Inverse spectral solution, modulation theory and linearized stability analysis of n-phase, quasi-periodic solutions of the nonlinear Schroedinger equation

The nonlinear Schroedinger (NLS) equation is studied under periodic boundary conditions. First, focus is on the exact theory of the periodic and quasi-periodic NLS solutions that can be expressed in terms of Riemann theta functions. Next, certain perturbations of these solutions are studied, namely; the slow modulation in space and time, perhaps in the presence of external perturbations, and perturbation of initial conditions (linearized instabilities). Floquet theory and scattering theory are used to investigate the periodic NLS spectrum of the AKNS linear system, and the Date technique is used to develop the inverse spectral transformation. Exact representations are found with a minimal set of parameters, for N-phase NLS wavetrains. Precise formulas are given for the physical wavenumbers and frequencies of these solutions. Multiphase averaging, a method due to Whitham and Flaschka, Forest, and McLaughlin, is then used to derive the modulation equations for N-phase NLS wavetrains. The modulation equations are then diagonalized, with explicit formulas for Riemann invariants and their characteristics speeds. The work of Ercolani, Forest, and McLaughlin on the sine-Gordon equations is followed to approach the linearized problems for all N-phase NS wavetrains, from both geometric and analytic viewpoints. Finally, the stable and unstable modes are characterized and the growth rates are computed explicitly