Probing wave dynamics in the modified fractional nonlinear Schrödinger equation: implications for ocean engineering

The nonlinear Schrödinger equation is used to model various phenomena, such as solitons self-focusing effects and rogue waves. In the ocean engineering, the modified nonlinear Schrödinger equation investigates the behavior of water waves, considering the complex interaction of dispersion nonlinearity, and dissipation effects. By introducing fractional derivatives to the model, the M-fractional conformable modified nonlinear Schrödinger equation allows for the investigation of fractional order effects, which can study more accurately the behavior of wave propagation in real-world ocean engineering. The novelty of our research lies in the application of of the M-fractional conformable derivative on the governed equation which represents an advancement in the existing work, which have used nonlinear Schrödinger equations without fractional derivatives. Two powerful techniques: the Jacobi elliptic function method and unified solver method are applied to attain solutions to the M-fractional modified nonlinear Schrödinger equation. The several results, including dark, bright, singular, periodic, and dark-bright soliton solutions are obtained which provide valuable insights into the complex behavior of water waves in ocean engineering. Additionally, 3D and contour graphs have been provided to visually illustrate the impact of the fractional order. We also illustrate these solutions at different values of the fractional order which explain how variations in this parameter affect wave propagation. These findings will contribute to the advancement of ocean engineering techniques, enhancing our ability to design and implement effective solutions for coastal protection, offshore structures, and marine renewable energy systems.