Data-driven parametric soliton-rogon state transitions for nonlinear wave equations using deep learning with Fourier neural operator

Abstract

Abstract In this paper, we develop the deep learning-based Fourier neural operator (FNO) approach to find parametric mappings, which are used to approximately display abundant wave structures in the nonlinear Schrödinger (NLS) equation, Hirota equation, and NLS equation with the generalized   -symmetric Scarf-II potentials. Specifically, we analyze the state transitions of different types of solitons (e.g. bright solitons, breathers, peakons, rogons, and periodic waves) appearing in these complex nonlinear wave equations. By checking the absolute errors between the predicted solutions and exact solutions, we can find that the FNO with the GeLu activation function can perform well in all cases even though these solution parameters have strong influences on the wave structures. Moreover, we find that the approximation errors via the physics-informed neural networks (PINNs) are similar in magnitude to those of the FNO. However, the FNO can learn the entire family of solutions under a given distribution every time, while the PINNs can only learn some specific solution each time. The results obtained in this paper will be useful for exploring physical mechanisms of soliton excitations in nonlinear wave equations and applying the FNO in other nonlinear wave equations.