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

Based on recent results obtained by the authors on the inverse scatteringmethod of the vector nonlinear Schr\ödinger equation with integrable boundaryconditions, we discuss the factorization of the interactions of N-solitonsolutions on the half-line. Using dressing transformations combined with amirror image technique, factorization of soliton-soliton and soliton-boundaryinteractions 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, basicaspects of the theory of set-theoretical reflection equations are introduced.