Oblique nonlinear Alfvén waves in strongly magnetized beam plasmas. Part 2. Soliton solutions and integrability

Abstract

Oblique propagation of MHD waves in warm multi-species plasmas with anisotropic pressures and different equilibrium drifts is described by a modified vector derivative nonlinear Schrödinger equation, if charge separation in Poisson's equation and the displacement current in Ampère's law are properly taken into account. This modified equation cannot be reduced to the standard derivative nonlinear Schrödinger equation, and hence requires a new approach to solitary-wave solutions, integrability and related problems. The new equation is shown to be integrable by the use of the prolongation method, and by finding a sufficient number of conservation laws, and possesses bright and dark soliton solutions, besides possible periodic solutions.

Oblique nonlinear Alfvén waves in strongly magnetized beam plasmas. Part 1. Nonlinear vector evolution equation

“…As we shall show in the following paper (Deconinck et al 1993), the extra term causes drastic changes in the way we have to look for solutions or determine the integrability. If we try to follow the same pattern as from (64) to (67), by introducing the notation <fio± = B Ox ±iB 0y (68) for the static field, we find that (62) becomes l t ) 0.…”

“…because of the complexities involved, is deferred to the following paper (Deconinck, Meuris & Verheest 1993). In particular, as we shall see, this allows, among other things, sub-Alfvenic solitary modes, which do not exist for the classical DNLS equation.…”

“…In § 2 we give the basic equations for a multi-species beam plasma, and derive in § 3 the general form of the MVDNLS equation for oblique propagation in cold plasmas. The modifications due to temperature and pressure effects are indicated in §4, and a short discussion with preliminary conclusions follows in §5, in order to highlight the essential differences with previous treatments and as a prelude to a more detailed investigation of the integrability of the MVDNLS equation (Deconinck et al 1993).…”

Complete integrability of a modified vector derivative nonlinear Schrödinger equation

“…Of course, for oblique propagation with α = 0, the circular polarization given by the DNLS is only apparent, since B ⊥ includes the static part B ⊥0 , and this shift leads in reality to elliptical polarization for the perpendicular wave field (Spangler and Plapp 1992). In addition, the MVDNLS has a class of stationary solitary wave solutions which the DNLS does not have, the subalfvénic modes, which have totally different properties compared to the known stationary solutions of the DNLS (Deconinck, Meuris and Verheest 1993b). …”

“…This method was adapted successfully to the nonlinear equation at hand (Deconinck, Meuris and Verheest 1993b). …”