Initial value problems for lattice equations

We describe how to pose straight band initial value problems for lattice equations defined on arbitrary stencils. In finitely many directions, we arrive at discrete Goursat problems and in the remaining directions we find Cauchy problems. Next, we consider (s1, s2)-periodic initial value problems. In the Goursat directions, the periodic solutions are generated by correspondences. In the Cauchy directions, assuming the equation to be multi-linear, the periodic solution can be obtained uniquely by iteration of a particularly simple mapping, whose dimension is a piecewise linear function of s1, s2. © 2009 IOP Publishing Ltd.