Electronic Journal of Qualitative Theory of Differential Equations 2003, No. 15, 1-19; http://www.math.u-szeged.hu/ejqtde/

On the structure of spectra of travelling waves

Peter L. Simon

Department of Applied Analysis,

Eötvös Loránd University Budapest, Hungary

E-mail: simonp@cs.elte.hu

October 19, 2003

Abstract

The linear stability of the travelling wave solutions of a general reaction-diffusion system is investigated. The spectrum of the corresponding second order differential operator L is studied. The problem is reduced to an asymptotically autonomous first order linear system. The relation between the spectrum of L and the corresponding first order system is dealt with in detail. The first order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of L is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

Keywords: linear stability of travelling waves, exponential dichotomies, Evans function

AMS classification: 35K57, 35B32.

1 Introduction

We investigate the stability of the travelling wave solutions of the system

∂_{τ} u = D∂_{xx}u + f(u),  (1)

where u : ℝ_{+} × ℝ → ℝ^{m}, D is a diagonal matrix with positive diagonal elements and f : ℝ^{m} → ℝ^{m} is a continuously differentiable function. The travelling wave solution of this equation has the form u(τ, x) = U(x − cτ), where U : ℝ → ℝ^{m}. For this function we have

DU′′(z) + cU′(z) + f(U(z)) = 0,  (z = x − cτ)  (2)

The stability of U can be determined by linearization. Putting u(τ, x) = U(x − cτ) + v(τ, x − cτ) in (1) the linearized equation for v takes the form

∂_{τ} v = D∂_{zz}v + c∂_{z}v + f′(U(z))v.  (3)

According to well known stability theorems (see e.g. [21]) the stability of the travelling wave solution U is determined by the spectrum of the second order differential operator

(LV)(t) = DV′′(t) + cV′(t) + Q(t)V (t),  (4)

defined in C_{0}(ℝ, ℂ^{m}) ∩ C^{2}(ℝ, ℂ^{m}), where Q(t) = f′(U(t)),

C_{0}(ℝ, ℂ^{m}) = {V : ℝ → ℂ^{m} | V is continuous, lim_{t→±∞} V (t) = 0},

endowed with the supremum norm ∥V ∥ = max_{ℝ} |V (t)|. It is assumed that Q : ℝ → ℂ^{m×m} is continuous, and the limits Q^{±} = lim_{t→±∞} Q(t) exist.

The complex number λ ∈ ℂ is called a regular value of L if the operator L−λI has bounded inverse that is defined in the whole space C_{0}(ℝ, ℂ^{m}). That is for any W ∈ C_{0}(ℝ, ℂ^{m}) there exists a unique solution of LV − λV = W in C_{0}(ℝ, ℂ^{m}) ∩ C^{2}(ℝ, ℂ^{m}), and there exists M > 0 such that for any W ∈ C_{0}(ℝ, ℂ^{m}) we have ∥V ∥ ≤ M∥W ∥. The spectrum σ(L) of L consists of non-regular values. The number λ is called an eigenvalue if L − λI has no inverse. The essential spectrum of L consists of those points of the spectrum which are not isolated eigenvalues with finite multiplicity. It is useful to introduce the first order system corresponding to equation LV − λV = W. Let x = (V, V′)^{T}, y = (0, W)^{T}, then the first order system is

ẋ(t) = A_{λ}(t)x(t) + y(t),  (5)

where

A_{λ}(t) = ( \begin{matrix}0 & I \\ D^{−1}(λI − Q(t)) & −cD^{−1} \end{matrix} ) .  (6)

The spectrum of L has been widely investigated. The position of the essential spectrum has been estimated, see e.g. [14], and Weyl’s lemma in [21]. This can be done using exponential dichotomies for the first order system (5), [5, 18]. This approach was also generalized to infinite dimensional systems, when A is a bounded operator on a Banach space [6] and also for unbounded operators [4]. Fredholm properties are also relevant when determining the spectrum of L [11, 18]. The relation between these two concepts is dealt with in [16, 18]. The determination of the isolated eigenvalues requires to solve a linear system with non-constant coefficients, which can be done in general only numerically. For the investigation of the isolated eigenvalues two concepts were introduced. The first was the Evans function [10], which is an analytic function on the complex plane, the zeros of which are the isolated eigenvalues of L. Later a topological invariant was introduced for the study of the spectrum [1]. These methods were applied for several systems in physics [3, 12, 17], chemistry [2, 7, 13, 19] and biology [10, 15].

