2018/02/28 by Jean Avan, Vincent Caudrelier, Nicolas Crampe · 1 citation
Mathematics · Physics and Astronomy · #hep-th #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1088/1751-8121/aac976
published as J. Phys. A51 (2018), 30LT01, Letter · 13 pages. Final version accepted for publication in J. Phys. A as a Letter
arxiv created 2018/06/01 · arxiv updated 2018/07/12
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 \bf 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 2× 2 for previously known Hamiltonians of type BCN and DN corresponding to constant and dynamical boundary matrices respectively.