NUMERICAL COMPUTATION OF AN EVANS FUNCTION IN THE FISHER/KPP EQUATION. 1. Introduction. The Fisher/Kolmogorov Petrovsky Pescounov (KPP) equation

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1 NUMERICAL COMPUTATION OF AN EVANS FUNCTION IN THE FISHER/KPP EQUATION K. HARLEY 1, R. MARANGELL 2, G. J. PETTET 1, AND M. WECHSELBERGER 2 Abstract. We demonstrate a geometrically inspired technique for computing an Evans function for the linearised operator about a travelling wave in the Fisher/KPP equation. We produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. We conclude with a new proof of stability of travelling waves of speed c 2 δ in the Fisher/KPP equation, and discuss extensions to travelling waves in higher order systems of PDEs. 1. Introduction The Fisher/Kolmogorov Petrovsky Pescounov (KPP) equation (1) u t = δu xx + u(1 u), δ > 0, is the classical example of a semilinear reaction-diffusion partial differential equation (PDE). First introduced by Luther in 1906 [27], it was named for Fisher [10], and for Kolmogorov, Petrovsky and Pescounov [22], who independently wrote seminal papers on the subject. It is widely applicable, and is the prototype of a PDE admitting travelling wave solutions [30]. By a travelling wave solution of the Fisher/KPP equation, we mean a solution of the form u(x ct), travelling from left to right with some positive (constant) wave propagation speed c. Luther originally used equation (1) to model and study travelling waves in chemical reactions, while Fisher and KPP used it to model the spread of an advantageous gene in a population. It qualitatively captures any system where both logistic growth and Fickian diffusion are present, and since its inception the Fisher/KPP equation has been used to 1 Mathematical Sciences School, Queensland University of Technology, Brisbane, QLD 4000 Australia. 2 School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006 Australia. Corresponding Author: robert.marangell@sydney.edu.au. 1

2 2 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER model flame propagation, cellular reaction to the introduction of toxins, as well as the voltage propagation through a nerve axon [3]. To study travelling wave solutions, we introduce a moving coordinate frame; setting z := x ct, and τ := t, equation (1) becomes (2) u τ = δu zz + cu z + u(1 u). Travelling waves u = û(z) will then satisfy the ordinary differential equation (ODE) (3) δu zz + cu z + u(1 u) = 0. Luther, Fisher, and Kolmogorov, Petrovsky and Pescounov, each showed that for δ = 1, a travelling wave of asymptotic speed c = 2 emerged for a variety of initial conditions, and that in some sense this wave was stable while other travelling waves were unstable. Kolmogorov, Petrovsky and Pescounov showed that the natural initial condition of a Heaviside function (or compact perturbations thereof), evolved to a travelling wave of speed c = 2, while Fisher argued from a heuristic viewpoint that the mathematically slowest positive wave was monotone and of speed c = 2, and that any non-monotone travelling waves would necessarily be negative and therefore fail to represent physical quantities, and hence could not exist (i.e. they must be unstable in the full PDE). After the original works listed above, the next stability results for the Fisher/KPP equation began to appear in the 1970 s and 1980 s [3, 4, 24, 28, 36, 37], where it was shown that the travelling waves in the Fisher/KPP equation were so-called pulled-fronts [15, 23] (though this terminology would not appear until later), and the waves of speed less than 2 δ were unstable to all perturbations [14]. It has been shown that the spatial asymptotic decay rate of the initial conditions affects the temporal asymptotic wave speed [15, 29]. This phenomenon is related to the fact that the continuous spectrum of the linearised operator about a travelling wave is in the right half plane for all wave speeds, but can be shown to be in the left half plane in an appropriate weighted space. This leads to either a restriction on the statement of stability [12, 13, 21, 36], or a so-called transient instability [34]. Consequently, one must either restrict analysis to localised perturbations [30, 36] or else care must be taken when trying to make precise an intuitive notion of stability [32]. Both of these approaches are beyond the scope of this article and indeed, the literature relating to stability of travelling waves in the Fisher/KPP equation is vast; see [30, 35, 38] and the references therein for a partial