The aim of the paper is to present a self-contained detailed study of the spectrum of L, and to fill the gap between the abstract results (on exponential dichotomies and on topological invariants) and the applications. The novelties of the paper are as follows.

• An algorithm is given for the determination of the exact position of the essential spectrum. The statements concerning the essential spectrum are proved by elementary methods. (Most of the known results give only sufficient conditions for the essential spectrum to lie in the left half plane.)

• All the theorems are proved in the finite dimensional case. The presentation does not need abstract techniques, hence for those applying the theory a self-contained method is shown. (According to the author’s knowledge a self-contained explanation, including the proofs, is only given for the case of unbounded operators [4]. The proof of the finite dimensional case must be compiled from different sources, e.g. [5, 8, 14, 16].)

• The relation between the spectrum of L and the corresponding first order system is dealt with in detail. (The standard reference in this context is [14] but the relation is not proved in that book.)

2 Relation between the spectrum of L and the first order system

Lemma 1 connects the determination of the spectrum of L with the study of system (5). In order to prove the lemma we will need the following Proposition.

Proposition 1 If W ∈ C_{0}(ℝ, ℂ^{m}) and V ∈ C_{0}(ℝ, ℂ^{m}) is the solution of LV − λV = W, then V′ ∈ C_{0}(ℝ, ℂ^{m}).

Proof We will show that V′ → 0 at +∞, it can be verified similarly, that V′ → 0 at −∞. Let us denote by z the real or imaginary part of the k-th coordinate of V, (k is arbitrary), and let d = D_{kk}. We will prove that lim_{t→+∞} ż(t) = 0 that implies the desired statement. First we prove in the case c ≠ 0. Since V and W tend to 0 at +∞, for any ε > 0 there exists t_{0} such that

−ε < dz̈(t) + cż(t) < ε for all t > t_{0}.

Dividing by d and multiplying by exp (ct/d) we obtain

−\frac{ε}{d}e^{\frac{ct}{d}} < (ż(t)e^{\frac{ct}{d}})′ < \frac{ε}{d}e^{\frac{ct}{d}} for all t > t_{0}.

Integrating in [t_{0}, t] we get

−\frac{ε}{c}(1 − e^{\frac{c(t_{0}−t)}{d}}) + ż(t_{0})e^{\frac{c(t_{0}−t)}{d}} < ż(t) < \frac{ε}{c}(1 − e^{\frac{c(t_{0}−t)}{d}}) + ż(t_{0})e^{\frac{c(t_{0}−t)}{d}}  for all t > t_{0}.  (7)

In the case c > 0, there exists t_{1} > t_{0} for which

−2\frac{ε}{c} < ż(t) < 2\frac{ε}{c} for all t > t_{1}

yielding lim_{+∞} ż = 0, what we wanted to prove.

In the case c < 0 we prove by contradiction. Assume that there exists α > 0 and a sequence t_{n} → ∞, such that |ż(t_{n})| = α. Let ε = −cα/2 and apply (7) for t_{0} = t_{n} when n is large enough. If ż(t_{n}) = α, then the inequality in the left hand side of (7) yields for t > t_{n} that

ż(t) > \frac{α}{2}(1 + e^{\frac{c(tn−t)}{d}}) → +∞

as t → +∞. This contradicts to the boundedness of z. If ż(t_{n}) = −α, then the inequality in the right hand side of (7) yields for t > t_{n} that

ż(t) < −\frac{α}{2}(1 + e^{\frac{c(tn−t)}{d}}) → −∞

as t → +∞. This contradicts to the boundedness of z.

