Analysis of soliton solutions with different wave configurations to the fractional coupled nonlinear Schrödinger equations and applications

In this research, we address the problem of solving (1 + 1)-dimensional fractional coupled nonlinear Schrödinger equations (FCNLSE) with beta derivatives, which are essential for understanding wave dynamics in various physical systems. These equations have significant importance in practical applications, particularly in the design of optical fiber networks, signal processing, and control systems, where precise modeling of wave behavior is crucial. To tackle this problem, we employ two powerful mathematical methodologies, the modified exponential function method and the rational exp(\(-\varphi\)(\(\eta\)))-expansion method. These methods are known for their ability to provide accurate analytical solutions to fractional nonlinear physical models, making them invaluable tools for solving complex mathematical problems. The growing popularity of fractional nonlinear partial differential equations stems from their versatile applicability, which extends to diverse domains of science and engineering. To approach the FCNLSE problem, we leverage a well-suited fractional complex wave transformation, effectively translating the original equation into a more tractable ordinary differential equation. This transformation sets the stage for the discovery of a wide range of solutions that encompass compactons, periodic cross-kink structures, peakons, as well as rational and cuspons solutions. These solutions are expressed in terms of rational, hyperbolic, trigonometric, and exponential functions, providing a rich mathematical tapestry to analyze and interpret. To enhance our comprehension of the physical significance of these solutions, we employ advanced visualization techniques, including the generation of three-dimensional, two-dimensional, and contour plots. These graphical representations offer a vivid insight into the dynamic behavior of the obtained solutions. Our findings not only emphasize the precision and effectiveness of the applied methodologies but also contribute significantly to the understanding of various physical phenomena. These novel solutions extend beyond previous efforts in the literature by introducing beta derivatives as a modeling tool for FCNLSE. Additionally, we uncover solution types that have not been previously reported, such as periodic cross-kink structures, expanding the landscape of possible solutions for FCNLSE.