Numerical computation of an Evans function for travelling waves
K. Harley, P. v Heijster, R. Marangell, G.J. Pettet, M. Wechselberger
NNUMERICAL COMPUTATION OF AN EVANS FUNCTION FORTRAVELLING WAVES
K. HARLEY , P. VAN HEIJSTER , R. MARANGELL † , G. J. PETTET ,AND M. WECHSELBERGER Abstract.
We demonstrate a geometrically inspired technique for computing Evansfunctions for the linearised operators about travelling waves. Using the examples ofthe F-KPP equation and a Keller-Segel model of bacterial chemotaxis, we produce anEvans function which is computable through several orders of magnitude in the spectralparameter and show how such a function can naturally be extended into the continuousspectrum. In both examples, we use this function to numerically verify the absence ofeigenvalues in a large region of the right half of the spectral plane. We also include a newproof of spectral stability in the appropriate weighted space of travelling waves of speed c ≥ √ δ in the F-KPP equation. Introduction
The main focus of this article is a geometrically inspired technique for numerically analysingthe spectral stability of a travelling wave. In particular, we illustrate a new method forcomputing an
Evans function for a linearised operator, linearised about a travelling wavesolution to a partial differential equation (PDE) in 1+1 independent variables. Evansfunctions first arose in the 1970’s [8] and are typically constructed using a matching con-dition (see, for example, [1, 11, 32]). They are analytic functions in some relevant regionof the complex λ -spectral plane, with the property that the multiplicity of their roots co-incide with the multiplicity of λ as an eigenvalue. A well-known obstacle in the numericalcomputation of the Evans function is the tendency of the associated eigenvalue ordinarydifferential equations (ODEs) to become stiff, an issue that can sometimes be overcome byworking in the exterior product space [2, 4, 5, 6]. Building on this, there have been twocomplementary directions to the study of (efficient) numerical computation of the Evans Mathematical Sciences School, Queensland University of Technology, Brisbane, QLD 4000 Australia. School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006 Australia. † Corresponding Author: email: [email protected]. a r X i v : . [ m a t h . SP ] M a y K. HARLEY, P. VAN HEIJSTER, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER function: Continuous Orthogonalization (see [17, 16, 39] and the references therein fordetails) and Grassmannian Spectral Shooting, or the
Riccati approach [3, 25, 26].It is this second point of view that we follow here. By following the Riccati approach,and using the geometric structure of the problem, we are led to an Evans function whosecomputation is efficiently tractable even through relatively large changes in the order ofthe spectral parameter. The geometric interpretation of the Riccati equation allows usto avoid the problem of finite time blowup of the relevant solutions to nonlinear ODEs.Furthermore, this approach is readily extendible to values of the spectral parameter whichlie in the continuous spectrum (also called the essential spectrum). In extending the Evansfunction into the continuous spectrum, we highlight another underlying relationship be-tween instabilities of a travelling wave and the geometry of the spectral problem.Our approach shows the power of topological/dynamical systems techniques to analyseanalytically and numerically the (spectral) stability of travelling waves that are pervasivein the mathematical biological literature. In this manuscript we illustrate this techniqueon two well-known models, both considered on an unbounded domain. The first is theFisher/Kolmogorov–Petrovsky–Piscounov (F-KPP) equation(1) u t = δu xx + u (1 − u ) , and the second is a Keller–Segel (K-S) model of bacterial chemotaxis u t = εu xx − αw ,w t = δw xx − β (cid:16) wu x u (cid:17) x . (2)The F-KPP equation was chosen because in this case, the travelling wave stability analysisbecomes analytically tractable (see Section 2.7). Further, the literature on stability oftravelling waves in eq. (1) is vast (see for example [28, 32, 34, 37] and the references thereinfor a partial list of the stability results). It thus provides a well-known backdrop againstwhich to verify our spectral calculations. The K-S model in eq. (2) was chosen in order tohighlight how to extend our methods beyond scalar PDEs. It is also convenient becauseexplicit solutions to eq. (2) can be found when ε = 0 [9]. We are thus able to omit atime-consuming step (numerically finding the solutions to the travelling wave ODEs) andfocus on setting up and analysing the linearised spectral problem. Explicit solutions arenot necessary for our methods to work (as the F-KPP example shows) and we discuss howthe stability analysis of travelling waves can be adapted to the K-S problem when ε (cid:54) = 0 inSection 4.2. GEOMETRICALLY INSPIRED EVANS FUNCTION 3
The F-KPP equation was first introduced by Luther in 1906 who originally used it to modeland study travelling waves in chemical reactions [27]. It was named for Fisher, and forKolmogorov, Petrovsky and Piscounov, who independently wrote seminal papers on theequation, using it to model the spread of a gene through a population [10, 24]. In thismanuscript, we assume the diffusion coefficient δ is strictly positive.Equation (2) was proposed in the 1970’s by Keller and Segel [22, 23] to describe chemo-tactically-driven cell migration in which a population of bacteria exhibits an advectiveflux in response to a gradient of a diffusible secondary species (i.e. nutrient); see [15, 35]and references therein for a current overview of PDE models with chemotaxis. In eq. (2),the bacteria population density is denoted by w ( x, t ) and the nutrient concentration by u ( x, t ). The model exhibits so-called logarithmic sensitivity and we assume a constantconsumption rate function. The diffusion of the nutrient is assumed to be much smallerthan the diffusion of the bacteria population: 0 ≤ ε (cid:28) δ . Finally, α > β > < δ < β .By a travelling wave solution of eqs. (1) or (2), we mean a solution to eq. (1) of the form u ( x − ct ), or a pair of solutions to eq. (2) of the form ( u ( x − ct ) , w ( x − ct )) travelling fromleft to right with some positive (constant) wave propagation speed c .To study travelling wave solutions, we introduce a moving coordinate frame; setting z := x − ct , and τ := t , eq. (1) becomes(3) u τ = δu zz + cu z + u (1 − u ) , while eq. (2) becomes u τ = εu zz − αw + cu z ,w τ = δw zz − β (cid:16) wu z u (cid:17) z + cw z . (4)Travelling waves u = ˆ u ( z ), or ( u, w ) = (ˆ u ( z ) , ˆ w ( z )) will then satisfy the ODEs:(5) δu zz + cu z + u (1 − u ) = 0 , or εu zz − αw + cu z = 0 ,δw zz − β (cid:16) wu z u (cid:17) z + cw z = 0 . (6) K. HARLEY, P. VAN HEIJSTER, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER
Once travelling waves to eqs. (1) and (2) have been found, we are next concerned withtheir stability. In particular, we are interested in the spectral stability of the travellingwaves. A full analysis of stability of travelling waves in the F-KPP and K-S equations iswell beyond the scope of this manuscript. However, in the F-KPP equation, it is knownthat travelling waves of speed c ≥ √ δ are spectrally and linearly stable relative to certainperturbations (or in certain weighted spaces) and that the travelling waves of speed 0 RM, GJP and MW gratefully acknowledge the partial supportof Australian Research Council grant ARC DP110102775. PvH gratefully acknowledgessupport under the Australian Research Council’s Discovery Early Career Researcher Awardfunding scheme DE140100741. KH also gratefully acknowledges support from an AustralianMathematical Society Lift-off Fellowship.2. Travelling waves in the F-KPP equation We use a dynamical systems approach to analyse the travelling wave problem of the F-KPPequation, i.e. we write eq. (5) as a system of first-order equations:(7) dudz = v , dvdz = 1 δ ( − cv − u (1 − u )) . There are two equilibria of eq. (7) in the uv -plane: one at (0 , 0) and one at (1 , Df ( u, v ) = (cid:32) − uδ − cδ (cid:33) . At the point (1 , Df ( u, v ) are(8) µ u = − c + √ c + 4 δ δ , µ s = − c − √ c + 4 δ δ . For