Existence of Traveling Waves in Infinite Range FPUT Lattices
Michael Herrmann, Karsten Matthies, Jan-Patrick Meyer
TL;DR
The paper addresses the existence of solitary traveling waves in infinite-range FPUT lattices with repulsive pair interactions. It develops a constrained variational framework on a cone of unimodal profiles and proves the existence of a one-parameter family of traveling-wave solutions $W_K$ with fixed energy $\|W_K\|_2^2=2K$ for $0<K<\nu^2/2$, yielding speeds $c_K$ tied to the variational structure. For power-law potentials $\Phi_m(r)=r^{-\alpha}$ with $\alpha>3/2$, the construction provides traveling waves across the admissible energy range, and the high-energy limit $K\to\nu^2/2$ causes the profile to concentrate and the speed to blow up, while small-energy behavior links to long-wave limits. The analysis advances traveling-wave theory beyond small amplitudes, addressing infinite-range interactions and establishing a rigorous variational route to nonlocal FPUT waves, with detailed high-energy asymptotics paralleling known results for finite-range and nearest-neighbor settings.
Abstract
We prove the existence of solitary waves in a lattice where all particles interact with each other by pair-wise repulsive forces that decay with distance. The variational existence proof is based on constrained optimization and provides a one-parameter family of unimodal solutions. We also describe the asymptotic behavior of large, fast, high-energy waves.