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 $\exp (-\varphi (\xi ))$-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.

In this paper, extended trial equation method (ETEM) is applied to find exact solutions of (1+1) dimensional nonlinear Ostrovsky equation. We constitute some exact solutions such as soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of this equation via ETEM. Then, we submit the results obtained by using this method.

The current manuscript is concerned with extracting an analytical approximate periodic solution of a damped cubic nonlinear Klein-Gordon equation. The Riemann-Liouville fractional calculus is utilized to obtain an analytic approximate solution. The Homotopy technique is absorbed in the multiple time-spatial scales. The approved scheme yields a generalization of the Homotopy equation; whereas, two different small parameters are adapted. The first parameter concerns with the temporal perturbation, simultaneously, the second one is accompanied by the spatial one. Therefore, the analytic approximate solution needs the two perturbation expansions. This approach conducts more advantages in handling the classical multiple scales method. Furthermore, the initial conditions are included throughout the multiple scale method to achieve a special solution of the governing equation of motion. The analysis ends up deriving two first-order equations within the extended variables and their actual solution is achieved. The procedure adopted here is very promising and powerful in managing similar numerous nonlinear problems arising in physics and engineering. Furthermore, the linearized stability of the corresponding ordinary Duffing differential equation is analyzed. Additionally, some phase portraits are shown.