Autoresonant excitation and evolution of nonlinear waves: The variational approach

It is shown that a large class of multidimensional nonlinear waves can be excited and controlled in adiabatically varying systems driven by an externally launched pump wave. The excitation proceeds via the trapping into the resonance, while later the nonlinear wave evolves by being phase locked with the pump wave in an extended region of space and/or time despite the variation of system's parameters. This automatic phase locking (autoresonance) yields a possibility of shaping the parameters of the nonlinear wave by varying the nonuniformity and/or time dependence of the parameters of the system. The multidimensional theory of the autoresonance for driven nonlinear waves is developed on the basis of the averaged variational principle and is illustrated by an example of a driven sine-Gordon equation.