Ion acoustic cnoidal waves in a dusty plasma with a critical dust density
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Ion Acoustic Cnoidal Waves in a Dusty Plasma with a Critical Dust Density V. V. Prudskikh Research Institute of Physics, Southern Federal University, pr. Stachki 194, RostovonDon, 344090 Russia Received October 5, 2008; in final form, January 12, 2009
Abstract—The propagation of nonlinear periodic ion acoustic waves in a dusty plasma is considered for con ditions in which the coefficient in the nonlinear equation that describes the quadratic nonlinearity of the medium is zero. An equation that accounts for the cubic nonlinearity of the system is derived, and its solution is found. The dependence of the phase velocity of a cnoidal wave on its amplitude and modulus is determined. In describing the effect of higher order nonlinearities on the properties of a dust ion acoustic wave, two cou pled equations for the first and secondorder potentials are obtained. It is shown that the nonlinear ion flux generated by a cnoidal wave propagating in a medium with a cubic nonlinearity is proportional to the fourth power of the wave amplitude. PACS numbers: 52.35.g, 52.35.Fp DOI: 10.1134/S1063780X09080054
1. INTRODUCTION The properties of plasmas containing electrons, ions, and also heavy dust grains (on the order of 10–12– 10–14 g) have been actively studied over the last 15 years. The presence of dust in the plasma gives rise to many new plasma modes, plasma instabilities, and nonlinear phenomena. Among the objects of dusty plasma research are dust ion acoustic waves, which were first considered by Shukla and Silin [1]. The pres ence of dust substantially modifies ion acoustic modes in comparison with those in a conventional electron– ion plasma. Thus, the phase velocity of dust ion acous tic waves is higher. As a result, the requirement on the extent to which the electron and ion components of a dusty plasma should be nonisothermal in order for such waves to exist is less stringent than that for a clas sical plasma. Investigation of the problem of the prop agation of nonlinear solitary dust ion acoustic waves [2] shows that the nonlinear coefficient in the relevant Korteweg–de Vries (KdV) equation can be not only positive, but negative as well, and that a solitary wave (soliton) can be either a compression or a rarefaction ion wave. This situation differs radically from that in an electron–ion plasma, where ion acoustic solitons can have only a positive potential. The equilibrium charge of a dust grain in a plasma is determined from the condition that the electron and ion fluxes to the grain surface are the same and is usually negative. The perturbations of the plasma density in the field of an electrostatic wave change the fluxes and, therefore, the grain charge—a circumstance that also influences the special behavior of waves in a dusty plasma. Moreover, in such a plasma, there exists a specific dissipation mechanism that arises from the nonadiabatic variation
of the charge of a dust grain in the field of the propa gating wave and that is responsible for the onset of dust ion acoustic shock waves [3].
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