Evans Function Research Articles

Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

We study the existence and stability of 2-bump solutions of the one-population homogenized Wilson–Cowan model, where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale. The existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a Heaviside function. A framework for analyzing the stability of 2-bumps is formulated, based on spectral theory for Fredholm integral operators. The stability method deforms to the standard Evans function approach for the translationally invariant case in the limit of no heterogeneity, in a way analogous to the single bump case for the homogenized model. The numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function. For the damped oscillating connectivity kernel, we give a concrete example of a 2-bump solution which is stable for all admissible values of the heterogeneity parameter.

Research Spotlights

The analysis of generalized eigenvalue problems is central to understanding a number of complex phenomena, including the stability of nonlinear waves. One generally seeks a characterization of a linearized spectrum in relation to the complex plane (e.g., eigenvalues strictly in the right half plane) from which one can deduce the stability of the system and the presence of features such as bifurcations of Hamiltonian--Hopf type. Two fundamentally different but useful tools for analyzing spectral stability include the Krein signature and the Evans function. The Krein signature is helpful in investigating the stability of purely imaginary eigenvalues (i.e., whether eigenvalues will move toward the right half plane under perturbations), while the Evans function can be used to detect eigenvalue locations. The paper “Graphical Krein Signature Theory and Evans--Krein Functions,” by Richard Kollár and Peter Miller, highlights a graphical interpretation of the Krein signature and more specifically stresses the utility of this graphical interpretation. On the computational side, the graphic interpretation is used to adapt the notion of an Evans function to an Evans--Krein function. The new generalization allows one to calculate the Krein signature in a way that is easy to incorporate into existing simulation capabilities that are already capable of evaluating an Evans function. This is in contrast to the traditional Evans function which cannot generally be used to directly deduce the Krein signature. In addition to this computational utility, the graphical interpretation of the Krein signature has nice theoretical properties as demonstrated by a set of proofs associated with index theorems for linearized Hamiltonians and includes relations to the well-known Vakhitov--Kolokolov criterion.

Graphical Krein Signature Theory and Evans--Krein Functions

Two concepts, evidently very different in nature, have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and, hence, their potential for moving into the right half-plane leading to dynamical instability under perturbation of the system; and (ii) the Evans function, an analytic function detecting the location of eigenvalues. One might expect these two concepts to be related, but unfortunately examples demonstrate that there is no way in general to deduce the Krein signature of an eigenvalue from the Evans function, for example, by studying derivatives of the latter. The purpose of this paper is to recall and popularize a simple graphical interpretation of the Krein signature well known in the spectral theory of polynomial operator pencils. Once esta...

TRAVELING WAVE SOLUTIONS OF NONLINEAR SCALAR INTEGRAL DIFFERENTIAL EQUATIONS ARISING FROM SYNAPTICALLY COUPLED NEURONAL NETWORKS

Consider the following nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. These model equations generalize many important nonlinear scalar integral differential equations aris ing from synaptically coupled neuronal networks. The kernel functions K and W represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), lateral inhibitions (modeled with Mexican hat kernel functions), lateral excitations (modeled with upside down Mexican hat kernel functions), but also synaptic couplings which may change sign for finitely many times or even infinitely many times. The function H = H(u − ) represents the Heaviside step function, which is defined by H(u − ) = 0 for all u . The functions and represent probability density functions defined on (0,1). The parameter c > 0 represents the speed of an action potential and the parameter > 0 represents a constant delay. In these equations, u = u(x, t) stands for the membrane potential of a neuron at position x and time t. The positive constants > 0 and > 0 represent synaptic rates. The positive constants > 0 and > 0 represent thresholds for excitation of neurons. The function f = f(u) represents the sodium currents in neuronal networks. The positive constant w 0 > 0 is to be given. The authors will establish the existence and stability of traveling wave solutions of these nonlinear scalar integral differential equations by coupling together speed index functions, stability index func tions (often called Evans functions, that is, complex analytic functions), implicit function theorem, intermediate value theorem, mean value theorem, global strong maximum principle for Evans func tions, linearized stability criterion and many other important techniques in dynamical systems. They will find sufficient conditions satisfied by the synaptic couplings, by the probability density functions, by the synaptic rate constants and by the thresholds so that the traveling wave solutions and their wave speeds exist, and the stability of the traveling wave solutions is true. The main results obtained in this paper greatly improve many previous results.

Exponential stability of the traveling fronts for a viscous Fisher-KPP equation

This paper is concerned with the stability of traveling frontsolutions for a viscous Fisher-KPP equation. By applying geometricsingular perturbation method, special Evans function estimates,detailed spectral analysis and $C_0$ semigroup theories, eachtraveling front solution with wave speed $c<-2\sqrt{f^\prime(0)}$ isproved to be locally exponentially stable in some appropriateexponentially weighted spaces.

