Localized structures and front propagation in systems with a conservation law

Knobloch, Edgar

doi:10.1093/imamat/hxw029

Cited by 35 publications

(28 citation statements)

References 48 publications

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“…Slanted snaking of stationary localized states has been reported before [15,16,24] and likewise attributed to the presence of a conserved quantity [25,26], following earlier work of Matthews and Cox [27,28]. The conserved quantity responsible for slanted snaking in the present system is the volume of fluid in the container.…”

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confidence: 67%

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Slanted snaking of localized Faraday waves

et al. 2017

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We report on an experimental, theoretical and numerical study of slanted snaking of spatially localized parametrically excited waves on the surface of a water-surfactant mixture in a Hele-Shaw cell. We demonstrate experimentally the presence of a hysteretic transition to spatially extended parametrically excited surface waves when the acceleration amplitude is varied, as well as the presence of spatially localized waves exhibiting slanted snaking. The latter extend outside the hysteresis loop. We attribute this behavior to the presence of a conserved quantity, the liquid volume, and introduce a universal model based on symmetry arguments, which couples the wave amplitude with such a conserved quantity. The model captures both the observed slanted snaking and the presence of localized waves outside the hysteresis loop, as demonstrated by numerical integration of the model equations.PACS numbers: 45.20.Ky

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confidence: 67%

“…The resulting expression is to be interpreted as describing the width-averaged data. Using symmetry arguments or multiscale analysis [26][27][28] one finds that the simplest set of equations for the amplitudes A and B takes the (scaled) form…”

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confidence: 99%

Slanted snaking of localized Faraday waves

et al. 2017

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“…Metastable localized states are found only on the lower branch in each of the three swallow-tail regions, and these regions are a reflection of slanted snaking that is expected of localized structures in mass-conserving systems [30,43,44]. Indeed, if we replot the data shown in figure 3(a) in terms of the L 2 norm U  of the order parameter U(x) we find a slanted snaking diagram ( figure 3(b)) of the type that is present in other conserved system such as the conserved Swift-Hohenberg equation [30] and binary fluid convection in a porous medium [45], a system in which temperature gradients and fluid flow redistribute a conserved amount of solute [46].…”

Section: Branches Of Spatially Extended and Localized Statesmentioning

confidence: 96%

Spatially localized quasicrystalline structures

et al. 2018

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Soft matter systems have been observed to self-assemble, over a range of system parameters, into quasicrystalline structures. The resulting quasicrystals (QCs) may minimize the free energy, and be in thermodynamic coexistence with the liquid state. At such state points, the likelihood of finding the presence of spatially localized states with quasicrystalline structure within the liquid is increased. Here we report the first examples of metastable spatially localized QCs of varying sizes in both two and three dimensions. Implications of these results for the nucleation of quasicrystalline structures are discussed. Our conclusions apply to a broad class of soft matter systems and more generally to continuum systems exhibiting quasipatterns.

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“…The figure indicates the linear stability properties of each state shown, with thin solid lines indicating linearly stable states (or local free energy minima) and thin dashed lines indicating unstable states. Note that the localized states coexist with the stable uniform state, a possibility that only arises because of mass conservation [63], and that they represent the global free energy minimum over a large part of their range of existence ( figure 9(b)). The inset of figure 9(b) indicates that the transition between successive global minima occurs via swallow-tail-like structures.…”

Section: Bifurcations In Finite Domains: Crystallization In 1dmentioning

confidence: 99%

First order phase transitions and the thermodynamic limit

et al. 2019

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We consider simple mean field continuum models for first order liquid-liquid demixing and solidliquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn-Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid-liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit.However, in a system of finite size or when a finite time horizon is considered, metastable states often play an important role and even unstable states may be crucial, as transient states, for extended time periods. The full set of states and their dependence on the various control parameters is conveniently presented in the form of bifurcation diagrams, well known in the context of dynamical systems and pattern formation theory [3][4][5]. The place of thermodynamic phase diagrams is taken by 'morphological phase diagrams' or state diagrams and stability diagrams [3,4,6]. The notion of a Maxwell point is often used in the context of pattern formation in nonconserved systems [7][8][9][10][11] to indicate equal energy states because of its dynamical significance [12,13]. In this context this notion applies equally to finite and infinite systems [7,8] although for finite systems it lacks the thermodynamic relevance as the condition for phase coexistence. In the context of buckling the corresponding concept is the Maxwell load [14].In this paper we show and discuss how the discontinuities in the TL represented by the Maxwell construction arise from the bifurcation diagrams relating stable, metastable and unstable steady states in finite-size systems. We focus on two systems: (i) phase decomposition of a binary liquid mixture and (ii) the liquid to crystalline solid phase transition. We investigate the transitions that occur in the context of the most basic mean-field continuum models for these two different phase transitions, namely, the Cahn-Hilliard equation [15][16][17] and the phase field crystal (PFC) model (or conserved Swift-Hohenberg equation) [18][19][20].Some aspects related to this question have been considered previously, in particular in relation to the nature of some of the states that can arise in finite-size systems in the two-phase region. References [21-24] describe theory and computer simulation results for atomistic models exhibiting gas-liquid, liquid-hexatic and hexaticsolid phase transitions that indicate how the Maxwell construction develops as the system size increases or the temperature decreases. For example, figures 3-5 of [23] compare Monte-Carlo computer simulation results in a finite three-dimensional domain (see also figures 2 and 3 of [22]) with the results from a capillary drop type model, also in a finite domain, with a mean-field expression for the chemi...

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Localized structures and front propagation in systems with a conservation law

Cited by 35 publications

References 48 publications

Slanted snaking of localized Faraday waves

Slanted snaking of localized Faraday waves

Spatially localized quasicrystalline structures

First order phase transitions and the thermodynamic limit

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