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, Tome 55 (2005) no. 6, pp. 2117-2127.

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.

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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     author = {Iliev, Plamen},
     title = {On the heat kernel and the Korteweg--de Vries hierarchy},
     journal = {Annales de l'Institut Fourier},
     pages = {2117--2127},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     doi = {10.5802/aif.2154},
     zbl = {1078.35103},
     mrnumber = {2187948},
     language = {en},
     url = {www.numdam.org/item/AIF_2005__55_6_2117_0/}
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Iliev, Plamen. On the heat kernel and the Korteweg--de Vries hierarchy. Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2117-2127. doi : 10.5802/aif.2154. http://www.numdam.org/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., Volume 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., Volume 30 (1977), pp. 95-148 | Article | MR 649926 | Zbl 0338.35024

[3] G. Andrews; R. Askey; R. Roy Special Functions, Encyclopedia of Mathematics and Its Applications, Volume 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., Volume 219 (2000), pp. 45-64 | Article | MR 1791911 | Zbl 0984.37084

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

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

[7] E. Date; M. Jimbo; M. Kashiwara; T. Miwa; M. Jimbo and T. Miwa Transformation groups for soliton equations (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, Advanced Series in Mathematical Physics, Volume 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., Volume 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), Discourses Math. Appl., Volume 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), Volume 54, Part 3, Amer. Math. Soc., Providence, RI, 1993 | MR 1216627 | Zbl 0791.58092

[12] F. A. Grünbaum; P. Iliev Heat kernel expansions on the integers, Math. Phys. Anal. Geom., Volume 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 (Translated from the 1992 Japanese original and edited by Atsushi Nagai, Jon Nimmo and Claire Gilson (with a foreword by Jarmo Hietarinta and Nimmo)) Volume 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, Volume 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., Volume 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., Volume 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., Volume 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, Volume 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) (1994), pp. 163-267 | Zbl 0850.81049