On the Variational Approach for Analyzing the Stability of Solutions of Evolution Equations

The eigenvalue problems arise in the analysis of stability of traveling waves or rest state solutions are currently dealt with, using the Evans function method. In the literature, it had been shown that, use of this method is not straightforward even in very simple examples. Here an extended \variational method to solve the eigenvalue problem for the higher order dierential equations is suggested. The extended method is matched to the well known variational iteration method. The criteria for validity of the eigenfunc- tions and eigenvalues obtained is presented. Attention is focused to nd eigenvalue and eigenfunction solutions of the Kuramoto-Slivashinsky and (K(p,q)) equation.

Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals

Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh’ theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are well-controlled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schrödinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHugh–Nagumo system. We also relate the contour method to the well-known Evans function and show that our approach provides an alternative to evaluating and computing its zeros.

Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one

We study the stability/instability of the subsonic travelling waves of the Nonlinear Schr\"odinger Equation in dimension one. Our aim is to propose several methods for showing instability (use of the Grillakis-Shatah-Strauss theory, proof of existence of an unstable eigenvalue via an Evans function) or stability. For the later, we show how to construct in a systematic way a Liapounov functional for which the travelling wave is a local minimizer. These approaches allow to give a complete stability/instability analysis in the energy space including the critical case of the kink solution. We also treat the case of a cusp in the energy-momentum diagram.

Stability of viscous detonations for Majda’s model

Using analytical and numerical Evans-function techniques, we examine the spectral stability of strong-detonation-wave solutions of a version of Majda’s scalar model for a reacting gas mixture with an Arrhenius-type ignition function. We introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane, and we use a numerical approximation of the Evans function to search for possible unstable eigenvalues in this region. Our results show, for the parameter values tested, that these waves are spectrally stable. Combining these numerical results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [G. Lyng, M. Raoofi, B. Texier, K. Zumbrun, Pointwise Green function bounds and stability of combustion waves, J. Differential Equations 233 (2) (2007) 654–698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonation-wave solutions of a Majda-type model without a smallness assumption. Notably, our results indicate that, for this simplified, scalar model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to “galloping” or “pulsating” solutions as is observed in the full reactive Navier–Stokes equations. This answers in the negative, for this model, a question posed by Majda as to whether such scalar detonation models capture this aspect of detonation behavior.

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation

In this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized Kuramoto–Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch–She–Thual, Bar–Nepomnyashchy, Chang–Demekhin–Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader’s convenience, we include in an appendix the corresponding treatment of the Swift–Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto–Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.

Analysing the stability of premixed rich hydrogen–air flame with the use of two-step models

In this paper we investigate the thermal-diffusive instability of premixed adiabatic flames in rich hydrogen–air mixtures at normal ambient conditions. Several models which feature the same two-step global kinetics with chain-branching and recombination reaction steps are considered. These global kinetic steps are assumed to be controlled by different elementary reactions. The flame speed and structure are investigated numerically by using the shooting-relaxation algorithms. The stability is studied by means of the Evans function method and by direct integration of the governing partial differential equations. It is demonstrated that the two-step models are capable of accurately predicting the speed and structure of combustion waves as well as flame stability and frequency of pulsations of unsteady combustion waves.

Existence and Stability of Traveling Pulses in a Neural Field Equation with Synaptic Depression

We examine the existence and stability of traveling pulse solutions in a continuum neural network with synaptic depression and smooth firing rate function. The existence proof relies on geometric singular perturbation theory and blow-up techniques, as one needs to track the solution near a point on the slow manifold that is not normally hyperbolic. The stability of the pulse is then investigated by computing the zeros of the corresponding Evans function. This study predicts that synaptic depression leads to the formation of stable traveling pulses with algebraic decay along their back. This characteristic feature differs from the exponential decay of traveling pulses of neural field models with linear adaptation.

Gasless Combustion Fronts with Heat Loss

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.

Pulses in a Gierer--Meinhardt Equation with a Slow Nonlinearity

In this paper, we study in detail the existence and stability of localized pulses in a Gierer--Meinhardt equation with an additional “slow” nonlinearity. This system is an explicit example of a general class of singularly perturbed, two component reaction-diffusion equations that goes significantly beyond well-studied model systems such as Gray--Scott and Gierer--Meinhardt. We investigate the existence of these pulses using the methods of geometric singular perturbation theory. The additional nonlinearity has a profound impact on both the stability analysis of the pulse---compared to Gray--Scott/Gierer--Meinhardt-type models a distinct extension of the Evans function approach has to be developed---and the stability properties of the pulse: several (de)stabilization mechanisms turn out to be possible. Moreover, it is shown by numerical simulations that, unlike the Gray--Scott/Gierer--Meinhardt-type models, the pulse solutions of the model exhibit a rich and complex behavior near the Hopf bifurcations.

Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems u t = d u xx − uf (v ), v t = v xx + uf (v ) with d > 0 and d ≠ 1, where f (v ) has super-linear or linear degeneracy at v = 0. By applying Lyapunov-Schmidt decomposition method in some appropriate exponentially weighted spaces, we obtain the existence and continuous dependence of wave fronts with some critical speeds and with exponential spatial decay for d near 1. By applying special phase plane analysis and approximate center manifold theorem, the existence of traveling waves with algebraic spatial decay or with some lower exponential decay is also obtained for d > 0. Further, by spectral estimates and Evans function method, the wave fronts with exponential spatial decay are proved to be spectrally or linearly stable in some suitable exponentially weighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to be locally exponentially stable in some exponentially weighted spaces and Lyapunov stable in C unif (ℝ) space, if the initial perturbation of the waves is small in both the weighted and unweighted norms; the perturbation of the waves also stays small in L 1 (ℝ) norm and decays algebraically in C unif (ℝ) norm, if the initial perturbation is in addition small in L 1 norm.

Whitham's modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky Equation

We study the spectral stability of periodic wave trains of the Korteweg-de Vries-Kuramoto-Sivashinsky equation which are, among many other applications, often used to describethe evo- lution of a thin liquid film flowing down an inclined ramp. More precisely, we show that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to side-band perturbations. Here, we use a direct Bloch ex- pansion method and spectral perturbation analysis instead of Evans function computations. We first establish, in our context, the now usual connection between first-order expansion of eigenvalues bi- furcating from the origin (both eigenvalue 0 and Floquet parameter 0) and the first-order Whitham's modulation system: the hyper- bolicity of such a system provides a necessary condition of spectral stability. Under a condition of strict hyperbolicity, we show that eigenvalues are indeed analytic in the neighborhood of the origin and that their expansion up to second order is connected to a vis- cous correction of the Whitham's equations. This, in turn, pro- vides new stability criteria. Finally, we study the Korteweg-de Vries limit: in this case, the domain of validity of the previous expansion shrinks to nothing and a new modulation theory is needed. The new modulation system consists of the Korteweg-de Vries modula- tion equations supplemented with a source term: relaxation limit in such a system provides, in turn, some stability criteria.

Spectral Behavior of Combustion Fronts with High Exothermicity

We study fronts in a reaction-diffusion model for high Lewis number combustion processes with the reaction rate in the form of an Arrhenius law. By a combination of spectral energy estimates and Evans function computation, we study the unstable spectrum of a class of combustion fronts with high exothermicity, where the previously known results do not apply and interesting spectral behavior is observed.

Instability of solitons under flexure and twist of an elastic rod

We study the stability of planar soliton solutions of equations describing the dynamics of an infinite inextensible unshearable rod under three-dimensional spatial perturbations. As a result of linearization about the soliton solution, we obtain an inhomogeneous scalar equation. This equation leads to a generalized eigenvalue problem. To establish the instability, we must verify the existence of an unstable eigenvalue (an eigenvalue with a positive real part). The corresponding proof of the instability is done using a local construction of the Evans function depending only on the spectral parameter. This function is analytic in the right half of the complex plane and has at least one zero on the positive real axis coinciding with an unstable eigenvalue of the generalized spectral problem.

Linear stability analysis for travelling waves of second order in time PDE's

We study travelling waves φc of second order in time PDE's . The linear stability analysis for these models is reduced to the question of the stability of quadratic pencils in the form , where .If is a self-adjoint operator, with a simple negative eigenvalue and a simple eigenvalue at zero, then we completely characterize the linear stability of φc. More precisely, we introduce an explicitly computable index , so that the wave φc is stable if and only if . The results are applicable both in the periodic case and in the whole line case.The method of proof involves a delicate analysis of a function , associated with , whose positive zeros are exactly the positive (unstable) eigenvalues of the pencil . We would like to emphasize that the function is not the Evans function for the problem, but rather a new object that we define herein, which fits the situation rather well.As an application, we consider three classical models—the ‘good’ Boussinesq equation, the Klein–Gordon–Zakharov (KGZ) system and the fourth order beam equation. In the whole line case, for the Boussinesq case and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable travelling waves (and for all powers of p in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander et al (2012, personal communication) and Alexander and Sachs (1995 Nonlinear World 2 471–507), which apply to the case p = 2. For the beam equation, we provide an implicit formula (depending only on the function , which works for all p and for both the periodic and the whole line cases.Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves. Informally, we have found that in all the examples that we have looked at, our theory applies, whenever the Grillakis–Shatah–Strauss (GSS) theory applies. We believe that the results in this paper (or a variation thereof) will enable the linear stability analysis as well as asymptotic stability analysis for most models in the form .

Saddle–node bifurcation of viscous profiles

Traveling wave solutions of viscous conservation laws, that are associated to Lax shocks of the inviscid equation, have generically a transversal viscous profile. In the case of a non-transversal viscous profile we show by using Melnikov theory that a parametrized perturbation of the profile equation leads generically to a saddle–node bifurcation of these solutions. An example of this bifurcation in the context of magnetohydrodynamics is given. The spectral stability of the traveling waves generated in the saddle–node bifurcation is studied via an Evans function approach. It is shown that generically one real eigenvalue of the linearization of the viscous conservation law around the parametrized family of traveling waves changes its sign at the bifurcation point. Hence this bifurcation describes the basic mechanism of a stable traveling wave which becomes unstable in a saddle–node bifurcation.