General Theory of Solitons

摘要

A soliton is a concept which describes various physical phenomena ranging from solitary waves on a water surface to ultra-short optical pulses in an optical fiber. The main feature of solitons is that they can propagate long distances without visible changes. From a mathematical point of view, a soliton is a localized solution of a partial differential equation describing the evolution of a nonlinear system with an infinite number of degrees of freedom. Solitons are usually attributed to integrable systems. In this instance, solitons remain unchanged during interactions, apart from a phase shift. They can be viewed as 'modes' of the system, and, along with radiation modes, they can be used to solve nitial-value problems using a nonlinear superposition of the modes. However, in the recent years, the notion of solitons has been extended to various systems which are not necessarily integrable. Following this new trend, we extend the notion of solitons and include a wider range of systems in our treatment. These include dissipative systems, Hamiltonian systems and a particular case of them, viz. integrable systems.