Beijing Institute of Mathematical Sciences and Applications

This course focuses on the rigorous analysis of long-time asymptotic behavior for one of the most fundamental integrable models—the nonlinear Schrödinger (NLS) equation. Building upon the inverse scattering transform, the course systematically introduces the Deift–Zhou nonlinear steepest descent method and develops a comprehensive analytical framework based on the deformation and analysis of associated Riemann–Hilbert (RH) problems, bridging initial value problems to precise asymptotic descriptions.
The course covers both defocusing and focusing NLS equations, with particular emphasis on asymptotic structures in different space–time regions, including the dispersive (non-soliton) region and the soliton region. Through techniques such as jump matrix factorization, g-function mechanisms, phase function analysis, and the construction of local model problem, the course reveals key phenomena in long-time dynamics, including dispersive decay, oscillatory modulation, and soliton contributions.
On top of this classical analytical framework, the course further incorporates emerging data-driven methodologies to explore the discovery and characterization of asymptotic structures. By embedding integrable constraints—such as Lax pairs, conservation laws, and RH consistency conditions—into machine learning architectures, we discuss how data-driven approaches can complement rigorous analysis, enabling the identification of asymptotic regimes, phase transitions, and reduced models in complex nonlinear systems.