Finally, let us consider the case c = 0. Then from the differential equation we get that z̈ tends to zero at infinity. According to the Landau-Kolmogorov inequality (see e.g. [9]) ∥ż∥ ≤ 4∥z̈∥∥z∥. Defining ∥ · ∥ as the supremum norm on [T, +∞) for T large enough, we get that ż → 0 at infinity. □

Lemma 1 (i) If system (5) has a unique solution x ∈ C_{0}(ℝ, ℂ^{2m}) for any y ∈ C_{0}(ℝ, ℂ^{2m}) and x depends continuously on y, (i.e. there exists M > 0, such that ∥x∥ ≤ M(∥y∥ for all y ∈ C_{0}(ℝ, ℂ^{2m})), then λ is a regular value of L.

(ii) If λ is a regular value of L, then for any differentiable function y ∈ C_{0}(ℝ, ℂ^{2m}), for which ẏ ∈ C_{0}(ℝ, ℂ^{2m}), there exists a unique solution x ∈ C_{0}(ℝ, ℂ^{2m}) of system (5); and there exists M > 0 such that for any y satisfying the above conditions the corresponding unique solution x satisfies ∥x∥ ≤ M(∥y∥ + ∥ẏ∥).

Proof (i)

Let W ∈ C_{0}(ℝ, ℂ^{m}), and y = (0, W)^{T} ∈ C_{0}(ℝ, ℂ^{2m}). Then there exists a unique solution x ∈ C_{0}(ℝ, ℂ^{2m}) of (5). Let V = (x_{1}, …, x_{m})^{T}, U = (x_{m+1}, …, x_{2m})^{T}, then V ∈ C_{0}(ℝ, ℂ^{m}) and U = V′, hence V is twice differentiable, and LV − λV = W. The continuity follows from ∥x∥ ≤ M∥y∥, namely

∥V ∥ ≤ ∥x∥ ≤ M∥y∥ = M∥W ∥.

(ii)

First we show that for any y ∈ C_{0}(ℝ, ℂ^{2m}) which is differentiable and ẏ ∈ C_{0}(ℝ, ℂ^{2m}) there exists a unique solution x ∈ C_{0}(ℝ, ℂ^{2m}) of (5). Let x = (V, U)^{T}, y = (y_{1}, y_{2})^{T}, where V, U, y_{1}, y_{2} : ℝ → ℂ^{m}, then system (5) takes the form

V̇ = U + y_{1}
U̇ = D^{−1}(λI − Q)V − cD^{−1}U + y_{2}

The differentiability of y_{1} implies that V is twice differentiable and V̈ = U̇ + ẏ_{1}, hence from the second equation

DV̈ + cV̇ + QV − λV = Dẏ_{1} + cy_{1} + y_{2}.

Since λ is a regular value of L, this equation has a unique solution V ∈ C_{0}(ℝ, ℂ^{m}), and there exists M_{1} > 0 for which

∥V ∥ ≤ M_{1}(∥ẏ_{1}∥ + ∥y_{1}∥ + ∥y_{2}∥).  (8)

Moreover, according to Proposition 1 V̇ ∈ C_{0}(ℝ, ℂ^{m}), yielding U = V̇ − y_{1} ∈ C_{0}(ℝ, ℂ^{m}). Thus x ∈ C_{0}(ℝ, ℂ^{2m}) what we wanted to prove.

Now we show the continuous dependence of x on y. Let us denote the maximum point of V̇_{k} (for some k = 1, …, m) by t_{k}. Since V̇_{k} → 0 at ±∞, the maximum point is not at infinity, hence V̈_{k}(t_{k}) = 0. Therefore using (8) we get that there exists M_{2} > 0 for which

|V̇_{k}(t_{k})| ≤ M_{2}(∥ẏ_{1}∥ + ∥y_{1}∥ + ∥y_{2}∥).

Similar inequality can be proved for the minimum of V̇_{k}, and for all coordinates of V, hence there exists M_{3} > 0 for which ∥V̇ ∥ ≤ M_{3}(∥ẏ_{1}∥ + ∥y_{1}∥ + ∥y_{2}∥). Since U = V̇ − y_{1}, there exists M_{4} > 0 for which ∥U∥ ≤ M_{4}(∥ẏ_{1}∥ + ∥y_{1}∥ + ∥y_{2}∥). Thus with M = 2(M_{1} + M_{4}) we have ∥x∥ ≤ M(∥y∥ + ∥ẏ∥). □

Now we turn to the study of general first order systems, the special case of which is system (5).

