Nanopteron-stegoton traveling waves in spring dimer Fermi-Pasta-Ulam-Tsingou lattices

Abstract

We study the existence of traveling waves in a spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. This is a one-dimensional lattice of identical particles connected by alternating nonlinear springs. As in the work of Faver and Wright on the mass dimer, or diatomic, lattice, we find that the lattice equations in the long wave scaling are singularly perturbed, and we apply a method of Beale to produce nanopteron traveling waves with wave speed slightly greater than the lattice’s speed of sound. The nanopteron wave profiles are the superposition of an exponentially decaying term (which itself is a small perturbation of a KdV sech 2 \operatorname {sech}^2 -type soliton) and a periodic term of very small amplitude. Further generalizing the spring forces from the mass dimer case, we allow the springs’ nonlinearity to contain higher order terms beyond the quadratic. This necessitates the use of composition operators to phrase the long wave problem, and these operators require delicate estimates due to the characteristic superposition of different function types from Beale’s ansatz. Unlike the diatomic case, the value of the leading order term in the traveling wave profiles alternates between particle sites, so that the spring dimer traveling waves are also “stegotons”, in the terminology of LeVeque and Yong. This behavior is absent in the mass dimer and confirms the approximation results of Gaison, Moskow, Wright, and Zhang for the spring dimer.