From Hamiltonian to zero curvature formulation for classical integrable boundary conditions

TL;DR

This paper bridges the Hamiltonian formalism and zero curvature approach for classical integrable systems with boundaries, providing a unified framework and explicit Lax pairs for the Toda chain with boundary conditions.

Contribution

It introduces a method to derive zero curvature equations from Hamiltonian formalism for boundary conditions, including a boundary Semenov-Tian-Shansky formula and explicit Lax pairs.

Findings

Unified Hamiltonian and zero curvature formulations for boundary conditions.

Derived Lax pairs for Toda chain with boundary matrices.

Demonstrated the approach on finite Toda chain with known Hamiltonians.

Abstract

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 $2\times 2$ for previously known Hamiltonians of type $BC_N$ and $D_N$ corresponding to constant and dynamical boundary matrices respectively.