Soliton resolution conjecture for FPUT lattices

Let an FPUT nearest-neighbour lattice in one dimension be a small perturbation of the linear harmonic lattice, the Toda lattice, or any integrable lattice, and suppose its initial data are asymptotically periodic in space. The variables are the lattice position $n$ and time $t$.

Soliton resolution conjecture. Asymptotically, the $(n,t)$-space splits into regions separated by straight lines through the origin. Some regions are periodic, with period equal to that of the background, while others exhibit modulated oscillations with frequency of order $1/t$ and amplitude and phase varying slowly with $n/t$. These periodicity and modulation regions are open cones bounded by half-lines emerging from the origin for positive times. Solitons, namely travelling waves with constant shape and speed, may also occur; their regions are small in $1/t$ and lie around some of these half-lines, whose slopes are the soliton speeds. If the initial-data background is constant, the modulated-oscillation region does not occur.

This conjecture describes the expected long-time decomposition of small perturbations of integrable one-dimensional lattices into periodic, modulated, and soliton regions. Numerical computations support it for small perturbations of completely integrable lattices, while larger perturbations may exhibit more complicated behaviour, potentially including chaos. For the exact Toda lattice, complete proofs are known.