Dynamics of front solutions in a specific reaction-diffusion system in one dimension

Ichiro, Shin; Ikeda, Hideo; Kawana, Takeyuki

doi:10.1007/bf03167516

Cited by 18 publications

(12 citation statements)

References 16 publications

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“…The fast reduced system (FRS) of (2.2) is obtained in the limit ε → 0, 6) and (v, q, w, r) = (v * , q * , w * , r * ) constants. It is independent of γ(ξ), therefore, this FRS coincides with the FRS of (2.2) 1 and (2.2) 2 .…”

Section: Existence Of Pinned 1-front Solutionsmentioning

confidence: 99%

“…In [9] pinning, rebound and penetration phenomena are studied in a system that is assumed to be close to a drift bifurcation, i.e., the bifurcation at which a traveling front bifurcates from the standing front. The analysis is based on a center manifold approach that has been developed to describe the weak interactions of fronts [6]. From the analytical point of view, the main differences between the present work and [9,10,11] is that (i), we use geometrical singular perturbation theory to explicitly construct the leading order profile of 1-front and 1-pulse, or 2-front, solutions and determine explicit stability conditions (in a three-component model), see Theorems 2.1, 3.1, and 4.1; (ii), our methods enable us to go beyond the setting of weak interactions and thus to consider N -front patterns (N > 1) that interact in a semi-strong fashion [3,22], see also section 5.…”

Section: Introductionmentioning

confidence: 99%

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Pinned fronts in heterogeneous media of jump type

et al. 2010

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In this paper, we analyze the impact of a (small) heterogeneity of jump type on the most simple localized solutions of a three-component FitzHugh-Nagumo-type system. We show that the heterogeneity can pin a 1-front solution, which travels with constant (non-zero) speed in the homogeneous setting, to a fixed, explicitly determined, distance from the heterogeneity. Moreover, we establish the stability of this heterogeneous pinned 1-front solution. In addition, we analyze the pinning of 1-pulse, or 2-front, solutions. The paper is concluded with simulations in which we consider the dynamics and interactions of N -front patterns in domains with M heterogeneities of jump type (N = 3, 4, M ≥ 1).

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Section: Existence Of Pinned 1-front Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pinned fronts in heterogeneous media of jump type

et al. 2010

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“…[25]). For spatially extended systems on the infinite line, it is then often possible to perform a centre manifold reduction, valid near the Hopf bifurcation point, to develop a weakly nonlinear normal form theory for large-scale oscillatory drift instabilities (see [9], [8] and the references therein). In contrast to this normal form theory, we emphasize that our Stefan problem with moving sources provides a description of large-scale oscillatory drift instabilities for values of τ 0 not necessarily close to the Hopf bifurcation point.…”

Section: In the Subregime O(1)mentioning

confidence: 99%

Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

Chen

Ward

2009

Eur. J. Appl. Math

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The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

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“…For the analysis, we do not deal with the original PDEs of (1), but with finite-dimensional ODEs that describe the pulse motion. Our study is inspired in part by the work of Ei et al [11] [32]. In [11], they employed the following nonlinearities for f and g of (1):…”

Section: Historical Background To the Present Studymentioning

confidence: 99%

“…Our study is inspired in part by the work of Ei et al [11] [32]. In [11], they employed the following nonlinearities for f and g of (1):…”

Section: Historical Background To the Present Studymentioning

confidence: 99%

Reduction approach to the dynamics of interacting front solutions in a bistable reaction–diffusion system and its application to heterogeneous media

Nishi

Nishiura

Teramoto

2019

Physica D: Nonlinear Phenomena

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The dynamics of pulse solutions in a bistable reaction-diffusion system are studied analytically by reducing partial differential equations (PDEs) to finitedimensional ordinary differential equations (ODEs). For the reduction, we apply the multiple-scales method to the mixed ODE-PDE system obtained by taking a singular limit of the PDEs. The reduced equations describe the interface motion of a pulse solution formed by two interacting front solutions. This motion is in qualitatively good agreement with that observed for the original PDE system. Furthermore, it is found that the reduction not only facilitates the analytical study of the pulse solution, especially the specification of the onset of local bifurcations, but also allows us to elucidate the global bifurcation structure behind the pulse behavior. As an application, the pulse dynamics in a heterogeneous bump-type medium are explored numerically and analytically. The reduced ODEs clarify the transition mechanisms between four pulse behaviors that occur at different parameter values.

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Dynamics of front solutions in a specific reaction-diffusion system in one dimension

Cited by 18 publications

References 16 publications

Pinned fronts in heterogeneous media of jump type

Pinned fronts in heterogeneous media of jump type

Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

Reduction approach to the dynamics of interacting front solutions in a bistable reaction–diffusion system and its application to heterogeneous media

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