Relativistic Nonlinear Whistler Waves in Cold Magnetized Plasmas

Starting from the Vlasov-Maxwell equations describing the dynamics of various species in a quasi-neutral plasma immersed in an external solenoidal magnetic field and utilizing a technique known as the hydrodynamic substitution, a relativistic hydrodynamic system of equations governing the dynamics of various species has been obtained. Based on the method of multiple scales, a system comprising three nonlinear Schrodinger equation for the transverse envelopes of the three basic whistler modes, has been derived. Using the method of formal series of Dubois-Violette, a traveling wave solution of the derived set of coupled nonlinear Schrodinger equations in both the relativistic and the non relativistic case has been obtained. An intriguing feature of our description is that whistler waves do not perturb the initial uniform density of plasma electrons. The plasma response to the induced whistler waves consists in transverse velocity redistribution, which follows exactly the behaviour of the electromagnetic whistlers. This property may have an important application for transverse focusing of charged particle beams in future laser plasma accelerators. Yet another interesting peculiarity are the selection rules governing the nonlinear mode coupling. According to these rules self coupling between modes in the non relativistic regime is absent, which is a direct consequence of the vector character of the interaction governed by the Lorentz force.