De Toda à KdV

Dario Bambusi (1) , Thomas Kappeler (2) , Thierry Paul (3)

1 Dipartimento de Matematica [Milano]
2 Institut für Mathematik [Zürich]
3 DMA - Département de Mathématiques et Applications - ENS Paris

We consider the large number of particles limit of a periodic Toda lattice for a family of initial data close to the equilibrium state. We show that each of the two edges of the spectra of the corresponding Jacobi matrices is up to an error, determined by the spectra of two Hill operators, associated to this family. We then show that the spectra of the Jacobi matrices remain almost constant when the matrices evolve along the two limiting KdV equations. Finally we prove that the Toda actions, when appropriately renormalized, converge to the ones of KdV .

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Dario Bambusi, Thomas Kappeler, Thierry Paul. De Toda à KdV. 2009. ⟨hal-00358604⟩