Study on the simplified MCH equation and the combined KdV–mKdV equations with solitary wave solutions

Abstract

This paper is concerned with the traveling wave solutions and analytical treatment of the simplified MCH equation and the combined KdV–mKdV equations. Based on a polynomial about x and t, the rational solutions are investigated. The abundant soliton and periodic wave solutions are obtained by using a direct function. The hyperbolic-type and trigonometric-type solutions are given by utilizing the improved tan(ϕ/2)-expansion method. The dynamic properties of these derived results are shown in some three-dimensional, density and 2D plots. Traveling wave solutions are utilized to depict water waves in the mentioned equations, and that is used in physical science to model quantum field theory, dust-acoustic waves, ion acoustic waves, which is taken into account through the application of the improved tan(ϕ/2)-expansion technique. In addition, the recommended technique allowed us to produce some dynamical wave patterns of kink, kink single soliton, single soliton, compacton, periodic shape and other structures are developed, which are shown using three-dimensional, density and 2D plots to more clearly illustrate the physical layout. The method is one of the proficient and effective approaches which have swiftly developed in order to searching appropriate responses to partial differential equations with nonlinear sciences.