Features of Discrete Integrability
23
Fig. 1 2D lattice
The conditions we consider are of the form:
Q = p 1 · u 0,0 u 1,0 u 0,1 u 1,1 + p 2 · u 0,0 u 1,0 u 0,1 + p 3 · u 0,0 u 1,0 u 1,1 + p 4 · u 1,0 u 0,1 u 1,1
+p 5 · u 0,0 u 0,1 u 1,1 + p 6 · u 0,0 u 0,1 + p 7 · u 1,0 u 0,1 + p 8 · u 0,1 u 1,1
(1)
+p 9 · u 0,0 u 1,0 + p 10 · u 0,0 u 1,1 + p 11 · u 1,0 u 1,1 + p 12 · u 0,1
+p 13 · u 0,0 + p 14 · u 1,0 + p 15 · u 1,1 + p 16 = 0
Notice that the multilinear nature of the relation implies that, for all cells, any of the
four corner values can be rationally expressed in terms of the three others.
3 A Few Standard Features of Integrable Quad Equations
3.1 Discrete Lax Pairs
Lax pairs are a characteristic feature of integrable differential equations, and
they have been extremely important in the development of the subject. For 1 +
1 dimensional partial differential equations, they become rather zero curvature
equations, and these have a straightforward discrete form. A discrete Lax pair is
a pair of matrices L(u), M(u) such that
ˆ
L · M ˜
M · L,
(2)
where ˆ
L = L( ˆ
u) (shift along the horizontal direction), ˜
M = M( ˜
u) (shift along
the vertical direction), and means proportionality, is equivalent to the defining