Search

In this investigation, the exact solutions of variable coefficients of generalized Zakharov-Kuznetsov (ZK) equation and the Gardner equation are studied with the help of an extended generalized G′/G expansion method. The main objective of this study is to establish the closed-form solutions and dynamics of analytical solutions to the generalized ZK equation and the Gardner equation. The generalized ZK equation and the Gardner equation govern the behavior of nonlinear wave phenomena in the presence of magnetic field in plasma dynamics, turbulence, bottom topography, and quantum field theory. We construct innovative solutions to the models under consideration using various computing tools and a recently developed extended generalized G′/G expansion technique. The extended generalized G′/G expansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial of G′/G. The derived solutions for both the equations are the hyperbolic, trigonometric, and rational functions. The obtained solutions have shock/kink waves and multisoliton, which depict the dynamical representations of the acquired solutions through the three-dimensional surface plots and the contour plots.

The extended generalized G′G–expansion is well defined and an efficient technique, which is used to obtain the exact traveling wave solutions to the governing nonlinear equations with constant coefficients as well as variable coefficients. In this paper, the modified Korteweg–de Vries (mKdV) equation and Burgers equation with variable coefficients are investigated through the extended generalized G′G–expansion method, which are exceptional cases of the nonlinear evolution equations widely used in a two-layer fluid system, in fluid-filled elastic tubes, in an atmospheric and oceanic dynamical system, traffic flow, turbulence in fluid dynamics, dusty plasma, ion-acoustic waves in a plasma system. New families of exact closed-form solutions are obtained in hyperbolic, trigonometric, and rational function solutions with the available free constants. With the help of computerized symbolic computation work, the newly formed closed-form solutions are validated by back substituting them into the equations using the computational mathematical software. Furthermore, the graphical representations of all these obtained solutions are discussed and demonstrated by giving the suitable best values of arbitrary functions and constants via three-dimensional surface and density plots. The dynamics of the solution profiles demonstrate the annihilation of 3D kink-type soliton waves, shock waves, double solitons, and multi-soliton wave structures.