Solitons and periodic processes as solutions of functional equations

There exists a class of soliton-type difference equations with shifted arguments which (in the limit when the shift goes to zero) can be reduced to the PDEʹs also of soliton type. Such difference equations have continuous and discrete solutions and all important features of the relevant limiting soliton-type pde. This statement will be illustrated by example of the discrete sine-Gordon equation and its dispersion relation. As for direct methods, also for the formalism of dispersion relations, the addition property of involved functions plays the essential role. By the term addition property usually the factorization of the product of two shifted functions is understood. We report here and prove the generalized addition property for the product of an arbitrary number of the Riemann theta functions. This formula seems to be useful for the analysis of quasi-periodic and soliton-type processes in N + 1 space-time since the addition property for exponential functions can be obtained by a standard limiting procedure.