Direct linearization of nonlinear difference-difference equations

Lax pair for the Adler (lattice Krichever–Novikov) system

“…Furthermore, the dependence on these parameters can be exploited to obtain continuum limits of the lattice equations to semi-continuous and fully continuous equations that are compatible with the original fully discrete equation, cf. [17,18].…”

Searching for integrable lattice maps using factorization

“…x 00 x 10 p 1 + x 00 x 01 p 5 (p 1 p 3 + p 2 ) + (x 00 x 11 + x 10 x 01 )p 2 + x 10 x 11 p 6 + x 01 x 11 p 3 (p 5 p 6 − p 2 ) = 0 (16) This has quadratic growth q 5 is all directions. This result illustrated the role of parametrization in factorization: there are several ways to reparametrize p 3 , p 5 so that combinations p 5 (p 1 p 3 + p 2 ) and p 3 (p 5 p 6 − p 2 ) look simpler, but this could easily lead to one quadratic rather than two linear factors at the corner.…”

Relativistic collisions as Yang–Baxter maps

“…This is a discrete version of the potential modified KdV equation originally presented in [12] and corresponds under a gauge tranformation to H 3 equation in [1] for δ = 0. The latter equation follows directly from the YB map (10) by setting x = w k+1,l w k,l , y = w k+1,l+1 w k+1,l , u = w k+1,l+1 w k,l+1 and v = w k,l+1 w k,l .…”