all values of c there is one positive eigenvalue and one negative eigenvalue. Thus, (1 , , Df ( u, v ) are(9) µ s = − c + √ c − δ δ , µ ss = − c − √ c − δ δ . K. HARLEY, P. VAN HEIJSTER, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER -50 -40 -30 -20 -10 0 10 20 30 40 5000.20.40.60.81 -10 -5 0 5 10 15 20-0.8-0.6-0.4-0.200.20.40.60.810 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.010 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8-0.6-0.4-0.200.20.40.6 1 Figure 1. Phase portraits and wave profiles for travelling waves in the F-KPP equation for various values of the wave speed c . In the figures on theleft, we have c ≥ √ δ , (the plots show c = 3 , , , , and 11, with δ = 1),while on the right, 0 ≤ c < √ δ ( c = 0 . , . , . , and 1, with δ = 1). Thetop figures are the wave profiles while the bottom figures show that theyindeed form heteroclinic connections in the phase portrait of eq. (7) andthat the origin is a stable node when c ≥ √ δ and a stable focus when0 < c < √ δ .For c ≥ √ δ these are two real (distinct or equal), negative eigenvalues so (0 , 0) is a(possibly degenerate) node. For 0 < c < √ δ , it is a stable focus. It is easy to see from therelated phase portrait that for any value of c > , 0) to (0 , c ≥ δ this orbit remains negative in v , corresponding to a familyof monotone travelling waves satisfying ˆ u ( −∞ ) = 1 and ˆ u (+ ∞ ) = 0. When c < δ , thereis a family of non-monotone travelling waves. See Figure 1.2.1. The spectral problem. Travelling wave solutions to (1) are steady state solutionsto (3). Once a travelling wave ˆ u ( z ) is found for a fixed c , we wish to consider how eq. (3) GEOMETRICALLY INSPIRED EVANS FUNCTION 7 behaves relative to perturbations (in the moving frame) about the travelling wave. Wemake the ansatz u ( z, τ ) = ˆ u ( z ) + p ( z, τ ), with p ( z, τ ) in an appropriate Banach space,substitute into eq. (3) and consider only the first-order perturbative terms to give theformal (linearised) equation for p :(10) p τ = δp zz + cp z + (1 − u ) p . Let H ( R ) denote the usual Sobolev space of functions from R to R which are squareintegrable and with first (weak) derivative also being square integrable. We define the(linear) operator L : H ( R ) → H ( R ) by L p := ( δ∂ zz + c∂ z + (1 − u )) p . Letting I be the identity map on H ( R ), we have the following definition: Definition 2.1. We say that a λ ∈ C is in the spectrum of the operator L if the operator L − λ I is not invertible on (some dense subset of) H ( R ). The set of all such λ ∈ C will bedenoted as σ ( L ).The operator L − λ I on H ( R ) is equivalent to the operator T ( λ ) : H ( R ) × L ( R ) →H ( R ) × L ( R ) given by(11) T ( λ ) (cid:32) pq (cid:33) := (cid:18) dd z − A ( z ; λ ) (cid:19) (cid:32) pq (cid:33) := (cid:32) pq (cid:33) (cid:48) − (cid:32) λ − uδ − cδ (cid:33) (cid:32) pq (cid:33) . Here we have defined (cid:48) := dd z and the matrix A ( z ; λ ), and we further define A ± ( λ ) := lim z →±∞ A ( z ; λ ) = (cid:32) λ ∓ δ − cδ (cid:33) . The (spatial) eigenvalues of A + ( λ ) are(12) µ u + ( λ ) := − c + (cid:112) c + 4 δ ( λ − δ , µ s + ( λ ) := − c − (cid:112) c + 4 δ ( λ − δ , and those of A − ( λ ) are(13) µ u − ( λ ) := − c + (cid:112) c + 4 δ ( λ + 1)2 δ , µ s − ( λ ) := − c − (cid:112) c + 4 δ ( λ + 1)2 δ . As it will often be convenient, when there is no ambiguity we will drop the arguments inthe eigenvalues of the matrices A ± ( λ ), writing instead µ s,u ± as appropriate. We remark alsothat in the case that λ = 0, we have that µ s,u − (0) = µ s,u , and that µ s,u + (0) = µ ss,s frombefore. Further, we have that the (spatial) eigenvectors of A ± ( λ ) are (1 , µ u,s ± ) (cid:62) . Lastly, we K. HARLEY, P. VAN HEIJSTER, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER denote the subspaces spanned by the various eigenvectors of A ± ( λ ) as ξ u,s ± ( λ ), respectively,again with the possibility of dropping the argument when convenient.2.2. The Continuous Spectrum. We claim that the spectrum of the operator L nat-urally falls into two parts: the continuous spectrum and the point spectrum. The pointspectrum will be values λ ∈ σ ( L ) such that ( L − λ ) is a Fredholm operator of index zero.The continuous spectrum will be the complement of the point spectrum (in σ ( L )). For thedescription of the continuous spectrum we follow [32, 33], while in order to best describethe point spectrum of L we follow [1, 18, 30]. There is no discrepancy with our choices,however, and equivalent statements for the point and continuous spectrum are found in allof [18, 20, 30, 32, 33]. We refer the reader to [20, 32] for a rigorous proof of the equivalenceof all such definitions as well as the fact that our definition of the spectrum can indeed bebroken up into the sets defined as the point and continuous spectrum, as given below. Definition 2.2. We recall that the signature of a matrix M , is the triple ( n , n , n ) wherethe n j ’s are the dimensions of the positive, negative and null space of M respectively. Thesignature will either be denoted by a triple of integers (e.g, (2 , , , + , − )). See Figure 6. Definition 2.3. We define the continuous spectrum of the operator L , denoted σ c ( L ) orsometimes just σ c , to be the set (in C ) of those λ for which the signatures of A + ( λ ) and A − ( λ ) are not equal.We note that one can track the real part of the eigenvalues of A ± ( λ ) and that only one ofthe signs of Re (cid:0) µ u ± (cid:1) will change as λ is varied. Further, in order for the sign of Re (cid:0) µ u ± ( λ ) (cid:1) to change, there must be a λ where Re (cid:0) µ u ± ( λ ) (cid:1) = 0. Writing ω | ω | for the sign of a realnumber ω , we have the following: Corollary 2.1. The set σ c ( L ) can be written as σ c ( L ) := (cid:40) λ ∈ C (cid:12)(cid:12)(cid:12)(cid:12) Re (cid:0) µ u − (cid:1) | Re (cid:0) µ u − (cid:1) | (cid:54) = Re (cid:0) µ u + (cid:1) | Re (cid:0) µ u + (cid:1) | (cid:41) . The equations defining the boundary of the continuous spectrum are important in theirown right and are the so-called dispersion relations . These are where at least one of theeigenvalues of A + ( λ ) or A − ( λ ) is purely imaginary and are given parametrically by(14) λ = − δk ± ick for k ∈ R . GEOMETRICALLY INSPIRED EVANS FUNCTION 9 Here ik would be the imaginary eigenvalue of A ± ( λ ). This describes two parabolas, openingleftward and intersecting the real axis at ± 1. The complex plane minus the continuousspectrum is composed of two disjoint sets: C \ σ c = Ω (cid:116) Ω . We define the sets in accordancewith Definition 2.3 (see Figure 2):Ω := (cid:8) λ ∈ C \ σ c | < Re (cid:0) µ u + ( λ ) (cid:1) < Re (cid:0) µ u − ( λ ) (cid:1)(cid:9) , Ω := (cid:8) λ ∈ C \ σ c | Re (cid:0) µ u + ( λ ) (cid:1) < Re (cid:0) µ u − ( λ ) (cid:1) < (cid:9) . Eigenvalues. For a λ (cid:54)∈ σ c we ask whether there are any nontrivial functions in thekernel of T ( λ ). That is, can we find a nontrivial solution in H ( R ) × L ( R ) to the firstorder system(15) (cid:32) pq (cid:33) (cid:48) = (cid:32) λ − uδ − cδ (cid:33) (cid:32) pq (cid:33) = A ( z ; λ ) (cid:32) pq (cid:33) ?Any such solution must decay to 0 as z → ±∞ and as the next proposition illustrates,there is only one way that this can be realised. Proposition 2.1. For λ ∈ C \ σ c , if ( p, q ) is a solution to (15) such that ( p, q ) ∈ H ( R ) × L ( R ) then (16) lim z →−∞ (cid:32) pq (cid:33) → ξ u − and lim z →∞ (cid:32) pq (cid:33) → ξ s + . That is, ( p, q ) decays to the stable subspace ξ s + of A + ( λ ) as z → + ∞ and the unstablesubspace ξ u − of A − ( λ ) as z → −∞ . A rigorous proof of this proposition can be found in [18, 20, 30]. An intuitive reasoningbehind why the proposition should be true is the following: For a λ ∈ Ω , as z → ∞ , thesystem (15) behaves like (cid:32) pq (cid:33) (cid:48) = A + ( λ ) (cid:32) pq (cid:33) and since we are in the region Ω , we have only one stable direction. Thus, if ( p, q ) → ξ s + . The same is true as z →−∞ because we must have that the solution decays to zero along the subspace ξ u − . Thisargument also shows that for any λ ∈ Ω no solutions decay to zero as z → −∞ . Definition 2.4. We will say that λ ∈ Ω is a (temporal) eigenvalue with eigenfunction p if we can find such a solution to (15) which is in H ( R ) × L ( R ). − − − − − − − − Ω Ω σ c Re( λ )Im( λ ) Figure 2. The continuous spectrum of the linearised operator around atravelling wave (c.f. eq. (11) and Definition 2.3). The boundary is given byeq. (14). For the above picture we chose δ = 1 and c = 5 √ we have0 < Re (cid:0) µ u + (cid:1) < Re (cid:0) µ u − (cid:1) , in Ω we have Re (cid:0) µ u + (cid:1) < Re (cid:0) µ u − (cid:1) < σ c we have Re (cid:0) µ u + (cid:1) < < Re (cid:0) µ u − (cid:1) .We next exploit the linearity of (15). For a fixed λ ∈ Ω and for each z ∈ R , let Ξ u ( z ; λ )be the linear subspace of solutions which decay to ξ u − as z → −∞ and let Ξ s ( z ; λ ) be thelinear subspace of solutions which decay to ξ s + as z → + ∞ . We note that in our examplewe can (for any fixed λ ) view Ξ u ( z ; λ ) and Ξ s ( z ; λ ) as (line) bundles over R . This justifiescalling Ξ u ( z ; λ ) ‘the unstable manifold’ and Ξ s ( z ; λ ) ‘the stable manifold’. What we meanby this is that Ξ u ( z ; λ ) is the manifold of solutions that decay as z → −∞ to the unstablesubspace of A − ( λ ) (and similarly for Ξ s ( z ; λ )). We can evaluate Ξ u and Ξ s at a fixed value z and if they are linearly dependent then we will have an eigenvalue. This is because ofuniqueness of solutions to ODEs; if they agree at one z then they must agree for all z ∈ R and so we have (a linear subspace) of solutions which decay as z → ±∞ . GEOMETRICALLY INSPIRED EVANS FUNCTION 11 Let w u ( z ; λ ) = (cid:32) w u ( z ; λ ) w u ( z ; λ ) (cid:33) and w s ( z ; λ ) = (cid:32) w s ( z ; λ ) w s ( z ; λ ) (cid:33) be two solutions in Ξ u and Ξ s , respectively. These are two vectors in C and we know that λ is an eigenvalue if and only if they are linearly dependent for some (and hence every) z ∈ R . For convenience, we choose z = 0. We have therefore shown the following: Proposition 2.2. The complex number λ ∈ Ω is an eigenvalue if and only if (17) D ( λ ) := det (cid:32) w u (0; λ ) w s (0; λ ) w u (0; λ ) w s (0; λ ) (cid:33) = 0 . Definition 2.5. The function D ( λ ) defined in Proposition 2.2 is called an Evans function .2.4. The Riccati equation. For Definition 2.5, we only compare two possible appropri-ately decaying solutions to the ODE (15). In the following, we are interested in whetheror not a pair of subspaces intersect, rather than the particulars of any given solution. Definition 2.6. The set of (complex) one-dimensional subspaces in complex two-space iscalled complex projective space and is denoted C P .Complex projective space can be given the structure of a complex manifold of one complexdimension and is topologically equivalent to the Riemann sphere, which we denote by S .A line in C through the origin is determined by a pair of complex numbers denoted [ p : q ]that are not both zero. We can write down all the lines where p (cid:54) = 0 as [1 : η ] and we seeright away that this is (equivalent to) a copy of the complex plane. Similarly, we write allthe lines where q (cid:54) = 0 as [ τ : 1] and so this too is equivalent to a copy of the complex plane.Further, for any line except for two (where p or q = 0), we have that η = 1 τ . These are thetypical charts on C P . For a given two-dimensional system of linear first-order ODEs, weget an equivalent (nonlinear, non-autonomous) flow on C P : the so-called Riccati equation .We obtain an expression (on each chart) for the Riccati equation by simply differentiatingthe defining relations of η and τ and using eq. (15). We get η (cid:48) = (cid:18) qp (cid:19) (cid:48) = 1 δ ( λ − u ) − cδ η − η ,τ (cid:48) = (cid:18) η (cid:19) (cid:48) = − η (cid:48) η = 1 + cδ τ − δ ( λ − u ) τ , (18) being two first-order non-autonomous nonlinear ODEs. Further, we have that ξ u ± and ξ s ± will be fixed points of these systems. To see this in coordinates, we have that in the η chart, ξ u,s ± is given by the eigenvalues µ u,s ± , while in the τ chart, they are the multiplicative inverses,a feature of eq. (15) that will not in general be true for an arbitrary two-dimensional systemof first order ODEs. We also have that λ will be an eigenvalue if and only if we can find aheteroclinic connection between ξ u − and ξ s + . In terms of the η chart, this is a heteroclinicconnection between µ u − and µ s + (and between their multiplicative inverses in the τ chart). Remark 2.1. By writing out the real and imaginary parts of the flow in the η and τ chartsand by considering the flow direction on the real axis, one can show that there cannot be aheteroclinic connection in the case of a spectral parameter with non-zero imaginary part.Further, by applying techniques used in [19] one can similarly show that there are no real,positive eigenvalues. In Section 2.7, we exploit this idea to prove the absence of eigenvaluesin the case of travelling waves in the F-KPP equation.We determine a related Evans function by letting η u ( z ; λ ) and η s ( z ; λ ) be the solutionswhich decay to ξ u − and ξ s + , respectively, in the η chart. Moreover, suppose that η s,u ( z ; λ )is finite for all z ∈ R . This corresponds to w s,u ( z ; λ ) being non-zero or staying in a singlechart. This requirement is not necessary and we discuss what happens (see section 2.6) ifwe need to leave the chart, below, but we include it here for convenience. We define a newfunction E η ( λ ) := D ( λ ) w u (0; λ ) w s (0; λ )= w u (0; λ ) w s (0; λ ) − w u (0; λ ) w s (0; λ ) w u (0; λ ) w s (0; λ )= η s (0; λ ) − η u (0; λ ) . (19)We define the functions τ u ( z ; λ ) and τ s ( z ; λ ) and the corresponding Evans function in the τ chart similarly. Here, though, we have E τ ( λ ) := D ( λ ) w u (0; λ ) w s (0; λ )= w u (0; λ ) w s (0; λ ) − w u (0; λ ) w s (0; λ ) w u (0; λ ) w s (0; λ )= τ u (0; λ ) − τ s (0; λ ) . (20)Note that for λ ∈ Ω , the function E η ( λ ) is zero if and only if η s (0; λ ) = η u (0; λ ). Byuniqueness of solutions to ODEs, we therefore have that η u ( z ; λ ) = η s ( z ; λ ) for all z ∈ R GEOMETRICALLY INSPIRED EVANS FUNCTION 13 and, hence, a heteroclinic connection between ξ s + and ξ u − exists. This will be true if andonly if λ ∈ Ω is an eigenvalue. The same argument holds for E τ ( λ ), i.e., we will have aneigenvalue if and only if E τ ( λ ) = 0.We only need to calculate the η ’s to compute the Evans functions. Since τ = 1 η , we have E τ ( λ ) = τ u (0; λ ) − τ s (0; λ ) = 1 η u (0; λ ) − η s (0; λ )= η s (0; λ ) − η u (0; λ ) η u (0; λ ) η s (0; λ )= E η ( λ ) η u (0; λ ) η s (0; λ ) , (21)so knowing how to compute E η ( λ ) is enough to compute E τ ( λ ).We are interested in where the function D ( λ ), and hence E η ( λ ), is equal to zero in theregion Ω , assuming that w u,s (0; λ ) (cid:54) = 0 for any λ ∈ C . To investigate this we exploit theanalyticity (or continuity) of E η ( λ ) for λ ∈ Ω . We appeal to a theorem from complexanalysis (see, for example, [7]), which says that if f ( λ ) is a meromorphic function on somesimply connected domain Ω ⊆ C with no zeros or poles on a closed curve γ ( t ) ⊆ Ω in thecomplex plane, then, letting N denote the number of zeros of f inside γ ( t ) and P denotethe number of poles inside γ ( t ), we have(22) N − P = 12 πi (cid:73) γ ˙ f ( λ ) f ( λ ) dλ , where ˙ denotes ddλ .The assumption that w u,s (0; λ ) (cid:54) = 0 for any λ ∈ Ω means that we stay in a single chart foreach λ ∈ Ω . Thus, we have that E η ( λ ) will not just be meromorphic but analytic , that is P ≡ 0, leading to the following: Proposition 2.3. Let γ ( t ) : [0 , → C be a simple closed curve in the complex plane,oriented counterclockwise and let D ( λ ) , E η ( λ ) and w s,u (0; λ ) be defined as above. Supposethat w s,u (0; ( γ ( t ))) (cid:54) = 0 for all t ∈ [0 , . Then, (cid:73) γ ( t ) ˙ E η ( λ ) E η ( λ ) dλ = (cid:73) γ ( t ) ˙ D ( λ ) D ( λ ) dλ . That is, the number of zeros of E η ( λ ) is the same as the number of zeros of the Evansfunction. For every value of λ ∈ Ω , the assumption that w u,s (0; λ ) (cid:54) = 0 is consistent with thebehaviour of numerical solutions to the F-KPP equation. Further, N may be thought of asthe winding number of E η ( λ ): the number of times E η ( λ ) (and hence D ( λ )) winds aroundthe origin (with a counter clockwise orientation), counted with sign, as we traverse γ ( t ).Hence we can visually determine the number of zeros of D ( λ ) in a closed contour in C