Short communication Lie symmetry analysis of electron–electromagnetic wave interaction under condition of the anomalous Doppler effect N.M. Ryskin ⇑ Saratov State University, Saratov 410012, Russia article info Article history: Received 6 August 2013 Received in revised form 21 January 2014 Accepted 1 February 2014 Available online 12 February 2014 Keywords: Lie symmetry analysis Anomalous Doppler effect Self-similar solutions Pulse compression abstract Lie symmetry analysis is applied for a problem of interaction of electron cyclotron oscilla- tors with a slow electromagnetic wave under condition of the anomalous Doppler effect. This analysis reveals scaling invariance of the system and existence of self-similar solutions which describe amplification of a short electromagnetic pulse with its subsequent com- pression. The results of theoretical analysis are confirmed by numerical simulations. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction Lie symmetry analysis is a powerful tool for finding exact solutions of nonlinear problems [1–4]. Many examples of appli- cation to physical problems have been demonstrated. Unfortunately, the benefits of this formalism are still undervalued in microwave electronics where one have to deal with various nonlinear problems of electron beam — electromagnetic wave interaction. In this article, Lie symmetry analysis of beam–wave interaction under condition of the anomalous Doppler (AD) effect is presented. Exact self-similar solutions describing amplification of a short electromagnetic pulse with subsequent compression are derived and analyzed. The AD effect occurs when an electromagnetic oscillator moves with a superluminal velocity. In such a situation, emission of radiation is accompanied by transition of the oscillator to a higher energy level and simultaneous deceleration of its axial motion [5–7]. AD radiation of electron cyclotron oscillators is of particular interest in microwave electronics. In Fig. 1, sche- matic drawing of an anomalous Doppler cyclotron resonance maser (AD–CRM) is presented. The electron beam drifting along the external constant magnetic field B 0 with the constant velocity v k interacts with the traveling electromagnetic wave (EMW) under the AD cyclotron resonance condition x  kv k ¼x H ; ð1Þ where x H ¼ eB 0 =mc is the electron cyclotron frequency, e, m, and c are the electron charge, mass, and Lorentz-factor, respec- tively. From Eq. (1) it is clear that the resonance condition holds true when v k is larger than the phase velocity v ph ¼ x=k of the slow EMW. Either dielectric-loaded or periodically corrugated metallic waveguide may be used to retard the EMW. The electron beam is initially rectilinear, i.e. the electrons are injected in the waveguide having only axial velocity. During the interaction with the EMW, the electrons lose their translational energy and gain the rotational one. In such a case, a http://dx.doi.org/10.1016/j.cnsns.2014.02.001 1007-5704/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Tel.: +7 8452514311; fax: +7 8452523864. E-mail address: RyskinNM@info.sgu.ru Commun Nonlinear Sci Numer Simulat 19 (2014) 2942–2950 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns