Extend the dressing Zakharov–Shabat method to Zh-reduced Lax operators

Develop an extension of the dressing Zakharov–Shabat method to the classes of Lax operators with Zh reductions that arise in real Hamiltonian forms of affine Toda field theories, accommodating dressing factors with 2h pole singularities.

Background

The dressing Zakharov–Shabat method is a central technique for constructing solutions of integrable systems from Lax pairs. For the RHF of ATFT considered, the Lax operators involve Zh reductions.

The authors note that extending the dressing method to these Zh-reduced operators is nontrivial because such reductions require dressing factors with 2h pole singularities, and explicitly identify this as an open problem.