Narrow Results

The aim of this study is to give a deep investigation into the dynamics of the simplified modified Camassa–Holm equation (CHe) for shallow water waves. Taking advantage of the semi-inverse method, we develop the variational principle, based on which the Hamiltonian of the system is extracted. By means of the Galilean transformation, the governing equation is transformed into a planar dynamical system. Then, the bifurcation analysis is presented via employing the theory of the planar dynamical system. Correspondingly, the quasi-periodic and chaotic behaviors of the system are also discussed by introducing two different kinds of perturbed terms. Finally, the variational method is based on the variational principle and Ritz method, and the Kudryashov method is used to construct the diverse solitary wave solutions, which include the bright solitary, dark solitary, kink solitary and the bright–dark solitary wave solutions. The graphic depictions of the obtained diverse solitary wave solutions are presented to elucidate the physical properties. The findings of this research enable us to gain a deeper understanding of the nonlinear dynamic characteristics of the considered equation.

The Kudryashov technique is employed to extract several classes of solitary wave solutions for the Joseph–Egri equation. The stability of the achieved solutions is tested. The numerical solution of this equation is also investigated. We also present the accuracy and the stability of the numerical schemes. Some two- and three-dimensional figures are shown to present the solutions on some specific domains. The used methods are found useful to be applied on other nonlinear evolution equations.

In this paper, time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations have been solved numerically utilizing the Kansa method, in which the multiquadrics are taken as radial basis function. To attain this, a numerical scheme based on finite difference and Kansa method has been proposed. In addition, the soliton solutions have been obtained by employing Kudryashov method and tanh method for comparison purpose with the obtained numerical solutions. The numerical examples are given to demonstrate the accuracy and applicability of the proposed method.

In this paper, a novel meshless numerical scheme to solve the time-fractional Oskolkov–Benjamin–Bona–Mahony–Burgers-type equation has been proposed. The proposed numerical scheme is based on finite difference and Kansa-radial basis function collocation approach. First, the finite difference scheme has been employed to discretize the time-fractional derivative and subsequently, the Kansa method is utilized to discretize the spatial derivatives. The stability and convergence analysis of the time-discretized numerical scheme are also elucidated in this paper. Moreover, the Kudryashov method has been utilized to acquire the soliton solutions for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme.

In this paper, Kudryashov and modified Kudryashov methods are implemented for the first time to compute new exact traveling wave solutions of the space-time fractional $(3+1)$-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation and Calogero–Bogoyavlenskii–Schiff and Bogoyavlensky Konopelchenko (CBS-BK) equation. With the help of wave transformation, the aforementioned fractional differential equations are converted into nonlinear ordinary differential equations. The purpose of this paper is to devise novel exact solutions for the space-time-fractional $(3+1)$-dimensional CBS and the space-time-fractional CBS-BK equations by utilizing the Kudryashov and modified Kudryashov techniques. The solutions, thus, acquired are demonstrated in figures by choosing appropriate values for the parameters. The solutions derived take the form of various wave patterns, including the kink type, the anti-kink type and the singular kink wave solutions. The obtained solutions are indeed beneficial to analyze the dynamic behavior of fractional CBS and CBS-BK equations in describing the interesting physical phenomena and mechanisms. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. The techniques presented here are very simple, efficacious and plausible and hence can be employed to attain new exact solutions for fractional PDEs.

In this paper, we present the exact solutions of the Majda–Biello system. This system describes the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves with a substantial midlatitude projection, in the presence of suitable horizontally and vertically sheared zonal mean flows. The methods used to construct the exact solutions are the Kudryashov method and Jacobi elliptic function method. These two methods yield solitary wave solutions and periodic wave solutions.

This paper deals with the numerical solution of the time-fractional Gilson–Pickering equation using the Kansa method, in which the multiquadrics were utilized as the radial basis function. To achieve this, a meshless numerical scheme based on the finite difference along with the Kansa method has been presented. First, the finite difference approach has been utilized to discretize the temporal derivative, and subsequently, the Kansa method is employed to discretize the spatial derivatives. The stability and convergence analysis of the numerical scheme are also elucidated in this paper. Furthermore, the soliton solutions have been acquired by implementing the Kudryashov method for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme.

In this paper, we apply two powerful methods, namely, the new extended auxiliary equation method and the generalized Kudryashov method for constructing many exact solutions and other solutions for the higher order dispersive nonlinear Schrödinger’s equation to secure soliton solutions in quadratic-cubic medium. Various solutions of the resulting nonlinear ODE are obtained by using the above two methods.