Abstract for Timothy Faver

Timothy Faver (Kennesaw State University)

Friday, March 31, 2023

Mass and spring dimer FPUT nanopterons with exponentially small, nonvanishing ripples

Abstract: Mass and spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattices are known to possess nanopteron traveling waves in relative displacement coordinates. Nanopterons are the superposition of a localized core and a small-amplitude periodic ripple; while it has been established that the rippleā€™s amplitude is small beyond all algebraic orders of the long wave parameter, more precise upper, and lower, bounds have not been known until recently. Methods from spatial dynamics enable the construction of nanopterons whose ripple amplitudes are both exponentially small and nonvanishing. I will give an overview of the spatial dynamics method and the various adaptations and generalizations needed to apply it to FPUT lattices, as well as interesting ancillary results obtained in the original position coordinates. I will also discuss challenges in extending the method to nonsymmetric dimers and FPUT lattices with more complicated material heterogeneities than the dimer structure, as well as connections of these FPUT solution techniques to other, diverse lattice problems, as well as interesting auxiliary techniques applicable to certain systems of nonlocal differential equations.