inthe case of F-KPP for quite large values of λ .As can be seen in Figure 3, there are no eigenvalues in Ω in the right half plane with | λ | ≤ × . It is evident that the winding number of E η ( λ ), and hence D ( λ ), is zero.2.5. Extending E η ( λ ) into the continuous spectrum. Since the goal of Evans functioncomputations is to numerically infer stability or otherwise, we need to concern ourselveswith values of the spectral parameter in the right half plane (that is, with Re ( λ ) ≥ . To this end, we need to consider values of λ inside σ c , the continuousspectrum, and re-visit our definition of (temporal) eigenvalues. We proceed in the manneroutlined in [18] and [30].Using Definition 2.4 for all λ with Re ( λ ) ≥ 0, if λ ∈ σ c , then the matrix A − ( λ ) has two(spatial) eigenvalues, both with negative real parts. This implies that every solution ofeq. (15) decays to zero as z → + ∞ . In particular, any solution which decays to 0 as z → −∞ will decay to zero as z → + ∞ , so, if we were to just require the existence of asolution decaying as z → ±∞ , we would see that every λ ∈ σ c would be an eigenvalue.Moreover, it is straightforward to see that these solutions are indeed in H ( R ) × L ( R ).This would seem to suggest that every travelling wave is spectrally unstable, and thelinearised operator, linearised about every wave has eigenvalues with positive real part.This is at odds with with numerical experiments as well as known stability results: forexample it has been known since its inception that the F-KPP wave of speed c = 2 √ δ is stable relative to many compactly supported perturbations [24], and moreover a widevariety of initial profiles will evolve in time to this wave (or at least, a closely related one)[36]. So in some sense we would like to say this wave is ‘stable’ but we would also like toreconcile this notion with the idea that the linearised operator about a stable travellingwave should not have eigenvalues in the right half plane. We are thus motivated to makethe following amendment to Definition 2.4 Definition 2.7. For a λ ∈ C with Re ( λ ) ≥ 0, we say that λ is an ‘eigenvalue’ if there isa solution to the Riccati equation that decays to ξ u − as z → −∞ and to ξ s + as z → + ∞ . GEOMETRICALLY INSPIRED EVANS FUNCTION 15 − − − − − · − − − − · Re( E η ( λ )) I m ( E η ( λ )) -1.326875 -1.326874 − · − Re( E η ( λ )) I m ( E η ( λ )) · − − · Re( λ ) I m ( λ ) − − . − . − . − . · − − − · − Re( λ ) I m ( λ ) Figure 3. Top: A plot of the function E η ( λ ), defined in eq. (19), for thecontour in the spectral parameter (bottom). The wave speed c = 2 . δ = 1. As can be clearly seen, E η ( λ ) does not wind around theorigin, even though the curve of spectrum does. This confirms that λ = 0 isnot an eigenvalue in the sense of Definition 2.7 and also numerically confirmsthe results in Section 2.7. Moreover, the spectral radius is quite large in thiscase (5 × ), while the function E η ( λ ) remains relatively well-behaved,even through fairly large changes of scale in the spectral parameter. Remark 2.2. We remark that the apparent contradiction which led to Definition 2.7 canbe resolved by the introduction of so-called weighted spaces . This amounts to restrictingperturbations to those which decay faster than a given prescribed rate ν , (in this examplethe space is denoted H ν ). Subsequently the spectrum is shifted, and one chooses ν (ifpossible) so that the spectrum is shifted into the negative half plane. Thus there are noeigenvalues in the continuous spectrum with eigenfunctions in H ν . The wave is then saidto be stable relative to these weighted perturbations (provided of course that there are noother eigenvalues with positive real part and with eigenfunctions in this weighted space).We claim that in the F-KPP travelling wave case, the presence of an ‘eigenvalue’ corre-sponds exactly to weighted instability for all weight functions which shift the continuousspectrum into the left half plane. The right edge of the continuous spectrum is moved to thepoint λ = 1 + cν + δν in the weighted space H ν . This will be to the left of the right edge ofthe continuous spectrum in the unweighed space H provided ν ∈ (cid:16) − c −√ c − δ δ , − c + √ c − δ δ (cid:17) .Moreover, this will be in the left half plane only if c < δ .Consider λ ∈ σ c ( L ) (on the unweighted space H ), with Re ( λ ) > 0. If λ is not an‘eigenvalue’, then all such solutions to eq. (15) will decay exactly like e − µ u + z as z → + ∞ ,and thus there can be no eigenfunctions in the weighted spaces H ν for ν which shift thecontinuous spectrum to the left. However, if λ is an ‘eigenvalue’ then this indicates thatthere will be a solution with a decay rate faster than any weight function which will movethe continuous spectrum to the left. Thus it will remain an ‘eigenvalue’ for all weightedspaces with weights ν shifting the continuous spectrum to the left. We can thereforeconclude that there is a point in the spectrum which will not be moved into the left halfplane in any such weighted space. Remark 2.3. Definition 2.7 follows the definition of ‘eigenvalue’ from [30]. The rootsof E η ( λ ) and E τ ( λ ) will detect the values λ where we have a solution decaying with the maximal exponential rate as z → ±∞ . It is obvious that this definition agrees exactly withour definition of an eigenvalue in the region Ω . Inside the continuous spectrum we willnot allow our eigenfunction to decay to 0 in just any fashion, it needs to decay along the(now strongly) stable subspace ξ s + . Since we will be primarily interested with the zeros of E η,τ ( λ ) and given the discussion in Remark 2.2, we drop the quotation marks, and simplyrefer to any such λ as a (temporal) eigenvalue of the linearised operator L .With Definition 2.7, eigenvalues still correspond exactly to zeros of E η ( λ ). Moreover, it isstraightforward to see that we can still define E η ( λ ) to be analytic as we extend λ into σ c . GEOMETRICALLY INSPIRED EVANS FUNCTION 17 We can, in fact, use some analysis of the Riccati equation on the η chart of C P to see whenexactly we get a zero of E η,τ ( λ ). To begin with, we fix a λ ∈ σ c with Re ( λ ) ≥ µ u − and µ s + . For a general λ we have that the unstableorbit in the η chart coming from µ u − (viewed as a subspace in C ) will tend towards thesteady state solution µ sc + := µ u + (here, because we are in σ c we denote µ sc + := µ u + to notethat it is in fact a stable fixed point of the Riccati flow on the η chart).Recall that µ sc + = − c + (cid:112) c + 4 δ ( λ − δ and µ s + = − c − (cid:112) c + 4 δ ( λ − δ , which are different points in the chart of C P for all values of λ except when c +4 δ ( λ − 1) =0, that is, except for λ = 1 − c / (4 δ ). At this value of λ , we have µ s + = µ sc − and so whatwas a heteroclinic connection (of the Riccati flow on the η chart of C P ) between the fixedpoints µ u − and µ sc + , is also a heteroclinic connection between the fixed points µ u − and µ s + .Consequently, we have a zero of E η ( λ ) and this value of λ will be an eigenvalue accordingto Definition 2.7.We observe that if c ≥ δ then the largest root, say ˜ λ , of the function E η ( λ ) is real andnegative. However, as c (cid:38) √ δ , we have that ˜ λ tends towards 0 and if c < √ δ , then E η ( λ ) has a real, positive root. Thus, we have an eigenvalue ˜ λ in the right half plane, which(evidently) destabilises the travelling wave. This corresponds with numerical experimentsas well as the analytic results proven in [12].2.6. Switching Charts. Suppose that for some fixed z we had that the solution of ourRiccati equation | η u ( z ; λ ) | → ∞ , implying that the corresponding solution in the τ chartmust tend to 0. Given the uniqueness of solutions to ODEs on manifolds, we can find avalue z < z such that | η u ( z ; λ ) | < ∞ and so consider the corresponding initial valueproblem in the τ chart where τ ∗ ( z ; λ ) = 1 η u ( z ; λ ) . Evolving the τ problem from z to anew z > z (noting along the way that τ ∗ ( z ; λ ) = 0), we can then consider the solutionof the Cauchy problem on the η chart with initial condition η u ( z ; λ ) = 1 τ ∗ ( z ; λ ) . In thisway, we have moved beyond the singularity of our Riccati solution. The impact that thisstrategy has upon our previously defined Evans functions needs to be explored. Given thatwe are no longer in the case where the number of poles of E η ( λ ) (the value P above) is 0, − − 50 50 100 − − − µ u + µ s + µ u − µ s − zη − − 50 50 100 − − µ u − µ s − µ sc + µ s + zη Figure 4. Plots of the relevant solutions to the Riccati equation, eq. (18),in the η chart for two (real) values of λ . On the left we have λ = 2 andit is obvious that a heteroclinic connection does not exist. On the right, λ = 0 . 