New Lax pairs of the Toda lattice and the nonlinearization under a higher-order Bargmann constraint

By starting from a new discrete spectral problem, the Toda lattice is derived through the discrete zero curvature equation. Applying the discrete variational identity to the spectral problem will also reach to the bi-Hamiltonian structure of the Toda lattice. Under a higher-order Bargmann symmetry constraint, the new Lax pairs and the adjoint Lax pairs are nonlinearized into integrable symplectic maps and finite-dimensional Liouville integrable Hamiltonian systems. Finally, a Backlund transformation of the Toda lattice is obtained. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3693975]