Session III.4 - Foundations of Numerical PDEs
Wednesday, June 21, 17:30 ~ 18:00
The periodic KdV equation: Computing with nonlinear Fourier series
Thomas Trogdon
University of Washington, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The integrability of the Korteweg-de Vries (KdV) equation affords an inverse scattering transform (IST) that "solves" the associated initial-value problem. First, historically, this fact allowed one to determine a class of explicit solutions, including both soliton solutions and (almost) periodic so-called finite-genus solutions. Then the IST allowed one to compute the long-time behavior of the solution to the Cauchy problem on the line using the Deift-Zhou method of nonlinear steepest descent. More recently, the IST has been used to compute solutions of the Cauchy problem on the line in the entire space-time plane.
In this talk, I will discuss a recent development that allows one to compute finite-genus solutions when the genus is large and to then effectively approximate the solution of the KdV equation with periodic initial data. This new approach uses a Riemann-Hilbert problem posed on (possibly thousands of) intervals and gives a natural interpretation as computing a nonlinear Fourier series approximation of the solution under consideration. Implications for dispersive quantization will be discussed.
Joint work with Deniz Bilman (University of Cincinnati), Ken McLaughlin (Tulane University) and Patrik Nabelek (Oregon State University).