Nonlinear stochastic wave behavior: optical soliton solutions and energy analysis of dispersive Maccari system via hybrid analytical methods

The nonlinear fractional Maccari system (NLFMS) under study is presented in this paper to derive exact solitary wave solutions via EDAM and NPREM. The Maccari system depicts the nonlinear interaction of several short-waves with a long-wave field in a dispersive environment. In terms of physical phenomena, the Maccari system entails three wave packets of rapidly traveling waves and a slowly varying long wave in a real-valued background field. The short-wave components interact with the long wave through the interaction of their energies, and at the same time, the long wave interferes with the short-wave propagation behavior. These interactions result in the formation of localized waves, wavefront modulation, and interference patterns, which play crucial roles in soliton theory and nonlinear wave mechanics. The analysis is done by employing the concept of the Riemann-Liouville fractional derivative, along with appropriate traveling wave transformations, to convert the fractional partial differential equation model into an ordinary differential equation. Through the aid of the Mathematica software program, various solitonic structures such as kink, anti-kink, lump, JEF, and WEF solutions are derived from both EDAM and NPREM. For investigating the physical admissibility and stability of the derived wave structures, the Energy Balance Method (EBM) is applied. It is shown that the derived waves have the property of conserving energy and hence are physically relevant and stable. In contrast to perturbation methods, the EBM approach analyzes the nonlinear dynamics of waves without any simplification assumptions.