Soliton formation in magnetized Vlasov plasmas

Soliton formation at the upper hybrid frequency has been discussed recently on the basis of warm two-fluid theory. In this paper we report the results of a reductive perturbation theory analysis of the Vlasov equation describing the nonlinear modulation of electrostatic waves in a warm magnetized plasma. This approach allows a treatment of soliton formation with scale lengths of the order of, or less than, the electron Larmor radius r_L for both upper hybrid and Bernstein waves. A three-dimensional generalized nonlinear Schrödinger equation has been found. Contributions to the nonlinear frequency shift derive from three distinct processes. One contribution describes the generation of the second harmonic governed by (ω + ω) → 2w where ω is the frequency of the electrostatic carrier; a second arises from the self-interaction between the high-frequency waves (ω + ω → ω + ω) while the third contribution describes the nonlinear coupling between the carrier and the slow background plasma motion.

Propagation orthogonal to the magnetic field is studied and leads to a simplified nonlinear Schrodinger equation with cubic nonlinearity, predicting soliton formation at the upper hybrid and cyclotron harmonic frequencies. Expressions for the nonlinear frequency shift coefficient are presented for the upper hybrid and Bernstein modes in the low temperature limit and the relative strengths of the three competing nonlinear processes are compared. For kr_L ≥ 1·0, numerical analysis shows the dominant nonlinearity to be that arising from the nonlinear wave-particle interaction. The stability of the soliton solution is determined by the relative signs of the dispersion and the nonlinear frequency shift coefficient. Some properties of upper hybrid and Bernstein solitons are examined. In particular the variation of the normalized dispersion and nonlinear frequency shift coefficient with the basic plasma parameters (kr_L)² and (ω_pe/Ω, is presented, where (ope, fi are the electron plasma frequency and electron gyrofrequency respectively. Regions of instability are identified for both upper hybrid and Bernstein modes. The connection between reductive perturbation theory and previous warm fluid theories is also established.