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 nonintegrabl e. With regard to the stationary system we present a rigorous proof of its nonintegrability by means of the Melnikov method. Concerning stationary loc alized states, we identify the parameter regions for which the origin of th e stationary map represents a hyperbolic equilibrium point. We show the exi stence of transversal intersections of the stable and unstable manifolds of the hyperbolic point. The associated homoclinic orbit is used to excite st anding bright two-soliton-like excitations on the double chain. We compute both, analytically as well as numerically, the dynamical energy exchange ra te between the two AL strings when on each of them a single AL soliton is l aunched. It is shown that the soliton interaction depends on the distance b etween the solitons and their mutual phase relation. There exist distinct e nergy exchange regimes ranging from suppressed to pronounced energy exchang e. In the latter case directed energy flow from one chain into the other ta kes 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 li mit with no mutual coupling survive as moving breathers under the action of the nonintegrable coupling, and thus experience no lattice pinning. The on ly pinned solution we obtained resulted from the homoclinic orbit derived f rom the stationary system. As an interesting dynamical feature we observe t hat a single soliton may split into two moving breathing states of differen t amplitudes as well as different velocities.