Discrete breathers — Advances in theory and applications

Flach, Sergej; Gorbach, Andrey V.

doi:10.1016/j.physrep.2008.05.002

Cited by 941 publications

(879 citation statements)

References 394 publications

(711 reference statements)

Supporting

Mentioning

838

Contrasting

Unclassified

Order By: Relevance

“…An example is given in [26, Section 2.2 & 2.3, pg. [9][10][11][12][13][14][15][16][17][18], where the minimization of a linear energy functional under a nonlinear constraint verified conditions for the co-existence of breather profiles. For instance, this alternative approach for (2.1) will show the existence of a regime for the hopping parameter α where an upper bound for the power is valid (see Remark II.3).…”

Section: Finite Dimensional Latticesmentioning

confidence: 99%

“…Partly also due to these applications, the DNLS has been a focal point of numerous mathematical/computational investigations in its own right, a number of which has been summarized in [8][9][10][11][12][13] and is related to models used in numerous other settings including micromechanical cantilever arrays [14] and DNA breathing dynamics [15], among others.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

2012

View full text Add to dashboard Cite

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.

show abstract

Section: Finite Dimensional Latticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

2012

View full text Add to dashboard Cite

show abstract

“…Such solutions can be generated from localized initial conditions or may arise from modulational instabilities of periodic waves [12,13,26]. Although such properties are classical in the context of anharmonic Hamiltonian lattices [9], energy localization is particularly strong in the DpS equation. Indeed, it is proved in [3] that the solution of (1) does not disperse for any nonzero initial condition in ℓ2(Z) (i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The case i = 1 yields the so-called even-parity modes (or site-centered solutions), and the case i = 2 corresponds to odd-parity modes (bond-centered solutions). These solutions were numerically computed in [12,26,13] for p = 5/2 (see also [9,23] and references therein in the case p = 4 with an additional on-site cubic nonlinearity).…”

Section: Theorem 1 the Stationary Dps Equationmentioning

confidence: 99%

Breather Solutions of the Discrete p-Schrödinger Equation

James

Starosvetsky

2013

Nonlinear Systems and Complexity

View full text Add to dashboard Cite

Summary. We consider the discrete p-Schrödinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order α = p − 1 > 1. Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even-or oddparity reflectional symmetries. We derive in addition analytical approximations for the breather profiles and the corresponding intersecting stable and unstable manifolds, valid on a whole range of nonlinearity orders α. In the limit of weak nonlinearity (α → 1 + ), we introduce a continuum limit connecting the stationary DpS and logarithmic nonlinear Schrödinger equations. In this limit, breathers correspond asymptotically to Gaussian homoclinic solutions. We numerically analyze the stability properties of breather solutions depending on their even-or odd-parity symmetry. A perturbation of an unstable breather generally results in a translational motion (traveling breather) when α is close to unity, whereas pinning becomes predominant for larger values of α.

show abstract

“…While ILMs are spatially localized nonlinear excitations that occur in perfect crystals [28][29][30] , in this model a local enhancement of the nonlinearity associated with off-centring Pb atoms 31 instigates the formation of ILMs on cooling [25][26][27] . The ILM frequency then slows below that of the TO phonon into the frequency gap between the TA and TO phonons, and eventually vanishes as PNRs become static [25][26][27] .…”

mentioning

confidence: 97%

Phonon localization drives polar nanoregions in a relaxor ferroelectric

et al. 2014

View full text Add to dashboard Cite

Relaxor ferroelectrics exemplify a class of functional materials where interplay between disorder and phase instability results in inhomogeneous nanoregions. Although known for about 30 years, there is no definitive explanation for polar nanoregions (PNRs). Here we show that ferroelectric phonon localization drives PNRs in relaxor ferroelectric PMN-30%PT using neutron scattering. At the frequency of a preexisting resonance mode, nanoregions of standing ferroelectric phonons develop with a coherence length equal to one wavelength and the PNR size. Anderson localization of ferroelectric phonons by resonance modes explains our observations and, with nonlinear slowing, the PNRs and relaxor properties. Phonon localization at additional resonances near the zone edges explains competing antiferroelectric distortions known to occur at the zone edges. Our results indicate the size and shape of PNRs that are not dictated by complex structural details, as commonly assumed, but by phonon resonance wave vectors. This discovery could guide the design of next generation relaxor ferroelectrics.

show abstract

Discrete breathers — Advances in theory and applications

Cited by 941 publications

References 394 publications

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

Breather Solutions of the Discrete p-Schrödinger Equation

Phonon localization drives polar nanoregions in a relaxor ferroelectric

Contact Info

Product

Resources

About