One-dimensional nonlinear Schrödinger equation: A nonlinear dynamical approach

We have studied a time-independent nonlinear Schr\"odinger equation of the tight-binding form on a one-dimensional lattice. The real and complex wave functions as solutions to the equation are considered separately for different physical problems. For each case, an area-preserving map for the discrete nonlinear Schr\"odinger equation is introduced and analyzed. The bounded solutions can be organized in hierarchies composed of periodic, quasiperiodic, as well as chaotic orbits on the phase plane of the nonlinear map. A ``stability-zone'' diagram, where the bounded orbits exist, is displayed in the parameter space, serving as ``phase diagram'' of the nonlinear Schr\"odinger equation under appropriate boundary conditions. Studies of the stability zone yield useful information for the physical problems considered. The periodic orbits and their stabilities can be obtained by a convergent perturbation method. Finally, we remark on several physical problems where these results might be applicable. In particular, we discuss the stabilities of the large polaron solution in the Holstein model.