Topological Coherent Modes for Nonlinear Schr\"odinger Equation

Nonlinear Schr\"odinger equation, complemented by a confining potential, possesses a discrete set of stationary solutions. These are called coherent modes, since the nonlinear Schr\"odinger equation describes coherent states. Such modes are also named topological because the solutions corresponding to different spectral levels have principally different spatial dependences. The theory of resonant excitation of these topological coherent modes is presented. The method of multiscale averaging is employed in deriving the evolution equations for resonant guiding centers. A rigorous qualitative analysis for these nonlinear differential equations is given. Temporal behaviour of fractional populations is illustrated by numerical solutions.