Hamiltonian and variational formulations of equations describing weakly nonlinear magnetohydrodynamic (MHD) wave interactions in one Cartesian space dimension are discussed. For wave propagation in uniform media, the wave interactions of interest consist of (a) three-wave resonant interactions in which high-frequency waves may evolve on long space and time scales if the wave phases satisfy the resonance conditions; (b) Burgers self-wave steepening for the magnetoacoustic waves, and (c) mean wave field effects, in which a particular wave interacts with the mean wave field of the other waves. The equations describe four types of resonant triads: slow–fast magnetoacoustic wave interaction, Alfvén–entropy wave interaction, Alfvén–magnetoacoustic wave interaction, and magnetoacoustic–entropy wave interaction. The formalism is restricted to coherent wave interactions. The equations are used to investigate the Alfvén-wave decay instability in which a large-amplitude forward propagating Alfvén wave decays owing to three-wave resonant interaction with a backward-propagating Alfvén wave and a forward-propagating slow magnetoacoustic wave. Exact solutions of the equations for Alfvén–entropy wave interactions are also discussed.

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0ê∥. Numerically and analytically, we find energy spectra E± ∼ kn±⊥, such that n+ + n− = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k∥ = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

Dispersion characteristics of surface waves in a semibounded plasma flow with relativistic velocity u0 parallel to the plasma–vacuum interface are presented. The plasma is considered to be cold and collisionless, which allows us to take into account only the electron motion. It is shown that – in contrast to the bulk waves, which are invariant with respect to Lorentz transformations and whose spectrum is independent of the flow velocity – the surface waves are not invariant, which leads to a dependence of their spectrum on the flow velocity and, correspondingly, to non-reciprocity. The latter peculiarity is due to the fact that the boundary conditions are not invariant.

Surface eigenmodes in a dusty plasma, where the dust is confined to a particular region, modelled as a uniform slab with non-uniform smooth boundary layers, are discussed. Inside this region, electron depletion occurs, while outside it, the plasma is just a normal uncontaminated plasma. In addition, the dust component can flow with respect to the background plasma, as a first approximation to cometary tails, where there is a notable difference between the fast flowing solar wind and the slow moving dust tail, viewed in the comet frame. The equilibrium densities and flow are non-uniform and the description involves a space-dependent dielectric function, which indicates the possibility of singular points, where eigenmodes resonate with local plasma oscillations. Surface eigenmodes on the slab are obtained analytically in the limit of long wavelengths, and these give rise to two distinct frequency domains, both having three different wave solutions. One of these refers to plasma surface waves, unaffected by the dust flow. The other two lead to convective surface modes, which become unstable for large flow speeds, akin to Buneman-type instabilities. A detailed study is made of the resonant processes in the boundary layers, including damping and growth rates.