Existence of genuine q-gap breathers with vanishing tails

Determine whether the time-periodic Fermi–Pasta–Ulam–Tsingou lattice defined by m · ün + c · u̇n + k(t) un = F(un+1 − un) − F(un − un−1) with time-periodic stiffness k(t) and polynomial interaction force F(w) = K2 w − K3 w^2 + K4 w^3 admits genuine q-gap breathers—temporally localized, spatially periodic solutions whose oscillatory tails asymptotically vanish as t → ±∞—for parameter regimes corresponding to wavenumber band gaps.

Background

The paper rigorously establishes the existence of generalized q-gap breathers in the time-periodic FPUT lattice, characterized by temporally localized profiles with small but nonzero oscillatory tails due to the presence of neutrally stable modes. Using normal form transformations and invariant manifold techniques, the authors construct homoclinic orbits with tails that can be made arbitrarily small over large but finite time intervals, and provide multiple-scale approximations via an amplitude equation.

Because neutral directions prevent a generic transverse intersection of stable and unstable manifolds in the full 2N-dimensional phase space, the solutions proved are not tail-free. The authors explicitly highlight the open question of whether true (genuine) q-gap breathers with tails decaying to zero exist in this setting.