In this paper, our aim is to further expand the use of the two-variable $(G^{\prime }/G,1/G)$-expansion approach to a new coupled KdV and Z-K system, which has various significant applications in different fields of applied sciences. The KdV equation, along with shallow-water waves and long internal waves in oceans, basically explains how long, one-dimensional waves propagate in a variety of physical conditions. The study of coastal waves on the basis of the ocean is done using the Zakharov–Kuznetsov (Z-K) equation and this model is utilized to illustrate ion-acoustic wave propagation. By using this method, different forms of analytical solutions of the new coupled KdV (NCKdV) system and the new coupled Z-K (NCZ-K) system, such as solitons, multi-peak solitons, solitary waves, trigonometric, hyperbolic and rational functions and other wave solutions are constructed. The significant features of multi-peak solitons induced by the higher-order effects, including velocity variations, localization or periodicity attenuation and state transitions, are revealed. When the localization disappears then the multi-peak soliton becomes a periodic wave. The constructed solutions are also presented graphically having their applications in engineering, etc. The stability of the solution is examined by utilizing modulation instability. The results obtained show that the proposed technique is universal and efficient. In addition, this technique can also be applied to lots of other new coupled systems arising in other areas of applied sciences.

In this paper, we study the Fokas–Lenells (F-L) equation in the presence of perturbation terms and we construct exact solutions. The modified extended direct algebraic method (MEDAM) is utilized, and soliton solutions, solitary wave solutions and elliptic function solutions are achieved. The physical meaning of the geometrical shapes for some of the obtained results is investigated for various choices of the free parameters that appear in the results. The stability of the model is investigated by using the standard linear stability analysis, which verifies that the exact solutions are stable. The achieved solutions and computational work show that the current method is powerful and influential.

This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable $(G^{\prime }/G,1/G)$-expansion and generalized exp($-\varphi \left(\xi \right)$)-expansion techniques are presented to construct the ion-acoustic wave results of a (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) model. As a result, the novel soliton and other wave solutions in a variety of forms, including kink- and anti-kink-type breather waves, dark and bright solitons, kink solitons, and multi-peak solitons, etc., are attained. With the help of software, the solitary wave results (that signify the electrostatic potential field), electric and magnetic fields, and quantum statistical pressures are also constructed. These solutions have numerous applications in various areas of physics and other areas of applied sciences. Graphical representations of some of the obtained results, and the electric and magnetic fields as well as the electrostatic field potential are also presented. These results demonstrate the effectiveness of the presented techniques, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields.