Evans Function Research Articles

Computing Evans functions numerically via boundary-value problems

The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.

Stability of travelling waves in a <i>Wolbachia</i> invasion

Numerous studies have examined the growth dynamics of Wolbachia within populations and the resultant rate of spatial spread. This spread is typically characterised as a travelling wave with bistable local growth dynamics due to a strong Allee effect generated from cytoplasmic incompatibility. While this rate of spread has been calculated from numerical solutions of reaction-diffusion models, none have examined the spectral stability of such travelling wave solutions. In this study we analyse the stability of a travelling wave solution generated by the reaction-diffusion model of Chan & Kim [ 4 ] by computing the essential and point spectrum of the linearised operator arising in the model. The point spectrum is computed via an Evans function using the compound matrix method, whereby we find that it has no roots with positive real part. Moreover, the essential spectrum lies strictly in the left half plane. Thus, we find that the travelling wave solution found by Chan & Kim [ 4 ] corresponding to competition between Wolbachia-infected and -uninfected mosquitoes is linearly stable. We employ a dimension counting argument to suggest that, under realistic conditions, the wavespeed corresponding to such a solution is unique.

Balanced flux formulations for multidimensional Evans-function computations for viscous shocks

The Evans function is a powerful tool for the stability analysis of viscous shock profiles; zeros of this function carry stability information. In the one-dimensional case, it is typical to compute the Evans function using Goodman’s integrated coordinates (1986); this device facilitates the search for zeros of the Evans function by winding number arguments. Although integrated coordinates are not available in the multidimensional case, we show here that there is a choice of coordinates which gives similar advantages.

Stability analysis of combustion waves for competitive exothermic reactions using Evans function

• Analysed a two-step adiabatic competitive exothermic reaction scheme. • Analysed the stability of the combustion waves using the Evan function method. • Determined the threshold values of parameters for the onset of pulsating behaviour. • Showed that the system exhibits pulsating and period-doubling behaviour. We consider travelling wave solutions of a reaction-diffusion system corresponding to an adiabatic two-step competitive exothermic reaction scheme. In such a scheme, a combustion process is assumed to be lumped into two different exothermic reactions. Although the rate constants of the reactions are distinct, both reactions occur simultaneously and feed on the same reactant. The travelling wave solutions are obtained via the shooting-relaxation method. The linear stability analysis is conducted using the Evans function technique and the compound matrix method. Further, threshold values of parameters corresponding to Hopf points are established. It is shown that the system exhibits pulsating behaviour when the parameter values are greater than the threshold values. The onset of instability is found for a broad range of parameter values. Two different numerical methods are then used to obtain solutions from the governing partial differential equations to validate the results.

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier--Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients $\mu$, $\mu +\eta$, and $\nu=\kappa/c_v$ in the physical ratios predicted by statistical mechanics, with Mach number $M>1.035$. Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced may be used in principle to decide stability for any $\gamma$-law gas, $\gamma>1$, and arbitrary $\mu>|\eta|\ge 0$, $\nu>0$ or, indeed, for shocks of much more general models and equations, including in particular viscoelasticity, combustion, and magnetohydrodynamics (MHD).

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut=uxx+α[βH(u−θ)−u]−w, wt=ε(u−γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u], under different conditions on the model constants.To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0<2(1+αγ)θ<αβγ; the existence and stability of two standing wave fronts if 2(1+αγ)θ=αβγ and γ2ε>1; the existence and instability of two standing wave fronts if 2(1+αγ)θ=αβγ and 0<γ2ε<1; the existence and instability of an upside down standing pulse solution if 0<(1+αγ)θ<αβγ<2(1+αγ)θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0<2θ<β; the existence and stability of two standing wave fronts if 2θ=β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0<θ<β<2θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u]−w0, where w0=α(β−2θ)>0 is a positive constant, if 0<2θ<β.To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear differential operator, if and only if λ0 is a zero of the Evans function. The stability, instability and bifurcations of the nonlinear waves follow from the zeros of the Evans functions.A very important motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.

Traveling waves and breathers in an excitatory-inhibitory neural field.

We study existence and stability of traveling activity bump solutions in an excitatory-inhibitory (E-I) neural field with Heaviside firing rate functions by deriving existence conditions for traveling bumps and an Evans function to analyze their spectral stability. Subsequently, we show that these existence and stability results reduce, in the limit of wave speed c→0, to the equivalent conditions developed for the stationary bump case. Using the results for the stationary bump case, we show that drift bifurcations of stationary bumps serve as a mechanism for generating traveling bump solutions in the E-I neural field as parameters are varied. Furthermore, we explore the interrelations between stationary and traveling types of bumps and breathers (time-periodic oscillatory bumps) by bridging together analytical and simulation results for stationary and traveling bumps and their bifurcations in a region of parameter space. Interestingly, we find evidence for a codimension-2 drift-Hopf bifurcation occurring as two parameters, inhibitory time constant τ and I-to-I synaptic connection strength w[over ¯]_{ii}, are varied and show that the codimension-2 point serves as an organizing center for the dynamics of these four types of spatially localized solutions. Additionally, we describe a case involving subcritical bifurcations that lead to traveling waves and breathers as τ is varied.

