Novel traveling wave solutions of generalized seventh-order KdV equation and related equation

In this paper, we acquire novel traveling wave solutions of the generalized seventh-order Korteweg–de Vries equation and the seventh-order Kawahara equation as a special case with physical interest. Primarily, we use the advanced exp(-φ(ξ))-expansion method to find new exact solutions of the first equation, by considering two auxiliary equations. Then, we attain some exact solutions of the seventh-order Kawahara equation by using this method with another auxiliary equation, and also using the modified (G/G) -expansion method, where G satisfies a second-order linear ordinary differential equation. Additionally, utilizing the recent scientific instruments, the 2D, 3D, and contour plots are displayed. The solutions obtained in this paper include bright solitons, dark solitary wave solutions, and multiple dark solitary wave solutions. It is shown that these two methods provide an effective mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering.