Zakharov–Kuznetsov–Burgers equation for dust ion acoustic waves

The nonlinear wave structures of small, but finite amplitude dust ion acoustic waves in a magnetized dusty plasma consisting of cold positive ions, isothermal electrons and variable charged stationary dust particles are investigated using reductive perturbation theory. The basic set of fluid equations is reduced to Zakharov–Kuznetsov–Burgers (ZKB) equation. The presence of charging process give rise to three cases. The first case arises when the charging process lead to originate anomalous dissipation, which makes possible existence of a new kind of shocks related to this dissipation. Case two, in the absence of dissipation (or if the dissipation is weak) the balance is then between nonlinear and dispersion effects, which can result in the formation of a symmetrical solitary waves. Case three considers the dissipation and dispersion at the same footing, i.e. we cannot neglect either dissipation or dispersion. Exact solution of the ZKB equation is obtained, for the first time, using a improved modified extended tanh-function method. Then, all possible cases of ZKB equation are covered.