Semi-discrete Lagrangian 2-forms and the Toda hierarchy

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is an 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. Semi-discrete Lagrangian multiforms provide a new way to relate differential-difference equations and partial differential equations. We discuss this relation in the context of the Toda lattice, which is part of an integrable hierarchy of differential-difference equations, each of which involves a derivative with respect to a continuous variable and a number of lattice shifts. We use the theory of semi-discrete Lagrangian multiforms to derive PDEs in the continuous variables of the Toda hierarchy, which hold as a consequence of the differential-difference equations, but do not involve any lattice shifts. As a second example, we briefly discuss the semi-discrete potential Korteweg-de Vries equation, which is related to the Volterra lattice.