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The aim of this work is to investigate new exact solutions of Tzitzéica type equations. We utilize the Painlevé transformation to transform the aforesaid non-linear evolution equations into ordinary differential equations. Then, the simplest equation method is employed for securing some real and complex solutions of the Tzitzéica equation, the Tzitzéica–Dodd–Bullough equation and the Dodd–Bullough–Mikhailov equation. After the execution of the simplest equation method, we obtain many new results more simply and reliably than the other approaches executed on these equations. The solutions are obtained and verified through soft computations. Also, the dynamics of some solutions are presented via three types of graphs including 2D, 3D and contour graphs. [ABSTRACT FROM AUTHOR]

In this paper, we have established the new exact solitons solutions of the nonlinear Biswas–Arshed equation which are used to study soliton dynamics in optical fibers, pivotal cloud foundry (PCF) and metamaterials via the prominent approach, known as auxiliary equation method. The obtained solutions disclosed that the suggested method is the essential addition for studying the exact solution of nonlinear Biswas–Arshed equation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the interaction of the rational function, hyperbolic as well as exponential function for the (3+1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation with the aid of Hirota's bilinear approach. We find several families of lump-type solutions. This method is a powerful and advantageous mathematical tool for establishing abundant lump solutions of nonlinear partial differential equations. In order to illustrate their dynamic properties, some figures are plotted with determined parameters. [ABSTRACT FROM AUTHOR]

In this work, the researchers for computing an exact solution of the (2 + 1) dimensions Kundu–Mukherjee–Naskar (KMN) equation used a newly developed technique, namely, the new extended direct algebraic method. This powerful method gives an exact solution to the KMN equation considered in this paper. The results are new and attest to the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we use the new defined direct algebraic method based on some particular Riccati equations to find exact solutions to a Kaup–Newell model equation. Several new solutions which represent long waves parallel to the magnetic fields have been obtained. Many solutions in generalized hyperbolic and triangular function forms, exponential or logarithmic, are expressed explicitly. Most of the solutions determined in the study are new in the related literature. [ABSTRACT FROM AUTHOR]

In this paper, the fractional reduced differential transform method (FRDTM) is used to obtain the series solution of time-fractional seventh-order Sawada–Kotera (SSK) and Lax's KdV (LKdV) equations under initial conditions (ICs). Here, the fractional derivatives are considered in the Caputo sense. The results obtained are contrasted with other previous techniques for a specific case, α = 1 revealing that the presented solutions agree with the existing solutions. Further, convergence analysis of the present results with an increasing number of terms of the solution and absolute error has also been studied. The behavior of the FRDTM solution and the effects on different values α are illustrated graphically. Also, CPU-time taken to obtain the solutions of the title problems using FRDTM has been demonstrated. [ABSTRACT FROM AUTHOR]