Dynamic analysis on optical pulses via modified PINNs: Soliton solutions, rogue waves and parameter discovery of the CQ-NLSE

Abstract

Under investigation in this paper is the cubic–quintic nonlinear Schrödinger equation, which describes the propagation of optical on resonant-frequency fields in the inhomogeneous fiber. According to abundant previous researches on the model, exact soliton solutions and rogue wave solutions have been derived through Darboux transformation. The modulation instability phenomenon has been analyzed to evaluate the ability of an initially perturbed plane wave to split into localized energy packets when propagating in a dispersive and nonlinear medium.

Numerical solutions with high accuracy are needed in fields of production and engineering. Nonetheless, the data acquisition costs of the optical pulse transmission system is high, which will limit the accuracy and the efficiency of typical numerical and data-driven methods. With the physical knowledge embedded into neural networks in the form of loss function, the problem of big data dependence has been solved. For dynamic analysis on optical pulses with small amount of known information, we strive to obtain high accuracy numerical solutions. Considering the case that the cubic–quintic nonlinear Schrödinger equation is converted to the Kundu–Eckhaus equation with simplified coefficient constraints through variable transformation, we construct modified physics-informed neural networks, where conversions on the input and output are attached to deep neural networks. Training networks with the given initial and boundary data, we effectively derive the expected soliton and rogue wave solutions, where the approximated one-soliton, two-soliton, first-order and second-order rogue waves are included. In general, the modified network reaches low prediction errors with small data available. With the coefficients of equations, the weights and the bias of networks combined as parameters to be trained, we deduce the corresponding value of condition settings for different systems. Moreover, we simulate diverse localized waves in the context of nonlinear electrical transmissions with different environment settings and compare the evolution process to reach conclusions on the parameter discovery.