Universal Localized Relaxation Mode of Uniformly Driven Kinks in Damped Multistable Systems

Abstract

Linear arrays of damped multistable systems in a constant driving field F are considered in the continuum limit. The existence of a universal localized relaxation mode ("inertia mode") of the driven kinks is explicitly proven. This mode, of frequency OJ = -ir)/m 9 collapses in the undamped (77 -0) free (F-+0) chain into the Goldstone mode of the corresponding "free kink," and in a chain without inertia (m -*0) it relaxes instantaneously.PACS numbers: 63.20.Pw Linear arrays of uniformly 1 " 10 or periodically 11 " 15 driven and linearly damped multistable systems (mainly modeled by the sine-Gordon or </ > 4 potential) have attracted in recent years increased attention. As a result of the interplay and competition between nonlinearity, damping, and driving force, fascinating effects can occur. 1 " 15 My aim in the present paper is to describe a new interesting phenomenon of this kind, which consists of the occurrence of a universal, localized, smooth relaxation mode of the uniformly driven kinks (domain walls) in linearly damped and tightly coupled multistable lattice-dynamical and similar systems. As mentioned in the Abstract, the existence of this universal relaxation mode is intrinsically connected to the inertia of the damped multistable chain. On this ground, it will be referred to hereafter in this paper as the "inertia mode" of the driven kinks.Let us consider therefore a linear chain of particles subjected to a constant external force F and described by the classical HamiltonianHere m is the particle mass, <$> { (t) represents the displacement of the i th particle at the time t from the site x { of a reference lattice, 0 f =B(p./dt 9 k is the strength of the harmonic coupling between neighboring particles, and V(<t> { ) is a general "multistable" on-site potential. We assume that the chain deformation changes gradually from one lattice site to the next (tightly coupled particles), so that the continuum theory applies. We also assume that each particle is subjected to a damping force proportional to its velocity. In this way we obtain the following equation of motion for the displacement field <fi(x 9 1):m'<p+rii -kl 2 (/)" = F-dV/d<p.Here rj is the damping coefficient, I is the lattice constant, the prime denotes B/dx 9 and all the quantities are expressed in physical units.In the absence of the driving field and dissipation, Eq. (1) describes a multistable Hamiltonian system. The stable steady states 4> n° correspond to degenerate minima 16 of the potential V(<p) 9 i.e., they are degenerate classical vacuums 17 of the displacement field. These uniform configurations are equally favored and are separated by transition regions called (static) kinks, solitons, or domain walls. Because of the "relativistic" invar iance of the corresponding equation of motion (1), the kinks separating two adjacent domains can also uniformly move along the chain, with any velocity smaller than the velocity of sound c = (kl 2 /m) l/2 .If a constant external field F is switched on, then, for limited intensities, 4 F <i ? raax , th...