Soliton collisions in the discrete nonlinear Schrödinger equation

We report analytical and numerical results for on-site and intersite collisions between solitons in the discrete nonlinear Schrödinger model. A semianalytical variational approximation correctly predicts gross features of the collision, viz., merger or bounce. We systematically examine the dependence of the collision outcome on initial velocity and amplitude of the solitons, as well as on the phase shift between them, and location of the collision point relative to the lattice; in some cases, the dependences are very intricate. In particular, merger of the solitons into a single one, and bounce after multiple collisions are found. Situations with a complicated system of alternating transmission and merger windows are identified too. The merger is often followed by symmetry breaking (SB), when the single soliton moves to the left or to the right, which implies momentum nonconservation. Two different types of the SB are identified, deterministic and spontaneous. The former one is accounted for by the location of the collision point relative to the lattice, and/or the phase shift between the solitons; the momentum generated during the collision due to the phase shift is calculated in an analytical approximation, its dependence on the solitons' velocities comparing well with numerical results. The spontaneous SB is explained by the modulational instability of a quasiflat plateau temporarily formed in the course of the collision.