Lax ROMs and Operator Atlases for Nonlinear Wave Dynamics

Reduced-order modeling is not only about keeping fewer degrees of freedom. It is also about choosing the right coordinates that match the dynamics. In nonlinear wave problems, that becomes especially important when coherent structures interact with radiation, shed energy into a wake, or move through different dynamical regimes.

In this talk, I will describe a framework for reduced modeling built from operators that adapt to the evolving state. In integrable problems, Lax pairs provide a natural starting point for this idea. Away from exact integrability, they still suggest useful local coordinates and a principled way to separate coherent structure from dispersive background. This leads to Lax reduced-order models and, more broadly, to operator atlases: families of local reduced models that track the dynamics as the solution changes.

I will focus on the basic mathematical picture, the computational construction of these models, and what the numerics show in dispersive wave equations, especially in a perturbed version KdV. The goal is to show how ideas from integrable systems can inform practical reduced models for nonlinear wave dynamics well beyond the classical exactly solvable setting.