Semi-discrete Lagrangian 2-forms and the Toda hierarchy

Duncan Sleigh; Mats Vermeeren

doi:10.1088/1751-8121/aca451

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arXiv · 2204.13063

Semi-discrete Lagrangian 2-forms and the Toda hierarchy

Duncan Sleigh·Mats Vermeeren

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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.

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Duncan Sleigh, Mats Vermeeren. 2022-04-27. Semi-discrete Lagrangian 2-forms and the Toda hierarchy. https://doi.org/10.1088/1751-8121%2Faca451

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