3 First order systems

Now for short let C_{0} = C_{0}(ℝ, ℂ^{n}) endowed with the supremum norm ∥x∥ = max_{ℝ} |x(t)|, and let A : ℝ → ℂ^{n×n} be a continuous function for which the limits

A^{±} = lim_{t→±∞} A(t)

exist. Let us consider the first order system

ẋ(t) = A(t)x(t) + y(t).  (9)

Our aim is to give necessary and sufficient condition for the existence and uniqueness of a solution x ∈ C_{0} of (9) for any y ∈ C_{0}, and for the continuous dependence of x on y.

Since A is continuous, system (9) has solutions for any y ∈ C_{0}, that can be given by the variation of constants formula as

x(t) = Ψ(t)x_{0} + ∫_{0}^{t} Ψ(t)Ψ(s)^{−1}y(s)ds,  (10)

where Ψ is the fundamental system of solutions of the homogeneous equation satisfying Ψ(0) = I, i.e. the n columns of the matrix Ψ(t) are n independent solutions of the homogeneous system

ẋ(t) = A(t)x(t).  (11)

Hence the question is that for a given y ∈ C_{0} does there exist a unique x_{0} ∈ ℂ^{n}, such that x ∈ C_{0} (x is given by (10)), and that does x depend continuously on y.

The dimension of the stable, unstable and central subspaces of the matrices A^{±} play important role. Let us denote the number of eigenvalues (with multiplicity) of A^{+} with positive, negative, zero real part by n_{u}^{+}, n_{s}^{+}, n_{c}^{+}, respectively. We define n_{u}^{−}, n_{s}^{−}, n_{c}^{−} similarly using A^{−}.

First we show that the continuous dependence is violated when n_{c}^{+} > 0 or n_{c}^{−} > 0. The main point in these cases is to prove the existence of a bounded solution of the homogeneous equation, which does not tend to zero.

Theorem 1 Let us assume that at least one of the following two conditions holds:

(a) n_{c}^{+} > 0 and ∫_{0}^{+∞} |A(t) − A^{+}| < ∞

(b) n_{c}^{−} > 0 and ∫_{−∞}^{0} |A(t) − A^{−}| < ∞.

Then the solution of (9) does not depend continuously on y, in the sense that there is no M > 0 for which ∥x∥ ≤ M(∥y∥ + ∥ẏ∥) holds for any differentiable function y ∈ C_{0}, for which ẏ ∈ C_{0}.

Proof We prove in case (a), the other case is similar. According to Theorem 1.10.1 of [8] there exists z_{0} ∈ ℂ^{n}, such that the solution t ↦ Ψ(t)z_{0} of (11) is bounded in [0, +∞) but does not tend to zero as t → +∞. Then there exist a > 0 and a sequence t_{k} → +∞, such that |Ψ(t_{k})z_{0}| = a for all k = 1, 2, . . .. Let h_{k} : ℝ → ℝ be continuously differentiable functions satisfying the following conditions

h_{k}|_{(−∞,0]} = 0, |h_{k}| ≤ 1, |ḣ_{k}| ≤ 1, ∫_{0}^{t_{k}} h_{k} = 1, ∫_{0}^{+∞}h_{k} = 0, lim_{+∞} h_{k} = 0 lim_{+∞} ḣ_{k} = 0.

Let y_{k}(t) = h_{k}(t)Ψ(t)z_{0}, then y_{k} ∈ C_{0}, y_{k} is differentiable and

ẏ_{k}(t) = ḣ_{k}(t)Ψ(t)z_{0} + h_{k}(t)Ψ̇(t)z_{0} = ḣ_{k}(t)Ψ(t)z_{0} + h_{k}(t)A(t)Ψ(t)z_{0}.

Hence ẏ_{k} → 0 at +∞, since h_{k} → 0, ḣ_{k} → 0 and A(t) is bounded. On the other hand, there exists M_{1} > 0 for which ∥ẏ_{k}∥ ≤ M_{1} for all k = 1, 2, . . .. Let

x_{k}(t) = ∫_{0}^{t} Ψ(t)Ψ(s)^{−1}y_{k}(s)ds

Then x_{k} is a solution of (9) belonging to y_{k}, and x_{k}(t) = Ψ(t)z_{0} ∫_{0}^{t} h_{k}(s)ds, hence x_{k} ∈ C_{0}. However,

∥x_{k}∥ = max_{t∈ℝ} |x_{k}(t)| ≥ |x_{k}(t_{k})| = t_{k}a.

