On the integrable symplectic map and the $N$-soliton solution of the Toda lattice

Three different types of polynomial expansions of the spectral function are used to introduce the Hamiltonian system and the symplectic map associated to the Toda lattice. The integrability of the symplectic map and the Darboux coordinates are discussed. Using the Darboux coordinates, the symplectic map is linearized, and the inversion problem is derived. Finally, inversion is used to provide the $N$-soliton solution for the Toda lattice.