Soliton interactions in the vector NLS equation

Collisions of solitons for two coupled and N-coupled NLS equation are investigated from various viewpoints. By suitably employing Manakov's well-known formulae for the polarization shift of interacting vector solitons, it is shown that the multisoliton interaction process is pairwise and the net result of the interaction is independent of the order in which such collisions occur. Further, this is shown to be related to the fact that the map determining the interaction of two solitons with nontrivial internal degrees of freedom (e.g. vector solitons) satisfies the Yang-Baxter relation. The associated matrix factorization problem is discussed in detail. Soliton interactions are also described in terms of linear fractional transformations, and the problem of existence of a solution for a basic three-collision gate, which has recently been introduced, is analysed.