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

Abstract

We extend the action for evolution equations of KdV and mKdV type which was derived by Capel and Nijhoff to the case of not periodic, but only equivariant phase space variables, introduced by Faddeev and 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.