ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 8, Numbers l and 2, Winter and Spring 1978 CIRCULAR POLARIZED NONLINEAR ALFVËN WAVES—A NEW TYPE OF NONLINEAR EVOLUTION EQUATION IN PLASMA PHYSICS MIKI WADATI, HEIJI SANUKI*, KIMIAKI KONNO** AND YOSHI-HIKO ICHIKAWA* ABSTRACT. A nonlinear evolution equation is derived for Alfvén waves propagating along the magnetic field in a cold plasma. The equation provides new types of solitary waves. The phase of a soli- tary wave is coupled nonlinearly with its amplitude, and the propa- gation velocity is restricted within the range determined by the as- ymptotic amplitude and the wave number. 1. Introduction. As early as in 1942, Alfvén [1] recognized that a hydromagnetic wave propagates in an incompressible, perfectly con- ducting fluid in the presence of a strong magnetic field. This Alfvén wave is nothing but the low frequency limit of electromagnetic waves propagating in a plasma. Generation and propagation of the Alfvén waves in a gaseous plasma has been investigated experimentally [2], and has attracted renewed interest as one of the useful ways to heat a plasma [3]. Alfvén wave propagation in solid state plasmas provides in- formation on the effective masses of carriers [4], Large amplitude in- compressible magnetic field perturbation observed in the solar wind has been attributed to propagation of the Alfvén wave [5]. Now, turning to the studies of nonlinear wave propagation in plasmas, we have seen the remarkable success of theoretical and experi- mental investigations of the ion acoustic solitary waves. Using reductive perturbation theory, Washimi and Taniuti [6] have predicted existence of the Korteweg-deVries type soliton for the ion acoustic mode. Their prediction has been experimentally confirmed by Ikezi et al. [7]. Sys- tematic ordering of dispersive effects and nonlinear steepening effects in the reductive perturbation theory [8] provides a rigorous procedure for reducing the hyperbolic system of nonlinear partial differential equations to a single nonlinear evolution equation; namely, the Korte- weg-deVries equation for a weakly dispersive system and the nonlinear Schrödinger equation for a strongly dispersive system. In the case of Alfvén waves, however, Kakutani and Ono [9] have noticed that it is necessary to modify the expansion scheme of the re- ductive perturbation theory so as to be consistent with steady state soli- tary wave solution of Kazantsev [10]. Thus, they have been led to the conclusion that the Alfvén wave is governed by the modified Korteweg- Copyright © 1978 Rocky Mountain Mathematics Consortium 323