Spatially Periodic Multipulse Patterns in a Generalized Klausmeier--Gray--Scott Model

Semiarid ecosystems form the stage for a plethora of vegetation patterns, a feature that has been captured in terms of mathematical models since the beginning of this millennium. To study these patterns, we use a reaction-advection-diffusion model that describes the interaction of vegetation and water supply on gentle slopes. As water diffuses much faster than vegetation, this model operates on multiple timescales. While many types of patters are observed in the field, our focus is on two-dimensional stripe patterns. These patterns are typically observed on sloped terrains. In the present idealized setting they correspond to solutions of the model that are spatially periodic in one direction---the $x$-direction---and are extended trivially in the perpendicular $y$-direction. The existence of long wavelength patterns in our model is established analytically using methods from geometric singular perturbation theory, in which a correct parameter scaling is crucial. Subsequently, an Evans function approach yields statements about their stability. Previous work has shown numerically that in order for stripe patterns to be stable, a sufficient slope is needed. The instability of the two-dimensional patterns in parameter regimes corresponding to gentle slopes is confirmed analytically in this article, and we show that the ecological resilience of stripe patterns increases with increasing slope. A full stability proof for steep slopes is, however, beyond the scope of this asymptotic analysis. Since the main destabilization mechanism for the constructed two-dimensional stripes is via perturbations in the transverse direction, we provide a detailed overview of stability results of one-dimensional, spatially periodic patterns and show that they can be stable.

Numerical proof of stability of viscous shock profiles

We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number [Formula: see text], below which profiles are known by energy estimates to be stable, to the upper stability boundary of [Formula: see text], above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.

Stability of gap soliton complexes in the nonlinear Schrödinger equation with periodic potential and repulsive nonlinearity

The work is devoted to numerical investigation of stability of stationary localized modes ("gap solitons") for the one-dimentional nonlinear Schr\"odinger equation (NLSE) with periodic potential and repulsive nonlinearity. Two classes of the modes are considered: a bound state of a pair of in-phase and out-of-phase fundamental gap solitons (FGSs) from the first bandgap separated by various number of empty potential wells. Using the standard framework of linear stability analysis, we computed the linear spectra for the gap solitons by means of the Fourier collocation method and the Evans function method. We found that the gap solitons of the first and second classes are exponentially unstable for odd and even numbers of separating periods of the potential, respectively. The real parts of unstable eigenvalues in corresponding spectra decay with the distance between FGSs exponentially. On the contrary, we observed that the modes of the first and second classes are either linearly stable or exhibit weak oscillatory instabilities if the number of empty potential wells separating FGSs is even and odd, respectively. In both cases, the oscillatory instabilities arise in some vicinity of upper bandgap edge. In order to check the linear stability results, we fulfilled numerical simulations for the time-dependent NLSE by means of a finite-difference scheme. As a result, all the considered exponentially unstable solutions have been deformed to long-lived pulsating formations whereas stable solutions conserved their shapes for a long time.

Stability of traveling waves for autocatalytic reaction systems with strong decay

This paper is concerned with the spatial decay and stability of travelling wave solutions for a reaction diffusion system $u_{t}=d u_{xx}-uv$, $v_{t}=v_{xx}+uv-Kv^{q}$ with $q&gt;1$. By applying Centre Manifold Theorem and detailed asymptotic analysis, we get the precise spatial decaying rate of the travelling waves with noncritical speeds. Further by applying spectral analysis, Evans function method and some numerical simulation, we proved the spectral stability and the linear exponential stability of the waves with noncritical speeds in some weighted spaces.

Localized Patterns in a Three-Component FitzHugh–Nagumo Model Revisited Via an Action Functional

In this manuscript, we combine geometrical singular perturbation techniques and an action functional to revisit—and further study—the existence and stability of stationary localized structures in a singularly perturbed three-component FitzHugh–Nagumo model. In particular, the action functional replaces the Melnikov integral approach used in Doelman et al. (J Dyn Differ Equ 21:73–115, 2009) to explicitly derive existence conditions for stationary localized structures. In addition, the action functional also gives critical information on the stability of these stationary localized structures, thus circumventing a tedious Evans function computation. This highlights the strength of the action functional approach.

Oscillatory instabilities of gap solitons in a repulsive Bose–Einstein condensate

The paper is devoted to numerical study of stability of nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross–Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest (“fundamental”) gap solitons in the first and in the second spectral gap undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The number of unstable eigenvalues and the associated instability rates are described. Several stable and unstable more complex (non-fundamental) gap solitons are also discussed. The results obtained from the Evans function approach are independently confirmed using the direct numerical integration of the GPE.

Spectra of Short Pulse Solutions of the Cubic–Quintic Complex Ginzburg–Landau Equation near Zero Dispersion

We describe a computational method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic–quintic complex Ginzburg–Landau equation about numerically determined stationary solutions. We compare the results of the method to a formula for an edge bifurcation obtained using the small dissipation perturbation theory of Kapitula and Sandstede. This comparison highlights the importance for analytical studies of perturbed nonlinear wave equations of using a pulse ansatz in which the phase is not constant, but rather depends on the perturbation parameter. In the presence of large dissipative effects, we discover variations in the structure of the spectrum as the dispersion crosses zero that are not predicted by the small dissipation theory. In particular, in the normal dispersion regime we observe a jump in the number of discrete eigenvalues when a pair of real eigenvalues merges with the intersection point of the two branches of the continuous spectrum. Finally, we contrast the method to computational Evans function methods.

The effect of perturbed advection on a class of solutions of a non-linear reaction-diffusion equation

In this work, the traveling wave solutions of a one-dimensional reaction-diffusion equation with advection are studied. The traveling wave solutions are obtained using the G′/G-expansion method. The shock thickness and spectral stability have been discussed for the obtained solution in the parameter interval. The essential spectra of the perturbed and linearized differential operator about the traveling antikink and kink solutions at the equilibrium states are obtained. The point spectrum is calculated using Evans function with Lie midpoint method and Magnus method. It is shown that, for a symmetric potential well, the traveling kink and antikink solutions which connect the stable equilibrium states of the system are stable. It is observed that the perturbation on the advection exhibits contrasting effect on the solution properties (shock thickness and the eigenvalue) of kink and antikink solutions. Variation of the reaction coefficient leads to instability of the solutions, unlike the diffusion coefficient which enhances the stability. On the other hand, the variation of reaction and diffusion coefficients show the monotonic effect on the shock thickness of the traveling kink and antikink solutions. This study is expected to be useful in analyzing the slow or fast invasion and stability of the population movement in different steady states.

Combustion waves in hydraulically resistant porous media in a special parameter regime

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.

Geometric phase in the Hopf bundle and the stability of non-linear waves

We develop a stability index for the traveling waves of non-linear reaction–diffusion equations using the geometric phase induced on the Hopf bundle S2n−1⊂Cn. This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeros correspond to the eigenvalues of the linearization of reaction–diffusion operators about the wave. The stability of a traveling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way’s Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on C2 and sketch the proof of the method of geometric phase for Cn and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.

The instability of the Hocking–Stewartson pulse and its geometric phase in the Hopf bundle

This work demonstrates an innovative numerical method for counting and locating eigenvalues with the Evans function. Utilizing the geometric phase in the Hopf bundle, the technique calculates the winding of the Evans function about a contour in the spectral plane, describing the eigenvalues enclosed by the contour for the Hocking–Stewartson pulse of the complex Ginzburg–Landau equation. Locating eigenvalues with the geometric phase in the Hopf bundle was proposed by Way (2009), and proven by Grudzien et al. (2016). Way demonstrated his proposed method for the Hocking–Stewartson pulse, and this manuscript redevelops this example as in the proof of the method in Grudzien et al. (2016), modifying his numerical shooting argument, and introduces new numerical results concerning the phase transition.

Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

We compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles. Specifically, we obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient.

Existence and stability of solutions of the cubic complex Ginzburg-Landau equation with delayed Raman scattering.

We found two stationary solutions of the cubic complex Ginzburg-Landau equation (CGLE) with an additional term modeling the delayed Raman scattering. Both solutions propagate with nonzero velocity. The solution that has lower peak amplitude is the continuation of the chirped soliton of the cubic CGLE and is unstable in all the parameter space of existence. The other solution is stable for values of nonlinear gain below a certain threshold. The solutions were found using a shooting method to integrate the ordinary differential equation that results from the evolution equation through a change of variables, and their stability was studied using the Evans function method. Additional integration of the evolution equation revealed the basis of attraction of the stable solutions. Furthermore, we have investigated the existence and stability of the high amplitude branch of solutions in the presence of other higher order terms originating from complex Raman, self-steepening, and imaginary group velocity.