We consider a model system, consisting of two nonlinearly coupled partial differential equations,... more We consider a model system, consisting of two nonlinearly coupled partial differential equations, to investigate nonlinear convective instabilities of travelling waves. The system exhibits front solutions which are travelling waves that asymptotically connect two different spatially homogeneous rest states. In the coordinate frame that moves with the front, the rest state ahead of the front is asymptotically stable, while the rest state behind the front experiences a Turing instability: upon increasing a bifurcation parameter, spatially periodic patterns arise. We show that the front is nonlinearly convectively unstable with respect to perturbations that are exponentially localized at +∞. More precisely, in the co-moving coordinate frame, the corresponding solution converges pointwise to a translate of the front, and the perturbation to the front is transported toward −∞. The proof of this stability result (which appears to be the first of this kind for dissipative systems) is based...
We study the planar front solution for a class of reaction diffusion equations in multidimensiona... more We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear level, the spectrum is stabilized by using an exponential weight. A-priori estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturbations belong to the intersection of the exponentially weighted space with the original space without a weight. These estimates are then used to show that in the original norm, initially small perturbations to the front remain bounded, while in the exponentially weighted norm, they algebraically decay in time.
We study the planar front solution for a class of reaction diffusion equations in multidimensiona... more We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear level, the spectrum is stabilized by using an exponential weight. A priori estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturbations belong to the intersection of the exponentially weighted space with the original space without a weight. These estimates are then used to show that in the original norm, initially small perturbations to the front remain bounded, while in the exponentially weighted norm, they algebraically decay in time.
In this work we investigate the existence of non-monotone traveling wave solutions to a reaction-... more In this work we investigate the existence of non-monotone traveling wave solutions to a reaction-diffusion system modeling social outbursts, such as rioting activity, originally proposed in [4]. The model consists of two scalar values, the level of unrest u and a tension field v. A key component of the model is a bandwagon effect in the unrest, provided the tension is sufficiently high. We focus on the so-called tension inhibitive regime, characterized by the fact that the level of unrest has a negative feedback on the tension. This regime has been shown to be physically relevant for the spatiotemporal spread of the 2005 French riots. We use Geometric Singular Perturbation Theory to study the existence of such solutions in two situations. The first is when both u and v diffuse at a very small rate. Here, the time scale over which the bandwagon effect is observed plays a key role. The second case we consider is when the tension diffuses at a much slower rate than the level of unrest. In this case, we are able to deduce that the driving dynamics are modeled by the well-known Fisher-KPP equation.
In this paper we prove that a class of non self-adjoint second order differential operators actin... more In this paper we prove that a class of non self-adjoint second order differential operators acting in cylinders Ω × R ⊆ R d+1 have only real discrete spectrum located to the right of the right most point of the essential spectrum. We describe the essential spectrum using the limiting properties of the potential. To track the discrete spectrum we use spatial dynamics and bi-semigroups of linear operators to estimate the decay rate of eigenfunctions associated to isolated eigenvalues.
Gaseous detonation is one of the classical problems of combustion theory. In the past decade, the... more Gaseous detonation is one of the classical problems of combustion theory. In the past decade, there has been significant progress in understanding the phenomenon. The problem is, however, far from being completely resolved. We study a model of subsonic detonation that describes propagation of the combustion fronts in highly resistible media [5]. The assumption of high resistance of the medium provides a natural simplification of the system of governing equations while still preserving many of the qualitative features of the original system. The model reads:
We consider a model that describes a flow of a thin liquid film modified by insoluble surfactant ... more We consider a model that describes a flow of a thin liquid film modified by insoluble surfactant which is moving down a slope. For this model, parameter regimes were previously identified when the ...
We prove the existence of traveling fronts in diffusive Rosenzweig–MacArthur and Holling–Tanner p... more We prove the existence of traveling fronts in diffusive Rosenzweig–MacArthur and Holling–Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig–MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling–Tanner population model. In the second situation for the Rosenzweig–MacArthur model we pr...
In this paper we study the stability of fronts in a reduction of a well-known PDE system that is ... more In this paper we study the stability of fronts in a reduction of a well-known PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more ... more Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more complex structure often occur in natural and experimentally built systems. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. The existence, dynamic properties and bifurcations of those solutions are of interest. In particular, their stability is important for applications, as the waves that are observable are usually stable. When the waves are unstable, further investigation is warranted of the way the instability is exhibited, i.e. the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves. A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.
We study front solutions of a system that models combustion in highly hydraulically resistant por... more We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively stable.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015
For a wide range of parameters, we study travelling waves in a diffusive version of the Holling–T... more For a wide range of parameters, we study travelling waves in a diffusive version of the Holling–Tanner predator–prey model from population dynamics. Fronts are constructed using geometric singular perturbation theory and the theory of rotated vector fields. We focus on the appearance of the fronts in various singular limits. In addition, periodic travelling waves of relaxation oscillation type are constructed using a recent generalization of the entry–exit function.
