Application of soliton theory to plasma confinement and fluid transport: analytical and numerical results for the Akbota equation

Abstract In this paper, an analytical and numerical analysis of the integrable Akbota equation, which is a nonlinear wave equation applicable in the areas of plasma physics and fluid dynamics, is performed. With the help of traveling wave transformation, the governing set of equations is converted into a nonlinear ordinary differential equation; thereby, it becomes possible to find closed form nonlinear wave solutions using the enhanced modified extended tanh expansion method. Various families of exact solution structures, such as dark solitons, periodic waves, singular solitons, and other types of nonlinear waves, are found, depending upon the parameter values of the equation. In order to check the reliability of the obtained analytical solutions, the differential transformation method (DTM) is used to produce the corresponding numerical approximations. It is observed that there is a good agreement between the numerical solutions and the exact solutions, which have extremely low absolute errors for the cases under consideration, thereby indicating the correctness of the analytic solution obtained. An important feature of enhanced modified extended tanh expansion is that it can systematically generate many different solution families in a closed form, while differential transformation method is a useful numerical scheme to obtain the approximation of nonlinear solutions and confirm the correctness of the analytic solution obtained. The obtained traveling wave solutions offer an analytical understanding of nonlinear wave propagation in models related to plasma physics and fluid dynamics. Although the current study concentrates on the nondimensional model of the problem and does not analyze quantitatively the process of plasma confinement and fluid transport, the analytical solutions may be employed as reference solutions for the investigation of nonlinear wave behavior in physical models.