From Hamiltonian to zero curvature formulation for classical integrable boundary conditions Auteur(s): Avan Jean, Caudrelier Vincent, Crampé N. (Article) Publié: -J.phys.a, vol. 51 p.30LT01 (2018) Texte intégral en Openaccess : Ref HAL: hal-01730052_v1 Ref Arxiv: 1802.07593 Ref INSPIRE: 1656694 DOI: 10.1088/1751-8121/aac976 WoS: 000435723000001 Ref. & Cit.: NASA ADS Exporter : BibTex | endNote 5 Citations Résumé: We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1 + 1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin’s formula for the double-row transfer matrix is used to derive Hamilton’s equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BC N and D N corresponding to constant and dynamical boundary matrices respectively. |