Micropteron traveling waves in diatomic Fermi–Pasta–Ulam–Tsingou lattices under the equal mass limit

Small Mass Nanopteron Traveling Waves in Mass-in-Mass Lattices with Cubic FPUT Potential

“…In Proposition 4 we spell out exactly how one exploits the solvability condition in conjunction with the second equation in (2.17) to summon the separate equations for η 2 and a. This is all in line with the spirit of nanopteron constructions past in [1,11,13,14,29].…”

“…Less trivial is the following extension of the invertibility of H c over a range of weights. This is stated and proved as Proposition 5.1 in [13].…”

“…However, experience with traveling waves in heterogeneous FPUT lattices, in which the FPUT lattice no longer has identical springs and masses but either alternating masses (the mass dimer or diatomic lattice) or alternating spring potentials (the spring dimer), suggests that material heterogeneity in lattices induces the classical solitary wave to perturb into a nonlocal solitary wave. Specifically, we refer to the long wave limit studied by Faver and Wright [14] for the mass dimer and again by Faver [11] for the spring dimer, the small mass limit in mass dimers of Hoffman and Wright [25], and the equal mass limit of Faver and Hupkes [13], again for the mass dimer. We also note a variety of numerical studies of nanopeterons in mass and spring dimer lattices, including [19,21,35,36,43,48].…”

“…We have chosen the cubic FPUT potential V cubic from (1.4) to govern the interaction of the beads in the MiM lattice studied in this paper for two reasons. First, the simple "linear + quadratic" structure 2 that the spring forces consequently possess puts us in concert with a host of previous papers studying traveling waves in lattices, namely [11,13,14,16,25]. Second, we view the cubic FPUT potential as an approximation to the Hertzian potential with precompression.…”

“…Like the prototypical model problem of Amick and Toland [1], all of these solvability conditions hinge on the vanishing of an "integrate against a sinusoid" functional. Substantially more complicated solvability conditions appear in the small mass FPUT problem [25] and in the equal mass FPUT problem [13]. Both still require a functional of the form "integrate against a function γ " to vanish, but the precise function γ is no longer an exact sinusoid.…”