ar X iv : m at h - ph / 0 41 00 41 v 1 1 8 O ct 2 00 4 Resonant pumping in nonlinear Klein - Gordon equation and solitary packets of waves ∗

Solution of the nonlinear Klein-Gordon equation perturbed by small external force is investigated. The frequency of perturbation varies slowly and passes through a resonance. The resonance generates a solitary packets of waves. Full asymptotic description of this process is presented. Introduction This work is devoted to the problem on a generation of solitary packets of waves by a small external driving force. We propose a new approach for generation of solitary packets of waves. We demonstrate that for perturbed nonlinear Klein-Gordon equation. In our approach the wave packets appear due to passing of external driving force through resonance. After the resonance the envelope function of the wave packet is determined by nonlinear Schrödinger equation (NLSE). In the most important cases the envelope function is a sequence of solitary waves which are called solitons. The wave packets with the solitons as the envelope function are propagated without a deformation. The parameters of the solitons are obviously defined by the value of the driving force on a resonance curve. Here we give the mathematical basis for the proposed approach. This basis allows to derive explicit formulas which define parameters for the solitary packets of waves with respect to the external driving force. Generation of the solitary packets of waves by a small driving force is described in detail. The formulas for the asymptotic solution before, after and in the neighborhood of the resonance curve are obtained. Proposed approach is based on a local resonance phenomenon. The local resonance in linear ordinary differential equations was investigated in papers [2, 3]. Later this phenomenon was investigated in partial differential equations in linear case [4] and in weak nonlinear case [5, 6]. As was shown in these papers the amplitude of the wave which crosses the local resonance increases by linear This work was supported by grants RFBR 03-01-00716, Leading Scientific Schools 1446.2003.1 and INTAS 03-51-4286. Institute of Math. USC RAS; [email protected] Ufa State Petroleum Technical University; [email protected] Ufa State Petroleum Technical University; [email protected]