01 and we can see that the solution tending to µ u − as z → −∞ (theupper solution, blue online) tends towards µ sc + as z → + ∞ (the middlesolution, red online). Thus, while we have a traditional eigenvalue, we donot have an ‘eigenvalue’ according to Definition 2.7.our winding number calculation becomes(23) (cid:73) γ ( t ) ˙ E η ( λ ) E η ( λ ) dλ = (cid:73) γ ( t ) ˙ D ( λ ) D ( λ ) dλ − (cid:73) γ ( t ) ˙ w s (0; λ ) w s (0; λ ) dλ − (cid:73) γ ( t ) ˙ w u (0; λ ) w u (0; λ ) dλ . We elaborate on the meaning of this result in the following theorem: Theorem 2.1. Let γ ( t ) be a parametrised curve in the complex plane such that D ( λ ) isanalytic and has no zeros on γ ( t ) . Then, the winding number of E η along γ ( t ) is thenumber of eigenvalues of L inside that curve minus the number of poles of E η ( λ ) inside γ ( t ) .Proof. Rearranging the definition of E η ( λ ) we have D ( λ ) = w s (0; λ ) w u (0; λ ) E η ( λ ) . Choosing a curve γ ( t ) such that D ( λ ) has no zeros on γ ( t ) (that is, avoiding any eigenval-ues) and such that D ( λ ) is analytic on γ ( t ), then applying the chain rule to logarithmicdifferentiation and rearranging, we have eq. (23). E η has a pole exactly when either w u (0; λ )or w s (0; λ ) is zero, and eigenvalues of L are the zeros of D ( λ ). (cid:3) GEOMETRICALLY INSPIRED EVANS FUNCTION 19 In the case of the F-KPP equation, this theorem enables us to find the number of eigenvaluesinside any bounded contour, except those containing the so-called ‘absolute spectrum’where the function E η ( λ ) has a branching point of its domain. Following [20], we definethe absolute spectrum as the λ ∈ C such that the real parts of µ u,s + or µ u,s − coincide. Thesecan be determined as λ ≤ − c / (4 δ ) in the case of µ + , and λ ≤ − − c / (4 δ ) for µ − (notethat λ ∈ R in the absolute spectrum). This offers another mechanism for destabilisation ofthe waves as c (cid:38) √ δ , namely that the absolute spectrum moves into the right half plane.In this case, the loss of meromorphicity of E η ( λ ) coincides exactly with the leading edgeof the absolute spectrum.2.7. A proof of the absence of eigenvalues with positive real part. The proofproceeds in two parts. For the first part, we show that there are no eigenvalues with non-zero imaginary part. For the second, we show the absence of a real positive eigenvaluewhen c ≥ √ δ .2.7.1. No complex eigenvalues. Recalling eq. (18), we have that on the η chart of C P , thelinearisation is given by(24) η (cid:48) = 1 δ ( λ + 1 − u ) − cδ η − η . Writing η = α + iβ and λ = m + in with m (cid:54) = 0, we have that eq. (24) becomes (whenviewed on C ≈ R ) α (cid:48) = 1 δ (1 − αc + m − u ) − α + β ,β (cid:48) = − αβ − βcδ + nδ . (25)We see that on the line β = 0, we have that β (cid:48) = nδ and so the sign of β (cid:48) is the same asthe sign of the imaginary part of the eigenvalue parameter λ . Consequently, the flow ispointing towards the upper half plane when Im ( λ ) > λ ) < 0. Now we have that an eigenvalue λ on this chart is a value of λ such thatthere is a connection under this flow from µ u − to µ s + .Given the previous statement about the direction of the flow on the real axis of this chart,we claim that Im (cid:0) µ u − (cid:1) > (cid:0) µ s + (cid:1) < λ ) > 0, and the reverse inequalities ifIm ( λ ) < 0. Thus, a connection is impossible, as long as Im ( λ ) (cid:54) = 0. Proceeding directlywe have that µ u − = − c + (cid:112) c + 4 δ ( λ + 1)2 δ and that Im (cid:0) µ u − (cid:1) = κ δ sin (cid:18) 12 arg (cid:0) c + 4 δ (1 + λ ) (cid:1)(cid:19) , where κ := (cid:112) | c + 4 δ ( λ + 1) | > λ ) > 0, we have that 0 < arg (cid:0) c + 4 δ (1 + λ ) (cid:1) ≤ π and so 0 < sin (cid:0) arg (cid:0) c + 4 δ (1 + λ ) (cid:1)(cid:1) ≤ 1. In other words, if Im ( λ ) > 0, then so is Im (cid:0) µ u − (cid:1) . Thesame calculation shows that if Im ( λ ) < 0, then Im (cid:0) µ u − (cid:1) < µ s + has the opposite sign to that of Im ( λ ). Thus, we have shown thatthere are no connections possible on this η chart.Essentially the same calculation shows that there is no connection between ( µ u − ) − and( µ s + ) − on the τ chart and it is worth noting explicitly that the above calculation is inde-pendent of the real part of the spectral parameter, and so we conclude that there are noeigenvalues with non-zero imaginary part (i.e. any eigenvalues must be real). Note thatthis calculation is independent of the continuous spectrum and so in order to concludestability we will need to take the continuous spectrum into account.2.7.2. Real eigenvalues. To show that there are no real, positive eigenvalues when c ≥ √ δ we proceed as in [19], although here we avoid the formal machinery discussed therein.Recalling eq. (18), if λ is real and positive and if c ≥ √ δ , we are looking for a heteroclinicconnection on R P ≈ S , the unit circle. The key idea is to evaluate the Riccati equationon the unit circle at the point µ s + . We have the following: (26) η (cid:48) | η = µ s + = 1 δ ( λ + 1 − u ) − cδ µ s + − ( µ s + ) = 1 δ (2 − u ) , noting here that this is independent of λ and strictly positive if c ≥ √ δ (actually for all c > c ≤ √ δ , then µ s + is no longer real for all real non-negative values of λ ).Next, for each λ ≥ 0, denote the solution on R P ≈ S decaying to µ u − by (cid:96) ( z ). We observethat if λ is not an eigenvalue we have that lim z → + ∞ (cid:96) ( z ) = µ u + ( λ ) (or µ sc + ( λ ) if λ ≤ Proposition 2.4. Suppose that (cid:96) ( z ) crosses µ s + N j times for some fixed values λ j , j = { , } . Then, the number of eigenvalues in the interval ( λ , λ ) is equal to | N − N | .Proof. Suppose, without loss of generality, that N = N + 1. Equation (26) means that (cid:96) ( z ) can only cross µ s + in one direction. This, combined with the previous observationsabout the limit of (cid:96) ( z ) for λ not an eigenvalue, means that there must be a λ ∈ ( λ , λ ) GEOMETRICALLY INSPIRED EVANS FUNCTION 21 where lim z → + ∞ (cid:96) ( z ) = µ s + . Notably, this is the definition of an eigenvalue (Definition 2.7).Further, the fact that (cid:96) ( z ) can cross µ s + in only one direction means that for each eigenvalue λ ∈ ( λ , λ ), the difference | N − N | must increase by one. (cid:3) Given Proposition 2.4, it suffices to show that there are no crossings for λ on the positivereal line of (cid:96) ( z ) as z ranges over R for c ≥ √ δ . If λ = 0, we have that eq. (15) is theequation of variations along ˆ u ( z ) in the phase plane. Thus, the solution (cid:96) ( z ) is simply the(unit) tangent vector to the curve (ˆ u ( z ) , ˆ u (cid:48) ( z )) in the phase plane. As z ranges over R it isobvious that the tangent vector to the curve is never parallel to the eigenvector at positiveinfinity (1 , µ s + ). Next, for λ (cid:29) (cid:96) ( z ) ∼ µ u − , thesteady state solution. Thus, there are no crossings for λ (cid:29) 1. This completes the proofthat there are no eigenvalues on the positive real line and the proof of spectral stability ofthe positive travelling waves in the F-KPP equation. Remark 2.4. To the best of our knowledge this is a new proof of the absence of eigenval-ues with positive real part, and nonzero imaginary part of the linearised operator abouttravelling waves in the F-KPP equation of speed c ≥ √ δ .3. Travelling Waves in a Keller-Segel Model We now turn our attention to the application of the techniques from Section 2 to a systemof PDEs with one spatial and one temporal independent variable, and more than onedependent variable. We focus on the parameter regime of eq. (2) wherein explicit solutionscan be found to the travelling wave equation eq. (6), and so for the remainder of thissection, we set ε = 0 for unless otherwise specified.Setting ε = 0, eq. (4) becomes u τ = cu z − αw ,w τ = δw zz − β (cid:16) wu z u (cid:17) z + cw z . (27)As before, a travelling wave solution will be a stationary solution (¯ u ( z ) , ¯ w ( z )) to eq. (27).In [9], an explicit solution is given:¯ u ( z ) = (cid:16) u − /γr + σe − c ( z + z ∗ ) /δ (cid:17) − γ , ¯ w ( z ) = e − c ( z + z ∗ ) /δ [¯ u ( z )] βδ , (28) − − − . . . . z ¯ u, ¯ w Figure 5. A plot of the explicit solutions given in eq. (28) for parametervalues δ = 1, α = 1, β = 2, c = 2. The dashed front profile is u ( z ), whilethe solid pulse is w ( z ).with γ = δβ − δ > , σ = α ( β − δ ) c > , and z ∗ an integration constant coming from the translational invariance of the travellingwave solutions (owing to the fact that eq. (27) is autonomous). Without loss of generalitywe set z ∗ = 0. We remark that u r is the asymptotic limit of the chemical attractant u as x → ∞ and, without loss of generality, as in [13], we set u r = 1. See Figure 5 for a plot ofthe solutions ¯ u ( z ) and ¯ w ( z ) with explicit parameter values.3.1. The spectral problem. The steady state solutions in eq. (28) (using (cid:48) := ddz asbefore) solve the nonlinear ODEs0 = cu (cid:48) − αw, δw (cid:48)(cid:48) + αβc (cid:18) u (cid:48) w u − ww (cid:48) u (cid:19) + cw (cid:48) . (29)Formally, the linearisation of eq. (27) about the steady state solution (¯ u, ¯ w ) is given as(dropping the bars for notational convenience)(30) (cid:32) pq (cid:33) t = L (cid:32) pq (cid:33) , GEOMETRICALLY INSPIRED EVANS FUNCTION 23 where L is defined as the following linear operator: L := (cid:32) c∂ z − α L p L q (cid:33) , where L p := − βwu ∂ zz + (cid:18) βwu (cid:48) u − βw (cid:48) u (cid:19) ∂ z + (cid:32) βwu (cid:48)(cid:48) u − βw ( u (cid:48) ) u + βu (cid:48) w (cid:48) u (cid:33) , L q := δ∂ zz + (cid:18) c − βu (cid:48) u (cid:19) ∂ z + (cid:32) β ( u (cid:48) ) u − βu (cid:48)(cid:48) u (cid:33) . (31)We seek λ ∈ C for which L − λ I is not invertible in some appropriate Banach space. Here, H ν ( R ) × H ν ( R ) will suffice, for an appropriately chosen weight ν . For the time being, weset ν = 0 and just consider H ( R ) × H ( R ). The operator L − λ I is equivalent to theoperator T ( λ ) := ddz − A ( z, λ ) on the space H ( R ) × H ( R ) × L ( R ) where A ( z, λ ) is givenas A ( z, λ ) := λc αc 00 0 1 A B C , with A := (cid:18) βwc δu (cid:19) λ + (cid:18) βw (cid:48) cδu − βwu (cid:48) cδu (cid:19) λ + 2 βw ( u (cid:48) ) δu − βwu (cid:48)(cid:48) δu − βu (cid:48) w (cid:48) δu , B := (cid:18) αβwc δu + 1 δ (cid:19) λ + αβw (cid:48) cδu − αβwu (cid:48) cδu + βu (cid:48)(cid:48) δu − β ( u (cid:48) ) δu , C := − cδ + αβwcδu + βu (cid:48) δu . That is, we are looking for solutions in H × H × L to the linear, non-autonomous ODEs(32) pqr (cid:48) = λc αc 00 0 1 A B C pqr . Observing that the solutions given in (28) satisfylim z →−∞ ( u, w, u (cid:48) , w (cid:48) ) = (0 , , , 0) and lim z →∞ ( u, w, u (cid:48) , w (cid:48) ) = (1 , , , , and that u (cid:48) = αc w, and lim z →−∞ wu = c α ( β − δ ) , we have that the limits as z → ±∞ of A , B and C denoted A ± , B ± and C ± , respectively,are A + = 0 , B + = λδ , C + = − cδ and A − = βλ αδ ( β − δ ) − βc λαδ ( β − δ ) , B − = (cid:18) β − δδ ( β − δ ) (cid:19) λ − c βδ ( β − δ ) , C − = c ( β + δ ) δ ( β − δ ) . We denote lim z →±∞ A ( z, λ ) by A ± ( λ ).3.2. The continuous spectrum. We have that the continuous spectrum (defined as thevalues of λ ∈ C for which the signature of A + ( λ ) is not equal to the signature of A − ( λ )) isbounded by the so-called dispersion relations: the values of λ ∈ C such that either A + ( λ )or A − ( λ ) has a purely imaginary eigenvalue. The dispersion relations are λ = − δk + ick , and λ = ick , where ik, ( k ∈ R ) is the purely imaginary eigenvalue of A + , and (implicitly): − λ cδ + (cid:18) − k c + ik (cid:18) δ − β − δ (cid:19)(cid:19) λ − ck ( β + δ ) δ ( β − δ ) + i (cid:18) δk ( β − δ ) − βc kδ ( β − δ ) (cid:19) = 0 , (33)where ik, ( k ∈ R ) is the purely imaginary eigenvalue of A − . We remark that as λ onlyenters eq. (33) quadratically, an exact expression can be found for it in terms of the otherparameters:(34) λ ± := − δ ( β − δ ) k + ic ( β − δ ) k ± √ ∆2( β − δ ) , where the discriminant ∆ is given as(35) ∆ := (cid:0) δ ( β − δ ) (cid:1) k + (cid:0) βc (4 δ − β ) (cid:1) k + i (cid:0) βcδ ( β − δ ) k − βc k (cid:1) . The entire imaginary axis is one of the dispersion relations (and hence contained in thecontinuous spectrum, σ c ( L )), and, in general, there are points in the continuous spectrumwith real part λ > 0, see Figure 6 for an illustration.We also note that, as in the F-KPP case, the dispersion relations break up the spectralplane into distinct regions. With a slight abuse of notation, we call region to the right ofthe continuous spectrum Ω . That is:Ω := { λ | Re ( λ ) > Re ( ζ ) ∀ ζ ∈ σ c ( L ) } . GEOMETRICALLY INSPIRED EVANS FUNCTION 25 − − − − − − − − − − Ω Ω Ω Ω Ω Ω σ c σ c σ c σ c σ c σ c σ c Re( λ )Im( λ ) Figure 6. The continuous spectrum for the linearised operator L aboutthe waves in eq. (28). The imaginary axis is included in σ c ( L ). It is alsoevident that some points in the right half of the complex plane are in σ c ( L ).The parameter values for this figure are the same as in Figure 5.There are five more regions in the complex plane where the signature of A + ( λ ) is the sameas that of A − ( λ ). The two that are bounded we will denote by Ω and Ω , and the threeunbounded ones will be denoted Ω (containing an unbounded region of the negative realaxis), Ω and Ω . The continuous spectrum will be the remaining part of the complexplane: σ c := C \ (cid:83) Ω j . Figure 6 shows a plot of the dispersion relations, the regions Ω j and the continuous spectrum for explicit choices of the parameter values α , β , c and δ . Remark 3.1. We remark that it is not possible to weight the continuous spectrum com-pletely into the left half plane. This agrees with known results [29] about such travellingwaves, and suggests the presence of so-called absolute spectrum in the right half plane. Nu-merically, we were able to (for the parameter values used) determine that the absolute spec-trum in the right half plane was contained in a small region R := [0 , . × [4 i, − i ] \ B . (0),where B . (0) is the ball of radius r = 0 . 01 about the origin. See Figure 7. As some pointsin the absolute spectrum will coincide with branching points of the Evans function, wegenerally avoid computing the Evans function in this region. We leave the precise calcula-tion of the absolute spectrum as well as a full spectral analysis of travelling waves in theseKeller-Segel models for future work. − − − 50 0 − − Re( λ )Im( λ ) − − . . − Figure 7. The (numerically determined) boundary of the absolute spec-trum of L . It clearly contains points in the right half plane, though thesepoints are reasonably far from the origin. The parameter values used werethe same as in Figure 6.3.3. Eigenvalues. For λ ∈ C \ σ c we have that A ± ( λ ) are hyperbolic and we again denotethe stable and unstable subspaces of A ± ( λ ) as ξ s ± ( λ ) and ξ u ± ( λ ) (or as ξ s,u ± where conve-nient). Just as in Proposition 2.1, we have that for λ ∈ C \ σ c , the existence of a solutionto eq. (32) decaying to 0 as z → ±∞ puts a geometric constraint on the direction of decay.That is: Proposition 3.1. For λ ∈ C \ σ c , if ( p, q, r ) is a solution to eq. (32) such that ( p, q, r ) ∈H ( R ) × H ( R ) × L ( R ) , then (36) lim z →−∞ pqr → ξ u − and lim z →∞ pqr → ξ s + . That is, ( p, q, r ) decays to the stable subspace ξ s + of A + ( λ ) as z → + ∞ and the unstablesubspace ξ u − of A − ( λ ) as z → −∞ . Again, see [18, 20, 30] for proofs of this proposition. We call a λ for which such a solutionexists a (temporal) eigenvalue , with eigenfunction (cid:32) pq (cid:33) . Just as in the F-KPP case, wehave that eigenvalues are not possible for all values of λ ∈ C \ σ c . In particular, if λ ∈ Ω ,the unstable subspace of A − ( λ ) is zero-dimensional and, hence, the kernel of T ( λ ) (or GEOMETRICALLY INSPIRED EVANS FUNCTION 27 equivalently L ) is empty by Proposition 3.1. Further, as we will be primarily concernedwith spectral stability, and the regions Ω , Ω , Ω and Ω are all contained in the left halfof the complex plane, we again focus our attention on λ ∈ Ω , where A + ( λ ) will have aone-dimensional stable subspace