Reflection maps arising from the trigonometric Yang–Baxter map for the integrable discrete Manakov system

Determine and explicitly construct the set-theoretical solutions of the reflection equation (reflection maps) associated with the trigonometric Yang–Baxter map derived from breather interactions in the integrable discrete Manakov system, for example by formulating the model on the half-line with integrable boundary conditions and identifying the resulting parametric reflection maps of trigonometric type.

Background

The paper derives explicit transmission coefficients for fundamental and composite breathers in the integrable discrete Manakov (vector Ablowitz–Ladik) system and interprets breather interactions via a refactorization property, yielding a novel parametric Yang–Baxter map of trigonometric type. In integrable systems, set-theoretical Yang–Baxter maps on the full line often have boundary analogues: solutions of the set-theoretical reflection equation (reflection maps), naturally connected to integrable boundary conditions on the half-line.

The authors note that constructing reflection maps from their newly obtained trigonometric Yang–Baxter map is a natural follow-up and expect such maps to arise by extending half-line methods known for the Ablowitz–Ladik system. They explicitly defer this development, framing a concrete open direction to build and characterize the corresponding reflection maps of trigonometric type.