Perturbation theories of a discrete, integrable nonlinear schrödinger equation

We rederive the discrete inverse-scattering transform (IST) perturbation results for the time evolution of the parameters of a discrete nonlinear Schrödinger soliton from certain mathematical identities that can be viewed as conserved quantities in the discrete, integrable nonlinear Schrödinger equation in (1+1) dimension. This method significantly simplifies the derivation of the IST perturbation results. We also present a specific example for which the adiabatic IST perturbation results and the collective coordinate method results exactly coincide. This is achieved by establishing a correct Lagrangian formalism for soliton parameters via transforming dynamical variables that obey a deformed Poisson structure to ones that possess a canonical Poisson structure.