Stability of Traveling Waves in Partly Parabolic Systems

Ghazaryan, Anna; Latushkin, Yuri; Schecter, Stephen

doi:10.1051/mmnp/20138503

Cited by 7 publications

(8 citation statements)

References 64 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This wave represents a combustion front that leaves behind of it high temperature u − = 1/β and no fuel, while in front of it temperature is 0 and there is fuel, with concentration normalized to 1. As discussed in Paragraph 3.2 of [GLS3], Hypothesis 2.10 is true and Hypothesis 2.4 can be verified (partly numerically) for small β > 0.…”

Section: Stable Manifoldsmentioning

confidence: 70%

“…In this system, u is the temperature, v is the concentration of unburned fuel, g is the unit reaction rate, and β > 0 is a constant parameter. This system was a primary guiding example in [G,GLS1,GLSS,GLS2,GLS3]. One motivation for looking at this well-studied problem, in which the reactant does not diffuse, was heat-enhanced methods of oil recovery in which the reactant is coke contained in the rock formation, see [AY].…”

Section: Stable Manifoldsmentioning

confidence: 99%

“…For each β > 0 there is a unique c > 0 for which such a wave exists, cf. [GLS3,§3.2]. This wave represents a combustion front that leaves behind of it high temperature u − = 1/β and no fuel, while in front of it temperature is 0 and there is fuel, with concentration normalized to 1.…”

Section: Stable Manifoldsmentioning

confidence: 99%

See 2 more Smart Citations

Stable foliations near a traveling front for reaction diffusion systems

Latushkin¹,

Schnaubelt²,

Yang³

2017

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the differential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front.

show abstract

Section: Stable Manifoldsmentioning

confidence: 70%

Section: Stable Manifoldsmentioning

confidence: 99%

See 1 more Smart Citation

Stable foliations near a traveling front for reaction diffusion systems

Latushkin¹,

Schnaubelt²,

Yang³

2017

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

show abstract

“…Recently, the analysis on a calcium model (CKKONS model [2,19]), which is based on current theory for waves of intracellular calcium concentration, indicates that the structure of one-dimensional waves in this model is quite different from that of the FHN system. Further, the stability analysis of waves in the CKKONS model is more subtle than those for the FHN system (see [19,6]). Motivated by these previous works, one may expect that the theory of two-dimensional waves in the CKKONS model is different from that in the FHN system.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that most analytical works on Goldbeter's model are only for the reduced system which is a piecewise linear approximation to Goldbeter's model, and which is a phenomenological formulation, and so, does not involve many biological details.Recently, the analysis on a calcium model (CKKONS model [2,19]), which is based on current theory for waves of intracellular calcium concentration, indicates that the structure of one-dimensional waves in this model is quite different from that of the FHN system. Further, the stability analysis of waves in the CKKONS model is more subtle than those for the FHN system (see [19,6]). Motivated by these previous works, one may expect that the theory of two-dimensional waves in the CKKONS model is different from that in the FHN system.Previous studies [10,12,22,20] have demonstrated that the curvature relation of waves is crucial for the evolution of waves in two spatial dimensions.…”

mentioning

confidence: 99%

Curvature Dependence of Propagating Velocity for a Simplified Calcium Model

Zhang

Sneyd

Tsai

2014

SIAM J. Appl. Math.

View full text Add to dashboard Cite

It is known that curvature relation plays a key role in the propagation of two-dimensional waves in an excitable model. Such a relation is believed to obey the eikonal equation for typical excitable models (e.g., the FitzHugh-Nagumo (FHN) model), which states that the relation between the normal velocity and the local curvature is approximately linear. In this paper, we show that for a simplified model of intracellular calcium dynamics, although its temporal dynamics can be investigated by analogy with the FHN model, the curvature relation does not obey the eikonal equation. Further, the inconsistency with the eikonal equation for the calcium model is because of the dispersion relation between wave speed s and volume-ratio parameter γ in the closed-cell version of the model, not because of the separation of the fast and the slow variables as in the FHN model. Hence this simplified calcium model may be an unexpected excitable system, whose wave propagation properties cannot be always understood by analogy with the FHN model. Since the celebrated works of of Hodgkin and Huxley [8], FitzHugh [5] andNagumo [15], wave propagation in excitable systems has been the subject of a vast number of applied mathematics studies. In particular, the understanding of waves in the FitzHugh-Nagumo (FHN) system has provided a great insight into waves propagation in a wide array of biological and chemical systems [4,10,20,18], ranging from action potentials in neurons to chemical waves in the Belousov-Zhabotinskii reaction.Although the FHN system is important in wave propagation theory of excitable systems, not all waves in biological systems can be well understood by this classic model. For example, earlier works on Goldbeter's model [7,16,17], which is derived from past theories on the mechanism underlying calcium waves and oscillations, has revealed that Goldbeter's model is an excitable system, but the description of waves in this model is different from that in the FHN system. We remark that most analytical works on Goldbeter's model are only for the reduced system which is a piecewise linear approximation to Goldbeter's model, and which is a phenomenological formulation, and so, does not involve many biological details.Recently, the analysis on a calcium model (CKKONS model [2,19]), which is based on current theory for waves of intracellular calcium concentration, indicates that the structure of one-dimensional waves in this model is quite different from that of the FHN system. Further, the stability analysis of waves in the CKKONS model is more subtle than those for the FHN system (see [19,6]). Motivated by these previous works, one may expect that the theory of two-dimensional waves in the CKKONS model is different from that in the FHN system.Previous studies [10,12,22,20] have demonstrated that the curvature relation of waves is crucial for the evolution of waves in two spatial dimensions. Specifically, previous theories [10,12,22,20] on the classical excitable models including the FHN system suggest that the propagation of t...

show abstract

Stability of equilibrium solutions of a double power reaction‐diffusion equation with a Dirac interaction

Hernández

Mayorga

2020

Mathematische Nachrichten

View full text Add to dashboard Cite

In this paper, information about the instability of equilibrium solutions of a nonlinear family of localized reaction‐diffusion equations in dimension one is provided. More precisely, explicit formulas to the equilibrium solutions are computed and, via analytic perturbation theory, the exact number of positive eigenvalues of the linear operator associated to the stability problem is analyzed. In addition, sufficient conditions for blow up of the solutions of the equation are also discussed.

show abstract

Stability of Traveling Waves in Partly Parabolic Systems

Cited by 7 publications

References 64 publications

Stable foliations near a traveling front for reaction diffusion systems

Stable foliations near a traveling front for reaction diffusion systems

Curvature Dependence of Propagating Velocity for a Simplified Calcium Model

Stability of equilibrium solutions of a double power reaction‐diffusion equation with a Dirac interaction

Contact Info

Product

Resources

About