On the Existence of Generalized Breathers and Transition Fronts in Time-Periodic Nonlinear Lattices

TL;DR

This paper proves the existence of generalized q-gap breathers and transition fronts in time-periodic nonlinear lattices, combining rigorous analysis with numerical simulations to explore localized oscillatory solutions and their dynamics.

Contribution

It introduces the concept of generalized q-gap breathers with oscillating tails and derives an amplitude equation for small amplitudes, extending understanding of localized solutions in time-periodic lattices.

Findings

Existence of generalized q-gap breathers with small oscillating tails.

Derivation of an amplitude equation for breather dynamics.

Numerical validation of analytical results and transition fronts with damping.

Abstract

We prove the existence of a class of time-localized and space-periodic breathers (called q-gap breathers) in nonlinear lattices with time-periodic coefficients. These q-gap breathers are the counterparts to the classical space-localized and time-periodic breathers found in space-periodic systems. Using normal form transformations, we establish rigorously the existence of such solutions with oscillating tails (in the time domain) that can be made arbitrarily small, but finite. Due to the presence of the oscillating tails, these solutions are coined generalized q-gap breathers. Using a multiple-scale analysis, we also derive a tractable amplitude equation that describes the dynamics of breathers in the limit of small amplitude. In the presence of damping, we demonstrate the existence of transition fronts that connect the trivial state to the time-periodic ones. The analytical results are corroborated by systematic numerical simulations.