31 research outputs found
Micropterons, Nanopterons and Solitary Wave Solutions to the Diatomic Fermi-Pasta-Ulam-Tsingou Problem
We use a specialized boundary-value problem solver for mixed-type functional
differential equations to numerically examine the landscape of traveling wave
solutions to the diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) problem. By using a
continuation approach, we are able to uncover the relationship between the
branches of micropterons and nanopterons that have been rigorously constructed
recently in various limiting regimes. We show that the associated surfaces are
connected together in a nontrivial fashion and illustrate the key role that
solitary waves play in the branch points. Finally, we numerically show that the
diatomic solitary waves are stable under the full dynamics of the FPUT system
Mass and spring dimer Fermi-Pasta-Ulam-Tsingou nanopterons with exponentially small, nonvanishing ripples
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