Excitation of multiphase waves of the nonlinear Schrödinger equation by capture into resonances

A method for adiabatic excitation and control of multiphase ($N$-band) waves of the periodic nonlinear Schrödinger (NLS) equation is developed. The approach is based on capturing the system into successive resonances with external, small amplitude plane waves having slowly varying frequencies. The excitation proceeds from zero and develops in stages, as an $(N+1)$-band $(N=0,1,2,\dots )$, growing amplitude wave is formed in the $(N+1)\text{th}$ stage from an $N$-band solution excited in the preceding stage. The method is illustrated in simulations, where the excited multiphase waves are analyzed via the spectral approach of the inverse scattering transform method. The theory of excitation of 0- and 1-band NLS solutions by capture into resonances is developed on the basis of a weakly nonlinear version of Whitham’s averaged variational principle. The phenomenon of thresholds on the driving amplitudes for capture into successive resonances and the stability of driven, phase-locked solutions in these cases are discussed.