Features of Discrete Integrability
29
0
1
2
3
4
Fig. 5 Evolution on the square lattice
4 A Case Study
We will concentrate on one specific quad equation, taken from [44]. It does not
belong to the ABS list, as it is not symmetric. It does not verify the consistency
around the cube condition.
E =
u 1,0 + 1
u 0,0 − 1
−
u 1,1 − 1
u 0,1 + 1
= 0
( 8 )
The variable u n,m at site (n, m) is denoted u 0,0 with u k,l standing for u n+k,m+l as
shown in Sect. 2.
4.1 Symmetries
Equation (E) possesses continuous symmetries. These symmetries split into vertical
and horizontal symmetries (H and V ), as is usual for this type of quad equations.
Three points symmetries:
H 3 : ∂ t 1 u 0,0 = (u
2
0,0 − 1)(u 1,0 − u −1,0 )
(9)
V 3 : ∂ τ 1 u 0,0 = (u
2
0,0 − 1)
1
u 0,1 + u 0,0
−
1
u 0,0 + u 0,−1
(10)
29
0
1
2
3
4
Fig. 5 Evolution on the square lattice
4 A Case Study
We will concentrate on one specific quad equation, taken from [44]. It does not
belong to the ABS list, as it is not symmetric. It does not verify the consistency
around the cube condition.
E =
u 1,0 + 1
u 0,0 − 1
−
u 1,1 − 1
u 0,1 + 1
= 0
( 8 )
The variable u n,m at site (n, m) is denoted u 0,0 with u k,l standing for u n+k,m+l as
shown in Sect. 2.
4.1 Symmetries
Equation (E) possesses continuous symmetries. These symmetries split into vertical
and horizontal symmetries (H and V ), as is usual for this type of quad equations.
Three points symmetries:
H 3 : ∂ t 1 u 0,0 = (u
2
0,0 − 1)(u 1,0 − u −1,0 )
(9)
V 3 : ∂ τ 1 u 0,0 = (u
2
0,0 − 1)
1
u 0,1 + u 0,0
−
1
u 0,0 + u 0,−1
(10)