Data-driven solutions and parameter discovery of the extended higher-order nonlinear Schrödinger equation in optical fibers

In this paper, we investigate an extended higher-order nonlinear Schr & ouml;dinger equation that includes third- order and fourth-order dispersion terms. By considering higher-order effects, the equation can more accurately simulate and predict the propagation behavior of ultrashort pulses in optical fibers, e.g., pulse splitting, compression and stability. In the optical fiber sensors, the sensitivity and resolution of the sensor can be improved by utilizing nonlinear effects. The analysis of the extended higher-order nonlinear Schr & ouml;dinger equation helps to understand how nonlinear effects affect the sensing signal, thereby designing more accurate sensors. Specific coefficients of each term in the equation are studied for different scenarios, such as optical fiber design and signal propagation in optical communications. Under certain conditions, the extended higher-order nonlinear Schr & ouml;dinger equation can be degenerated into the Hirota equation or the Lakshmanan- Porsezian-Daniel equation. With the development of numerical algorithms, the utilization of deep neural networks for solving equations has become increasingly attractive. With given initial conditions and boundary conditions, the physical information is embedded into the neural network in the form of loss functions. We efficiently derive numerical solutions to the extended equation for one-soliton, two-soliton and rogue-wave cases by training weights and biases in physics-informed neural networks. Compared with traditional numerical methods, physics-informed neural networks yield lower prediction errors with less data volume, thus will reduce the cost and time consumptions for data sampling in practical engineering. Furthermore, we solve the inverse problem associated with the extended equation. This approach is critical for extracting fundamental parameters of optical systems from observable data, thereby deepening our understanding of complex optical behavior and helping to improve the performance of optical fiber communication technologies.