3 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION 3 list of results. We refer the reader to the aforementioned references for detailed, formal stability results for travelling wave solutions to the Fisher/KPP equation. The main focus of this article is a geometrically inspired technique for numerically computing an Evans function which can be readily extended to higher order systems of PDEs. Evans functions first arose in the 1970 s [9], and are typically constructed using a matching condition (see for example [1, 11, 35]). They are analytic functions in some relevant region of the complex λ-spectral plane, with the property that the multiplicity of its roots coincides with the multiplicity of λ as an eigenvalue. A well-known obstacle in the numerical computation of the Evans function is the tendency of the ODEs to become stiff, an issue that can be overcome by working in the exterior product space [2, 5, 6, 7]. Building on this, there have been two complementary directions to the study of (efficient) numerical computation of the Evans function: Continuous Orthogonalization (see [17, 16, 39] and the references therein for details) and Grassmannian Spectral Shooting, or the Riccati approach ([25, 26]). It is this second point of view that we will follow here, using the Fisher/KPP equation as the problem of interest for an illustration of this approach to assessing stability of travelling wave solutions. By following the Riccati approach we are lead to an Evans function computation that is tractable even through relatively large changes in the order of the spectral parameter (from 10 6 up to ). Furthermore, this approach is readily extendible to values of the spectral parameter which lie in the continuous spectrum. In extending the Evans function into the continuous spectrum, we highlight an underlying relationship between instabilities of a travelling wave and the geometry of the spectral problem. After setting up the (temporal) eigenvalue problem in Section 2, we introduce and apply the main geometric ideas in Section 3. We introduce the Riccati equation, and use it to construct a new, well-behaved function whose roots correspond to eigenvalues (an Evans function). We then extend this function into the continuous spectrum in the natural way, and show that its roots still correspond to instabilities. We then discuss a natural, geometric extension of this function, concluding with Theorem 3.1 which relates the number of eigenvalues inside a curve of spectrum to the number of zeros and poles of a meromorphic function along the curve (the winding number). In Section 4, we conclude with, to the best of our knowledge, a new proof of the lack of eigenvalues with positive real part of the linearised operator about a travelling wave in the Fisher/KPP equation.

4 4 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER 2. The spectral problem In this section we recall the existing analysis for the travelling wave equation (3), while explicitly identifying the travelling waves in which we are interested. We first write equation (3) as a system of equations: (4) du dz = v, dv dz = 1 ( cv u(1 u)). δ There are two equilibria of equation (4) in the uv-plane, one at (0, 0) and one at (1, 0). The Jacobian of (4) is given by ( ) 0 1 Df(u, v) =. 1+2u δ At the point (1, 0) the eigenvalues of Df(u, v) are given by (5) µ u 1 = c + c 2 + 4δ c δ, µ s 1 = c c 2 + 4δ For all values of c there is one positive eigenvalue and one negative eigenvalue, thus (1, 0) is a saddle point in the phase plane. At the point (0, 0), the eigenvalues of Df(u, v) are given by: (6) µ s 0 = c + c 2 4δ, µ ss 0 = c c 2 4δ. For c 2 δ these are two real (distinct or equal), negative eigenvalues so (0, 0) is a (possibly degenerate) node, while if 0 < c < 2 δ then it will be a stable focus. It is easy to see from the related phase portrait that for any value of c > 0 there will be a heteroclinic orbit connecting (1, 0) to (0, 0). When c 2 4δ this orbit will remain negative in u, corresponding to a family of monotone travelling waves satisfying û( ) = 1 and û(+ ) = 0. When c 2 < 4δ, there will be a family of non-monotone travelling waves. These waves (both monotone and not) are the focus of this article. Travelling wave solutions to (1) are steady state solutions to (2). Once a travelling wave û(z) is found for a fixed c, we wish to consider how equation (2) behaves relative to perturbations (in the moving frame) about the travelling wave. We make the ansatz u(z, τ) = û(z) + p(z, τ), with p(z, τ) in an appropriate Banach space, substitute into equation (2), and consider only the first order perturbative terms to give the formal (linearised) equation for p: (7) p τ = δp zz + cp z + (1 2û)p..

