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

Abstract

Nonlinear MHD waves propagating obliquely to the external magnetic field in warm multi-species plasmas with anisotropic pressures and different equilibrium drifts are treated without imposing the customary quasi-neutrality between the different species or neglecting the displacement current in Ampère's law. The wave magnetic field obeys a vector nonlinear evolution equation, which in the limits of parallel propagation or of both the neglect of the displacement current and the imposition of quasi-neutrality reduces to the vector formulation of the well-known derivative nonlinear Schrödinger equation.

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

“…19 • On the contrary, the same model plays a fundamental role in plasma physics. In warm multi-species plasmas with anisotropic pressures and different equilibrium drifts, coupled DNLSEs govern the oblique propagation of nonlinear magnetohydrodynamic waves relative to an external magnetic field under certain conditions, dropping the quasi-neutrality assumption [20]. If Alfvén waves in a magnetized plasma bear both right and left circular polarizations, the equations for the respective amplitudes lead to the system of DNLSEs in the form of Eq.…”

“…(1). For a weak noise level of 1%, the configurations of both the 'elevation-elevation' and 'elevation -four-petal' types exhibit reasonable robustness, in the sense that the growth phase of the 20 RWs persists in an essentially unaffected form on top of only a mildly disturbed background (Figs. 9 and 10).…”

“…Besides the technique of inverse scattering transform [15], the Hirota bilinear method is also applicable to these equations [17]. In addition to their great significance as examples of nonlinear dynamics in a general setting, these equations are also well known as relevant models for the properties of very narrow pulses in nonlinear optics [18,19], the propagation of Alfvén waves in magnetized plasmas [20,21], and the description of electromagnetic waves in an antiferromagnetic medium [22].…”

Real Lax spectrum implies spectral stability

“…The methods described in this paper can be applied to other equations not mentioned in Sections 4 and 6. Some other examples include: the KdV equation, Hirota's equation (or the mixed generalized NLS‐generalized mKdV equation) 23,34 , the Modified Vector dNLS equation, 35 the Massive Thirring Model, 36 the O nonlinear ‐model, 37 the complex reverse space‐time nonlocal mKdV equation, 38 and the reverse space‐time nonlocal generalized sine‐Gordon equations 38 …”

Complete integrability of a modified vector derivative nonlinear Schrödinger equation

“…The DNLS can account for oblique propagation, provided α is zero, i.e. imposing charge neutrality to all orders and neglecting the displacement current in Ampère's law (Deconinck, Meuris and Verheest 1993a). The bothersome third term in (1) also disappears if B ⊥0 is zero, in the case of strictly parallel propagation.…”