De Toda \`a KdV

TL;DR

This paper investigates the connection between the periodic Toda lattice and the KdV equation, showing spectral stability and convergence of actions in the large particle limit near equilibrium.

Contribution

It establishes the spectral correspondence and action convergence between Toda lattice and KdV in the large particle limit for near-equilibrium initial data.

Findings

Spectral edges of Jacobi matrices relate to Hill operators.

Spectra of Jacobi matrices are nearly constant along KdV evolution.

Toda actions converge to KdV actions after renormalization.

Abstract

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 .