Since ∥y_{k}∥ ≤ ∥Ψ(·)z_{0}∥ for all k = 1, 2, …, the inequality ∥x_{k}∥ ≤ M(∥y_{k}∥ + ∥ẏ_{k}∥) would mean t_{k}a ≤ M(∥Ψ(·)z_{0}∥ + M_{1}) for all k = 1, 2, …, which contradicts to t_{k} → +∞. □

In the further considerations we will assume n_{c}^{+} = 0 or n_{c}^{−} = 0. In this case we will use exponential dichotomies to answer the above question.

Definition 1 System (11) possesses an exponential dichotomy in the interval J if there exist a projection P and positive numbers K, L, α, β, such that

∥Ψ(t)P Ψ(s)^{−1}∥ ≤ Ke^{−α(t−s)}  for t ≥ s, t, s ∈ J  (12)
∥Ψ(t)(I − P)Ψ(s)^{−1}∥ ≤ Le^{−β(s−t)}  for s ≥ t, t, s ∈ J  (13)

We will show that the exponential dichotomy on ℝ implies the existence, uniqueness and continuous dependence of the solution of (9). It can be shown that system (11) possesses an exponential dichotomy on ℝ if A(t) is constant and it has no eigenvalues on the imaginary axis, that is when n_{c}^{+} = n_{c}^{−} = 0. For the time dependent case there is no exponential dichotomy on ℝ in general when n_{c}^{+} = n_{c}^{−} = 0. However, we will show that system (11) possesses exponential dichotomies on ℝ_{+} and on ℝ_{−}. If the projections of these two dichotomies are the same, then system (11) possesses an exponential dichotomy on ℝ as well, and the existence, uniqueness and continuous dependence follows.

We will use the following properties of exponential dichotomies.

Lemma 2 (i) Let P_{1} and P_{2} be projections, for which Im P_{1} = Im P_{2}. If system (11) possesses an exponential dichotomy in the interval J with the projection P_{1}, then it possesses an exponential dichotomy in the interval J with the projection P_{2} as well.

(ii) System (11) possesses an exponential dichotomy in any closed and bounded interval [a, b], with any projection.

(iii) If system (11) possesses an exponential dichotomy in the intervals (a, b] and [b, c) with the same projection P, then it possesses an exponential dichotomy in the interval (a, c) (here a can be −∞, and c can be +∞).

Proof

(i) This statement is proved in [5] pp. 16.

(ii) The function (t, s) ↦ ∥Ψ(t)P Ψ(s)^{−1}∥e^{α(t−s)} is continuous on the square [a, b] × [a, b], hence it has a maximum K, implying (12). Inequality (13) follows similarly. (iii) We only have to prove that (12) holds for t ∈ [b, c), s ∈ (a, b], and that (13) holds for s ∈ [b, c), t ∈ (a, b]. We will show only the first one, the proof of the second one is similar. Thus let a < s ≤ b ≤ t < c. Then

∥Ψ(t)P Ψ(s)^{−1}∥ = ∥Ψ(t)P Ψ(b)^{−1}Ψ(b)P Ψ(s)^{−1}∥ ≤ Ke^{−α(t−b)}Ke^{−α(b−s)} = K^{2}e^{−α(t−s)}.

The following Lemma can be proved using the results in [5].

Lemma 3 (i) If n_{c}^{+} = 0, then system (11) possesses an exponential dichotomy in [0, +∞), the projection of which is denoted by P^{+}. Moreover, dim(ker P^{+}) = n_{u}^{+}, dim(Im P^{+}) = n_{s}^{+}.

(ii) If n_{c}^{−} = 0, then system (11) possesses an exponential dichotomy in (−∞, 0], the projection of which is denoted by P^{−}. Moreover, dim(ker P^{−}) = n_{u}^{−}, dim(Im P^{−}) = n_{s}^{−}.

