A numerical study of nonlinear Alfvén waves and solitons

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Abstract
Finite‐amplitude Alfvén waves can be modeled by a nonlinear wave equation termed the derivative nonlinear Schrödinger equation. A computer program has been developed that solves the derivative nonlinear Schrödinger equation via the ‘‘split‐step’’ Fourier method. This program has been used to investigate a number of topics in the area of nonlinear Alfvén waves. When analytic envelope solitons are used as initial conditions, the wave packets propagate without distortion and with the expected speed–amplitude relation. When an arbitrary, amplitude‐modulated wave is used as an initial condition, the results depend strongly on the β of the plasma and the polarization of the wave. For a left circularly polarized wave in a β1, a collapse instability has been observed in which the wave amplitude increases and modulation scale decreases. For other combinations of polarization and value of β, the wave packet tends to broaden, eliminating the initial modulation.