Mass and spring dimer Fermi-Pasta-Ulam-Tsingou nanopterons with exponentially small, nonvanishing ripples

Abstract

We study traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattices in the long wave limit. Such lattices are known to possess nanopteron traveling waves in relative displacement coordinates. These nanopteron profiles consist of the superposition of an exponentially localized "core," which is close to a KdV solitary wave, and a periodic "ripple," whose amplitude is small beyond all algebraic orders of the long wave parameter, although a zero amplitude is not precluded. Here we deploy techniques of spatial dynamics, inspired by results of Iooss and Kirchg\"{a}ssner, Iooss and James, and Venney and Zimmer, to construct mass and spring dimer nanopterons whose ripples are both exponentially small and also nonvanishing. We first obtain "growing front" traveling waves in the original position coordinates and then pass to relative displacement. To study position, we recast its traveling wave problem as a first-order equation on an infinite-dimensional Banach space; then we develop hypotheses that, when met, allow us to reduce such a first-order problem to one solved by Lombardi. A key part of our analysis is then the passage back from the reduced problem to the original one. Our hypotheses free us from working strictly with lattices but are easily checked for FPUT mass and spring dimers. We also give a detailed exposition and reinterpretation of Lombardi's methods, to illustrate how our hypotheses work in concert with his techniques, and we provide a dialogue with prior methods of constructing FPUT nanopterons, to expose similarities and differences with the present approach.