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

Abstract

Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schrödinger 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 reection map, that satises a set-theoretical reection equation which we also introduce. Two classes of solutions for the reection map are constructed. Finally, basic aspects of the theory of the set-theoretical reection equation are introduced.

From Hamiltonian to zero curvature formulation for classical integrable boundary conditions

“…This formed the basis of a line of work initiated by Habibullin [6] which led eventually to the notion of nonlinear mirror image method for tackling the Inverse Scattering Method on the half-line, see e.g. [7,8,9,10,11,12]. Note also that relation (1) has been revived more recently within the framework of the so-called Unified Transform [13] where it is used to define the notion of linearizable boundary conditions.…”

Combinatorial solutions to the reflection equation

“…This set-theoretical equation together with the first examples of solutions first appeared in the work of Caudrelier and Zhang [4]. A more systematic study and a classification in a slightly different setting (i.e.…”

Relativistic collisions as Yang–Baxter maps

“…However, strictly speaking we have not proved the Liouville integrability of the transfer maps due to the lack of a suitable Poisson structure and this is an issue that we would like to further investigate in the future. In the same framework it will be interesting to study the integrability of the transfer dynamics as defined by Veselov in [24,25], as well as reflection maps [5] associated with relativistic collisions and fixed boundary initial value problems.…”