A wave coupling formalism for magnetohydrodynamic (MHD) waves in
a non-uniform background flow is used to study parametric instabilities of the
large-amplitude, circularly polarized, simple plane Alfvén wave in one Cartesian
space dimension. The large-amplitude Alfvén wave (the pump wave) is regarded
as the background flow, and the seven small-amplitude MHD waves (the backward
and forward fast and slow magnetoacoustic waves, the backward and forward
Alfvén waves, and the entropy wave) interact with the pump wave via wave coupling
coefficients that depend on the gradients and time dependence of the background
flow. The dispersion equation for the waves D(k,ω) = 0 obtained from the
wave coupling equations reduces to that obtained by previous authors. The general
solution of the initial value problem for the waves is obtained by Fourier and
Laplace transforms. The dispersion function D(k,ω) is a fifth-order polynomial in
both the wavenumber k and the frequency ω. The regions of instability and the
neutral stability curves are determined. Instabilities that arise from solving the
dispersion equation D(k,ω) = 0, both in the form ω = ω(k), where k is real, and
in the form k = k(ω), where ω is real, are investigated. The instabilities depend
parametrically on the pump wave amplitude and on the plasma beta. The wave
interaction equations are also studied from the perspective of a single master wave
equation, with multiple wave modes, and with a source term due to the entropy
wave. The wave hierarchies for short- and long-wavelength waves of the master wave
equation are used to discuss wave stability. Expanding the dispersion equation for
the different long-wavelength eigenmodes about k = 0 yields either the linearized
Korteweg–deVries equation or the Schrödinger equation as the generic wave equation at long-wavelengths. The corresponding short-wavelength wave equations are
also obtained. Initial value problems for the wave interaction equations are investigated.
An inspection of the double-root solutions of the dispersion equation for k,
satisfying the equations D(k,ω) = 0 and ∂D(k,ω) = ∂k = 0 and pinch point analysis
shows that the solutions of the wave interaction equations for hump or pulse-like
initial data develop an absolute instability. Fourier solutions and asymptotic analysis
are used to study the absolute instability.