Beijing Institute of Mathematical Sciences and Applications

This course provides an in-depth exploration of core mathematical methods in modern integrable systems theory and their application to the asymptotic analysis of nonlinear partial differential equations. The curriculum focuses on two powerful analytical tools: the D-bar Method and the Deift-Zhou Nonlinear Steepest Descent Method. Through a systematic study of the Nonlinear Schrödinger Equation (NLS)—a canonical integrable model—the course will demonstrate in detail how to employ these techniques to rigorously derive the long-time asymptotic behavior of solutions in different regions, such as the solitonless (non-soliton) region and the soliton region. Students will gain a thorough understanding of the core concepts and computational techniques of these advanced methods, uncovering the rich mathematical structure and asymptotic properties of both soliton and non-soliton solutions.