SOLITONS AND PERIODIC PROCESSES AS SOLUTIONS OF FUNCTIONAL EQUATIONS

There exists a class of soliton-type difference equations with shifted Aarguments which (in the limit when the shift goes to zero) can be Areduced to the PDEʹs also of soliton type. Such difference equations Ahave continuous and discrete solutions and all important features of Athe relevant limiting soliton-type pde. This statement will be Aillustrated by example of the discrete sine-Gordon equation and its Adispersion relation. As for direct methods, also for the formalism of Adispersion relations, the addition property of involved functions Aplays the essential role. By the term addition property usually the Afactorization of the product of two shifted functions is understood. AWe report here and prove the generalized addition property for the product of an arbitrary number of the Riemann theta functions. This Aformula seems to be useful for the analysis of quasi-periodic and Asoliton-type processes in N + I space-time since the addition property Afor exponential functions can be obtained by a standard limiting Aprocedure.