Analysis and Applied Mathematics Seminar
Zhimeng Ouyang
University of Chicago
Lattice Approximations to Nonlinear Dispersive Equations
Abstract: Lattice models play a pivotal role in the investigation of microscopic multi-particle systems, with their continuum limits forming the foundation of macroscopic effective theory. These models have found wide-ranging applications in condensed matter physics, numerical analysis, and analysis of PDEs.
In this talk, I will present our recent work on the continuum limits of some lattice models to the corresponding nonlinear dispersive equations. Using the integrable Ablowitz–Ladik system as a prototype, we establish that solutions of this discrete model converge to solutions of either the cubic nonlinear Schrödinger equation (NLS) or the modified Korteweg–de Vries equation (mKdV) in certain limiting regimes. Notably, we consider white-noise-like initial data which excites Fourier modes throughout the circle, and demonstrate convergence to a system of NLS/mKdV. This result suggests that a sole continuum equation may not suffice to encapsulate the lattice dynamics in such a low-regularity setting akin to thermal equilibrium.
I will also outline the framework of our proof and discuss its broader implications, including its extension to more general lattice approximations of dispersive PDEs. In particular, our approach provides new insights into constructing dynamics for the Landau–Lifshitz spin model in its Gibbs state.
Monday March 10, 2025 at 4:00 PM in 636 SEO