Initial value problems for the integrable discrete equations on quadgraphs
are investigated. We give a geometric criterion of when such a
problem is well-posed. In the basic example of the discrete KdV equation
an effective integration scheme based on the matrix factorization
problem is proposed and the interaction of the solutions with the localized
defects in the regular square lattice are discussed in details.
The examples of kinks and solitons on various quad-graphs, including
quasiperiodic tilings, are presented.
This pre-print has been submitted, and accepted, to the journal Acta Applicandae Mathematicae. The deinitive version: ADLER, V.E. and VESELOV, A.P., 2004. Cauchy problem for integrable discrete equations on quad-graphs. Acta Applicandae Mathematicae, 84(2),pp.237-262 is available at www.springerlink.com.