Soliton interaction for a nonlinear discrete double chain

We investigate solution behavior with an emphasis on the localization of a double chain built up from two coupled one-dimensional Ablowitz-Ladik (AL) lattices. Whereas each one-dimensional AL lattice is completely integrable, the AL-type coupling between them causes the system to become nonintegrable. With regard to the stationary system we present a rigorous proof of its nonintegrability by means of the Melnikov method. Concerning stationary localized states, we identify the parameter regions for which the origin of the stationary map represents a hyperbolic equilibrium point. We show the existence of transversal intersections of the stable and unstable manifolds of the hyperbolic point. The associated homoclinic orbit is used to excite standing bright two-soliton-like excitations on the double chain. We compute both, analytically as well as numerically, the dynamical energy exchange rate between the two AL strings when on each of them a single AL soliton is launched. It is shown that the soliton interaction depends on the distance between the solitons and their mutual phase relation. There exist distinct energy exchange regimes ranging from suppressed to pronounced energy exchange. In the latter case directed energy flow from one chain into the other takes place. Eventually almost all energy is stored in a single chain in the form of a breather solution showing a bias toward one-dimensional coherent excitation patterns. In general, the single solitons from the integrable limit with no mutual coupling survive as moving breathers under the action of the nonintegrable coupling, and thus experience no lattice pinning. The only pinned solution we obtained resulted from the homoclinic orbit derived from the stationary system. As an interesting dynamical feature we observe that a single soliton may split into two moving breathing states of different amplitudes as well as different velocities.