5 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION 5 Let H 1 (R) denote the usual Sobolev space of functions from R to R which are square integrable, and with first (weak) derivative also being square integrable. We define the (linear) operator L : H 1 (R) H 1 (R) by Lp := (δ zz + c z + (1 2û)) p. Letting I be the identity map on H 1 (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 λi is not invertible on (some dense subset of) H 1 (R). The set of all such λ C will be denoted as σ(l). The operator L λi on H 1 (R) is equivalent to the operator T (λ) : H 1 (R) L 2 (R) H 1 (R) L 2 (R) given by ( ) ( ) ( ) ( ) ( ) ( ) p d p p 0 1 p (8) T (λ) := A(z; λ) :=. q dz q q q λ 1+2û δ Here we have defined := d and the matrix A(z; λ), and we further define dz ( ) 0 1 A ± (λ) := lim A(z; λ) =. z ± We note that the (spatial) eigenvalues of A + (λ) are (9) µ u +(λ) := c + c 2 + 4δ(λ 1) and those for A (λ) are λ 1 δ c δ c δ, µ s +(λ) := c c 2 + 4δ(λ 1) (10) µ u (λ) := c + c 2 + 4δ(λ + 1), µ s (λ) := c c 2 + 4δ(λ + 1). As it will often be convenient, when there is no ambiguity we will drop the arguments in the eigenvalues of the matrices A ± (λ), writing instead µ s,u ± as appropriate. We remark also that in the case that λ = 0, we have that µ s,u (0) = µs,u 1, and that µs,u + (0) = µss,s 0 from before. Further we have that the (spatial) eigenvectors of A ± (λ) are given by (1, µ u,s ± )T. Lastly we 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. Remark 2.1. We claim that the spectrum of the operator L naturally falls into two parts, the continuous spectrum and the point spectrum. For the description of the continuous spectrum we follow [34, 35], while in order to best describe the point spectrum of L we follow [1, 18, 31]. There is no discrepancy with our choices however and equivalent statements,

6 6 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER for the point and continuous spectrum can be found in all of [18, 31, 35, 34]. We refer the reader to [35] for rigorous proof of the equivalence of all such definitions as well as the fact that our definition of the spectrum can indeed be broken up into the sets defined as the point and continuous spectrum as given below The Continuous Spectrum. Definition 2.2. We define the continuous spectrum of the operator L denoted σ c (L), or sometimes just σ c, to be the set (in C) of those λ for which one of the following holds: (1) A + (λ) and A (λ) have a different number of unstable eigenvalues. (2) A (λ) or A + (λ) has a purely imaginary eigenvalue. We note that one can track the real part of the eigenvalues of A ± (λ), and that only one of the signs of Re ( µ u ( ±) will change at a time. Further, in order for the sign of Re µ u ± (λ) ) to change, there must be a λ where Re ( µ u ±(λ) ) = 0. Writing ω for the sign of a real number ω ω, we have the following: Corollary 2.1. The closure of the set defined in part (1) of Definition 2.2 contains the set defined in part (2) as its boundary, and σ c (L) can be written as { σ c (L) := λ C Re ( ) µ u Re ( ) µ u Re ( ) } µ u + Re ( ) µ u. + The equations defining the boundary of the continuous spectrum are important in their own right, and are the so-called dispersion relations. In light of Corollary 2.1, those eigenvalues such that part (2) of Definition 2.2 holds, are given parametrically by (11) λ = δk 2 ± 1 + ick for k R. Here ik would be the imaginary eigenvalue of A ± (λ). This describes two parabolas, opening leftward and intersecting the real axis at ±1. The complex plane minus the continuous spectrum is composed of two disjoint sets C \ σ c = Ω 1 Ω 2. We define here the sets in accordance with Definition 2.2 (see Figure 1): Ω 1 := { λ C \ σ c 0 < Re ( µ u +(λ) ) < Re ( µ u (λ) )}, Ω 2 := { λ C \ σ c Re ( µ u +(λ) ) < Re ( µ u (λ) ) < 0 }.

