Resonant excitation and control of high order dispersive nonlinear waves

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Abstract
Autoresonant excitation of high order nonlinear waves with space–time varying parameters is investigated. A class of driven, two-component nonlinear waves described by the variational principle is studied in detail. The autoresonance in the system proceeds as an external eikonal pump wave excites a nonlinear daughter wave after crossing the linear resonance surface. Beyond the linear resonance, the pump and the daughter waves stay phase locked in an extended region of space–time despite the variation of the system’s parameters. The theory of the autoresonance is developed on the bases of the averaged variational principle and comprises a generalization of the formalism for scalar fields. The relation of the wave autoresonance problem to an associated two degrees of freedom problem in nonlinear dynamics is discussed. The conditions for the stable autoresonant solutions are (a) the adiabaticity of the driven system and (b) a sufficient nonlinearity. The theory is applied to the problem of resonant excitation and control of a Korteweg–de Vries (KdV) wave by means of launching an external pump wave with space–time varying frequency and wave vector. Numerical examples for temporal and spatial autoresonance in this system are presented.