Advancing Stability through Rigorous Computation (24frg009)


Vera Hur (University of Illinois Urbana-Champaign)

(École Polytechnique)


The Banff International Research Station will host the "Advancing Stability through Rigorous Computation" Focused Research Group workshop in Banff from July 14 - 21, 2024.

Hamiltonian partial differential equations (PDEs) are commonly used to describe various natural phenomena like wave motion. However, figuring out if these solutions are stable can be tricky and involves a lot of nuances. One important condition for stability is that when we study the equation's behavior around a solution, the spectrum of the resulting operator should only consist of points along the imaginary axis. While we have many ways to tell if something is unstable, there aren't any surefire methods to prove stability yet, except for a few special cases.

Excitingly, a growing area called validated numerics or rigorous computation is on the brink of offering new tools to help us understand the behavior of these types of equations. These tools can help us analyze operators that pop up when studying the stability of traveling wave solutions. We're currently close to proving that for a particular equation called the cubic Korteweg-de Vries (KdV) equation, even though it's not easy to solve directly, the spectrum of its linearized version around a traveling wave solution is entirely imaginary. If successful, this would be a groundbreaking result, as it would be the first time we've proven this kind of stability without needing the equation to be easily solvable.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. BIRS is supported by Canada’s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta’s Advanced Education and Technology, and Mexico’s Consejo Nacional de Ciencia y Tecnología (CONACYT).