By Robert S MacKay, Directeur de Recherche Cnrs Thierry Dauxois, Anna Litvak-Hinenzon
This e-book presents an creation to localised excitations in spatially discrete structures, from the experimental, numerical and mathematical issues of view. sometimes called discrete breathers, nonlinear lattice excitations and intrinsic localised modes; those are spatially localised time periodic motions in networks of dynamical devices. Examples of such networks are molecular crystals, biomolecules, and arrays of Josephson superconducting junctions. The publication additionally addresses the formation of discrete breathers and their strength position in strength move in such structures.
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Extra resources for Energy Localisation and Transfer (Advanced Series in Nonlinear Dynamics)
Sample text
Not only was the existence of regular motion on a two-dimensional torus found in both cases, but the tori intersections for the reduced and full problems were practically identical. 25 Thus we arrive at two conclusions: i) the breather-like object corresponds to a trajectory in the phase space of the full system which is for the times observed practically embedded on a two-dimensional torus manifold, thus being quasi-periodic in time; ii) the breather-like object can be reproduced within a reduced problem, where all particles but the central one and its two neighbors are fixed at their groundstate positions, thereby reducing the number of relevant degrees of freedom.
PM = 0. (51) So for N degrees of freedom we will search for zeros in 2N — 1 coupled equations of 2N — 1 variables. g. through energy conservation. If that will be not the case, we can not ensure that our procedure computes a PO. ,PM-2,PM-I,PM+I,PM+2, > •••>IrM-2>IrM-l'FM+l>FM+2' , (52) •••JPN) • " ' ^/v) ' (53) F = R{T) - R . (54) 36 S. Flach Given an initial guess RW expand dFn Fn(R) = Fn{RW) + Y, jnr\m(Rm - R%]) dR„ F{R) = F{Ri0)) +M(R- Mnm = dFn 8Rm ,HW R{0)) dRn(T). e. find an R such that F = 0: R = R{0) - A T 1 F ( £ ( 0 ) ) .
Fixing XQ = yo and varying K (see Ref. 16). The numerical scheme has been even used for a formal existence proof of breathers as homoclinic orbits. 16 5. Obtaining breathers up to machine precision: Part II So far we have searched for discrete breather periodic orbits as solutions of algebraic equations. The variables were either Fourier coefficients or simply the amplitudes at a given site. Also the methods of solving these equations have been quite special, using some particular properties of the system.