Discreteness effects on a sine-Gordon breather

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Abstract
We employ collective-variable theory to describe the dynamics of a breather excitation in its center-of-mass frame in continuous and discrete systems of one spatial dimension. The exact equations of motion for the collective variable and coupled phonon field are derived for any system which supports breatherlike excitations that have even spatial parity where the collective variable represents half the distance between the breather subkinks. We then specialize the theory to the sine-Gordon (SG) case. For the continuum SG system we derive the exact effective potential in terms of the collective variable and discuss the relativistic effects on the breather subkinks which are quite different than the relativistic effects on isolated boosted SG solitons. The effect of the subkink interaction is such that the subkinks do not Lorentz contract as they speed up when approaching the collision region (breather center). For the discrete SG breather system we derive expressions for the total energy, the effective potential which governs the motion of the breather subkinks, and a stability criterion for the breather’s position relative to the lattice sites. We compare with simulation and find good agreement in most cases. Using molecular-dynamics and Fourier-transform techniques, we show that discrete SG breathers spontaneously make remarkably sharp transitions from a short lifetime to a long lifetime. The breather lifetimes on each side of the transition differ by more than four orders of magnitude. We relate the existence of the transition to the structure of the system’s frequency spectrum. We also study the frequency spectrum of a ‘‘static’’ breather whose subkinks are trapped by the Peierls potential, yet are close enough to interact. For the latter case we illustrate the connection between collective-variable theory and standard perturbation analysis.