Semi-strong Interaction Research Articles

Localized pattern formation: semi-strong interaction asymptotic analysis for three components model

We investigate a three-component system involving the Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Our goal is to investigate the connection between homoclinic snaking and semi-strength interaction in a three-variable reaction–diffusion system. A two-parameter bifurcation diagram of homogeneous, periodic and localized patterns is obtained numerically, and a natural asymptotic scaling for semi-strong interaction theory is found where an activator source term a = O ( δ 1 ) and b = O ( δ 1 ) , with δ 1 ≪ 1 being the diffusion ratio. Under this regime, singular perturbation techniques are used to construct localized steady-state patterns, and new types of non-local eigenvalue problems (NLEP) are derived to determine the stability of these patterns to O ( 1 ) time-scale instabilities. We extend the scope of the NLEP by considering a general scenario where both time-scaling parameters are non-zero. All analytical results are found to agree with numerics. Further numerical results are presented on the location of various types of breathing Hopf instability for localized patterns.

Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems

AbstractSystems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.

Spikes and localised patterns for a novel Schnakenberg model in the semi-strong interaction regime

An analysis is undertaken of the formation and stability of localised patterns in a 1D Schanckenberg model, with source terms in both the activator and inhibitor fields. The aim is to illustrate the connection between semi-strong asymptotic analysis and the theory of localised pattern formation within a pinning region created by a subcritical Turing bifurcation. A two-parameter bifurcation diagram of homogeneous, periodic and localised patterns is obtained numerically. A natural asymptotic scaling for semi-strong interaction theory is found where an activator source term \[a = O(\varepsilon )\] and the inhibitor source \[b = O({\varepsilon ^2})\], with ε2 being the diffusion ratio. The theory predicts a fold of spike solutions leading to onset of localised patterns upon increase of b from zero. Non-local eigenvalue arguments show that both branches emanating from the fold are unstable, with the higher intensity branch becoming stable through a Hopf bifurcation as b increases beyond the \[O(\varepsilon )\] regime. All analytical results are found to agree with numerics. In particular, the asymptotic expression for the fold is found to be accurate beyond its region of validity, and its extension into the pinning region is found to form the low b boundary of the so-called homoclinic snaking region. Further numerical results point to both sub and supercritical Hopf bifurcation and novel spikeinsertion dynamics.

Stripe to Spot Transition in a Plant Root Hair Initiation Model

A generalised Schnakenberg reaction-diffusion system with source and loss terms and a spatially dependent coefficient of the nonlinear term is studied both numerically and analytically in two spatial dimensions. The system has been proposed as a model of hair initiation in the epidermal cells of plant roots. Specifically the model captures the kinetics of a small G-protein ROP, which can occur in active and inactive forms, and whose activation is believed to be mediated by a gradient of the plant hormone auxin. Here the model is made more realistic with the inclusion of a transverse co-ordinate. Localised stripe-like solutions of active ROP occur for high enough total auxin concentration and lie on a complex bifurcation diagram of single and multi-pulse solutions. Transverse stability computations, confirmed by numerical simulation show that, apart from a boundary stripe, these 1D solutions typically undergo a transverse instability into spots. The spots so formed typically drift and undergo secondary instabilities such as spot replication. A novel 2D numerical continuation analysis is performed that shows the various stable hybrid spot-like states can coexist. The parameter values studied lead to a natural singularly perturbed, so-called semi-strong interaction regime. This scaling enables an analytical explanation of the initial instability, by describing the dispersion relation of a certain non-local eigenvalue problem. The analytical results are found to agree favourably with the numerics. Possible biological implications of the results are discussed.

Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in $${\mathbb {R}}^2$$ R 2

The linear stability of steady-state periodic patterns of localized spots in $\R^2$ for the two-component Gierer-Meinhardt (GM) and Schnakenburg reaction-diffusion models is analyzed in the semi-strong interaction limit corresponding to an asymptotically small diffusion coefficient $\eps^2$ of the activator concentration. In the limit $\eps\to 0$, localized spots in the activator are centered at the lattice points of a Bravais lattice with constant area $|\Omega|$. To leading order in $\nu={-1/\log\eps}$, the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies $D={D_0/\nu}$, for some $D_0$ independent of the lattice and the Bloch wavevector $\kb$. From a combination of the method of matched asymptotic expansions, Floquet-Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems, an explicit analytical formula for the continuous band of spectrum that lies within an ${\mathcal O}(\nu)$ neighborhood of the origin in the spectral plane is derived when $D={D_0/\nu} + D_1$, where $D_1={\mathcal O}(1)$ is a de-tuning parameter. The periodic pattern is linearly stable when $D_1$ is chosen small enough so that this continuous band is in the stable left-half plane $\mbox{Re}(\lambda)<0$ for all $\kb$. Moreover, for both the Schnakenburg and GM models, our analysis identifies a model-dependent objective function, involving the regular part of the Bloch Green's function, that must be maximized in order to determine the specific periodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of $D$. From a numerical computation, it is shown within the class of oblique Bravais lattices that a regular hexagonal lattice arrangement of spots is optimal for maximizing the stability threshold in $D$.

Mathematical Modeling of Plant Root Hair Initiation: Dynamics of Localized Patches

A mathematical analysis is undertaken of a Schnakenberg reaction-diffusion system in one dimension with a spatial gradient governing the active reaction. This system has previously been proposed as a model of the initiation of hairs from the root epidermis Arabidopsis, a key cellular-level morphogenesis problem. This process involves the dynamics of the small G-proteins, Rhos of plants, which bind to form a single localized patch on the cell membrane, prompting cell wall softening and subsequent hair growth. A numerical bifurcation analysis is presented as two key parameters, involving the cell length and the overall concentration of the auxin catalyst, are varied. The results show hysteretic transitions from a boundary patch to a single interior patch, and to multiple patches whose locations are carefully controlled by the auxin gradient. The results are confirmed by an asymptotic analysis using semistrong interaction theory, leading to closed form expressions for the patch locations and intensities. A c...

First and Second Order Semistrong Interaction in Reaction-Diffusion Systems

Spatial scale separation often leads to sharp interfaces that can be fully localized pulses or transition layer fronts connecting different states. This paper concerns the asymptotic interaction laws of pulses and fronts in the so-called semistrong regime of strongly differing diffusion lengths for reaction-diffusion systems in one space dimension. An asymptotic expansion and matching approach is applied in a model independent common framework. First order semistrong interaction is introduced as a general interface interaction type. It is distinct from the semistrong interaction studied over the past decade, which is referred to as “second order” here. Laws of motion are derived for pulses as well as fronts in abstract systems with attention to the effect of symmetries. First order interaction for pulses is shown to be gradient-like under conditions that are numerically checked for a class of equations including the Gray--Scott and Schnakenberg models.

Front Interactions in a Three-Component System

The three-component reaction-diffusion system introduced in [C. P. Schenk et al., Phys. Rev. Lett., 78 (1997), pp. 3781–3784] has become a paradigm model in pattern formation. It exhibits a rich variety of dynamics of fronts, pulses, and spots. The front and pulse interactions range in type from weak, in which the localized structures interact only through their exponentially small tails, to strong interactions, in which they annihilate or collide and in which all components are far from equilibrium in the domains between the localized structures. Intermediate to these two extremes sits the semistrong interaction regime, in which the activator component of the front is near equilibrium in the intervals between adjacent fronts but both inhibitor components are far from equilibrium there, and hence their concentration profiles drive the front evolution. In this paper, we focus on dynamically evolving N-front solutions in the semistrong regime. The primary result is use of a renormalization group method to rigorously derive the system of N coupled ODEs that governs the positions of the fronts. The operators associated with the linearization about the N-front solutions have N small eigenvalues, and the N-front solutions may be decomposed into a component in the space spanned by the associated eigenfunctions and a component projected onto the complement of this space. This decomposition is carried out iteratively at a sequence of times. The former projections yield the ODEs for the front positions, while the latter projections are associated with remainders that we show stay small in a suitable norm during each iteration of the renormalization group method. Our results also help extend the application of the renormalization group method from the weak interaction regime for which it was initially developed to the semistrong interaction regime. The second set of results that we present is a detailed analysis of this system of ODEs, providing a classification of the possible front interactions in the cases of $N=1,2,3,4$, as well as how front solutions interact with the stationary pulse solutions studied earlier in [A. Doelman, P. van Heijster, and T. J. Kaper, J. Dynam. Differential Equations, 21 (2009), pp. 73–115; P. van Heijster, A. Doelman, and T. J. Kaper, Phys. D, 237 (2008), pp. 3335–3368]. Moreover, we present some results on the general case of N-front interactions.

