Momentum constraints in collective-variable theory

Boesch, R.; Willis, C. R.

doi:10.1103/physrevb.42.6371

Cited by 18 publications

(10 citation statements)

References 10 publications

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“…However, even though there was a series of papers that elaborated on and clarified the above issues (i.e. the resonances [18]- [19], the P-N oscillation [20]- [19], the stability of coherent soliton-like structures on diffusively coupled lattices [21] as well as the collective variable behavior of these structures [22]- [26]), not all of the story was unveiled. In particular, Braun, Peyrard and Kivshar observed much later ( [27]) that in a range of parameter values for the discrete sine Gordon equation (also known as the Frenkel-Kontorova model and originally proposed for the study of dislocations, [11]) that a discrete mode bifurcated from the bottom edge of the phonon spectrum.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of internal solitary wave modes from the essential spectrum

Jones

2000

Phys. Rev. E

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The bifurcation of internal modes from the phonon band in models supporting solitary wave solutions is currently one of the exciting phenomena in the field. We will present a number of analytical and semianalytical techniques for the detection, study, and understanding of these modes. We will see how they appear, without threshold, due to the discretization of the continuum equations. This perturbation is viewed in terms of a singular continuum approximation and analyzed by both perturbation theory and the Evans's function method. It is shown that these methods give equivalent results. Moreover, they are corroborated by mixed analyticalnumerical computations based on the recently developed discrete Evans's function method. The extent to which these predictions survive to strong discretizations is discussed. The results will be presented in the context of both the sine-Gordon and the 4 models.

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Section: Introductionmentioning

confidence: 99%

Bifurcation of internal solitary wave modes from the essential spectrum

Jones

2000

Phys. Rev. E

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“…Of particular relevance to our work are the more recent articles of Boesch, Willis and coworkers [59,54,8,9,11,10]. In this work the discrete sine-Gordon equation is solved with kink-like initial data on the lattice.…”

Section: Introductionmentioning

confidence: 99%

“…It is well-known that there are two static kink solutions, a high energy one centered on a lattice site and a low energy one centered between two consecutive lattice sites [48,27,15,59,54,8,9,11,7,34,30,31]. The low energy kink corresponds to the minimizer of the static Hamiltonian energy, h[u], displayed in (2.10).…”

Section: The Sine-gordon Equationmentioning

confidence: 99%

Dynamics of lattice kinks

Kevrekidis

Weinstein

2000

Physica D: Nonlinear Phenomena

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We consider a class of Hamiltonian nonlinear wave equations governing a field defined on a spatially discrete one dimensional lattice, with discreteness parameter, d = h −1 , where h > 0 is the lattice spacing. The specific cases we consider in detail are the discrete sine-Gordon (SG) and discrete φ 4 models. For finite d and in the continuum limit (d → ∞) these equations have static kink-like (heteroclinic) states which are stable. In contrast to the continuum case, due to the breaking of Lorentz invariance, discrete kinks cannot be "Lorentz boosted" to obtain traveling discrete kinks. Peyrard and Kruskal pioneered the study of how a kink, initially propagating in the lattice dynamically adjusts in the absence of an available family of traveling kinks. We study in detail the final stages of the discrete kink's evolution during which it is pinned to a specified lattice site (equilibrium position in the Peierls-Nabarro barrier). We find: (i) for d sufficiently large (sufficiently small lattice spacing), the state of the system approaches an asymptotically stable ground state static kink (centered between lattice sites).(ii) for d sufficiently small d < d * the static kink bifurcates to one or more time periodic states. For the discrete φ 4 we have: wobbling kinks which have the same spatial symmetry as the static kink as well as "g-wobblers" and "e-wobblers", which have different spatial symmetry. In the discrete sine-Gordon case, the "ewobbler" has the spatial symmetry of the kink whereas the "g-wobbler" has the opposite one. These time-periodic states may be regarded as a class of discrete breather / topological defect states; they are spatially localized and time periodic oscillations mounted on a static kink background.The large time limit of solutions with initial data near a kink is marked by damped oscillation about one of these two types of asymptotic states. In case (i) we compute the characteristics of the damped oscillation (frequency and d-dependent rate of decay). In case (ii) we prove the existence of, and give analytical and numerical evidence for the asymptotic stability of wobbling solutions. * Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 USA † Mathematical Sciences Research, Bell Laboratories -Lucent Technologies, and Department of Mathematics, University of Michigan -Ann Arbor, 600 Mountain Avenue, Murray Hill, NJ 07974-0636 USA 1The mechanism for decay is the radiation of excess energy, stored in internal modes, away from the kink core to infinity. This process is studied in detail using general techniques of scattering theory and normal forms. In particular, we derive a dispersive normal form, from which one can anticipate the character of the dynamics. The methods we use are very general and are appropriate for the study of dynamical systems which may be viewed as a system of discrete oscillators (e.g. kink together with its internal modes) coupled to a field (e.g. dispersive radiation or phonons). The approach is based on and extends an approach...

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“…To enforce them at later times, it suffices to require their time derivative (denoted with a dot) to vanish, namely,Ċ 0,1 ≡ 0. 42 This provides the following relationships between the first time derivative of ∆η(x, t) and that of the CVs:…”

Section: A Methodsmentioning

confidence: 99%

“…Indeed, adding such constraints by means of Lagrange multipliers to some hypothetical Lagrangian would leave the Euler-Lagrange equations of motion unchanged, the constraints acting as an ignorable null Lagrangian. 42 Thus, the same governing equations for the CVs and the residual as with d'Alembert's principle would be obtained, the constraints being put into action in both cases as above, namely, in a second step by substitutions in the governing equations. This shows that the present approach is fully consistent with that by Boesch et al, while being usable with the DPE for which no Lagrangian is available, mainly due to the "local" term in Eq.…”

Section: B Commentsmentioning

confidence: 99%

Equation of motion and subsonic-transonic transitions of rectilinear edge dislocations: A collective-variable approach

Pellegrini

2014

Phys. Rev. B

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A theoretical framework is proposed to derive a dynamic equation motion for rectilinear dislocations within isotropic continuum elastodynamics. The theory relies on a recent dynamic extension of the Peierls-Nabarro equation, so as to account for core-width generalized stacking-fault energy effects. The degrees of freedom of the solution of the latter equation are reduced by means of the collective-variable method, well-known in soliton theory, which we reformulate in a way suitable to the problem at hand. Through these means, two coupled governing equations for the dislocation position and core width are obtained, which are combined into one single complex-valued equation of motion, of compact form. The latter equation embodies the history dependence of dislocation inertia. It is employed to investigate the motion of an edge dislocation under uniform time-dependent loading, with focus on the subsonic/transonic transition. Except in the steady-state supersonic range of velocities-which the equation does not address-our results are in good agreement with atomistic simulations on tungsten. In particular, we provide an explanation for the transition, showing that it is governed by a loading-dependent dynamic critical stress. The transition has the character of a delayed bifurcation. Moreover, various quantitative predictions are made, that could be tested in atomistic simulations. Overall, this work demonstrates the crucial role played by core-width variations in dynamic dislocation motion.

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Momentum constraints in collective-variable theory

Cited by 18 publications

References 10 publications

Bifurcation of internal solitary wave modes from the essential spectrum

Bifurcation of internal solitary wave modes from the essential spectrum

Dynamics of lattice kinks

Equation of motion and subsonic-transonic transitions of rectilinear edge dislocations: A collective-variable approach

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