Small-amplitude periodic traveling waves in dimer Fermi-Pasta-Ulam-Tsingou lattices

TL;DR

This paper proves the existence of small-amplitude periodic traveling waves in dimer FPUT lattices without symmetry assumptions, using bifurcation theory and exploiting the lattice's gradient and translation invariance.

Contribution

It extends previous results by removing symmetry constraints, allowing for the analysis of more general dimer FPUT lattices with alternating masses and potentials.

Findings

Existence of small-amplitude periodic traveling waves in general dimer FPUT lattices.

Removal of symmetry assumptions broadens the class of lattices where such waves are proven to exist.

Utilization of gradient structure and translation invariance simplifies the bifurcation analysis.

Abstract

We prove the existence of small-amplitude periodic traveling waves in dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattices without assumptions of physical symmetry. Such lattices are infinite, one-dimensional chains of coupled particles in which the particle masses and/or the potentials of the coupling springs can alternate. Previously, periodic traveling waves were constructed in a variety of limiting regimes for the symmetric mass and spring dimers, in which only one kind of material data alternates. The new results discussed here remove the symmetry assumptions by exploiting the gradient structure and translation invariance of the traveling wave problem. Together, these features eliminate certain solvability conditions that symmetry would otherwise manage and facilitate a bifurcation argument involving a two-dimensional kernel.