Integrable Lattice Equations

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When mathematical models are simulated on a computer, the result is a system of partial difference equations, whose solutions evolve with discrete steps on a lattice in space and time. While many tools have been developed to study continuous equations, very few mathematical techniques are available for analysing non-linear lattice equations. We aim to develop techniques of solving the initial-value problem for a class of such equations. Our examples include integrable lattice equations that arise in the simulation of many physical problems ranging from the progression of shallow water waves to signals in an optical fibre.