Search

In this investigation, a different form of extended generalized G′G2-expansion technique with variable coefficient is proposed. The advantage of the extended generalized G′G2–expansion technique is that it can be used to solve nonlinear evolution equations with both constant and variable coefficients, whereas the basic G′G2 method can only be used with constant coefficients. The proposed technique is used to solve the Bogoyavlensky–Konopelchenko (BK) equation with variable coefficients, which depicts the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Further, the BK equation with variable coefficients is applied for stratified internal waves, shallow-water waves, ion-acoustic waves, and water propagation in a liquid. The hyperbolic, trigonometric, and rational form solutions of the BK equation are dynamically represented as the annihilation of three-dimensional kink waves, multi-soliton waves, single solitons, and so on. Furthermore, the derived solutions of the considered equation, which comprise arbitrary functional parameters and other constant parameters, can be used to enhance the advanced behaviours of physical situations.

In this paper 2+1–dimensional Kadomtsev–Petviashvili (KP) equation with variable coefficients is investigated through the extended generalized G′G–expansion technique. One of the most universal model is KP equation, which is used to explain the ion acoustic waves in plasma physics, to model two dimensional shallow water waves, and in ferromagnetic, Bose–Einstein condensation and string theory. The obtained exact solutions of KP equation are in the form of hyperbolic function, trigonometric function, and rational function. With the aid of symbolic computational software Mathematica, the three dimensional surface plots with corresponding contour plots are provided for the obtained closed from solutions, which are of the form of solitary waves, multi solitons and periodic solitary wave like dynamical structures.

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.