Autoresonant solutions of the nonlinear Schrödinger equation

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Abstract
Resonant driving of the nonlinear Schrödinger (NLS) equation by small-amplitude oscillations or waves with adiabatically varying frequencies and/or wave vectors is proposed as a method of excitation and control of wave-type solutions of the system. The idea is based on the autoresonance phenomenon, i.e., a continuous nonlinear phase locking between the solutions of the NLS equation and the driving oscillations, despite the space-time variation of the parameters of the driver. We illustrate this phenomenon in the examples of excitation of plane and standing waves in the driven NLS system, where one varies the driver parameters in time or space. The relation of autoresonance in these applications to the corresponding problems in nonlinear dynamics is outlined. One of these dynamical problems comprises a different type of multifrequency autoresonance in a Hamiltonian system with two degrees of freedom. The averaged variational principle is used in studying the problem of autoresonant excitation and stabilization of more general cnoidal solutions of the NLS equation.