ar X iv : m at h - ph / 0 50 30 19 v 1 9 M ar 2 00 5 The slowly passage through the resonances and wave packets with the different carriers ∗

Solution of the nonlinear Klein-Gordon equation perturbed by small external force is investigated. The perturbation is represented by finite collections of harmonics. The frequencies of the perturbation vary slowly and pass through the resonant values consecutively. The resonances lead to the sequence of the wave packets with the different fast oscillated carriers. Full asymptotic description of this process is presented. Introduction In this work we study the problem of a generation of sequences of solitary packets with different carriers in the optical fiber. The nonlinear Klein-Gordon equation is studied as a modeling equation. We consider this equation which is perturbed by a small external driving force with finite collection of modes. The wave packets appear due to passage through a resonance by different modes of the external force. After the passage through the whole resonances the solution contains the full collection of the solitary packages of waves with different carriers. The envelope functions of these packages satisfy to nonlinear Schrödinger equation (NLSE). In general the derivation of NLSE for small solutions of nonlinear equations is well known [1, 2, 3] and justified [4]. Our solution has a more complicate structure. Before all resonances the solution has an order of the perturbation and defined by the external force. After the passage through the resonance of the last mode of the perturbation the solution has the order of the square root of the order of the perturbation and satisfies NLSE. So we show the process of the resonant transformation of the solution and the appearance of the wave packets with the different carriers. Earlier the resonant generation of periodic waves by a small external force was investigated by a computer simulation [5]. The phenomenon of the generation and the scattering of the solitary waves in the case of nonlinear Schrodinger 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]