Mathematical and numerical aspects of nonlinear integrable systems

The modern history of integrable systems begins with the numerical discovery of solitons [Phys. Rev. Lett. 15 (1965), 240--243] for the Korteweg–de Vries equation. The importance of solitons stems from the fact that they exhibit particle-type interactions, and they also characterize the long term asymptotic behavior of the solution. Integrable systems provide the key to our understanding of these remarkable interactions. They appear in different areas of physics, both in the classical and quantum domains. The theory of integrable systems has been an active area of mathematics for the past forty years. Different aspects of the subject have fundamental relations with mechanics, applied mathematics, algebraic structures, theoretical physics, analysis, geometry, and so on.

The main focus in this proposal is on the mathematical and numerical aspects of nonlinear integrable systems. We plan to study the algebraic structure of explicit solutions for discrete integrable systems in terms of determinants，quasideterminants, and pfaffians. Meanwhile, attention will be paid to the connections among extrapolation, integrability, and orthogonality, and their interplays in numerical analysis. In particular, we will study the interrelations among discrete integrable systems, orthogonal polynomials (and their q-analogues) and extrapolation methods, and their applications to numerical algorithms.