The Korteweg-de Vries limit for the global dynamics of the Toda lattice

TL;DR

This paper proves that solutions of the Toda lattice with H^1 initial data converge globally in time to solutions of the KdV equation under the KdV scaling, establishing a long-wave limit.

Contribution

It rigorously demonstrates the global convergence of Toda lattice solutions to the KdV equation using harmonic analysis and integrable structure tools.

Findings

Global in time convergence of Toda lattice to KdV solutions.

Construction and conservation of mass and energy for Toda lattice.

Derivation of long-wave KdV limits for the Toda lattice.

Abstract

It has been observed that the dynamics of the Toda lattice can be well described by solutions of the Korteweg-de Vries (KdV) equation in the continuum limit. We show that, under the KdV scaling and a suitable translation, the solution of the Toda lattice with H^1 initial data converges to that of the KdV equation globally in time. Our proof relies on tools from harmonic analysis and also on the construction and the conservation of mass and energy of the Toda lattice, the latter of which are derived from the completely integrable structure of the Toda lattice. As a consequence, we obtain long-wave KdV limits for the Toda lattice.