We consider a model for the flow of a thin liquid film down an inclined plane in the presence of ... more We consider a model for the flow of a thin liquid film down an inclined plane in the presence of a surfactant. The model is known to possess various families of traveling wave solutions. We use a combination of analytical and numerical methods to study the stability of the traveling waves. We show that for at least some of these waves the spectra of the linearization of the system about them are within the closed left-half complex plane.
We consider a known model that describes formation of mussel beds on soft sediments. The model co... more We consider a known model that describes formation of mussel beds on soft sediments. The model consists of nonlinearly coupled pdes that capture evolution of mussel biomass on the sediment and algae in the water layer overlying the mussel bed. The system accounts for the diffusive spread of mussels, while the diffusion of algae is neglected and at the same time the tidal flow of the water is considered to be the main source of transport for algae, but does not affect mussels. Therefore, both the diffusion and the advection matrices in the system are singular. A numerical investigation of this system in some parameter regimes is known. We present a systematic analytic treatment of this model. Among other techniques we use Geometric Singular Perturbation Theory to analyze the nonlinear mechanisms of pattern and wave formation in this system.
Fronts are traveling waves in spatially extended systems that connect two dieren t spatially homo... more Fronts are traveling waves in spatially extended systems that connect two dieren t spatially homoge- neous rest states. If the rest state behind the front undergoes a supercritical Turing instability, then the front will also destabilize. On the linear level, however, the front will be only convectively unstable since perturbations will be pushed away from the front as it propagates.
We consider a system of two reaction di!usion equations with the KPP type nonlinearity which desc... more We consider a system of two reaction di!usion equations with the KPP type nonlinearity which describes propagation of pressure driven flames. It is known that the system admits a family of traveling wave solutions parameterized by their velocity. In this paper we show that these traveling fronts are stable under the assumption that perturbations belong to an appropriate weighted L2 space. We also discuss an interesting meta-stable pattern the system exhibits in certain cases.
For a model of gasless combustion with heat loss, we use geometric singular perturbation theory t... more For a model of gasless combustion with heat loss, we use geometric singular perturbation theory to show existence of traveling combustion fronts. We show that the fronts are nonlinearly stable in an appropriate sense if an Evans function criterion, which can be verified numerically, is satisfied. For a solid reactant and exothermicity parameter that is not too large, we verify numerically that the criterion is satisfied.
We consider a model system, consisting of two nonlinearly coupled partial differential equations,... more We consider a model system, consisting of two nonlinearly coupled partial differential equations, to investigate nonlinear convective instabilities of travelling waves. The system exhibits front solutions which are travelling waves that asymptotically connect two different spatially homogeneous rest states. In the coordinate frame that moves with the front, the rest state ahead of the front is asymptotically stable, while the rest state behind the front experiences a Turing instability: upon increasing a bifurcation parameter, spatially periodic patterns arise. We show that the front is nonlinearly convectively unstable with respect to perturbations that are exponentially localized at +∞. More precisely, in the co-moving coordinate frame, the corresponding solution converges pointwise to a translate of the front, and the perturbation to the front is transported toward −∞. The proof of this stability result (which appears to be the first of this kind for dissipative systems) is based...
We study the planar front solution for a class of reaction diffusion equations in multidimensiona... more We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear level, the spectrum is stabilized by using an exponential weight. A-priori estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturbations belong to the intersection of the exponentially weighted space with the original space without a weight. These estimates are then used to show that in the original norm, initially small perturbations to the front remain bounded, while in the exponentially weighted norm, they algebraically decay in time.
We study the planar front solution for a class of reaction diffusion equations in multidimensiona... more We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear level, the spectrum is stabilized by using an exponential weight. A priori estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturbations belong to the intersection of the exponentially weighted space with the original space without a weight. These estimates are then used to show that in the original norm, initially small perturbations to the front remain bounded, while in the exponentially weighted norm, they algebraically decay in time.
In this work we investigate the existence of non-monotone traveling wave solutions to a reaction-... more In this work we investigate the existence of non-monotone traveling wave solutions to a reaction-diffusion system modeling social outbursts, such as rioting activity, originally proposed in [4]. The model consists of two scalar values, the level of unrest u and a tension field v. A key component of the model is a bandwagon effect in the unrest, provided the tension is sufficiently high. We focus on the so-called tension inhibitive regime, characterized by the fact that the level of unrest has a negative feedback on the tension. This regime has been shown to be physically relevant for the spatiotemporal spread of the 2005 French riots. We use Geometric Singular Perturbation Theory to study the existence of such solutions in two situations. The first is when both u and v diffuse at a very small rate. Here, the time scale over which the bandwagon effect is observed plays a key role. The second case we consider is when the tension diffuses at a much slower rate than the level of unrest. In this case, we are able to deduce that the driving dynamics are modeled by the well-known Fisher-KPP equation.
