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.
Deutsche Forschungsgemeinschaft (DFG), Project Number VE 1211/1-1
- Mathematical Sciences
Published inJournal of Physics A: Mathematical and Theoretical
- VoR (Version of Record)
Rights holder© The Authors
Publisher statementThis is an Open Access Article. It is published by IOP Publishing under the Creative Commons Attribution 4.0 International Licence (CC BY). Full details of this licence are available at: https://creativecommons.org/licenses/by/4.0/