Autophasing of solitons
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Autophasing of Solitons S. V. Batalova, E. M. Maslovb*, and A. G. Shagalova a
b
Institute of Metal Physics, Russian Academy of Sciences, Ural Branch, Yekaterinburg, 620219 Russia Institute of Terrestrial Magnetism, Ionosphere, and Radiowave Propagation, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia *email: [email protected] Received November 10, 2008
Abstract—The effect of an external wave perturbation with a slowly varying frequency on a soliton of the nonlinear Schrödinger equation is investigated. The equations that describe the time evolution of the per turbedsoliton parameters are derived. The necessary and sufficient soliton phase locking conditions that relate the rate of change in the frequency of the perturbation, its amplitude, wave number, and phase to the initial values of parameters for the soliton have been found. PACS numbers: 05.45.Yv DOI: 10.1134/S1063776109050185
1. INTRODUCTION Autophasing (or, according to a different terminol ogy, autoresonance) is currently used as an efficient method for controlling the dynamics of nonlinear sys tems using an external pumping. The main idea of autophasing dates back to the works [1, 2] on the acceleration of relativistic particles. An adequate the ory of this effect was proposed in [3] for the simplest model of a nonlinear pendulum excited by a periodic pumping with a slowly varying frequency near the fre quency of the system’s linear resonance. If the pump ing amplitude exceeds some critical value dependent on the rate of change in frequency, then the pendulum phase turns out to be locked by the external pumping. This makes it possible to effectively control the pendu lum oscillation amplitude by gradually changing the pumping frequency. An important feature of the autoresonance effect is the possibility of exciting oscil lations in the system with a large amplitude, much larger (by several orders of magnitude) than that of the pumping itself, which is of particular interest for vari ous applications. At present, the idea of autophasing has been devel oped for a number of more complex physical systems (see, e.g., [4]) and, in particular, for nonlinear waves and solitons. In the latter case, the key problem was the excitation of largeamplitude nonlinear waves with given amplitude–phase characteristics using small perturbations of a special form included in the equa tion describing the process. Thus, for example, it was shown in [5, 6] that waves on water described by the Korteweg–de Vries equation could be effectively gen erated by perturbations like traveling waves. Subse quently, it was shown that the autophasing method could be used to generate multiphase periodic waves
described by the Korteweg–de Vries equation with full control of each of the generatedwave phases [7]. The generation of periodic waves in terms of the nonlinear Schrödinger equation (NSE) was investigated in [8, 9] for both singlephase and mulitiphase solutions. In the latter case, several stable scenarios for
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