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 : 10.5802/aif.2154
Classification : 35Q53, 35K05, 37K10
Keywords: Heat kernel expansions, KdV hierarchy, tau functions
Mot clés : Noyau de la chaleur, hiérarchie de KdV, fonctions tau
Iliev, Plamen 1

1 Georgia Institute of Technology, school of mathematics, Atlanta GA 30332-0160 (USA)
@article{AIF_2005__55_6_2117_0,
     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{\textquoteright}institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     doi = {10.5802/aif.2154},
     mrnumber = {2187948},
     zbl = {1078.35103},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2154/}
}
TY  - JOUR
AU  - Iliev, Plamen
TI  - On the heat kernel and the Korteweg--de Vries hierarchy
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 2117
EP  - 2127
VL  - 55
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2154/
DO  - 10.5802/aif.2154
LA  - en
ID  - AIF_2005__55_6_2117_0
ER  - 
%0 Journal Article
%A Iliev, Plamen
%T On the heat kernel and the Korteweg--de Vries hierarchy
%J Annales de l'Institut Fourier
%D 2005
%P 2117-2127
%V 55
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2154/
%R 10.5802/aif.2154
%G en
%F AIF_2005__55_6_2117_0
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/articles/10.5802/aif.2154/

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

[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

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

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

[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 | Zbl

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

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

[13] J. Hadamard Lectures on Cauchy's Problem, New Haven, Yale Univ. Press (1923) | JFM

[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

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

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

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

[19] H. P. McKean; P. van Moerbeke The spectrum of Hill's equation, Invent. Math., Volume 30 (1975), pp. 217-274 | DOI | MR | Zbl

[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 | Zbl

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

[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

Cité par Sources :