This manuscript investigates the fractional Phi-four equation by using $q$ -homotopy analysis transform method ( $q$ -HATM) numerically. The Phi-four equation is obtained from one of the special cases of the Klein-Gordon model. Moreover, it is used to model the kink and anti-kink solitary wave interactions arising in nuclear particle physics and biological structures for the last several decades. The proposed technique is composed of Laplace transform and $q$ -homotopy analysis techniques, and fractional derivative defined in the sense of Caputo. For the governing fractional-order model, the Banach’s fixed point hypothesis is studied to establish the existence and uniqueness of the achieved solution. To illustrate and validate the effectiveness of the projected algorithm, we analyze the considered model in terms of arbitrary order with two distinct cases and also introduce corresponding numerical simulation. Moreover, the physical behaviors of the obtained solutions with respect to fractional-order are presented via various simulations.

In this paper, we employ the sine-Gordon expansion method to shallow water wave models which are Kadomtsev-Petviashvili-Benjamin-Bona-Mahony and the Benney-Luke equations. We construct many new complex combined dark-bright soliton, anti-kink soliton solutions for the governing models. The 2D, 3D and contour plots are given under the suitable coefficients. The obtained results show that the approach proposed for these completely integrable equations can be used effectively.

This paper presents many new complex combined dark-bright soliton solutions obtained with the help of the accurate sine-Gordon expansion method to the B-type Kadomtsev-Petviashvili-Boussinesq equation with binary power order nonlinearity. With the use of some computational programs, we plot many new surfaces of the results obtained in this paper. In addition, we present the interactions between complex travelling wave patterns and their solitons.

This paper investigates the coupled nonlinear Hirota–Maccari system with the help of using an analytical approach, which is the extended sinh-Gordon equation expansion method (ShGEEM). Complex, solitary, and singular periodic traveling solutions are successfully gained to the nonlinear model considered. The constraint conditions that validate the existence of the reported soliton solutions are also given in a detailed manner. The two-dimensional (2D), three-dimensional, and contour graphs to some of the obtained solutions are presented via several computational programs. These simulations present deeper investigations about the wave distributions of the coupled nonlinear Hirota–Maccari system.

In this article, some new travelling wave solutions of the (3+1) dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation are obtained using the modified exponential function method. When the solution functions obtained are examined, it is seen that functions with periodic functions are obtained. Two and three dimensional graphs of the travelling wave solutions of the BLMP equation are drawn by selecting the appropriate parameters