Bäcklund Transformations and Integrable Discrete Equations.

Groups of discrete transformations between the solution spaces of an equation or related equations have numerous applications and originate, in essence, in the works of Darboux and other geometers of the last part of the nineteenth century. They have been particularly fruitful in soliton theory where they are a machine for generating multi-soliton solutions. However, they also generate lattices of equations which themselves have integrable structure.

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