Unraveling the dynamic complexity: exploring the (3+1)-dimensional conformable mKdV-ZK equation

The primary objective of this article is to construct novel solitary wave solutions for the nonlinear (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation with a conformable derivative by employing the subsidiary ordinary differential equation method and 1 φ ( ς ) , φ ′ ( ς ) φ ( ς ) method. The solutions obtained encompass diverse bright, dark, singular, and periodic solitary waves. Additionally, the article thoroughly discusses the sufficient conditions for the existence of these solutions. To visually represent the obtained solutions, we include 2D and 3D graphical images using the computational software Maple 18. We also included a comparison of fractional derivatives at different values of fractional order and conducted an analysis to understand the influence of nonlinear parameters on wave behaviors. The significance of these solutions lies in their wide-ranging applications in applied sciences and mathematical physics. Furthermore, the proposed methods are invaluable in efficiently solving nonlinear partial differential equations prevalent in engineering and physical science.