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.

A direct rational exponential scheme is offered to construct exact multi-soliton so-lutions of nonlinear partial differential EQUATIONs. We have considered the Calogero-Bogoyavlenskii-Schiff EQUATION and KDV EQUATION as two concrete examples to show efficiency of the method. As a result, one wave, two wave and three wave soliton solutions are obtained. Corresponding potential energy of the soliton are found. Fur-thermore, three-dimensional plots of the wave solutions and its potential functions are given to visualize the dynamics of the model and its energy. We also provided the corresponding density plot of the solutions to understand the real direction and par-ticles density in the waves which help to realize the elastic situations of the achieved solutions.

In this paper, we use the rational radial basis functions ( RRBFs) method to solve the Korteweg-de Vries (KDV) EQUATION, particularly when the EQUATION has a solution with steep front or sharp gradients. We approximate the spatial derivatives by RRBFs method then we apply an explicit fourth-order Runge-Kutta method to advance the resulting semi-discrete system in time. Numerical examples show that the presented scheme preserves the conservation laws and the results obtained from this method are in good agreement with analytical solutions.

In this paper, an efficient meshfree collocation scheme based on meshfree radial basis function is implemented for the numerical solution of 7th-order Korteweg-de Vires (KDV) EQUATIONs. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. The proposed numerical scheme is tested on several test problems. The efficiency and accuracy of the suggested scheme is analyzed via ||L|| and ||L||2 error norms.

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKDV) EQUATION with ansatz method and Galerkin , nite element method based on cubic B-splines over , nite elements. Propagation of single solitary wave is investigated to show the e, ciency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L1 error norms. Also, three invariants I1,I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann's theory is surveyed and indicated that our method is unconditionally stable.