The propagation characteristics of small-amplitude dust-acoustic (DA) solitary waves (SWs) and shocks are studied in an unmagnetized dusty plasma with a pair of trapped positive and negative ions. Using the standard reductive perturbation technique with two different scalings of stretched coordinates, the evolution equations for DA SWs and shocks are derived in the form of complex Korteweg-de Vries and Burgers' equations. The effects of the dust charge variation, the dust thermal pressure, and the ratios of the positive to negative ion number densities as well as the free to trapped ion temperatures on the profiles of SWs and shocks are analysed and discussed.

In this report, the investigation of the properties of dust acoustic (DA) solitary wave propagation in an adiabatic dusty plasma including the effect of the non-thermal ions and trapped electrons is presented. The reductive perturbation method has been employed to derive the modified Korteweg–de Vries (mK-dV) equation for dust acoustic solitary waves in a homogeneous, unmagnetized, and collisionless plasma whose constituents are electrons, singly charged positive ions, singly charged negative ions, and massive charged dust particles. The stationary analytical solution of the mK-dV equation is numerically analyzed and where the effect of various dusty plasma constituents DA solitary wave propagation is taken into account. It is observed that both the ions in dusty plasma play as a key role for the formation of both rarefactive as well as the compressive DA solitary waves and also the ion concentration controls the transformation of negative to positive potentials of the waves.

Using the well-known reductive perturbation technique, the three-dimensional (3D) Burgers equation and modified 3D Burgers equation have been derived for a plasma system comprising of non-thermal ions, Maxwellian electrons, and negatively charged fluctuating dust particles. The salient features of nonlinear propagation of shock waves in such plasmas have been investigated in detail. The different temperature non-thermal ions and Maxwellian electrons are found to play an important role in the shock waves solution. The analytical solution of the 3D Burgers equation and modified 3D Burgers equation ratifying the propagation of dust acoustic shock waves are derived using the well-known tanh method. On increasing the population of non-thermal ions, an enhancement in the amplitude of shock waves is seen for negatively charged dust particles. A striking dependence of amplitude and width of shock waves on the ratio of ion temperatures and densities are also reported. Finally we introduced a new stretching coordinate and perturbation for the nth-order nonlinear 3D Burgers equation and its solution by the use of the tanh method. We found that, due to higher nonlinearity, the amplitude of shock waves decreases while width remains constant for all plasma parameters considered in the present investigation. The features accounted here could be relevant in the case of different space and astrophysical plasmas and laboratory dusty plasma for negatively charged dust fluctuation.