Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

Herrmann, Michael; Matthies, Karsten; Schwetlick, Hartmut; Zimmer, Johannes

doi:10.1137/120877878

Cited by 13 publications

(10 citation statements)

References 24 publications

(29 reference statements)

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“…Next, let us compare our result (3)-(5) to previous work on localized waves in the FPU model. Existence of travelling waves with constant asymptotic values at infinity has been established via different approaches: variational methods [16,43,11,39,21], see also [42,22] for a generalization to nonconvex potentials, center manifold arguments [25], and comparison to KdV [12]. The latter two approaches are limited to small amplitude, but also deliver the waveform.…”

Section: Arxiv:12082805v1 [Math-ph] 14 Aug 2012mentioning

confidence: 99%

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Cnoidal Waves on Fermi–Pasta–Ulam Lattices

Friesecke¹,

Mikikits-Leitner²

2014

J Dyn Diff Equat

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We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion qn = V (q n+1 − qn) − V (qn − q n−1 ) with generic nearest-neighbour potential V . We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength −3 suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of [12] to a periodic setting and the spectral theory of the periodic Schrödinger operator with KdV cnoidal wave potential.

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Section: Arxiv:12082805v1 [Math-ph] 14 Aug 2012mentioning

confidence: 99%

“…Next, we recall from [12] how the system (24), (25) formally reduces to KdV as ε → 0. In the small ε regime the wave speed scaling (22) implies…”

Section: Renormalization and The Continuum Limitmentioning

confidence: 99%

Cnoidal Waves on Fermi–Pasta–Ulam Lattices

Friesecke¹,

Mikikits-Leitner²

2014

J Dyn Diff Equat

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“…Different analytical techniques could be applied to bilinear laws regularized by a small spinodal layer, discontinuous laws with two different slopes, the trilinear law (5) or friction laws including a time delay, see e.g. [34,57,58,[77][78][79][80][81][82] and references therein.…”

Section: Discussionmentioning

confidence: 99%

Solitary waves in the excitable Burridge–Knopoff model

2018

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The Burridge-Knopoff model is a lattice differential equation describing a chain of blocks connected by springs and pulled over a surface. This model was originally introduced to investigate nonlinear effects arising in the dynamics of earthquake faults. One of the main ingredients of the model is a nonlinear velocity-dependent friction force between the blocks and the fixed surface. For some classes of non-monotonic friction forces, the system displays a large response to perturbations above a threshold, which is characteristic of excitable dynamics. Using extensive numerical simulations, we show that this response corresponds to the propagation of a solitary wave for a broad range of friction laws (smooth or nonsmooth) and parameter values. These solitary waves develop shock-like profiles at large coupling (a phenomenon connected with the existence of weak solutions in a formal continuum limit) and propagation failure occurs at low coupling. We introduce a simplified piecewise linear friction law (reminiscent of the McKean nonlinearity for excitable cells) which allows us to obtain an analytical expression of solitary waves and study some of their qualitative properties, such as wave speed and propagation failure. We propose a possible physical realization of this system as a chain of impulsively forced mechanical oscillators. In certain parameter regimes, non-monotonic friction forces can also give rise to bistability between the ground state and limit-cycle oscillations and allow for the propagation of fronts connecting these two stable states.

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“…In this case we expect that KdV-type waves still exist, but exhibit oscillatory tails due to resonances with the continuous spectrum. A proof of this fact would involve sophisticated arguments lying beyond the scope of the present paper, but we refer to [14,27,20] for similar rigorous results on certain generalizations of one-dimensional FPU chains.…”

Section: Applications To Different Latticesmentioning

confidence: 91%

KdV-like solitary waves in two-dimensional FPU-lattices

Chen¹,

Herrmann²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We prove the existence of solitary waves in the KdV limit of two-dimensional FPU-type lattices using asymptotic analysis of nonlinear and singularly perturbed integral equations. In particular, we generalize the existing results by Friesecke and Matthies since we allow for arbitrary propagation directions and non-unidirectional wave profiles. Keywords:two-dimensional FPU-lattices, KdV limit of lattice waves, asymptotic analysis of singularly perturbed integral equations MSC (2010): 37K60, 37K40, 74H10in [15] to 1D lattices with nonlocal interactions, replacing the original analysis of Fourier poles in the complex plane by direct estimates on the real line. The KdV approximation can also be used to derive modulation equations for the large-scale dynamics of well-chosen initial data. We refer to [23,17,18,19] for results in the classical FPU setting, and to [12] for more complex atomic systems.Existence results on various types of waves in FPU chains can be found in the literature. The paper [21] provides a precise description of all subsonic and supersonic waves with small amplitudes via the center manifold reduction. In [11] the existence of solitary waves with large amplitudes was proved for the first time by applying the concentration-compactness principle, see also [1] for similar results under slightly different conditions and [4,13] for another variational approach yielding unimodal waves. In all these large-amplitude results the wave speed is in principle unknown. However, using mountain pass techniques as in [25,22] one also finds periodic and solitary waves with prescribed wave speed above a critical threshold. Finally, for certain potentials there even exist several types of heteroclinic waves, see [16,26,24].The investigations mentioned above concern exclusively 1D FPU lattices. In the practice of physics, two or more dimensional FPU-lattices are more relevant, since they provide simplified models for crystals and solids. However, to the best of our knowledge, no existence result for KdV-like waves in such FPU lattices has been obtained except for a degenerate case [5], where it is assumed that the wave is unidirectional and longitudinal as it moves along the horizontal direction. This assumption reduces the original two-dimensional system to a one-dimensional one and the geometric nonlinearity, which arises from the linear stress-strain relation due to the introduction of diagonal springs, enables one to apply the results from [7].In the present paper we show an existence result for 2D FPU lattices in a general framework, which covers different lattice geometries and allows for arbitrary propagation directions. Our asymptotic approach adapts some arguments developed in [15] but is more sophisticated due to the two-dimensional setting. The discussion of the 2D case in the present paper hints at similar results for 3D lattices, which have to be left for further investigations. Besides, the study of stability of the resulting solutions requires ideas lying beyond the scope of the present paper and will be...

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Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

Cited by 13 publications

References 24 publications

Cnoidal Waves on Fermi–Pasta–Ulam Lattices

Cnoidal Waves on Fermi–Pasta–Ulam Lattices

Solitary waves in the excitable Burridge–Knopoff model

KdV-like solitary waves in two-dimensional FPU-lattices

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