Semi-discrete Lagrangian 2-forms and the Toda hierarchy

TL;DR

This paper develops a variational framework for semi-discrete integrable systems, exemplified by the Toda hierarchy, extending Lagrangian multiforms to capture hierarchies of commuting equations in a unified variational principle.

Contribution

It introduces a semi-discrete Lagrangian multiform theory for integrable systems, deriving PDEs from hierarchies like the Toda lattice without lattice shifts.

Findings

Derived PDEs in continuous variables from the Toda hierarchy

Extended Lagrangian multiforms to semi-discrete systems

Connected semi-discrete Toda and KdV equations

Abstract

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is a natural extension of the ideas known by the names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which have previously been established in both the fully discrete and fully continuous cases. The main feature of these ideas is to capture a hierarchy of commuting equations in a single variational principle. Our main example to illustrate the new semi-discrete theory of Lagrangian multiforms is the Toda lattice. This ODE describes the evolution in continuous time of a 1-dimensional lattice of particles with nearest-neighbour interaction. It is part of an integrable hierarchy of ODEs, each of which involves a derivative with respect to a continuous variable and a number of lattice shifts. We will use the Lagrangian multiform theory to derive PDEs in the continuous variables of the Toda hierarchy, which do not involve any lattice shifts. As a second example, we briefly discuss the semi-discrete potential KdV equation, which is related to the Volterra lattice.