Newly formed solitary wave solutions and other solitons to the (3+1)-dimensional mKdV-ZK equation utilizing a new modified Sardar sub-equation approach

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg-de Vries-Zakharov-Kuznetsov (mKdV-ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron-positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV-ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark-bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV-ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the (3+1)-dimensional mKdV-ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.