7 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION Ω 1 Im(λ) 2 0 Ω 2 σ c Re(λ) Figure 1. The continuous spectrum of the linearised operator around a travelling wave (c.f. equation (8) and Definition 2.2). The boundary is given by equation (11). For the above picture we chose δ = 1 and c = 5. In 6 Ω 1 we have 0 < Re ( µ u +) < Re ( µ u ), in Ω2 we have Re ( µ u +) < Re ( µ u ) < 0 while in σ c we have Re ( µ u +) < 0 < Re ( µ u ) Eigenvalues. For a λ σ c we can ask whether there are any nontrivial functions in the kernel of T (λ). That is, can we find a nontrivial solution in H 1 (R) L 2 (R) to the first order system (12) ( ) ( p = q ) ( ) ( ) 0 1 = A(z; λ)? λ 1+2û δ c δ Any such solution must decay to 0 as z ±, there being only one way that this can be realised. Proposition 2.1. For λ C \ σ c, if (p, q) T is a solution to (12) such that (p, q) T H 1 (R) L 2 (R) then ( ) ( ) (13) p lim ξ u p z and lim q z q ξ+ s. p q p q

8 8 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER That is (p, q) T decays to the stable subspace ξ+ s of A + (λ) as z + and the unstable subspace ξ u of A (λ) as z. A rigorous proof of this proposition can be found in [18, 31]. An intuitive reasoning behind why the proposition should be true is the following: For a λ Ω 1, as z, the system (12) behaves like ( p q ) = A + (λ) ( and since we are in the region Ω 1, we have only one stable direction. Thus if (p, q) T 0, it must do so along the direction of the stable subspace ξ+. s The same is true as z, because we must have that the solution decays to zero along the subspace ξ. u This argument also shows that for any λ Ω 2 no solutions decay to zero as z. Definition 2.3. We will say that λ Ω 1 is a (temporal) eigenvalue with eigenfunction p if we can find such a solution to (12) which is in H 1 (R) L 2 (R). We next exploit the linearity of (12). For a fixed λ Ω 1, 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 the linear subspace of solutions which decay to ξ s + as z +. We note that in our example we can (for any fixed λ) view Ξ u (z; λ) and Ξ s (z; λ) as (line) bundles over R. This allows us to justify calling Ξ u (z; λ) the unstable manifold and Ξ s (z; λ) the stable manifold. What we mean by this is that Ξ u (z; λ) is the manifold of solutions which decay as z to the unstable subspace of A (λ) (and similarly for Ξ s (z; λ)). We can evaluate Ξ u and Ξ s at a fixed value z 0, and if they are linearly dependent then we will have an eigenvalue. This is because of uniqueness of solutions to ODEs, if they agree at one z 0 then they must agree for all z R and so we have (a linear subspace) of solutions which decay as z ±. Let w u (z; λ) = ( ) w1 u (z; λ) w2 u (z; λ) p q ) and w s (z; λ) =, ( ) w1 s (z; λ) w2 s (z; λ) be two solutions in Ξ u and Ξ s respectively. These are two vectors in C 2, and we know that λ is an eigenvalue if and only if they are linearly dependent for some (and hence every) z 0 R. For convenience we choose z 0 = 0. We have therefore shown the following: Proposition 2.2. The complex number λ Ω 1 is an eigenvalue if and only if ( ) w1 u (14) D(λ) := det (0; λ) ws 1 (0; λ) w2 u(0; λ) = 0. ws 2 (0; λ)

9 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION 9 Definition 2.4. The function D(λ) defined in Proposition 2.2 is called an Evans function. 3. The Riccati equation For Definition 2.4, we only compared two possible appropriately decaying solutions to the ODE (12). In the following we are interested in whether or not a pair of subspaces intersect, rather than the particulars of any given solution. Definition 3.1. The set of one (complex) dimensional subspaces in complex two space is called complex projective space and is denoted CP 1. Complex projective space can be given the structure of a complex manifold of one complex dimension and is topologically equivalent to the Riemann sphere, which we denote here by S 2. A line in C 2 through the origin is determined by a pair of complex numbers denoted [p : q] which are not both zero. We can write down all the lines where p 0 as [1 : η], and we see right away that this is (equivalent to) a copy of the complex plane. Similarly we write all the lines where q 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 the typical charts on CP 1. Definition 3.2. Given a linear 2 2 ODE, we get an equivalent flow on CP 1. This is called the Riccati equation. We can obtain an expression (on each chart) for the Riccati equation by simply differentiating the defining relations of η and τ and using equation (12). We get: ( ) q η = = 1 p δ (λ + 1 2û) c δ η η2, (15) ( ) 1 τ = = η η η 2 = 1 + c δ τ 1 δ (λ + 1 2û) τ 2. 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 µu,s ± while in the τ chart they are the multiplicative inverses, a feature of our particular 2 2 system that will not in general be true for an arbitrary system. We also have that λ will be an eigenvalue if and only if we can find a heteroclinic connection between ξ u and ξ s +. In terms of the η chart, this is a heteroclinic connection between µ u and µ s + (and between their multiplicative inverses in the τ chart).

