Yang-Baxter and reflection maps from vector solitons with a boundary

Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schrodinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on the half-line. Using dressing transformations combined with a mirror image technique, factorization of soliton-soliton and soliton-boundary interactions is proved. We discover a new object, which we call reflection map, that satisfies a set-theoretical reflection equation which we also introduce. Two classes of solutions for the reflection map are constructed. Finally, basic aspects of the theory of the set-theoretical reflection equation are introduced.