Some Novel Optical Pulses in Hydrodynamical Nonlinear Complex Equation using M-Truncated Fractional Derivative

Abstract

This study investigates soliton solutions and dynamic wave structures in the complex Ginzburg-Landau (CGL) equation, which is crucial for understanding wave propagation in various physical systems. We employ three analytical methods: the Kumar-Malik method, the generalized Arnous method, and the energy balance method to derive novel closed-form solutions. These solutions exhibit diverse solitonic phenomena, including multi-wave solitons, complex solitons, singular solitons, periodic oscillating waves, dark-wave, and bright-wave profiles. Our results reveal new families of exact solitary waves via the generalized Arnous method and diverse soliton solutions through the Kumar-Malik method, including hyperbolic, trigonometric, and Jacobi elliptic functions. Verification is ensured through back-substitution to the considered model using Mathematica software. Additionally, we plot the various graphs with the appropriate parametric values under the influence of the M-truncated fractional derivative to visualize the solution behaviors with varying parameter values. This research contributes significantly to understanding wave dynamics in physical oceanography, and the unique outcomes explored in this research will play a vital role for the forthcoming investigation of nonlinear equations.