The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations

Haine, Luc

doi:10.5802/aif.2140

The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations
[Les matrices spectrales des solitons de Toda et la solution fondamentale de versions discrètes de l'équation de la chaleur]

Haine, Luc ¹

¹ Université catholique de Louvain, institut de mathématique pure et appliquée, chemin du Cyclotron 2, 1348 Louvain-la-Neuve (Belgique)

Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 1765-1788.

Résumé
Abstract

A l’aide de la théorie de Sato, on calcule la matrice spectrale de Stieltjes associée à une matrice de Jacobi doublement infinie, donnant lieu à une solution $g$ -soliton du réseau de Toda. On utilise ce résultat pour donner un développement explicite de la solution fondamentale de versions discrètes de l’équation de la chaleur, en termes d’une série des $q$ -déformations de Jackson des fonctions de Bessel. Pour les solitons dits de Askey-Wilson, ce développement se réduit à une somme finie.

The Stieltjes spectral matrix measure of the doubly infinite Jacobi matrix associated with a Toda $g$ -soliton is computed, using Sato theory. The result is used to give an explicit expansion of the fundamental solution of some discrete heat equations, in a series of Jackson’s $q$ -Bessel functions. For Askey-Wilson type solitons, this expansion reduces to a finite sum.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2140

Classification : 35Q51, 37K20, 39A13
Keywords: Heat kernel, Toda lattice hierarchy
Mot clés : noyau de la chaleur, réseau de Toda

Affiliations des auteurs :

Haine, Luc ¹

¹ Université catholique de Louvain, institut de mathématique pure et appliquée, chemin du Cyclotron 2, 1348 Louvain-la-Neuve (Belgique)

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Haine, Luc. The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations. Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 1765-1788. doi : 10.5802/aif.2140. http://www.numdam.org/articles/10.5802/aif.2140/

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