By Robert S MacKay, Directeur de Recherche Cnrs Thierry Dauxois, Anna Litvak-Hinenzon

This booklet offers an creation to localised excitations in spatially discrete platforms, 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 ebook additionally addresses the formation of discrete breathers and their capability position in power move in such platforms.

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**Extra resources for Energy Localisation and Transfer (Advanced Series in Nonlinear Dynamics)**

**Sample text**

6 For time-reversal breathers it is possible to find an origin in time when xi(t) = xi(—t) , pi(t) = —pi(—t), which saves 50% of computational time. For time-reversal parity-invariant breathers xi(t + T/2) = —xi(t) , pi(t + T/2) = —pi(t) we may save 75% of computational time. Higher dimensional lattices may allow for further symmetries. Computing lattice permutational invariant breathers may substantially lower the computational effort by finding the irreducible breather section. At the same time even in the presence of additional symmetries breather Computational Studies of Discrete Breathers 39 solutions may be found which lack these symmetries.

23) k = — oo Thus we arrive at a set of coupled nonlinear algebraic equations for the Fourier coefficients A^i of the breather solution we search for: k2Q2bAkl = v2Akl + w2{2Akl - A M _ ! - Akil+1) + F{kf . (24) 26 S. Flach If a breather solution exists, then in its spatial tails all amplitudes are small. Thus we can assume that the nonlinear terms in (24) are negligible in the tails of a breather. We are then left with the linearized equations k2n2bAkl = v2Akl + w2(2Akl - Ak>t-i - AM+1) . (25) These equations are not much different from the linearization of the equations of motion as discussed in 1 which lead to the dispersion relation uiq for small amplitude plane waves.

34) ; being a homogeneous function of the coordinates. The equations of motion take the form Xl+V2Xi = -V2mxfm-1 -W2m(xi-Xi-1)2m'1 +W2m(xi+l~Xi)2m-1 . (35) These systems allow for time space separation for a sub-manifold of all possible trajectories: xi(t) = AtG(t) . (36) Inserting (36) into (35) we obtain G + v2G (37) Q2m- 1 -K = ^ [-V2mA]m-1 - w2m(Al - A , . ) 2 - " 1 + w2m(Al+1 - At)2™"1} . (38) Here K > 0 is a separation parameter, which can be chosen freely. The master function G obeys a trivial differential equation for an anharmonic oscillator G = - v 2 G - KG2m~l .