Abstract
Propagation of three-dimensional nonlinear ion-acoustic solitary waves and shocks in a homogeneous magnetised electron-positron-ion plasma is analysed. Modified extended mapping method is introduced to find ion-acoustic solitary wave solutions of the three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation. As a result, solitary wave solutions (which represent electrostatic field potential), electric fields, magnetic fields and quantum statistical pressures are obtained with the aid of Mathematica. These new exact solitary wave solutions are obtained in different forms such as periodic, kink and antikink, dark soliton, bright soliton, bright and dark solitary wave etc. The results are expressed in the forms of hyperbolic, trigonometric, exponential and rational functions. The electrostatic field potential and electric and magnetic fields are shown graphically. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical and physical problems.