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 in NC] to the case of not periodic, but only equivariant phase space variables, introduced in FV2]. The di erence 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.

Poisson brackets of mappings obtained as (q,−p) reductions of lattice equations

“…where we have assumed that one can solve v n+p+q uniquely from equation (9). For example, the (3, −2)-reduction is depicted in Figure 1.…”

Poisson structures for difference equations

“…, n − 1, and x n = F(x) (8) is a Poisson map. By now, many authors have studied similar problems from the point of view of cluster algebras [9,14], r-matrix approach [19], using three leg forms for ( p, p) reductions of maps in the ABS list [2], by considering symplectic structures [16] and many other [3,4,7,8,15,20,23,25,26].…”

“…Lemma 1. Suppose that M : R n → R n , x → x is a map of the form (7) and (8) where F(x) = φ(y 1 , y 2 , . .…”

“…Then M preserves the constant Poisson structure {x i , x j } = T i,j if and only if the map x) and v i = e xi preserves the quadratic Poisson structure {v i , v j } = T i,j v i v j . The map L is of the form (7) and (8) with G as in (10).…”

Free Massive Fermions�Inside the Quantum Discrete sine-Gordon Model

“…Integrable lattice systems of the above type had been thoroughly discussed e.g. in [BRST,NCP,S,V,FV2,CN,EK]) In [EK] it was shown that it is possible to derive the above equation as an equation of motion from an explicitly given action. Moreover -using covariant phase space techniques -it is possible to derive the symplectic structure belonging to the above model via variation of this action.…”