Dynamical Study of Nonlinear Fractional-order Schrödinger Equations with Bifurcation, Chaos and Modulation Instability Analysis

Our research on fractional-order nonlinear Schrödinger equations (FONSEs) reveals several new findings, which may contribute to our comprehension of wave dynamics and hold practical importance for the field of ocean engineering. We have employed an innovative approach to derive double periodic Weierstrass elliptic function solutions for FONSEs, thereby offering exact solutions for these equations. Additionally, we have observed that fractional derivatives significantly impact the dynamics of solitary waves, potentially holding significance for the design of ocean structures. Our research reveals the previously unknown phenomenon of oblique wave variations, which can impact the reliability and lifespan of offshore structures. Our findings highlight the significance of taking into account various fractional derivatives in future studies. Using the planar dynamical system technique, we gain a deeper understanding of the behavior of FONSEs, revealing critical thresholds and regions of chaotic behavior. Linear stability analysis provides a strong framework for studying the modulation instability of dynamical systems, shedding light on the conditions and mechanisms of modulated behavior. Applying this analysis to the FONSEs offers insights into the critical parameters, growth rates, and formation of modulated patterns, with potential implications for innovative research in ocean engineering.