Variational principle, conservation laws and exact solutions for dust ion acoustic shock waves modeling modified Burger equation

The nonlinear dust ion acoustic (DIA) shock waves have been investigated in one-dimensional, collisionless and unmagnetized dusty plasma consisting of mobile ion, vortex-like electron and charge fluctuating static dust particles. The basic set of fluid equations is reduced to the modified Burger equation by using the reductive perturbation method. The effects of dust grain charge fluctuation and vortex-like (trapped) electron are found to modify the properties of the DIA shock waves significantly. By the semi-inverse method, a variational principle, and conservation laws for the modified Burger equation are obtained. It is shown that the modified Burger equation is non-integrable using Painlevé analysis. A set of new exact solutions are obtained by the auto-Bäcklund transformations. Finally, we will study the physical meanings of solutions.