Discrete vector solitons: Composite solitons, Yang-Baxter maps and computation

Collisions of solitons for an integrable discretization of the coupled nonlinear Schrödinger equation are investigated. By a generalization of Manakov's well-known formulas for the polarization shift of interacting vector solitons, it is shown that the multisoliton interaction process is equivalent to a sequence pairwise interactions and, moreover, the net result of the interaction is independent of the order in which such collisions occur. Further, the order-invariance is shown to be related to the fact that the map that determines the interaction of two such solitons satisfies the Yang-Baxter relation. The associated matrix factorization problem is discussed in detail and the notion of fundamental and composite solitons is elucidated. Moreover, it is shown that, in analogy with the continuous case, collisions of fundamental solitons can be described by explicit fractional linear transformations of a complex-valued scalar polarization state. Because the parameters controlling the energy switching between the two components exhibit nontrivial information transformation, they can, in principle, be used to implement logic operations.