Dynamics of multiple higher-order pole solutions of a two-component Sasa-Satsuma equation based on Riemann–Hilbert approach and PINN algorithm

Abstract

Using the inverse scattering transform, we systematically present a Riemann–Hilbert (RH) approach to the Cauchy problem for a two-component Sasa–Satsuma equation with zero boundary conditions as $\left|x\right|\to \infty$. The direct scattering problem establishes the analyticity, symmetries of the Jost solutions and scattering matrix, and particularly, a detailed analysis of the discrete spectrum in the presence of multiple higher-order zeros of the determinant of analytic scattering coefficient. The inverse scattering problem is addressed through a corresponding $4\times 4$ matrix-valued RH problem associated with the residual conditions of these higher-order poles. The reconstruction formula for the solution is derived. Then, under reflectionless conditions, we settle the RH problem which can be transformed into a linear algebraic system, and obtain the formula of multiple higher-order pole solutions to the two-component Sasa–Satsuma equation. Subsequently, we employed the data obtained through the RH method to train the physics-informed neural network (PINN) and an improved version of PINN, enabling us to derive various types of data-driven solutions for the two-component Sasa–Satsuma equation. The results demonstrate that the algorithm shows outstanding training performance for both one-soliton and two-soliton solutions, as well as for more complex second-order and third-order pole solutions. Finally, we explore the data-driven parameters discovery problem in the context of one-soliton solution using the PINN algorithm. These findings highlight the robustness of the PINN framework in handling complex nonlinear equations and its potential for applications in various fields, including fluid dynamics and optical communications.