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In this work, a generalized (G ′ / G) -expansion method has been used for solving the nonlinear Chen–Lee–Liu equation. This method is a more common, general, and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations (NPDEs), where G = G (τ) follows the Jacobi elliptic equation [ G ′ (τ) ] 2 = H (G) , and we let H (G) be a fourth-order polynomial. Many new exact solutions such as the hyperbolic, rational, and trigonometric solutions with different parameters in terms of the Jacobi elliptic functions are obtained. The distinct solutions obtained in this paper clearly explain the importance of some physical structures in the field of nonlinear phenomena. Also, this method deals very well with higher-order nonlinear equations in the field of science. The numerical results described in the plots were obtained by using Maple. [ABSTRACT FROM AUTHOR]

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 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]