In a finite system this is equivalent to the Floquet matrix FE being given by: Multibreather and vortex breather stability in Klein--Gordon lattices: Phonobreathers are multibreathers without oscillators at rest. Comparison between the eigenvalues E of the Newton operator obtained numerically points , predicted by the symmetric MST, theorem 1, dashed lines and by the non—symmetric MST, theorem 2, continuous line , which is the right one to be used. Calculating the derivative of Eq. If the on site potential is hard, Fig. 
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Introduction to Quantum Mechanics. All these examples are interesting in themselves, but also illustrate the use of the MST, and show that it provides gordonsya powerful method of easy applicability to determine multibreathers stability. If we derive Eq. The solution u is stable if: Theorem 2 Non—symmetric MST The generalization of theorem 1 for non—symmetric potentials consists of constructing the perturbation matrix Q in the following way: Due to the possible forms of the bifurcations, if all the multipliers are isolated except the double 1, the system is structurally stable, i.
It will be denoted the extended form while the previous one is the reduced goedonsys.
Comparison between the eigenvalues E of the Newton operator gprdonsys numerically pointspredicted by the symmetric MST, theorem 1, dashed lines and by the non—symmetric MST, theorem 2, continuous linewhich is the right one to be used. Dark breathers are multibreathers with only one or a few oscillators at rest.
The Bloch functions Eq. We will briefly summarize the band theory in the next section, while we introduce the notation and some basic concepts, on which our theory is based.
Certainly, it is of not so straight- forward applicability, although for certain cases there are only a few matrix elements Qn m to calculate. For the hard on—site potential, the curvature of the bands is negative, and when the band moved upwards, the intersection points will be kept, and the system is be stable.
If the on—site potential is symmetric or we consider only in—phase multibreathers, it involves only some simple modifica- tions of the coupling matrix. Juan F R Archilla. The two simplest ver- sions correspond to time—reversible multibreathers, with linear coupling and homogeneous on—site potentials.
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For soft potentials and values of the gorconsys until about one half of the rest frequency, a good approximation is the multibreather frequency, result which we demonstrate is exact for the Morse potential. We can reobtain the missing function that spans ker N.
If the on site potential is hard, Fig. Demonstration of the stability or instability of multibreathers at low coupling The continuous gordoonsys correspond to excited oscillators and the dashed ones to oscillators at rest.
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The choice of this kind of potential relies in the fact that the expressions of the orbits are easy to manage. The inversion of Eq. The last integral in Eq. If gordonsyz are more than one the stability is undefined within first order perturbation theory. Intraband discrete breathers in disordered nonlinear systems I: The generalized version of the MST is of not so straightfor- ward applicability, although its complexity depends on the characteristics of the system and multibreather considered.
This gkrdonsys can be very easily extended, to the case when there are oscillators out of phase, at least for an even potential V u. Therefore, a bifurcation to an unstable solution can only take place in three different forms: Theorem 4 Two—site breathers Consider time—reversible, two—site breathers in an homogeneous Klein-Gordon system with attractive linear coupling, then: Numerical methods based on the anti—integrability concept.
We define the following matrices: Intrinsic Localised Modes in Nonlinear Lattices. Using C from Eq.
Equation 15 P P becomes: They provide p identical bands which q are easily calculated solv- ing the equation above. New Journal of Physics, 4:
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