Modulation Instability and Dynamical Analysis of New Abundant Closed-Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M-Fractional Derivative

In this work, we study the modulation instability (MI) and closed-form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation with a truncated M-fractional derivative. The mKdV-ZK equation can be used to describe the behavior of ion-acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved F-expansion scheme and unified solver techniques for the mKdV-ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three-dimensional (3D) plots with density plots and two-dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed-form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved F-expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell-shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena.