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, magneticfields 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. Theseresults demonstrate the efficiency and precision of the method that can be applied to many other mathematical and physical problems.

The generalised nonlinear Schrödinger equation (NLSE) of third order is investigated, which accepts one-hump embedded solitons in a single-parameter family. In this paper, we constructed analytical solutions in the form of solitary waves and solitons of third-order NLSE by employing the extended simple equation method and exp($−\Phi(\xi)$)-expansion method. In applied physics and engineering, the obtained exact solutions have important applications. The stability of the model is examined by employing modulational instability which verifies that all the achieved exact solutions are stable. The movements of exact solitons are also presented graphically, which assist the researchers to know the physical interpretation of this complex model. Several such types of problems arising in engineering and physics can be resolved by utilising these reliable, influential and effective methods.