Solitary Phenomena in Finite Dissipative Discrete Systems

Abstract

Idealized nonlinear systems which are assumed to be infinite, continuous and nondissipative are well known to exhibit a variety of solitary-wave phenomena [1]. For example, the space-time dependence of the relative phase ⌽(x,t) in an extended Josephson junction [2] is governed by a sine-Gordon-type equation. In the idealized case, the sine-Gordon equation is known to possess solitary solutions where a localized change of ⌽(x,t) by 2π propagates in an unchanged form with an arbitrary velocity v < c where c is the velocity of light in the junction. However, the existence and properties of such solitary solutions for a realistic system is an important unsolved problem [3,4]. Finite dissipation always exists in real systems, presumably balanced by driving forces. Also, finite boundaries may be important and the continuum approximation not always valid for discrete Josephson-junction arrays or for the pendulum systems used for analog simulations. The problem is not only of fundamental interest: a knowledge of the dynamic I–V characteristics and their dependencies on external magnetic fields and external electromagnetic waves is germane to the understanding of Josephson-junction devices [5] and SQUIDS [6]. An interesting case occurs when the driving (e.g., current) source is localized in space. Energy is injected into the system at some point and it has to be fed into the solitary mode at long distances.

Partially supported by the Commission for Basic Research of the Israeli Academy of Sciences