Fall 2026 • Weekly Seminars
For information on the related DUT Differential Equations Seminar (DDES), please check here.
This talk explains how the infinite-dimensional KAM theorem of Dolgopyat, Fayad, and Paradela applies to a concrete system of infinitely many weakly interacting particles. Starting from the physical Hamiltonian, we impose suitable conditions on the masses and potential, perform the heliocentric and scaling reductions, and pass to action–angle variables. The goal is to show how the concrete model is reduced step by step to the hypotheses of the abstract KAM theorem.
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For the infinite Fermi–Pasta–Ulam (FPU) system, general solutions can be approximated by counter-propagating waves associated with solutions to the Korteweg–de Vries (KdV) equation as the lattice mesh size goes to zero. The Toda lattice is a special case of the FPU system. We show that, by exploiting the conservation of the FPU Hamiltonian, the continuum limit from the FPU system to the KdV equation with $L^2$-level initial data holds in appropriate norms on an arbitrary time interval, thereby answering an open question posed by Hong, Kwak, and Yang (2021). Moreover, for the local-in-time continuum limit of the FPU system to the KdV equation, we lower the required Sobolev regularity to $H^s$ with $s > -3/4$. To establish this low-regularity continuum limit, we prove key trilinear estimates that are sharp up to the endpoint by combining linear estimates, the transversality of characteristic curves, and multilinear dispersive smoothing properties of the linear FPU flow. This is joint work with Ruoyuan Liu.