Wandering breathers and self-trapping in weakly coupled nonlinear chains: Classical counterpart of macroscopic tunneling quantum dynamics

Abstract

We present analytical and numerical studies of the phase-coherent dynamics of intrinsically localized excitations (breathers) in a system of two weakly coupled nonlinear oscillator chains. We show that there are two qualitatively different dynamical regimes of the coupled breathers, either immovable or slowly moving: the periodic transverse translation (wandering) of the low-amplitude breather between the chains and the one-chain-localization of the high-amplitude breather. These two modes of coupled nonlinear excitations, which involve a large number of anharmonic oscillators, can be mapped onto two solutions of a single pendulum equation, detached by a separatrix mode. We also show that these two regimes of coupled phase-coherent breathers are similar and are described by a similar pair of equations to the two regimes in the nonlinear tunneling dynamics of two weakly linked interacting (nonideal) Bose-Einstein condensates. On the basis of this profound analogy, we predict a tunneling mode of two weakly coupled Bose-Einstein condensates in which their relative phase oscillates around pi/2 mod pi. We also show that the magnitude of the static displacements of the coupled chains with nonlinear localized excitation, induced by the cubic term in the intrachain anharmonic potential, scales approximately as the total vibrational energy of the excitation, either a one- or two-chain one, and does not depend on the interchain coupling. This feature is also valid for a narrow stripe of several parallel-coupled nonlinear chains. We also study two-chain breathers which can be considered as bound states of discrete breathers, with different symmetry and center locations in the coupled chains, and bifurcation of the antiphase two-chain breather into the one-chain one. Bound states of two breathers with different commensurate frequencies are found in the two-chain system. Merging of two breathers with different frequencies into one breather in two coupled chains is observed. Wandering of the low-amplitude breather in a system of several, up to five, coupled nonlinear chains is studied, and the dependence of the wandering period on the number of chains is analytically estimated and compared with numerical results. The delocalizing transition of a one-dimensional (1D) breather in the 2D system of a large number of parallel-coupled nonlinear oscillator chains is described, in which the breather, initially excited in a given chain, abruptly spreads its vibrational energy in the whole 2D system upon decreasing the breather frequency or amplitude below the threshold one. The threshold breather frequency is above the cutoff phonon frequency in the 2D system, and the threshold breather amplitude scales as the square root of the interchain coupling constant. The delocalizing transition of the discrete vibrational breather in 2D and 3D systems of parallel-coupled nonlinear oscillator chains has an analogy with the delocalizing transition for Bose-Einstein condensates in 2D and 3D optical lattices.