Asymptotic solutions for localized vibrational modes in strongly anharmonic periodic systems

Page, J. B.

doi:10.1103/physrevb.41.7835

Cited by 431 publications

(204 citation statements)

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“…In that case, the assumptions made on the curve Γ u 0 are automatically satisfied. Indeed, condition (20) implies that the origin is a hyperbolic fixed point of T , since DT (0) has a pair of real eigenvalues Λ, Λ −1 given by (21). This yields the existence of a C 1 local unstable manifold of the origin, …”

Section: Homoclinic Solutions Of Generalized Dnls Equationsmentioning

confidence: 99%

“…The first one is reminiscent of the method of successive approximations [20,25], but presents some limitations since a convergence analysis is not available. The second one relies on formal continuum approximations which provide quantitatively correct results at least for p = 5/2 and p = 4 [17,26,13].…”

Section: Theorem 1 the Stationary Dps Equationmentioning

confidence: 99%

“…Condition (20) implies that the origin is a hyperbolic fixed point of T , since the Jacobian matrix DT (0) has a pair of real eigenvalues Λ, Λ −1 given by…”

Section: Heuristicsmentioning

confidence: 99%

“…Remark 3 In the work [22], the authors have considered the smooth situation when condition (20) is satisfied. In that case, the assumptions made on the curve Γ u 0 are automatically satisfied.…”

Section: Homoclinic Solutions Of Generalized Dnls Equationsmentioning

confidence: 99%

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Breather Solutions of the Discrete p-Schrödinger Equation

James

Starosvetsky

2013

Nonlinear Systems and Complexity

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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.

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Section: Homoclinic Solutions Of Generalized Dnls Equationsmentioning

confidence: 99%

Section: Theorem 1 the Stationary Dps Equationmentioning

confidence: 99%

“…Condition (20) implies that the origin is a hyperbolic fixed point of T , since the Jacobian matrix DT (0) has a pair of real eigenvalues Λ, Λ −1 given by…”

Section: Heuristicsmentioning

confidence: 99%

Section: Homoclinic Solutions Of Generalized Dnls Equationsmentioning

confidence: 99%

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Breather Solutions of the Discrete p-Schrödinger Equation

James

Starosvetsky

2013

Nonlinear Systems and Complexity

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“…Note that it does not much matter whether the injected breather is of even or odd parity ͑here we have injected an odd one͒. 24 As we will see in the following, although the initial temperature is extremely low, these decay times are orders of magnitude shorter than those of a breather injected in a zero temperature chain, the crucial difference being the presence of low wavelength phonons in the former but not in the latter. To support this description we have also observed a breather in a chain connected to a heat bath that is maintained at an extremely low but nonzero temperature.…”

Section: Mixed Arraysmentioning

confidence: 87%

Breathers and thermal relaxation in Fermi–Pasta–Ulam arrays

Reigada

Sarmiento

Lindenberg

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Breather stability and longevity in thermally relaxing nonlinear arrays depend sensitively on their interactions with other excitations. We review numerical results for the relaxation of breathers in Fermi-Pasta-Ulam arrays, with a specific focus on the different relaxation channels and their dependence on the interparticle interactions, dimensionality, initial condition, and system parameters. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1537090͔Breathers are highly localized oscillatory excitations in discrete nonlinear lattices that have been invoked as a possible way to store and transport vibrational energy in a large variety of physical and biophysical contexts. A particular scenario where the robustness and longevity of breathers has been a matter of considerable debate involves nonlinear arrays subject to thermal relaxation via the connection of surface sites to a cold environment. The important questions are these: Can breathers "created spontaneously or by design… survive for a long time in such a relaxing environment? If they can survive, can they move? We detail answers to these questions, one of which is rather unequivocal: Breathers that move do not live very long. So is another: Breathers are quite robust when they do not move. The more complicated question then deals with the conditions that allow breathers to remain stationary and undisturbed for a long time in a relaxing environment. We detail some conditions that lead to this outcome, and others that definitely do not.

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Self‐localized single‐anharmonic vibrational modes in two‐dimensional lattices

Fischer

1993

Annalen der Physik

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Asymptotic solutions for localized vibrational modes in strongly anharmonic periodic systems

Cited by 431 publications

References 4 publications

Breather Solutions of the Discrete p-Schrödinger Equation

Breather Solutions of the Discrete p-Schrödinger Equation

Breathers and thermal relaxation in Fermi–Pasta–Ulam arrays

Self‐localized single‐anharmonic vibrational modes in two‐dimensional lattices

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