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The primary objective of this study is to examine the behavior of the nonlinear Kaup-Newell equation. By employing the modified Kudryashov method and extended tanh function method, we have successfully derived novel solitary wave and periodic solutions. These new solutions are presented in trigonometric, hyperbolic and exponential function types. The proposed two approaches are efficient, direct and fascinating. These newly discovered solutions are illustrated using two-dimensional and three-dimensional graphs, incorporating suitable parameters values. These graphs are crucial for elucidating the dynamic properties of optical fibers. [ABSTRACT FROM AUTHOR]

The coupled Boussinesq equation (CBE) acts a key role in modeling the shallow water waves of two-layer fluid flow. However, it becomes powerless for the nonsmooth boundary. So, a fractal modification of the CBE is suggested in this paper. Aided by the semi-inverse method, we successfully construct its fractal generalized variational principle. In addition, three recent techniques, namely, the Sardar-subequation method, He's frequency formulation method and subequation method, combined with the two-scale fractal dimension transform, are utilized to find abundant wave solutions. By these methods, various solutions expressed by the generalized hyperbolic functions, generalized trigonometric functions, hyperbolic functions and cosine function are obtained. With the aid of the mathematical software, the three-dimensional contours and two-dimensional curves are plotted to present the dynamic behaviors of the solutions. The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the fractal nonlinear evolution equations. [ABSTRACT FROM AUTHOR]