The semistrong limit of multipulse interaction in a thermally driven optical system

We consider the semistrong limit of pulse interaction in a thermally driven, parametrically forced, nonlinear Schrödinger (TDNLS) system modeling pulse interaction in an optical cavity. The TDNLS couples a parabolic equation to a hyperbolic system, and in the semistrong scaling we construct pulse solutions which experience both short-range, tail–tail interactions and long-range thermal coupling. We extend the renormalization group (RG) methods used to derive semistrong interaction laws in reaction–diffusion systems to the hyperbolic–parabolic setting of the TDNLS system. A key step is to capture the singularly perturbed structure of the semigroup through the control of the commutator of the resolvent and a re-scaling operator. The RG approach reduces the pulse dynamics to a closed system of ordinary differential equations for the pulse locations.

Semistrong Pulse Interactions in a Class of Coupled Reaction-Diffusion Equations

Pulse-pulse interactions play central roles in a variety of pattern formation phenomena, including self-replication. In this article, we develop a theory for the semistrong interaction of pulses in a class of singularly perturbed coupled reaction-diffusion equations that includes the (generalized) Gierer--Meinhardt, Gray--Scott, Schnakenberg, and Thomas models, among others. Geometric conditions are determined on the reaction kinetics for whether the pulses in a two-pulse solution attract or repel, and ODEs are derived for the time-dependent separation distance between their centers and for their wave speeds. In addition, conditions for the existence of stationary two-pulse solutions are identified, and the interactions between stationary and dynamically evolving two-pulse solutions are studied. The theoretical results are illustrated in a series of examples. In two of these, which are related to the classical Gierer--Meinhardt equation, we find that the pulse amplitudes blow up in finite time. Moreover, ...

A model for η production andS 11-wave

In this paper we analyse both η-pion production and π-nucleonS11 phase shift. In particular we obtain numerical information about the latter from the πN→ηN total cross-section. We use extended Le Couteur-Newton formalism for only two channels. Experimental results are explained by a semi-strong interaction between πN channel and ηN channel.

SU3Breaking by an Isotriplet

We propose a model for SU3 breaking through a “fifth” semistrong interaction mediated by an isotriplet of bosons. This model yields a formula equivalent to the G-M-O mass formula, as well as a new sum rule which may explain the Σ-Λ mass difference. Possible candidates for this new interaction might be spinning pions. Assuming that spions have semistrong interaction and postulating that they break the µ-e symmetry by a direct coupling with the µνµ current, one could explain their nonobserved e2 decay.

Model ofCViolation in Semistrong Interactions

A simple model of $C$ violation in semistrong interactions is proposed. The basic semistrong interaction is an "equal" mixture of $C$-invariant and -noninvariant parts. The former part is the coupling between two neutral vector mesons (Sakurai's interaction), and the latter is the coupling between neutral vector and scalar mesons. The nature of this particular interaction and the possible experimental consequences are discussed. The small ${{K}_{2}}^{0}\ensuremath{\rightarrow}2\ensuremath{\pi}$ decay rate can be consistent with our model. Also the $\ensuremath{\rho}\ensuremath{\eta}\ensuremath{\pi}$ coupling and the $C$-violating effect in the ${\ensuremath{\Sigma}}^{0}\ensuremath{\rightarrow}\ensuremath{\Lambda}{e}^{+}{e}^{\ensuremath{-}}$ decay are calculated.