Spectrum of the M5-traveling waves

In this paper, we study the essential spectrum of the operator obtained by linearizing at traveling waves that occur in the one-dimensional version of the M5-model for mesenchymal cell movement inside a directed tissue made up of highly aligned fibers. We show that traveling waves are spectrally unstable in L2(ℝ; ℂ3) as the essential spectrum includes the imaginary axis. Tools in the proof include exponential dichotomies and Fredholm properties. We prove that a weighted space Lw2(ℝ; ℂ3) with the same function for the tree variables of the linearized operator is no suitable to shift the essential spectrum to the left of the imaginary axis. We find a pair of appropriate weight functions whereby on the weighted space Lwα2(ℝ; ℂ2) × Lwε2(ℝ; ℂ) the essential spectrum lies on {Reλ<0}, outside the imaginary axis.

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Stability of combustion waves in a simplified gas–solid combustion model in porous media

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2017.0185 ◽

2018 ◽

Vol 376 (2117) ◽

pp. 20170185 ◽

Cited By ~ 1

Author(s):

Fatih Ozbag ◽

Stephen Schecter

Keyword(s):

Weight Function ◽

Essential Spectrum ◽

Parabolic Systems ◽

Travelling Wave ◽

Weight Functions ◽

Combustion Model ◽

Combustion Waves ◽

Linearized System ◽

The Stability ◽

Linearized Operator

We study the stability of the combustion waves that occur in a simplified model for injection of air into a porous medium that initially contains some solid fuel. We determine the essential spectrum of the linearized system at a travelling wave. For certain waves, we are able to use a weight function to stabilize the essential spectrum. We perform a numerical computation of the Evans function to show that some of these waves have no unstable discrete spectrum. The system is partly parabolic, so the linearized operator is not sectorial, and the weight function decays at one end. We use an extension of a recent result about partly parabolic systems that are stabilized by such weight functions to show nonlinear stability. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

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Orbital stability of traveling waves for the one-dimensional Gross–Pitaevskii equation

Journal de Mathématiques Pures et Appliquées ◽

10.1016/j.matpur.2008.09.009 ◽

2009 ◽

Vol 91 (2) ◽

pp. 178-210 ◽

Cited By ~ 18

Author(s):

Patrick Gérard ◽

Zhifei Zhang

Keyword(s):

Traveling Waves ◽

Orbital Stability ◽

Pitaevskii Equation ◽

One Dimensional ◽

The One

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Numerical investigation of the propagation of shock waves in rigid porous materials: development of the computer code and comparison with experimental results

Journal of Fluid Mechanics ◽

10.1017/s0022112096007872 ◽

1996 ◽

Vol 324 ◽

pp. 163-179 ◽

Cited By ~ 21

Author(s):

A. Levy ◽

G. Ben-Dor ◽

S. Sorek

Keyword(s):

Porous Materials ◽

Initial Conditions ◽

Computer Code ◽

Experimental Results ◽

Numerical Code ◽

Materials Development ◽

Numerical Predictions ◽

Wide Range ◽

Dimensional Version ◽

The One

The governing equations of the flow field which is obtained when a thermoelastic rigid porous medium is struck head-one by a shock wave are developed using the multiphase approach. The one-dimensional version of these equations is solved numerically using a TVD-based numerical code. The numerical predictions are compared to experimental results and good to excellent agreements are obtained for different porous materials and a wide range of initial conditions.

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STABILITY OF STEADY STATE AND TRAVELING WAVES SOLUTIONS IN COUPLED MAP LATTICES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408020240 ◽

2008 ◽

Vol 18 (01) ◽

pp. 219-225 ◽

Cited By ~ 3

Author(s):

DANIEL TURZÍK ◽

MIROSLAVA DUBCOVÁ

Keyword(s):

Steady State ◽

Essential Spectrum ◽

Traveling Waves ◽

Traveling Wave ◽

Traveling Wave Solutions ◽

Linear Operators ◽

Coupled Map Lattices ◽

Basic Tool ◽

Wave Solutions ◽

The Stability

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

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ON TWO-DIMENSIONAL DYNAMICAL SYSTEMS ASSOCIATED WITH MEMBRANE KINETICS UNDERLYING CARDIAC ARRHYTHMIAS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409023792 ◽

2009 ◽

Vol 19 (05) ◽

pp. 1709-1732 ◽

Cited By ~ 1

Author(s):

B. M. BAKER ◽

M. E. KIDWELL ◽

R. P. KLINE ◽

I. POPOVICI

Keyword(s):

