By Oleg M. Braun, Yuri S. Kivshar
This booklet offers an summary of the fundamental strategies, tools and purposes of nonlinear low-dimensional good nation physics according to the Frenkel-Kontorova version and its generalizations. It permits a nonspecialist to procure foundations within the interdisciplinary recommendations and techniques of either stable nation physics and nonlinear technological know-how. It covers many very important subject matters comparable to the nonlinear dynamics of discrete systems,the dynamics of solitons and their interplay, commensurate and incommensurate structures, statistical mechanics of nonlinear structures, and nonequilibrium dynamics of interacting many-body platforms. additionally taken care of are the principlal equations on nonlinear solid-state physics, besides the homes and techniques for his or her learn.
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Additional info for The Frenkel-Kontorova model
Several results of those studies are presented in Figs. 4 below; these results show a variation of the kink’s velocity in the cases of moving and trapped states. , Ref. 33) deﬁned with the normalization ωmax dω ρ(ω) = 1. ωmin The function ρ(ω) tends to inﬁnity at the edges of the phonon spectrum band, ωmin and ωmax , respectively. Anharmonic lattice vibrations lead to the generation of higher-order harmonics in the phonon spectrum. e. 34) where ωk is replaced by ωtrav or ωtrap depending on the type of the kink’s motion in the chain, and the integer numbers k1 and k2 stand for the order of the resonance.
We discuss both moving and trapped kinks as well as the dynamics of kinks in the presence of disorder. Additionally, we consider the modiﬁcation of kink properties in the generalized FK models that include an on-site potential of a general form and anharmonic interactions between the particles in the chain. 1 The Peierls-Nabarro Potential Kinks are fundamental topological nonlinear modes of the FK model, and they are responsible for many important physical characteristics of a chain of coupled particles on a substrate.
1. 1) reduce to the SG equation. The latter condition can be rewritten in the form d = c0 /ω0 a, so that the parameter d characterizes the lattice discreteness , and it deﬁnes also the width of the kink. If the right end of the mechanical line makes only small excursions around the down position at θ = 0, then only small-amplitude waves are excited in the chain. If the initial disturbance is a complete rotation of the ﬁrst pendulum, this rotation propagates collectively along the chain in the form of a kink.