Non-linear constant-profile plane waves in a cold plasma under an applied magnetic field†

Periodic plane waves propagating in a homogeneous cold Vlasov plasma under the influence of an external magnetic field B₀ have been studied in this paper. Non-linear coupled differential equations for the two transverse components of the local magnetic field have been obtained for any angle α between B₀ and the wave vector. Starting from the previously obtained exact solution for α = 90° and the zeroth-order solution for α = 0, two perturbation methods are used to obtain first-order solutions for α = 0 and intermediate angles. A numerical example has been worked out in detail for a specific value of the field energy density; although it is not possible to match the two sets of results rigorously, they seem to converge smoothly to the same limit.