10 10 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER Remark 3.1. By writing out the real and imaginary parts of the flow in the η and τ charts, and by considering the flow direction on the real axis, one can show that there cannot be a heteroclinic connection in the case of a spectral parameter with non-zero imaginary part. Further, by applying techniques used in [20] one can similarly show that there are no real positive eigenvalues. In Section 4 we exploit this idea to prove the absence of eigenvalues in the case of travelling waves in the Fisher/KPP equation. We determine a related Evans function by letting η u (z; λ) and η s (z; λ) be the solutions which 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 1 (z; λ) being non-zero or staying in a single chart. This requirement is not necessary, and we discuss what happens if we need to leave the chart below, but we include it here for convenience. We define a new function (16) D(λ) E η (λ) := w1 u(0; λ)ws 1 (0; λ) = wu 1 (0; λ)ws 2 (0; λ) wu 2 (0; λ)ws 1 (0; λ) w1 u(0; λ)ws 1 (0; λ) =η s (0; λ) η u (0; λ). 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 (17) D(λ) E τ (λ) := w2 u(0; λ)ws 2 (0; λ) = wu 1 (0; λ)ws 2 (0; λ) wu 2 (0; λ)ws 1 (0; λ) w2 u(0; λ)ws 2 (0; λ) =τ u (0; λ) τ s (0; λ). Note that for λ Ω 1, the function E η (λ) is zero if and only if η s (0; λ) = η u (0; λ). By uniqueness of solutions to ODE s, we will therefore have that η u (z; λ) = η s (z; λ) for all z R, and hence a heteroclinic connection between ξ s + and ξ u exists. This will be true if and only if λ Ω 1 is an eigenvalue. The same argument holds for E τ (λ), i.e. we will have and eigenvalue if and only if E τ (λ) = 0.

11 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION 11 We only need to calculate the η s to compute the Evans functions. Since τ = 1, we have η (18) E τ (λ) =τ u (0; λ) τ s (0; λ) = = ηs (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) 1 η u (0; λ) 1 η s (0; λ) so knowing how to compute E η (λ) is enough to compute E τ (λ) Zeros of the Evans function. We are interested in where the function D(λ) and hence E η (λ) is equal to zero in the region Ω 1, assuming that w u,s 1 (0; λ) 0 for any λ C. To investigate this we can exploit the analyticity (or continuity) of E η (λ) for λ Ω 1. We appeal to a theorem from complex analysis (see for example [8]), which says that if f(λ) is a meromorphic function on some simply connected domain Ω C with no zeros or poles on a curve γ(t) Ω in the complex plane, then, letting N denote the number of zeros of f inside γ(t), and P denote the number of poles inside γ(t), we have: (19) N P = 1 f(λ) 2πi f(λ) dλ where denotes d dλ. The assumption that w u,s 1 (0; λ) 0 for any λ Ω 1 means that we stay in a single chart for each λ Ω 1. Thus we have that E η (λ) will be not just meromorphic but analytic, that is P 0, leading to the following: Proposition 3.1. Let γ(t) : [0, 1] C be a simple closed curve in the complex plane, oriented counterclockwise and let D(λ), E η (λ), and w s,u 1 (0; λ) be defined as above. Suppose that w s,u 1 (0; (γ(t))) 0 for all t [0, 1]. Then γ(t) E η (λ) E η (λ) dλ = γ γ(t) Ḋ(λ) D(λ) dλ. That is, the number of zeros of E η (λ) is the same as the number of zeros of the Evans function. For every value of λ Ω 1, the assumption that w u,s 1 (0; λ) 0 is consistent with the behaviour of numerical solutions to the Fisher/KPP equation. Further, N may be thought

