Investigating exact solutions, sensitivity, and chaotic behavior of multi-fractional order stochastic Davey–Sewartson equations for hydrodynamics research applications

Our research aims to explore the fractional derivative of the Stochastic Davey–Stewartson equation (FDSDSEs) and make contributions in seven aspects: Firstly, we discovered previously unknown exact solutions for FDSDSEs, including Weierstrass elliptic function solutions. These findings provide new insights for our understanding and analysis of these equations. Secondly, we analyzed the effect of noise on stochastic solutions. We found that higher noise intensity can disrupt patterns and result in flatter surfaces. However, we observed that multiplicative noise can stabilize the solution to some extent. This reveal the role of noise in stochastic dynamics and has important implications for wave behavior in the ocean environment. Thirdly, we emphasized the impact of fractional derivatives on noise dynamics. Our research indicates the importance of considering the order of derivatives α in analysis and modeling. Different order derivatives have different effects on noise, which is significant for ocean researchers in selecting appropriate derivatives to study stochastic systems. Fourthly, we compared the effects of different fractional derivatives on noise in stochastic solutions. We found that choosing appropriate derivative orders can more accurately describe wave behavior in the ocean environment. This discovery has important implications for improving wave models and predicting ocean waves. Fifthly, we studied the influence of tilted wave variations in FDSDSE solutions. We observed spiral-like patterns, providing new insights into wave propagation in the ocean. This is significant for understanding wave behavior in the ocean and relevant ocean engineering applications. Sixthly, we explored the phase diagrams and chaotic behavior of FDSDSE through sensitivity analysis and perturbation factors. This enhances our understanding of practical applications and control strategies in ocean hydrodynamics research and is also important for delving into the dynamic behavior and phase diagrams of these equations. Finally, we compared our research results with well-known wave phenomena in the ocean through visualization, highlighting the practical relevance of our study. These comparisons involve random propagation of ocean solitary waves, twisting propagation of ocean waves, and rotating propagation of solitary waves, providing some inspiration for advancing the fields of ocean science and engineering.