The laminations of a crystal near an anti-continuum limit

Knibbeler, Vincent; Mramor, Blaž; Rink, Bob

doi:10.1088/0951-7715/27/5/927

Cited by 9 publications

(6 citation statements)

References 22 publications

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“…The anti-integrable limit also can be used to show that certain higher-dimensional symplectic maps [MM92] and higher-order recurrence relations modeling crystal lattices have analogues of cantori [KMR14]. However, Aubry-Mather theory has not been generalized to these cases.…”

Section: B Cantorimentioning

confidence: 99%

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Thirty years of turnstiles and transport

Meiss

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the action of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions. The problem of transport in dynamical systems is to quantify the motion of collections of trajectories between regions of phase space with physical significance, for example, to determine chemical reaction rates, mixing rates in a fluid, or particle confinement times in an accelerator or fusion plasma device. For Hamiltonian systems with a phase space containing regular (elliptic islands and tori) and irregular (chaotic) components, transport is impeded by partial barriers bounding resonance zones or by remnants of invariant tori. In the former case the barriers are formed from the broken separatrices of an unstable periodic orbit, and in the latter, they are formed from similar stable and unstable manifolds of a remnant torus-a cantorus. Thirty years ago, MacKay, Meiss, and Percival discovered that cantori form robust partial barriers for area-preserving maps: in any region of phase space the most resistant barriers are the cantori. In this review, I discuss this work and survey subsequent developments. While much has been done, there are still many open questions.

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Section: B Cantorimentioning

confidence: 99%

“…higher-order recurrence relations modeling crystal lattices have analogues of cantori [KMR14]. However, Aubry-Mather theory has not been generalized to these cases.…”

Section: Figmentioning

confidence: 99%

Thirty years of turnstiles and transport

Meiss

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…A proof based on Newton iteration shows that it is possible to find solutions of (1.1) for small constants ρ, as continuations of the known solutions of the problem (3.11). This is the content of the following theorem, the proof of which is very similar to those in [26,29]. We provide a proof for the reader's convenience in the appendix.…”

Section: The Anti-continuum Limitmentioning

confidence: 76%

“…Similarly as in the proof of corollary 4.7, we may choose a sequenceñι with ni ≤ñi ≤ Now we are ready to prove our main theorem. The final step of the proof follows ideas from [26]. Theorem 4.11.…”

Section: The Dirichlet Problem At Infinitymentioning

confidence: 99%

Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups

Mramor¹

2018

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen-Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by Γ-convergence.

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“…The ideas of proof are almost the same, except Lemma 6, as those in [Bangert, 1989]. These ideas may be used to discuss similar questions [Bangert, 1987[Bangert, , 1989, see also the proof of Theorem 5.2 in [Knibbeler et al, 2014]. Farina and Valdinoci [2012] improved the conclusions in [Bangert, 1989] for the autonomous PDE case.…”

Section: Introductionmentioning

confidence: 90%

Minimizers with Bounded Action for the High-Dimensional Frenkel–Kontorova Model

Miao

Wang

Qin

2015

Int. J. Bifurcation Chaos

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In Aubry-Mather theory for monotone twist maps or for one-dimensional Frenkel-Kontorova (FK) model with nearest neighbor interactions, each global minimizer (minimal energy configuration) is naturally Birkhoff. However, this is not true for the one-dimensional FK model with non-nearest neighbor interactions or for the high-dimensional FK model. In this paper, we study the Birkhoff property of minimizers with bounded action for the high-dimensional FK model.

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The laminations of a crystal near an anti-continuum limit

Cited by 9 publications

References 22 publications

Thirty years of turnstiles and transport

Thirty years of turnstiles and transport

Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups

Minimizers with Bounded Action for the High-Dimensional Frenkel–Kontorova Model

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