12 12 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER of as the winding number of E η (λ), the number of times E η (λ) (and hence D(λ)) winds around the 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 in the case of Fisher/KPP for quite large values of λ. As can be seen in Figure 2, there are no eigenvalues in Ω 1 in the right half plane with λ It is evident that the winding number of E η (λ) and hence D(λ) is zero Extending E η (λ) into the continuous spectrum. Since the goal of Evans function computations is to numerically infer stability or otherwise, we need to concern ourselves with values of the spectral parameter in the right half plane (that is with Re (λ) 0), not just those in Ω 1. To this end we need to consider values of λ inside σ c, the continuous spectrum and re-visit our definition of (temporal) eigenvalues. We proceed in the manner outlined in [18] and [31]. Using Definition 2.3 for all λ with Re (λ) 0, then if λ σ c, the matrix A (λ) has two (spatial) eigenvalues both with negative real parts. This implies that every solution of equation (12) 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 a solution 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 1 (R) L 2 (R). This would seem to suggest that every travelling wave is unstable - an outcome clearly inconsistent with numerical solutions or indeed what is known analytically for travelling wave solutions to the Fisher/KPP equation. This motivates the following amendment to Definition 2.3. Definition 3.3. For a λ C with Re (λ) 0, we say that λ is an eigenvalue if there is a solution to the Riccati equation that decays to ξ u as z and to ξ s + as z +. Remark 3.2. We note here that this follows the definition of eigenvalue from [31]. The roots of 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 with our definition of an eigenvalue in the region Ω 1. Inside the continuous spectrum we will not 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 since we can compare the known stability results for the Fisher/KPP

13 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION Im(Eη(λ)) Im(Eη(λ)) Re(E η (λ)) Re(E η (λ)) 1 2 Im(λ) Re(λ) 10 6 Im(λ) Re(λ) Figure 2. Top: A plot of the function E η (λ) defined in equation (16) for the contour in the spectral parameter (bottom). The wave speed c = 2.4, and in these figures δ = 1. As can be clearly seen, E η (λ) does not wind around the origin, even though the curve of spectrum does. Moreover, the spectral radius is quite large in this case ( ), while the function E η (λ) remains relatively well-behaved, even through fairly large changes of scale in the spectral parameter.

14 14 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER equation, we drop the quotation marks, and simply refer to any such λ as an (temporal) eigenvalue of the linearised operator L. With Definition 3.3, eigenvalues will still correspond exactly to zeros of E η (λ). Moreover, it is straightforward to see that we can still define E η (λ) to be analytic as we extend λ into σ c. We can in fact use some analysis of the Riccati equation on the η-chart of CP 1 to see when exactly we get a zero of E η,τ (λ). To begin with, we fix a λ σ c with Re (λ) 0 and seek a heteroclinic connection between µ u and µ s +. For a general λ we will have that the unstable orbit in the η-chart coming from µ u (viewed as a subspace in C 2 ) will tend towards the steady state solution µ sc + := µ u + (here, because we are in σ c we denote µ sc + := µ u + to note that it is in fact a stable fixed point of the Riccati flow on the η-chart). Recall that µ sc + = c + c 2 + 4δ(λ 1) and µ s + = c c 2 + 4δ(λ 1) which will be different points in the chart of CP 1, for all values of λ except for when c 2 + 4δ(λ 1) = 0, that is except for λ = 1 c 2 /(4δ). At this value of λ we have µ s + = µ sc, and so what was a heteroclinic connection (of the Riccati flow on the η-chart of CP 1 ) between the fixed points µ u and µ sc + will also be a heteroclinic connection between the fixed points µ u and µ s +, and so we will have a zero of E η (λ), and this value of λ will be an eigenvalue according to Definition 3.3. We observe that if c 2 4δ then the largest root, say λ, of the function E η (λ) will be real and negative. However as c 2 δ, we have that λ will tend towards 0 and if c < 2 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 experiments, as well as the analytic results proven in [14]., 3.3. Switching Charts. Suppose that for some fixed z 0 we had that the solution of our Riccati equation η u (z; λ), implying that the corresponding solution in the τ chart must tend to 0. Given the uniqueness of solutions to ODEs on manifolds, we can find a value z 1 < z 0 such that η u (z 1 ; λ) < and so consider the corresponding initial value problem in the τ chart where τ 1 (z 1 ; λ) = η u (z 1 ; λ). Evolving the τ problem from z 1 to a new z 2 > z 0 (noting along the way that τ (z 0 ; λ) = 0), we can then consider the solution of the Cauchy problem on the η chart with initial condition η u 1 (z 2 ; λ) = τ. In this (z 2 ; λ) way we have moved beyond the singularity of our Riccati solution. The impact that this

