It is known that long waves in spatially periodic polymer
Fermi-Pasta-Ulam-Tsingou lattices are well-approximated for long, but
not infinite, times by suitably scaled solutions of Korteweg-de Vries
equations. It is also known that dimer FPUT lattices possess nanopteron
solutions, i.e., traveling wave solutions which are the superposition of
a KdV-like solitary wave and a very small amplitude ripple. Such
solutions have infinite mechanical energy. In this article we
investigate numerically what happens over very long time scales (longer
than the time of validity for the KdV approximation) to solutions of
diatomic FPUT which are initially suitably scaled (finite energy) KdV
solitary waves. That is we omit the ripple. What we find is that the
solitary wave continuously leaves behind a very small amplitude
“oscillatory wake.” This periodic tail saps energy from the solitary
wave at a very slow (numerically sub-exponential) rate. We take this as
evidence that the diatomic FPUT “solitary wave” is in fact
quasi-stationary or metastable.
In this casual-style presentation, I discuss the research above in the context of the mathematical "type" of wave examined in the project, and potential applications to real world events such as tsunamis.