Alfvén Solitons

Abstract

The theory of MHD soliton formation parallel to the magnetic field, as described by the DNLS equation and related models, is reviewed. The known results on the physical relevance, solutions and mathematical properties of the DNLS equation, are briefly described. New results on the extension of the theory to finite beta and three space dimensions are presented. A hybrid fluid and kinetic treatment of the case of finite temperature based on the guiding center plasma model, leads to an additional nonlocal term due to resonant particles. The combination of modulational instability and resonant particle damping leads to the conclusion that parallel weakly dispersive Alfvén waves are nonlinearly damped in a finite beta plasma.

Fluid models for kinetic effects on coherent nonlinear Alfvén waves. I. Fundamental theory

“…A fully kinetic calculation was undertaken by Rogister [5] for the case of a high-beta plasma. For parallel propagation, his results counside with equations obtained later by Mjølhus and Wyller [12,6] and Spangler [13,14]. The effect of Landau damping appears in the DNLS via an additional cubic term which is an integral operator over space.…”

“…We first use one-fluid MHD equations with parallel dissipation coefficients µ and χ which are constant and independent of mode frequency and wave-number. Later we replace constant χ with the integral operator representation of Hammet and Perkins [17], and obtain a modifield DNLS similar to that obtained by Mjølhus and Wyller [12,6] and Spangler [13,14]. However, our approach yields expressions for coefficients of the nonlinear terms which are much simpler than theirs and allows clear, unambiguous physical interpretation of the results.…”

“…Thus, with the notable exceptions of Ref. [6,[12][13][14], the theoretical 'lore' of nonlinear Alfvén waves in a collisionless, β ∼ 1 plasma is built upon a conceptual paradigm constructed for a collisional, β < 1 system. This state of affairs is due, in part, to the intractability and unwieldiness of straightforward kinetic analysis of nonlinear Alfvén waves.…”

“…The other attempts to include Landau damping in the nonlinear Alfvén wave evolution used kinetics to calculate second-order density (or pressure) perturbation, but the DNLS-like equation (for b) was obtained from MHD equations. Mjølhus and Wyller [12,6] used a guiding center formalism and Spangler [13,14] used the full Vlasov equation. Both methods predict a nonlocal integral term, and the coefficients of the nonlinear terms coinside with those of Rogister [5].…”

N-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

“…For example, in space plasma, when the DNLS equation was generalized to a multi-dimensional one [27,28], it was shown that the dark solitons are not stable in presence of transverse perturbations [28].…”