Optical Solitons and Analysis of Chaotic Nature for the Temporal M‐Fractional Yajima–Oikawa Model in Shortwave and Longwave

This study proposes a comprehensive study on fractional soliton solutions and chaotic nature for the temporal M‐fractional Yajima–Oikawa (YO) model in shortwave and longwave regimes. Utilizing the new Jacobian elliptic function method, the optical soliton solutions are examined with diverse categories, including kinky‐periodic wave, kink with bell wave, breather wave, bright bell wave, periodic multibreather wave, interaction of kink and bell wave, antikinky periodic wave, double periodic wave, interaction between antikink dark bell wave, and dark bell wave. Furthermore, the impact of the temporal M‐fractional parameter on the obtained soliton solutions is examined and compared with its original form. There is no previous documentation of these soliton structures in the existing literature, making them novel contributions. First, the chaotic nature of the YO model is investigated using two‐ and three‐dimensional phase portraits, and its sensitivity to initial conditions is also presented. This behavior is essential for understanding modulation, stability, and energy transfer in nonlinear media. It also sheds light on real‐world phenomena like wave turbulence, optical pulse propagation, and fluid and plasma system instability. To demonstrate the dynamic characteristics of the derived solutions, three‐dimensional surface plots and 2D wave profiles are produced utilizing Maple. This study manifests the aspects of energy localization and modulation instability in complex media, which were illuminated by the M‐fractional parameter’s considerable effects on wave propagation and stability compared with classical models. The results validate these approaches’ efficacy, simplicity, and versatility, illustrating their potential applicability in addressing additional nonlinear partial differential equations (PDEs) pertinent to optical wave dynamics. Moreover, this study contributes some new phenomena to advance the concept of nonlinear optical research and communication technology.