Soliton Particles

Abstract

It is well known that the majority of solvable field and lattice nonlinear evolution equations have the so called N-soliton solutions u _N , which asymptotically, i.e. for t → ±∞, decompose into a sum of single solitons s _>i , that is extended objects of permanent shape, moving at a constant speed. Their dynamic behaviour has been studied extensively and solitons have been found to be stable against mutual collisions and to behave like particles. These useful properties make them attractive for a description of not only a wide class of physical phenomena [41],[114],[115],[173],[193], but also biological [57]and others [192],[174]. In this chapter we discuss the time independent decomposition of N-soliton solutions into a sum of extended objects being closely related to the eigenfunctions of the discrete part of the spectrum of a recursion hereditary operator. These objects will be called soliton particles (interacting solitons) [19],[21],[87]. Moreover, we present the analytic form of soliton particles, their equations of motion with the multi-Hamiltonian structure and other algebraic properties. Finally we present multisoliton perturbation theory constructed in a purely algebraic way.