Symmetry analysis and similarity electrostatic waves in a nonuniform dusty magnetoplasma

We apply the group theory to a (3+1)-dimensional nonlinear system relevant for the low-frequency electrostatic waves in a nonuniform dusty magnetoplasma. In correspondence with the generators of the symmetry group allowed by the system, new types of similarity reductions are performed. Some new exact solutions are obtained, which can be in the form of solitary waves, shock waves and periodic waves. Especially, our solutions indicate that the system may have time-dependent nonlinear shears. Some explicit similarity electrostatic wave solutions with time periodic nonlinear shears are displayed graphically.