Numerically exact computation of q-gap breathers via Poincaré map roots

Develop and demonstrate a numerical method that computes q-gap breather solutions of the time-periodic Fermi–Pasta–Ulam–Tsingou lattice as numerical roots of the appropriate time-2T Poincaré map to a user-prescribed tolerance, thereby producing numerically exact q-gap breathers and validating their properties against the analytical approximations derived in the paper.

Background

While the paper provides rigorous existence of generalized q-gap breathers with small tails and tractable asymptotic approximations from multiple-scale analysis, it does not furnish numerically exact solutions obtained as roots of a discrete-time map (e.g., the time-2T Poincaré map).

The authors explicitly flag the need for computational algorithms that can reliably find q-gap breather solutions to a specified tolerance, which would bridge theory and numerics and enable systematic parameter continuation and validation.