15 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION 15 η η µ u 1 µ u µ u + z µ sc + z µ s + 2 µ s + 2 µ s 3 µ s Figure 3. Plots of the relevant solutions to the Riccati equation, equation (15), in the η chart for two (real) values of λ. On the left we have λ = 2 and it 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 (the upper solution - blue online) tends towards µ sc + as z + (the middle solution - red online). Thus while we have a traditional eigenvalue, we do not have an eigenvalue according to Definition 3.3. strategy has upon our previously defined Evans functions needs to be explored. Given that we are no longer in the case where the the number of poles of E η (λ) (the value P above) is 0, our winding number calculation becomes (20) γ(t) E η (λ) E η (λ) dλ = γ(t) Ḋ(λ) D(λ) dλ ẇ s 1 γ(t) w1 s (0; λ) (0; λ)dλ ẇ u 1 γ(t) w1 u We elaborate on the meaning of this result in the following theorem. (0; λ) (0; λ)dλ. Theorem 3.1. Let γ(t) be a parametrised curve in the complex plane such that D(λ) is analytic and has no zeros on γ(t). Then the winding number of E η along γ(t) is the number 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 1(0; λ)w u 1 (0; λ)e η (λ). Choosing a curve γ(t) such that D(λ) has no zeros on γ(t) (that is avoiding any eigenvalues), and such that D(λ) is analytic on γ(t), then applying the chain rule to logarithmic

16 16 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER differentiation and rearranging we have equation (20). E η has a pole exactly when either w1 u(0; λ) or ws 1 (0; λ) is zero, and eigenvalues of L are the zeros of D(λ). In the case of the Fisher/KPP equation this theorem will enable us to find the number of eigenvalues inside any bounded contour, except those containing the so-called absolute spectrum. Following [35], we define the absolute spectrum as the λ C such that the real parts of µ u,s + or µu,s coincide. These can be determined as λ 1 c2 /(4δ) in the case of µ + and λ 1 c 2 /(4δ) for µ (note that λ R in the absolute spectrum). This offers another mechanism for destabilisation of the waves as c 2 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 edge of the absolute spectrum. 4. A proof of the absence of eigenvalues with positive real part. The proof proceeds in two parts. For the first part, we show that there are no eigenvalues with positive real part and non-zero imaginary part. For the second, we will show the absence of a real positive eigenvalue when c 2 δ No complex eigenvalues. Recalling equation (15) we have that on the η chart of CP 1 the linearisation is given by (21) η = 1 δ (λ + 1 2û) c δ η η2. Writing η = α+iβ and λ = m+in with m 0, we have that equation (21) becomes (when viewed on C R 2 ) (22) α = 1 δ (1 αc + m 2û) α2 + β 2, β = 2αβ βc δ + n δ. We see that on the line β = 0 we have that β = n δ and so the sign of β is the same as the sign of the imaginary part of the eigenvalue parameter λ, so the flow is pointing towards the upper half plane when Im (λ) > 0 and towards the lower half plane when Im (λ) < 0. Now we have that an eigenvalue λ on this chart is a value of λ such that there 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 ( µ u ) > 0 and Im ( µ s + ) < 0 if Im (λ) > 0, and the reverse inequalities

