Exploring modulation instability, PINN-predicted solutions, and dynamic behavior of the Hirota equation

The Hirota equation serves as a significant integrable mathematical model for nonlinear phenomena, and its study is of great significance in fields such as nonlinear optics and fluid dynamics. In this paper, we numerically solve the Hirota equation using a physics-informed neural network (PINN) and conduct a modulation instability analysis, successfully determining the parameter range of unstable waves and providing a theoretical basis for the existence of rogue waves. PINN embeds the governing equations as soft constraints directly into the neural network and minimizes the loss function to enable the PINN model to learn solutions that conform to physical laws from limited training data. In this study, the one-, two-, and three-soliton solutions as well as the first- and second-order rogue wave solutions of the Hirota equation are successfully simulated using PINN. The PINN model exhibits high accuracy in predicting the complex dynamics of the Hirota equation, and the predicted waveforms and propagation characteristics are highly consistent with the corresponding exact analytical solutions. The numerical solutions obtained using PINN provide a new approach for analyzing and solving equations with complex interactions or general initial and boundary conditions. This study confirms that PINN, as a dataefficient and meshless computational tool, can be utilized to compute numerical solutions of nonlinear partial differential equations and investigate the dynamic structures inherent in complex nonlinear equations, providing a highly promising computational method for numerical research in nonlinear science.