Computational soliton solutions for the fractional nonlinear dynamical model arising in water wave

This manuscript is dedicated to the comprehensive exploration of solitary wave solutions for the fractional couple Drinfeld-Sokolov-Wilson equation, which is a versatile mathematical model that finds applications in various branches of physics, including nonlinear acoustics and fluid mechanics. The new extended direct algebraic method is employed as a powerful analytical tool throughout the study. A general algorithm that is essential for the analysis of the models stated is introduced in the manuscript. The travelling wave transformation is used to convert these models into ordinary differential equations, which makes the analysis easier to handle. The study yields a diverse set of solitary wave solutions in the form of dark, dark-bright, bright-dark, singular, periodic, mixed trigonometric, and rational forms. Also, by using the Hamiltonian property, validation of the solutions is conducted, which confirms the accuracy and stability of segregated solitary wave solutions. The discovered results are provided not only in numerical form but also with insightful physical interpretations, which contribute to a deeper comprehension of the complex dynamics these mathematical models depict. The utilization of the new extended direct algebraic method and the broad spectrum of obtained solutions contribute to the depth and significance of this research in the field of nonlinear wave equations.