Autoresonance of coupled nonlinear waves

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Abstract
Adiabatic passage of weakly coupled nonlinear waves with space-time varying parameters through resonance is investigated. Slow evolution equations describing this wave interaction problem are obtained via Whitham’s averaged variational principle. Autoresonant solutions of these equations are found and, locally, comprise adiabatically varying quasiuniform wave train solutions of the decoupled problem. At the same time, the waves are globally phase locked in an extended region of space-time despite the variation of the system’s parameters. Conditions for entering and sustaining this multidimensional autoresonance are the internal resonant excitation of one of the coupled waves and sufficient adiabaticity and nonlinearity of the problem. These conditions have their origin in a similar adiabatic resonance problem in nonlinear dynamics. The theory is illustrated by an example of the autoresonance in a system of coupled sine-Gordon equations.