A dusty plasma system containing Boltzmann electrons, mobile ions and charge fluctuating stationary dust has been considered. The nonlinear propagation of the dust-ion-acoustic waves in such a dusty plasma has been investigated by employing the reductive perturbation method. It has been shown that the dust charge fluctuation is a source of dissipation and is responsible for the formation of the dust-ionacoustic shock waves. The basic features of such dust-ion-acoustic shock waves have been identified. The implications of our results in space and laboratory dusty plasmas are discussed.It has been first theoretically [1] shown that, due to the conservation of equilibrium charge density n i0 e − n e0 e = z d0 n d0 e and the strong inequality n e0 n i0 [where n s0 is the equilibrium particle number density of the species s with s = e, i, d for electrons, ions, dust particles, z d0 is the equilibrium number of electrons residing onto the dust grain surface, and e is the magnitude of the electronic charge], a dusty plasma (with negatively charged static dust grains) supports low frequency dust-ion-acoustic (DIA) waves with phase speed much smaller (larger) than electron (ion) thermal speed. The dispersion relation (a relation between the wave frequency ω and the wave number k) of the linear DIA waves is [1] ω 2 = (n i0 /n e0 )k 2 C 2is the ion-acoustic speed (with T e being the electron temperature and m i being the ion mass) and λ De = (k B T e /4πn e0 e 2 ) 1/2 is 0375-9601/$ -see front matter

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The basic features and multi-dimensional instability of electrostatic solitary waves propagating in an ultra-relativistic degenerate dense magnetized plasma have been investigated by the reductive perturbation method and the small-k perturbation expansion technique. The Zakharov–Kuznetsov (ZK) equation has been derived, and its numerical solutions for some special cases have been analysed to identify the basic features (viz. amplitude, width, instability, etc.) of these electrostatic solitary structures. The implications of our results in some compact astrophysical objects, particularly white dwarfs and neutron stars, have been briefly discussed.

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A rigorous theoretical investigation has been made of the dust ion-acoustic (DIA) solitary and shock waves in an unmagnetized dusty plasma (containing vortexlike electrons, mobile ions, and charge fluctuating static dust) by reductive perturbation method. The effects of dust grain charge fluctuation and vortexlike (trapped) electron are found to modify the properties of the DIA solitary and shock waves significantly. The implications of these results for some space and astrophysical dusty plasma systems, especially planetary ring systems, are briefly mentioned.