | Zbl 05135868

[17] M. Kac Can one hear the shape of a drum?, Amer. Math. Monthly, Tome 73 (1966), pp. 1-23 | Article | MR 201237 | Zbl 0139.05603

[18] H. P. Mckean; I. Singer Curvature and the eigenvalues of the Laplacian, J. Diff. Geom., Tome 1 (1967), pp. 43-69 | MR 217739 | Zbl 0198.44301

[19] H. P. Mckean; P. Van Moerbeke The spectrum of Hill's equation, Invent. Math., Tome 30 (1975), pp. 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., Tome 5 (1982), pp. 259-271 | MR 730247 | Zbl 0528.58020

[21] R. Schimming An explicit expression for the Korteweg-de Vries hierarchy, Z. Anal. Anwendungen, Tome 7 (1988), pp. 203-214 | MR 951118 | Zbl 0659.35089

[22] P. Van Moerbeke; O. Babelon Et Al. Integrable foundations of string theory, Lectures on integrable systems, CIMPA-Summer school at Sophia– Antipolis (1991), Singapore: World Scientific (1994), pp. 163-267 | Zbl 0850.81049