Dynamical Behavior ◽

Basins Of Attraction ◽

Two Dimensional ◽

Stimulus Period ◽

Smooth Maps ◽

Periodic Stimulation ◽

Piecewise Smooth Maps ◽

Dimensional Version ◽

Bifurcation Law ◽

The One

We study the orbits, stability and coexistence of orbits in the two-dimensional dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation — as a function of (a) kinetic parameters (two amplitudes, two rate constants) and (b) stimulus period. The original paper focused mostly on the one-dimensional version of this model (one amplitude, one rate constant), whose orbits, stability properties, and bifurcations were analyzed via the theory of skew-tent (hence unimodal) maps; the principal family of orbits were so-called "n-escalators", with n a positive integer. The two-dimensional analog (motivated by experimental results) has led to the current study of continuous, piecewise smooth maps of a polygonal planar region into itself, whose dynamical behavior includes the coexistence of stable orbits. Our principal results show (1) how the amplitude parameters control which escalators can come into existence, (2) escalator bifurcation behavior as the stimulus period is lowered — leading to a "1/n bifurcation law", and (3) the existence of basins of attraction via the coexistence of three orbits (two of them stable, one unstable) at the first (largest stimulus period) bifurcation. We consider the latter result our most important, as it is conjectured to be connected with arrhythmia.

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Simple Modelling of Extracellular Matrix Alignment in Dermal Wound Healing I. Cell Flux Induced Alignment

Journal of Theoretical Medicine ◽

10.1080/10273669808833018 ◽

1998 ◽

Vol 1 (3) ◽

pp. 175-192 ◽

Cited By ~ 14

Author(s):

Luke Olsen ◽

Philip K. Maini ◽

Jonathan A. Sherratt ◽

Ben Marchant

Keyword(s):

Extracellular Matrix ◽

Wound Healing ◽

Cell Movement ◽

Collagen Fibre ◽

Cell Types ◽

Travelling Wave ◽

Generic Model ◽

Dimensional Version ◽

Healing Wounds ◽

Dermal Wound

We present a generic model to investigate alignment due to cell movement with spefic application to collagen fibre alignment in wound healing. In particular, alignment in two orthogonal directions is considered. Numerical simulation are presented to show how alignment is affected by key parameter min the model. from a travelling wave analysis of a simplified one-dimensional version of the model we derive a first order ordinary differential equation to describe the time evolution of aligment. We conclude that in the wound healing context,faster healing wounds result in more aligment and hence more serve scarring. It is shown how the model can be extended to included orientation dependent Kinetics,multipkle cell types and several extracellular matrix materials.

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On the spectra of non-self-adjoint realisations of second-order elliptic operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015353 ◽

1981 ◽

Vol 90 (1-2) ◽

pp. 71-105 ◽

Cited By ~ 5

Author(s):

W. D. Evans

Keyword(s):

Spherical Shell ◽

Essential Spectrum ◽

Weighted Space ◽

Elliptic Operators ◽

Second Order ◽

Sectorial Operator ◽

Image Position ◽

Second Order Elliptic Operators ◽

Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

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Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves

European Journal of Applied Mathematics ◽

10.1017/s0956792507007218 ◽

2007 ◽

Vol 18 (6) ◽

pp. 679-708 ◽

Cited By ~ 29

Author(s):

RACHEL LEVY ◽

MICHAEL SHEARER ◽

THOMAS P. WITELSKI

Keyword(s):

Free Surface ◽

Traveling Waves ◽

Surfactant Concentration ◽

Piecewise Linear ◽

Traveling Wave Solutions ◽

Liquid Films ◽

Insoluble Surfactant ◽

Small Parameters ◽

Gravity Driven ◽

The One

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

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On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension

Journal of Mathematical Imaging and Vision ◽

10.1007/s10851-019-00905-z ◽

2019 ◽

Vol 61 (9) ◽

pp. 1276-1300

Author(s):

Niels Chr Overgaard

Keyword(s):

Regularization Parameter ◽

Total Variation Regularization ◽

Functions Of Bounded Variation ◽

Signal Restoration ◽

Uniqueness Of Solutions ◽

Taut String ◽

Fundamental Estimate ◽

Dimensional Version ◽

String Algorithm ◽

The One

Abstract We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string interpretation plays an essential role in the proof of the fundamental estimate. This estimate implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the corrupted signal as the regularization parameter vanishes. It can also be used to prove semi-group properties of the denoising model. Finally, it is indicated how the methods developed can be applied to related problems such as the fused lasso model, isotonic regression and signal restoration with higher-order total variation regularization.

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