An official website of the United States government
We provide the first experimental realization of wavenumber bandgap (𝑞−gap) breathers. Experiments are obtained in the setting of a time-periodic phononic lattice where the model and experiment exhibit good qualitative agreement. 𝑞−gap breathers are localized in time and periodic in space, and are the counterparts to the classical breathers found in space-periodic systems. We derive an exact condition for modulation instability that leads to the opening of wavenumber bandgaps in which the 𝑞−gap breathers can arise. The 𝑞−gap breathers become more narrow and larger in amplitude as the wavenumber goes further into the bandgap. In the presence of damping, these structures acquire a nonzero, oscillating tail. The controllable temporal localization that 𝑞−gap breathers make possible has potential applications in the creation of phononic frequency combs, energy harvesting or acoustic signal processing.
more »
« less
This content will become publicly available on March 31, 2026
On the Existence of Generalized Breathers and Transition Fronts in Time-Periodic Nonlinear Lattices
We prove the existence of a class of time-localized and space-periodic breathers (called q-gap breathers) in nonlinear lattices with time-periodic coe!cients. 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.
more »
« less
- Award ID(s):
- 2107945
- PAR ID:
- 10608333
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM Journal on Applied Dynamical Systems
- Volume:
- 24
- Issue:
- 1
- ISSN:
- 1536-0040
- Page Range / eLocation ID:
- 894 to 928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We investigate the spectral properties of one-dimensional spatially modulated nonlinear phononic lattices, and their evolution as a function of amplitude. In the linear regime, the stiffness modulations define a family of periodic and quasiperiodic lattices whose bandgaps host topological edge states localized at the boundaries of finite domains. With cubic nonlinearities, we show that edge states whose eigenvalue branch remains within the gap as amplitude increases remain localized, and therefore appear to be robust with respect to amplitude. In contrast, edge states whose corresponding branch approaches the bulk bands experience de-localization transitions. These transitions are predicted through continuation studies on the linear eigenmodes as a function of amplitude, and are confirmed by direct time domain simulations on finite lattices. Through our predictions, we also observe a series of amplitude-induced localization transitions as the bulk modes detach from the nonlinear bulk bands and become discrete breathers that are localized in one or more regions of the domain. Remarkably, the predicted transitions are independent of the size of the finite lattice, and exist for both periodic and quasiperiodic lattices. These results highlight the co-existence of topological edge states and discrete breathers in nonlinear modulated lattices. Their interplay may be exploited for amplitude-induced eigenstate transitions, for the assessment of the robustness of localized states, and as a strategy to induce discrete breathers through amplitude tuning.more » « less
-
Abstract We study—experimentally, theoretically, and numerically—nonlinear excitations in lattices of magnets with long-range interactions. We examine breather solutions, which are spatially localized and periodic in time, in a chain with algebraically-decaying interactions. It was established two decades ago (Flach 1998Phys. Rev.E58R4116) that lattices with long-range interactions can have breather solutions in which the spatial decay of the tails has a crossover from exponential to algebraic decay. In this article, we revisit this problem in the setting of a chain of repelling magnets with a mass defect and verify, both numerically and experimentally, the existence of breathers with such a crossover.more » « less
-
In this article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrödinger equations subject to spatially localized driving and damping. They provide an alternate description of the metastable behavior in such lattice systems which agrees with previous predictions for the evolution of metastable states while providing more accurate approximations to these states. We analyze the stability of these breathers, finding a very small positive eigenvalue whose eigenvector lies almost tangent to the surface of the cylinder formed by the family of breathers. This causes solutions to slide along the cylinder without leaving its neighborhood for very long times.more » « less
-
We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.more » « less
