Multiperiodic processes of soliton type with application to physics

Abstract

The aim of the present review paper is to show how periodic processes and localized excitations can be simultaneously incorporated into nonlinear equations of soliton type. Starting from perturbative approach it is shown that for all classes of soliton type equations i.e. partial derivative, ordinary derivative and functional equations there can exist exact solutions in form of solitons on the background of quasiperiodic process. These solutions are useful if one wants to discuss a signal with noise in nonlinear systems. From mathematical point of view the simplest way to handle such situations in soliton type equations is the dispersion equation formalism which proves also that a noise makes a dominant part of solution. The role of modular transformation and its relation with transformations on Riemann surfaces suitable for a description of quasi-periodic processes in soliton type equations is also discussed. As an application, a fully nonlinear approach to a long Josephson junction and its consequences are discussed.