Autowave propagation for general reaction diffusion systems

Molotkov, I. A.; Vakulenko, Sergey

doi:10.1016/0165-2125(93)90005-z

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…where b is the Burgers vector, |b| is the norm of the Burgers vector ρ m is the dislocation density, v d is the dislocation velocity, and D is a diffusivity of the strain ε. This heuristic approach was later shown to be the 1-D solution obtained from the coarse-graining of the time-evolution gradient flow dynamics of dislocations which results in a fractional reactiondiffusion equation (Monneau and Patrizi, 2012). The wellstudied jerky flow phenomenon associated with PLC band formation is now understood as the result of initially independent, statistically distributed non-local-scale dislocation mills ultimately coalescing and propagating into a macroscopic band.…”

Section: Oscillatory Deformation Bands In Plastic Deformation Of Materials With Internal Structurementioning

confidence: 99%

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Regenauer-Lieb

Schrank

et al. 2021

Solid Earth

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Abstract. Theoretical approaches to earthquake instabilities propose shear-dominated source mechanisms. Here we take a fresh look at the role of possible volumetric instabilities preceding a shear instability. We investigate the phenomena that may prepare earthquake instabilities using the coupling of thermo-hydro-mechano-chemical reaction–diffusion equations in a THMC diffusion matrix. We show that the off-diagonal cross-diffusivities can give rise to a new class of waves known as cross-diffusion or quasi-soliton waves. Their unique property is that for critical conditions cross-diffusion waves can funnel wave energy into a stationary wave focus from large to small scale. We show that the rich solution space of the reaction–cross-diffusion approach to earthquake instabilities can recover classical Turing instabilities (periodic in space instabilities), Hopf bifurcations (spring-slider-like earthquake models), and a new class of quasi-soliton waves. Only the quasi-soliton waves can lead to extreme focussing of the wave energy into short-wavelength instabilities of short duration. The equivalent extreme event in ocean waves and optical fibres leads to the appearance of “rogue waves” and high energy pulses of light in photonics. In the context of hydromechanical coupling, a rogue wave would appear as a sudden fluid pressure spike. This spike is likely to cause unstable slip on a pre-existing (near-critically stressed) fault acting as a trigger for the ultimate (shear) seismic moment release.

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Section: Oscillatory Deformation Bands In Plastic Deformation Of Materials With Internal Structurementioning

confidence: 99%

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Regenauer-Lieb

Schrank

et al. 2021

Solid Earth

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“…This geologically more relevant situation is described in Molotkov and Vakulenko (1993). The authors describe generalised reaction-diffusion systems and find that the wave behaviour depends on only three parameters.…”

Section: Reaction-diffusion Length/time Scalesmentioning

confidence: 99%

“…a bistable or multistable (for several reactions) region with a stable stationary mode and a mode for the nucleation of propagating autowaves above a critical activation threshold; ii) in the activated state, the wavefront separates two regions, a local region characterised by the particular THMC diffusional length scale L d affected by the reactions R i , and a large region at > L d which is outside the reaction-diffusion wave; iii) for long time scales, the wavefield is governed by characteristic self-oscillatory motions which for bistable systems are described by just three parameters. For multistable systems chaotic oscillations are expected (Molotkov and Vakulenko, 1993). For the analysis of this complicated system we will propose to use perturbation theory and illustrate the approach through some basic concepts of signal processing.…”

Section: Reaction-diffusion Length/time Scalesmentioning

confidence: 99%

“…A fully populated diffusion matrix provides the opportunity to extend the postulate of coupling different scale processes in earthquake mechanics (Ohnaka, 2003). Cross-diffusion, in excitable media (Molotkov and Vakulenko, 1993), can lead to the formation of self-supporting standing wave solutions (Berenstein and Beta, 2012). The integrable focusing non-linear Schrödinger equation bears similarities but cross-diffusion waves describe a much broader class of waves and hence offer a richer solution for real-life applications (Tsyganov and Biktashev, 2014).…”

Section: Cross-diffusion Wavesmentioning

confidence: 99%

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Cross-Diffusion Waves as a trigger for multiscale, multiphysics Instabilities: Application to earthquakes

Regenauer‐Lieb

Schrank

et al. 2020

Preprint

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Abstract. Theoretical approaches to earthquake instabilities propose shear-dominated instabilities as a source mechanism. Here we take a fresh look at the role of possible volumetric instabilities preceding a shear instability. We investigate the phenomena that may prepare earthquake instabilities using the coupling of Thermo-Hydro-Mechano-Chemical reaction-diffusion equations in a THMC diffusion matrix. We show that the off-diagonal cross-diffusivities can give rise to a new class of waves known as cross-diffusion waves. Their unique property is that for critical conditions cross-diffusion waves can funnel wave energy into a quasi-stationary wave focus from large to small-scale. The equivalent extreme event in ocean waves and optical fibres leads to the appearance of rogue waves and high energy pulses of light in lasers. In the context of hydromechanical coupling, a rogue wave would appear as a sudden fluid pressure spike on the future fault plane. This is here interpreted as a trigger for the ultimate (shear) seismic moment release.

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“…The asymptotic approach we use here is described in [14]. Mathematical details and validation of the pertinent analysis can be found in [15].…”

Section: Asymptotic Analysismentioning

confidence: 99%

On metastable deflagration in porous medium combustion

Bessonov

Gordon

Sivashinsky

et al. 2005

Applied Mathematics Letters

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A regime of low velocity deflagration in hydraulically resisted flows such as those occurring in porous beds is discussed. An asymptotic expression for the deflagration velocity is derived. The obtained dependency elucidates the mechanism controlling the gradual enhancement of the propagation velocity prior to the abrupt transition from slow to fast combustion. This enhancement is caused by the drag-induced diffusion of pressure ahead of the advancing front. The time of transition from the slow to fast propagation is estimated.

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Autowave propagation for general reaction diffusion systems

Cited by 5 publications

References 7 publications

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Cross-Diffusion Waves as a trigger for multiscale, multiphysics Instabilities: Application to earthquakes

On metastable deflagration in porous medium combustion

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