and where A − ( λ ) will have a two-dimensional unstablesubspace.The Evans function in this case is set up similarly. The main difference is that now wehave a two-dimensional subspace at −∞ . Letting Ξ u and Ξ s denote the unstable andstable manifolds respectively, we have that Ξ s ( z ; λ ) is a (complex) line bundle (over R )again while Ξ u ( z ; λ ) will be a complex vector bundle of rank 2.We let w s ( z ; λ ) = w s ( z ; λ ) w s ( z ; λ ) w s ( z ; λ ) and w uj ( z ; λ ) = w uj, ( z ; λ ) w uj, ( z ; λ ) w uj, ( z ; λ ) , j = 1 , w s ( z ; λ ) ∈ Ξ s and w uj ( z ; λ ) a pair of linearly independentsolutions to eq. (32) in Ξ u and define the Evans function: D ( λ ) := det w u , (0; λ ) w u , (0; λ ) w s (0; λ ) w u , (0; λ ) w u , (0; λ ) w s (0; λ ) w u , (0; λ ) w u , (0; λ ) w s (0; λ ) . Just as in the F-KPP case, it is clear that λ ∈ Ω is an eigenvalue if and only if D ( λ ) = 0.3.4. The Riccati Equation. Because we are interested in the evolution of subspacesunder the flow of a linear ODE, rather than the behaviour of explicit solutions to eq. (32),it is natural to look at how subspaces evolve under the flow described in eq. (32). Sincewe have a one-dimensional stable subspace ξ s + as z → + ∞ , we need to understand howthe flow from eq. (32) leads to a flow on the set of one-dimensional subspaces in C , i.e.,the complex projective plane C P . Likewise, since we have a two-dimensional unstablesubspace ξ u − as z → −∞ , we need to translate the flow from eq. (32) to a flow on the spaceof two-dimensional subspaces in C . This space is called the complex Grassmannian oftwo planes in three space and is denoted Gr (2 , C P or Gr (2 , C P or Gr (2 , Pl¨ucker relations forGr(2,3)) for each coordinate on each chart. For C P this is done in the following way (totally analogous to the C P case). A line in C is determined by a triple of numbers [ p : q : r ] not all zero and subject to the fact that forany complex number ζ , the triple [ ζp : ζq : ζr ] represents the same line as [ p : q : r ]. Thus,for example, we can write down all the lines where q (cid:54) = 0 as [ η : 1 : η ]. Here, η := pq and η := rq . Differentiating and using eq. (32) leads to an expression for the Riccati equationon this chart: η (cid:48) = λc η + αc − η η η (cid:48) = A η + B + C η − η . (37)Again, the stable subspace of A + ( λ ) will be a point in this chart (usually, if not, useanother chart), with a one complex dimensional stable manifold, evolving under the Riccatiequation. In this chart of C P , we will denote such a solution as [ η s ( z ; λ ) : 1 : η s ( z ; λ )].For Gr (2 , Gr (2 , → C P . For a pair ofvectors in C , v = ( v , v , v ) and w = ( w , w , w ), we have that v and w are linearlyindependent (i.e. they span a two-plane) provided that not all of K i,j := ( v i w j − v j w i ) for1 ≤ i < j ≤ K , K , K ) that must not be all zeroif v and w span a plane. Further, the plane spanned by ζ v and ζ w for ζ , ∈ C will bethe same as that spanned by v and w and will produce the triple ζ ζ ( K , K , K ). Itis thus clear that we can represent a two-plane in three-space as a triple [ K : K : K ]in C P .If v and w are linearly independent solutions to eq. (32), then by using the product rule,the plane spanned by them in the Pl¨ucker coordinates will solve the linear ODE(38) K K K (cid:48) = λc B λc + C αc −A C K K K = A ∧ A K K K where the last A ∧ A means the exterior product of the matrix A with itself.The idea now is to use the Riccati equation for eq. (38) to write down how the linear flowgiven by eq. (32) behaves on pairs of subspaces. From this perspective it is clear that wehave three charts from which to choose for the Pl¨ucker embedding of Gr (2 , 3) (on whichthe unstable manifold will be a curve) and we have three for the Pl¨ucker embedding of Gr (1 , 3) (on which the stable manifold will be a curve). Suppose for concreteness, that K (cid:54) = 0. Then by setting κ = − K K and κ = K K , and using eq. (38) we have that κ and κ will satisfy the nonlinear ODEs κ (cid:48) = A + (cid:18) C − λc (cid:19) κ − κ κ κ (cid:48) = B − αc κ + C κ − κ (39)The unstable subspace of A − ( λ ) will be a point on this chart (usually) and it has a one-dimensional unstable manifold, denoted in coordinates on this chart as [1 : κ u ( z ; λ ) : − κ u ( z ; λ )].All that remains is how to relate η , and κ , to D ( λ ). Proceeding as we did in the F-KPP case, suppose that the solution w s stays in the same chart (of C P ) for all z andthat the pair of solutions ( w u , w u ) stay on the same chart (of Gr (2 , z . Byway of example, suppose it is in the two charts for which we have written expressions forthe Riccati equation, eqs. (37) and (39), respectively. Then, in particular, we have that w s ( z ; λ ) (cid:54) = 0 and the matrix W u := (cid:32) w u , ( z ; λ ) w u , ( z ; λ ) w u , ( z ; λ ) w u , ( z ; λ ) (cid:33) is invertible for all z (becausewe are in the charts where q (cid:54) = 0 and where K (cid:54) = 0). Defining K u ( z ; λ ) := det W u = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w u , ( z ; λ ) w u , ( z ; λ ) w u , ( z ; λ ) w u , ( z ; λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = w u ( z ; λ ) w u ( z ; λ ) − w u ( z ; λ ) w u ( z ; λ ) (cid:54) = 0We have that the matrix w u , ( z ; λ ) w u , ( z ; λ ) w s ( z ; λ ) w u , ( z ; λ ) w u , ( z ; λ ) w s ( z ; λ ) w u , ( z ; λ ) w u , ( z ; λ ) w s ( z ; λ ) (cid:32) ( W u ) − w ( z ; λ ) (cid:33) = η s ( z ; λ )0 1 1 κ u ( z ; λ ) κ u ( z ; λ ) η s ( z ; λ ) (40)is well defined for all values of z . Evaluating at z = 0 and taking determinants gives(41) D ( λ ) K u (0; λ ) w s (0; λ ) = η s (0; λ ) − κ u (0; λ ) − η s (0; λ ) κ u (0; λ ) . Define the function E q ( λ ) := η s (0; λ ) − κ u (0; λ ) − η s (0; λ ) κ u (0; λ ) . The subscripts indicates that the q coordinate of Ξ s and the K coordinate of Ξ u are both (cid:54) = 0. Since each of the solutions that we are tracking stay in the same chart, E q ( λ ) = 0 if and only if D ( λ ) = 0 . Again, provided the solutions η s , and κ u , stay in the same charts,we can use the argument principle to determine the number of zeros E q ( λ ) has for anyprescribed curve in the region Ω .3.5. Switching charts and extending into the continuous spectrum. Just as in theF-KPP case, should a singularity of the solution of the Riccati equation appear, we caninterpret this as the solution leaving the chart. Then we can switch to a different chart bythe same method described earlier: namely choose a value z for which the solution is notsingular, use this as an initial condition on a different chart and evolve the solution on saidchart beyond the point of singularity. Then, if desired, one can switch back to the originalchart.It is also worth noting, that as we are only ever tracking a finite number of solutions tothe Riccati equation on compact manifolds, it is always possible to find at least one set ofcharts (one for C P and one for Gr (2 , z (though this is not necessarily always one of the canonical charts).That is, it is always possible to choose charts so that the solutions used in the shootingfor the Evans function stay bounded for all values of the independent variable. For theparameter values considered in this example, we found that the charts [ pq : 1 : rp ], and[1 : K K : K K ] would suffice for all λ with Re ( λ ) ≥ E q ( λ ) (or its analog on any pair of charts from C P and Gr (2 , λ for which we can find a solution to eq. (32) decaying tozero but for which we can find a solution to eq. (32) decaying in a specific, geometric way.As we vary λ across the dispersion relation curves into the continuous spectrum, we cancontinuously track ξ s + ( λ ) and ξ u − ( λ ). This gives a straightforward continuation of E q ( λ )(or its analogs on other charts) into the continuous spectrum (though not the absolutespectrum).3.6. Stability Analysis. In this section we numerically establish that there is no pointspectrum of the operator L with real part between 0 and 10 , except possibly in theregion R := [0 , . × [4 i, − i ] \ B . (0). We also show that λ = 0 is an eigenvalue ofmultiplicity 2. For this analysis, the parameter values chosen were the same as in [14],namely α = 1 , β = 2 , c = 2 , and δ = 1. GEOMETRICALLY INSPIRED EVANS FUNCTION 31 Using the