17 THE EVANS FUNCTION IN THE FISHER/KPP EQUATION 17 for when Im (λ) < 0. Thus a connection is impossible, as long as Im (λ) 0. Proceeding directly we have that µ u = c + c 2 + 4δ(λ + 1), and that Im ( ( µ u ) κ 1 = δ sin 2 arg ( c 2 + 4δ(1 + λ) ) ) where κ := µ s (λ) > 0 and by arg, we mean the principle argument of the complex number. If Im (λ) > 0, we have that 0 < arg ( c 2 + 4δ(1 + λ) ) π, and so 0 < sin ( 1 2 arg ( c 2 + 4δ(1 + λ) )) 1. In other words, if Im (λ) > 0 then so is Im ( µ u ). The same calculation shows that if Im (λ) < 0, then Im ( µ u ) < 0 as well. Similarly, the imaginary part of µ s + has the opposite sign as that of Im (λ). Thus we have shown that there are no connections possible on this η chart. Essentially the same calculation shows that there is no connection between (µ u ) 1 and (µ s +) 1 on the τ chart, and so we conclude that there are no eigenvalues with nonzero imaginary part. Note that this calculation is independent of the continuous spectrum, and so in order to conclude stability we will need to take the continuous spectrum into account Real eigenvalues. To show that there are no real positive eigenvalues when c 2 δ we proceed as in [20], though here we avoid the formal machinery of the Maslov index (see Section 4.3). Recalling equation (15), if λ is real and positive and if c 2 δ we are looking for a heteroclinic connection on RP 1 S 1 the unit circle. The key idea is to evaluate the Riccati equation on the unit circle at the point µ s +. We have the following: (23) η η=µ s + = 1 δ (λ + 1 2û) c δ µs + (µ s +) 2 = 1 (2 2û), δ noting here that this is independent of λ and strictly positive if c 2 δ (actually for all c > 0, but if c 2 δ, µ s + is no longer real for all real non negative values of λ). Next, for each λ 0, denote the solution on RP 1 S 1 decaying to µ u by l(z). We observe that if λ is not an eigenvalue we have that lim l(z) = z + µu +(λ) (or µ sc + (λ) if λ 1 to remain consistient with earlier notation). The implication of equation (23) is the following: Proposition 4.1. Suppose that l(z) crosses µ s + N j times for some fixed values λ j, j = {1, 2}. Then the number of eigenvalues in the interval (λ 1, λ 2 ) is equal to N 1 N 2. Proof. Suppose, without loss of generality, that N 1 = N Equation (23) means that l(z) can only cross µ s + in one direction. This combined with the previous observations

18 18 K. HARLEY, R. MARANGELL, G. J. PETTET, AND M. WECHSELBERGER about the limit of l(z) for λ not an eigenvalue, means that there must be a λ (λ 1, λ 2 ) where lim l(z) = z + µs +. Notably, this is the definition of an eigenvalue (Definition 3.3). Given Proposition 4.1, it suffices to show that there are no crossings for λ on the positive real line of l(z) as z ranges over R for c 2 δ. If λ = 0, we have that equation (12) is the equation of variations along û(z) in the phase plane. Thus the solution l(z) is simply the (unit) tangent vector to the curve (û(z), û (z)) T in the phase plane. As z ranges over R it is obvious that the tangent vector to the curve is never parallel to the eigenvector at positive infinity (1, µ s +) T. Next, for λ 1 we observe that equation (12) is hyperbolic, so l(z) µ u, the steady state solution. Thus there are no crossings for λ 1. This completes the proof that there are no eigenvalues on the positive real line, and the proof of spectral stability of the positive travelling waves in the Fisher/KPP equation Concluding Remarks. To the best of our knowledge this is a new proof of the absence of eigenvalues of the linearised operator about travelling waves in the Fisher/KPP equation of speed c 2 δ. In order to generalise the type of argument used in Section 4.2 to systems of equations, the so-called Maslov index needs to be introduced. The Maslov index is the winding number of a curve in the space of Lagrangian planes Λ(n). One computes the Maslov index of a curve locally by computing the crossing form of the curve with respect to the train of a fixed Lagrangian plane. The aim in the general case is the same as here, show that the crossing form is independent (or at least) monotone with respect to the spectral parameter λ, and then determine the number of (signed) crossings for some natural choices of λ (see [19, 33] for more details and calculations regarding this). In the low dimensions here everything collapses to the circle, Λ(1) S 1 RP 1 and the train of the subspace (1, µ s +) T is just the point itself, so there was no need to describe the more complicated machinery. References [1] J. Alexander, R. Gardner, and C. K. R. T. Jones. A topological invariant arising in the stability analysis of travelling waves. Journal fur die reine und angewandte Mathematik (Crelles Journal), 410: , [2] L. Allen and T. J. Bridges. Numerical exterior algebra and the compound-matrix method. Numerische Mathematik, 92: , [3] D. Aronson and H. Weinberger. Nonlinear diffusion in population genetics, combustion and nerve propagation. In J. A. Goldstein, editor, Partial Differential Equations and Related Topics, number 466 in Lecture Notes in Mathematics. Springer Verlag, New York, 1975.

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