Analytical soliton solutions and physics-informed neural network validation for the generalized higher-order nonlinear Schrödinger equation

Abstract

This paper examines the generalized higher-order nonlinear Schrödinger equation (GHONLSE), which is an underlying model used to investigate the wave propagation in nonlinear dispersive media such as optical fibers and plasma systems, in which the higher-order dispersion and nonlinear interaction effects are subject to important influence on the wave dynamics. The GHONLSE, which builds on the classical nonlinear Schrödinger model, can support a wide range of nonlinear wave modes, such as localized solitons, periodic waves, and breather-type oscillatory waves. Analytical solutions of the governing equation with accurate traveling-waves are obtained by building on the extended generalized Riccati equation mapping method, which offers a systematic method to obtain several families of analytical solutions with different morphological properties. Five representative solutions are chosen out of such families to represent a wide variety of dynamical behaviors, such as smooth localized solitons, periodic oscillations, and multi-peak structures with strong nonlinear interactions. In order to confirm and further study such analytical solutions, a physics-informed neural network (PINN) framework is designed and run in MATLAB, in which the governing GHONLSE is directly incorporated into the loss function, which allows the neural network to be trained to learn solutions that satisfy the differential equation and the specified analytical constraints. The comparison of the analytical solutions and the PINN reconstructions shows high correlation in terms of the morphology and amplitude distributions of the waves and propagation characteristics in the computational domain. In the error analysis, neural approximation is highly accurate to smooth and periodic structures, and smaller error is found near the steep gradient or local spike areas, which is naturally more difficult to represent in neural representation. The convergence behavior of the training also proves the effective and stable learning of the nonlinear dynamics of the GHONLSE. In general, the findings form a single analytical-computational model that combines the precise solution methodologies with physics-aware machine learning to offer a strong and physically sound methodology in examining pertinent nonlinear wave behavior in high-order nonlinear evolution equations.