Symmetries of ${\mathbb{Z}}_{N}$ graded discrete integrable systems

Abstract

We recently introduced a class of Z N graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). We discuss differential-difference equations which then we interpret as symmetries of the discrete systems. In particular, we present nonlocal symmetries which are associated with the 2D Toda lattice.

A discrete Darboux-Lax scheme for integrable difference equations

“…Generalized symmetries are (integrable) differential-difference equations involving shifts in one lattice direction and are compatible with the P∆E. Their Lax pair is semi-discrete and its discrete part coincides with the one of the two equations of the fully discrete Lax pair (3), see for instance [12]. This relation allows us to extend the Darboux and Bäcklund transformations for the P∆E to corresponding ones for the differential-difference equations and employ the Bäcklund transformation and its superposition principle in the construction of solutions for the symmetries.…”

“…• The Bianchi commuting diagram, aka superposition principle, for the auto-Bäcklund transformation (12) follows from the permutation of four Darboux matrices according to the diagram in Figure 2. More precisely, starting with a solution f of (6) we can construct two new solutions f and f using the Bäcklund transformations B(f , f , g 1 ; ε 1 ) = 0 and B(f , f , g 2 ; ε 2 ) = 0, respectively.…”

The Lattice Sine-Gordon Equation as a Superposition Formula for an NLS-Type System

“…In the preceding calculations, we derived the symmetries (1.8) and (1.11) in a way that highlights the connection with the lattice Liouville equation and its integrals. One could employ different approaches, e.g., calculate them directly or extract from previously published articles [7,8]. Also, it is worth mentioning that the integrals (1.6) are related to the conserved densities of equations (1.8) and (1.11), i.e., ∂ x ln w l,m , ∂ t ln w l,m ∈ Im(S l − 1), which is in parallel with the continuous case.…”

On consistent systems of difference equations

“…It is not difficult now to see that the denominator of the second symmetry shifted forward in the second direction is the defining function of (11). Thus we could have derived system (8) in two different ways without any reference to the quad equation (4). A very interesting observation is that the lowest order symmetries of system (8) are given by the first flow in ( 14) and the second one in (12), i.e.…”

“…Thus we could have derived system (8) in two different ways without any reference to the quad equation (4). A very interesting observation is that the lowest order symmetries of system (8) are given by the first flow in ( 14) and the second one in (12), i.e. by the symmetries of (11) and (13) which do not degenerate on the solutions of the overdetermined system (8).…”