Riemann-hilbert problem and physics-informed neural networks method for the nonlocal Sasa-Satsuma equation

This paper investigates N-soliton and data-driven solutions of a novel nonlocal Sasa-Satsuma (SS) equation by applying the Riemann-Hilbert problem and the physics-informed neural networks (PINN) method. Based on the zero curvature formulation with arbitrary order spatial and temporal spectral matrixes, the novel nonlocal SS equation is constructed from the nonlocal integrable SS hierarchies possessing the bi-Hamiltonian structures and the Liouville integrability. Analyzing properties of the Jost matrix, the Riemann-Hilbert problem and some novel symmetry constraints are derived for obtaining the N-soliton solutions of the nonlocal SS equation. Moreover, the dynamic characteristics of these one- and two-soliton solutions are visually displayed in some figures. Finally, the data-driven solutions of the nonlocal SS equation are availably learned via the PINN approach combining with the spatial and temporal nonlocal terms. And the results show the error range between the predicted data-driven solutions and the exact solutions, which indicate the effectiveness of the method.