In this paper we prove that a class of non self-adjoint second order differential operators actin... more In this paper we prove that a class of non self-adjoint second order differential operators acting in cylinders Ω × R ⊆ R d+1 have only real discrete spectrum located to the right of the right most point of the essential spectrum. We describe the essential spectrum using the limiting properties of the potential. To track the discrete spectrum we use spatial dynamics and bi-semigroups of linear operators to estimate the decay rate of eigenfunctions associated to isolated eigenvalues.
Gaseous detonation is one of the classical problems of combustion theory. In the past decade, the... more Gaseous detonation is one of the classical problems of combustion theory. In the past decade, there has been significant progress in understanding the phenomenon. The problem is, however, far from being completely resolved. We study a model of subsonic detonation that describes propagation of the combustion fronts in highly resistible media [5]. The assumption of high resistance of the medium provides a natural simplification of the system of governing equations while still preserving many of the qualitative features of the original system. The model reads:
We consider a model that describes a flow of a thin liquid film modified by insoluble surfactant ... more We consider a model that describes a flow of a thin liquid film modified by insoluble surfactant which is moving down a slope. For this model, parameter regimes were previously identified when the ...
We prove the existence of traveling fronts in diffusive Rosenzweig–MacArthur and Holling–Tanner p... more We prove the existence of traveling fronts in diffusive Rosenzweig–MacArthur and Holling–Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig–MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling–Tanner population model. In the second situation for the Rosenzweig–MacArthur model we pr...
In this paper we study the stability of fronts in a reduction of a well-known PDE system that is ... more In this paper we study the stability of fronts in a reduction of a well-known PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more ... more Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more complex structure often occur in natural and experimentally built systems. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. The existence, dynamic properties and bifurcations of those solutions are of interest. In particular, their stability is important for applications, as the waves that are observable are usually stable. When the waves are unstable, further investigation is warranted of the way the instability is exhibited, i.e. the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves. A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.
We study front solutions of a system that models combustion in highly hydraulically resistant por... more We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively stable.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015
For a wide range of parameters, we study travelling waves in a diffusive version of the Holling–T... more For a wide range of parameters, we study travelling waves in a diffusive version of the Holling–Tanner predator–prey model from population dynamics. Fronts are constructed using geometric singular perturbation theory and the theory of rotated vector fields. We focus on the appearance of the fronts in various singular limits. In addition, periodic travelling waves of relaxation oscillation type are constructed using a recent generalization of the entry–exit function.
We consider a model for the flow of a thin liquid film down an inclined plane in the presence of ... more We consider a model for the flow of a thin liquid film down an inclined plane in the presence of a surfactant. The model is known to possess various families of traveling wave solutions. We use a combination of analytical and numerical methods to study the stability of the traveling waves. We show that for at least some of these waves the spectra of the linearization of the system about them are within the closed left-half complex plane.
We consider a known model that describes formation of mussel beds on soft sediments. The model co... more We consider a known model that describes formation of mussel beds on soft sediments. The model consists of nonlinearly coupled pdes that capture evolution of mussel biomass on the sediment and algae in the water layer overlying the mussel bed. The system accounts for the diffusive spread of mussels, while the diffusion of algae is neglected and at the same time the tidal flow of the water is considered to be the main source of transport for algae, but does not affect mussels. Therefore, both the diffusion and the advection matrices in the system are singular. A numerical investigation of this system in some parameter regimes is known. We present a systematic analytic treatment of this model. Among other techniques we use Geometric Singular Perturbation Theory to analyze the nonlinear mechanisms of pattern and wave formation in this system.
Fronts are traveling waves in spatially extended systems that connect two dieren t spatially homo... more Fronts are traveling waves in spatially extended systems that connect two dieren t spatially homoge- neous rest states. If the rest state behind the front undergoes a supercritical Turing instability, then the front will also destabilize. On the linear level, however, the front will be only convectively unstable since perturbations will be pushed away from the front as it propagates.
We consider a system of two reaction di!usion equations with the KPP type nonlinearity which desc... more We consider a system of two reaction di!usion equations with the KPP type nonlinearity which describes propagation of pressure driven flames. It is known that the system admits a family of traveling wave solutions parameterized by their velocity. In this paper we show that these traveling fronts are stable under the assumption that perturbations belong to an appropriate weighted L2 space. We also discuss an interesting meta-stable pattern the system exhibits in certain cases.
For a model of gasless combustion with heat loss, we use geometric singular perturbation theory t... more For a model of gasless combustion with heat loss, we use geometric singular perturbation theory to show existence of traveling combustion fronts. We show that the fronts are nonlinearly stable in an appropriate sense if an Evans function criterion, which can be verified numerically, is satisfied. For a solid reactant and exothermicity parameter that is not too large, we verify numerically that the criterion is satisfied.
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Papers by Anna Ghazaryan