Growth and decay of discrete nonlinear Schroedinger breathers interacting with internal modes or standing-wave phonons

Magnus Johansson; Serge Aubry

doi:10.1103/physreve.61.5864

Home/Archive/arXiv

arXiv · nlin/0002016

Growth and decay of discrete nonlinear Schroedinger breathers interacting with internal modes or standing-wave phonons

Magnus Johansson·Serge Aubry

View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the long-time evolution of weakly perturbed single-site breathers (localized stationary states) in the discrete nonlinear Schroedinger (DNLS) equation. The perturbations we consider correspond to time-periodic solutions of the linearized equations around the breather, and can be either (i) spatially localized, or (ii) spatially extended. For case (i), which corresponds to the excitation of an internal mode of the breather, we find that the nonlinear interaction between the breather and its internal mode always leads to a slow growth of the breather amplitude and frequency. In case (ii), corresponding to interaction between the breather and a standing-wave phonon, the breather will grow provided that the wave vector of the phonon is such that the generation of radiating higher-harmonics at the breather is possible. In other cases, breather decay is observed. This condition yields a limit value for the breather frequency above which no further growth is possible. We also discuss another mechanism for breather growth and destruction which becomes important when the amplitude of the perturbation is non-negligible, and which originates from the oscillatory instabilities of the nonlinear standing-wave phonons.

Explore related subjects

nlin.PS cond-mat nlin.CD physics.optics

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Magnus Johansson, Serge Aubry. 2000-02-11. Growth and decay of discrete nonlinear Schroedinger breathers interacting with internal modes or standing-wave phonons. https://doi.org/10.1103/physreve.61.5864

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Is there an accelerating nonspreading wave packet in the Schrödinger equation with higher even-order dispersions?

Probably not. For Airy-type initial data, although nearly all lobes accelerate, the principal lobe remains essentially stationary. Moreover, the overall wave packet gradually disperses.

nlin.PS↗

arXiv2026

Routes to chaos in a mass-conserving two-species reaction-diffusion model

Mass-conserving reaction-diffusion systems with two species correspond to a seemingly simple case where pattern formation occurs under the influence of a conservation law. Here, we first revisit their linear stability behavior and point out that generically two instabilities can occur: a stationary large-scale mass-conserving (Cahn-Hilliard) instability and an instability that combines features of a stationary large-scale non-mass-conserving (Allen-Cahn) instability and a oscillatory large-scale mass-conserving (conserved-Hopf) instability. We term it an Allen-Cahn-Hopf instability. Second, we investigate the nonlinear dynamics for a specific model related to the formation of cell polarization where only a Cahn-Hilliard instability can occur, i.e., all primary bifurcations are stationary. We analyze how secondary and further bifurcations subsequently give rise to various oscillatory states. The emerging rich spectrum of spatiotemporal behavior includes several period-doubling cascades related to different forms of spatial and temporal symmetry breaking. Beside regular states, three types of low-dimensional spatiotemporal chaos occur and involve transitions like fusion and an outer crises. Our results demonstrate the importance of nonlinear interactions in the dynamics of mass-conserving reaction-diffusion systems, and show that even a simple two-species system with primary bifurcations of Cahn-Hilliard type can show complex spatiotemporal behavior.

nlin.PS↗

arXiv2026

Duck hunting with quantum mechanics

We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.

nlin.PS↗