Ricatti Evans functions outlined in this section, we can numerically verify thatthere are no eigenvalues (in the sense of Definition 2.7) for a large region in the right halfcomplex plane (out to | λ | < ), both within and without of the continuous spectrum.We first compute the Evans function E q ( λ ) on a spectral curve consisting of the righthalf of an annulus (including the imaginary axis) with inner radius r = 4 and outer radius r = 10 . We can visually inspect that there is no winding of the Evans function aroundthe origin, and thus conclude that there are no eigenvalues of the operator L in this region,see Figure 8.We next compute the function E q ( λ ) for λ ∈ C on the boundary of the half disc of radius r = 4 shifted to the right by 0 . L in this region either. Figures 8 and 9 allow us to concludethat all eigenvalues of the the operator L in the right half plane either have norm greaterthat 10 or else lie in the region R := [0 , . × [4 i, − i ] \ B . (0) in the complex plane.In order to evaluate the function E q ( λ ) reasonably efficiently, one needs to be sufficientlyfar enough away from the absolute spectrum. For the parameters considered in this man-uscript, it was found that the absolute spectrum is not the entire region [0 , . × [ − i, i ],but is bounded away from the origin (see Figure 7). We were thus able to evaluate theEvans function E q ( λ ) for λ on the boundary of a small disc (radius r = 10 − ) about theorigin. We found that on this boundary the function E q ( λ ) wound around the origin twotimes, and so we conclude that λ = 0 is an eigenvalue of multiplicity 2. See Figures 10and 11. 4. Summary of Results and Concluding Remarks We have illustrated how to use the underlying geometry of the spectral problem in orderto facilitate computation of the spectrum of a linearised operator about a travelling wavein a PDE with 1+1 independent variables. The geometric interpretation of the Riccatiequations allows us to handle the blow-up of solutions to nonlinear ODEs. We have thusused these solutions to develop new Evans functions, and used them to numerically verifythe spectral stability of travelling waves in the F-KPP equation, and the absence of eigen-values in a large region of the complex plane for the the explicit travelling waves in theK-S system when ε = 0. We have also shown in this case that λ = 0 is an eigenvalue ofmultiplicity 2. · · · · · − · − · · · Re( λ )Im( λ ) − − − , − , − , 000 0 − , − , , , Re( E q ( λ ))Im( E q ( λ )) − − − − − − − − Figure 8. A plot of the function E q ( λ ) (top) for λ on the closed curveon the bottom. We have that 4 < | λ | < . As can be seen in the top rightfigure, the image of the Evans function clearly does not wind around theorigin. We conclude that there are no eigenvalues in this region. The topright picture is a zoomed in plot of the Evans function nearer to 0, whilethe bottom right picture is a zoomed in plot of the curve in the spectralplane.The Evans functions we have produced are seemingly very well behaved in comparison tomore naive attempts at computing them. They are reasonably easy to compute for largevalues of the spectral plane, and their winding around the origin can be visually inspectedin both the examples that we have shown. Finally, our methods are fairly general, weare able to develop the corresponding Riccati Equations and Evans functions for a generalclass of non-self adjoint operators, and we can compute the Evans functions for a large GEOMETRICALLY INSPIRED EVANS FUNCTION 33 − − Re( λ )Im( λ ) − − − − − − Re( E q ( λ ))Im( E q ( λ )) − . . . − − . . Figure 9. A plot of the function E q ( λ ) (left) for λ on the closed curveon the right. The central inset shows that there is no winding of the Evansfunction in this region about the origin either. − . 01 0 . − . . Re( λ )Im( λ ) − . . − . . Re( E q ( λ ))Im( E q ( λ )) Figure 10. A plot of the function E q ( λ ) (left) for λ on the closed curveon the right. It is clear that the Evans function winds around the origin,suggesting that λ = 0 is an eigenvalue.set of values in the spectral plane and also, regardless of the dimensions of the stable andunstable subspaces at ±∞ , ξ u − and ξ s + .4.1. Summary of stability results. We have verified that the continuous spectrum ofthe linearised operator L , linearised about travelling waves in the F-KPP equation canbe weighted to the left half plane, Further we have explicitly verified that there are noeigenvalues in the sense of Definition 2.7 with Re ( λ ) ≥ 0, in the F-KPP travelling waveswith wave speed c > √ δ . We have provided a new proof of spectral stability of thetravelling waves of speed c > √ δ to the F-KPP equation. Since the operator is sectorial,we can therefore confirm linear stability of the F-KPP travelling waves [21]. π π π π π π − π − π π π θ Arg( E q ) Figure 11. A plot of the argument of the function E q ( λ ) for λ =10 − e πit as t passes through a domain of length 1.The argument goesthrough a change of 4 π suggesting that 0 is an eigenvalue of multiplicity2.For the K-S system when ε = 0 and for the explicit solutions in eq. (28) and parametersconsidered, we were unable to weight the continuous spectrum into the left half plane. Thisis consistent with known results about the system [29] and suggests the presence of absolutespectrum with positive real part. The absolute spectrum appears to be bounded away fromthe origin, and therefore enters the right half plane at some point on the imaginary axis(for the parameter values used in this work, we numerically found this to be between ± i and ± i , see Figure 7). It is unclear what effects this has on the dynamics of the travellingwaves, and further study is required.We have verified that for the linear operator linearised about the Keller–Segel waves ¯ u and¯ w in eq. (28), there are no eigenvalues with 0 ≤ Re ( λ ) ≤ except possibly in the region R := [0 , . × [4 i, − i ] \ B . (0) . We have also numerically shown that 0 is an eigenvalueof multiplicity 2.4.2. Future Work: The K-S system in the case when ε (cid:54) = 0 . If we return to eq. (2)and consider 0 < ε (cid:28) 1, travelling waves are still known to exist (see for example [38] andthe references therein) though no explicit formula for them is known. Further it was shownin [13] that the travelling wave solutions in this case, say ¯ u ε ( z ) and, ¯ w ε ( z ) are perturba-tions of ¯ u and ¯ w from eq. (28). One could then linearise around (¯ u ε ( z ) , ¯ w ε ( z )), and bycomputing the asymptotic limits of the functions, their derivatives and appropriate ratiosof them, determine the dispersion relations, and subsequently the continuous spectrumof the linearised operator. We conjecture (as is typical in these types of travelling wave GEOMETRICALLY INSPIRED EVANS FUNCTION 35 examples) that the inclusion of a nonzero diffusion term in the first equation of eq. (2)will lead to the resulting linearised operator being sectorial. In this instance however, weexpect to see absolute spectrum in the right half plane, though the impact of this on theexplicit dynamics as in the ε = 0 case may not be clear.We then aim to repeat the procedure outlined above to numerically investigate whetherthere were eigenvalues for the linearised system. Numerically finding ¯ u ε ( z ) and ¯ w ε ( z ) is abit time consuming, and as this manuscript was primarily to provide examples illustratingour methods, we have, in the interest of expediency, elected to focus on the model whereexplicit solutions are known.Provided that one can numerically find the solutions (¯ u ε , ¯ w ε ) however, it is not difficult toextend our methods to compute a similar Evans function and determine the presence (orlack thereof) of eigenvalues in the right half plane. The emerging Riccati equations willdetermine a flow on Gr ( k, k planes in C , where k is determinedby the dimensions of the stable and unstable subspaces of the asymptotic end states of theoperator for λ in the region equivalent to Ω (i.e. to the right of the continuous spectrum).Further, the expressions for the Riccati equations are found in much the same way as forthe Keller–Segel and F-KPP models, one must just use a different Pl¨ucker embedding foreach separate k appearing in the problem. 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