Doubly discrete Lagrangian systems related to the Hirota and Sine-Gordon equation

TL;DR

This paper extends the action principle for KdV and MKdV type evolution equations to non-periodic, equivariant phase space variables, deriving doubly discrete versions of sine-Gordon and Hirota equations with associated symplectic structures.

Contribution

It introduces a new formulation of discrete integrable systems using equivariant phase space variables, linking reduction techniques to discrete Lagrangian systems.

Findings

Derived doubly discrete sine-Gordon and Hirota equations.

Established symplectic structures for these discrete systems.

Connected phase space reduction to integrable difference equations.

Abstract

We extend the action for evolution equations of KdV and MKdV type which was derived in [Capel/Nijhoff] to the case of not periodic, but only equivariant phase space variables, introduced in [Faddeev/Volkov]. The difference of these variables may be interpreted as reduced phase space variables via a Marsden-Weinstein reduction where the monodromies play the role of the momentum map. As an example we obtain the doubly discrete sine-Gordon equation and the Hirota equation and the corresponding symplectic structures.