The weakly nonlinear dynamics of dust ion-acoustic waves (DIAWs) are investigated in a dusty plasma consisting of hot ion fluid, variable charge stationary dust grains and non-thermally distributed electrons. The Korteweg-de Vries equation, as well as the Korteweg-de Vries-Burgers equation, are derived on the basis of the well-known reductive perturbation theory. It is shown that, due to electron non-thermality and finite ion temperature, the present dusty plasma model can support compressive as well as rarefactive DIA solitary waves. Furthermore, there may exist collisionless DIA shock-like waves which have either monotonic or oscillatory behavior, the properties of which depend sensitively on the number of fast non-thermal electrons. The results complement and provide new insights into previously published results on this problem (Mamun, A. A. and Shukla, P. K. 2002 IEEE Trans. Plasma Sci. 30, 720).

It is shown that when the non-isothermal trapped electron current is rederived based on the orbit-limited motion theory, the variable dust charge can be expressed in terms of the Lambert function. One can then take advantage of this new transcendental function to illustrate how variable-charge nonlinear trapped dust modes can be investigated semi-analytically.

A weakly nonlinear analysis is carried out to derive a Korteweg–de Vries Burgers-like equation for small but finite amplitude dust ion-acoustic (DIA) waves in a charge varying dusty plasma with non thermally distributed electrons. The correct expression for the nonthermal electron charging current is used. Interestingly, it may be noted that due to electron nonthermality and finite equilibrium ion streaming velocity, the present dusty plasma model can admit compressive as well as rarefactive DIA solitary waves. Furthermore, there may exist DIA shocks which have either monotonic or oscillatory behavior and the properties of which depend sensitively on the number of fast nonthermal electrons. Our results should be useful to understand the properties of localized DIA waves that may occur in space dusty plasmas.

Weakly nonlinear dust ion-acoustic (DIA) shock waves are investigated in a dusty plasma with nonthermal electrons. A modified Korteweg–de Vries equation with a cubic nonlinearity is derived. Due to the net negative dust charge μZd and electron nonthermality, the present plasma model can admit compressive and rarefactive weak DIA shock waves. The effect of increasing μZd is to lower the critical nonthermal parameter βc above which only rarefactive DIA shock waves are admitted. Our investigation may help to understand the nonlinear structures observed in the auroral acceleration regions.