Analysis of nonlinear Schrödinger equations with anticubic nonlinearity in optical fiber systems via physics informed neural network (PINN)

Abstract

The higher order nonlinear phenomena such as the anticubic nonlinearity have a major influence on the evolution, stability and localization of optical pulses and play a significant role in the evolution of nonlinear optical waves in dispersive media. Therefore, effective numerical approaches for solving these nonlinear evolution equations are required to understand the intricate wave dynamics in optical fiber communication systems. In this work, a physics-informed neural network (PINN) architecture based on the DeepXDE library is developed to solve the non-linear Schrödinger equation with the anti-cubic nonlinearity. The proposed framework employs a fully connected neural network and directly integrates the underlying physical rules into the learning process without any meshing discretization. We explore the robustness of the proposed model using six representative initial conditions such as hyperbolic tangent, sine, cosine, Gaussian, super-Gaussian and hyperbolic secant profiles and study the effect of inverse nonlinear interaction parameter 𝑏 1 on the wave dynamics systematically. The numerical results show that the proposed PINN can well capture the spatiotemporal evolution of confined and periodic nonlinear wave structures. Contour and 3D surface graphs exhibit the effect of 𝑏 1 on the localization of the wave and its amplitude modulation. The numerical findings show that, the proposed PINN effectively captures the spatiotemporal evolution of constrained periodic nonlinear wave structures. Graphs of contours and three-dimensional surfaces reveal how 𝑏 1 affects the localization and amplitude modulation of the wave. The broad generalizability and steady learning tendency of the recommended framework are supported by consistent outcomes from the convergence experiment. An additional validation of the PINN prediction was conducted independently, employing the identical parameters, computational domain, governing equation, and boundary conditions as a conventional finite-difference (FD) numerical solution for a standard Gaussian initial condition. Since the PINN and FD intensity profiles agree well at 𝑡 = 2 , there is more evidence that the proposed framework is trustworthy. The results demonstrate that the proposed PINN is an effective and robust mesh-free computing approach for nonlinear wave development, and it finds applications in nonlinear optics, optical fiber communication, and other nonlinear systems.