Proof We prove only (i), the second statement can be verified similarly. Since n_{c}^{+} = 0, system ẋ = A_{+}x with constant coefficients possesses an exponential dichotomy in [0, +∞) the projection of which, denoted by P^{+}, has an n_{s}^{+} dimensional range and n_{u}^{+} dimensional kernel, see [5] pp.10. For any δ > 0 there exists t_{0} > 0, such that ∥A(t) − A_{+}∥ < δ for t > t_{0}. Hence according to proposition 1 of [5] on pp. 34 system (11) possesses an exponential dichotomy in [t_{0}, +∞) with projection P^{+}. Applying statements (ii) and (iii) of Lemma 2 we can complete the proof. □

Let us introduce the following subspaces.

E_{s}^{+} = Im P^{+}, E_{u}^{+} = ker P^{+} = Im (I − P^{+}), E_{s}^{−} = Im P^{−}, E_{u}^{−} = ker P^{−} = Im (I − P^{−}).

According to the next Proposition the subspace E_{s}^{+} consists of those initial conditions x_{0}, for which the solution of the homogeneous equation tend to 0 as t → +∞. Similarly, the subspace E_{u}^{−} consists of those initial conditions x_{0}, for which the solution of the homogeneous equation tend to 0 as t → −∞.

Proposition 2 (i) Assume n_{c}^{+} = 0. Then lim_{t→+∞} Ψ(t)x_{0} = 0 if and only if x_{0} ∈ E_{s}^{+}.

(ii) Assume n_{c}^{−} = 0. Then lim_{t→−∞} Ψ(t)x_{0} = 0 if and only if x_{0} ∈ E_{u}^{−}.

Proof We prove only (i), the second statement can be verified similarly. For x_{0} ∈ E_{s}^{+} we have P^{+}x_{0} = x_{0}, hence setting s = 0 in (12)

|Ψ(t)x_{0}| ≤ Ke^{−αt}|x_{0}| t > 0.

Thus for x_{0} ∈ E_{s}^{+} we have proved lim_{t→+∞} Ψ(t)x_{0} = 0.

Let us assume now lim_{t→+∞} Ψ(t)x_{0} = 0. Since x_{0} = P^{+}x_{0} + (I − P^{+})x_{0} and P^{+}x_{0} ∈ E_{s}^{+} we have

0 = lim_{t→+∞} Ψ(t)x_{0} = lim_{t→+∞} Ψ(t)P^{+}x_{0} + lim_{t→+∞} Ψ(t)(I − P^{+})x_{0} = lim_{t→+∞} Ψ(t)(I − P^{+})x_{0}  (14)

Applying (13) to Ψ(s)(I − P^{+})x_{0} and setting t = 0 we get

|(I − P^{+})x_{0}|L^{−1}e^{βs} ≤ |Ψ(s)(I − P^{+})x_{0}| s > 0.

Therefore (14) can be satisfied only in the case (I − P^{+})x_{0} = 0, which means that x_{0} ∈ E_{s}^{+}, what we had to prove. □

Corollary 1 Let us assume n_{c}^{+} = n_{c}^{−} = 0. The homogeneous equation (11) has a unique solution in C_{0} (it is x ≡ 0) if and only if dim(E_{s}^{+} ∩ E_{u}^{−}) = 0.

Proof Let us assume dim(E_{s}^{+} ∩ E_{u}^{−}) = 0. If for a solution x = Ψ(·)x_{0} of (11) we have lim_{±∞} x = 0, then according to Proposition 2 x_{0} ∈ E_{s}^{+} ∩ E_{u}^{−}, hence x_{0} = 0, that is x ≡ 0.

Let us now assume dim(E_{s}^{+} ∩ E_{u}^{−}) > 0. Let x_{0} ∈ E_{s}^{+} ∩ E_{u}^{−}, x_{0} ≠ 0. Then x = Ψ(·)x_{0} is a nonzero solution of (11) and lim_{±∞} x = 0. □

We will need the following three lemmas. The first is a generalization of Proposition 2 to the inhomogeneous equation.

Lemma 4 Let y ∈ C_{0}, x_{0} ∈ ℂ^{n} and let x be given by (10).

(i) Assume n_{c}^{+} = 0. Then lim_{+∞} x = 0 if and only if

(I − P^{+})x_{0} = − ∫_{0}^{+∞}(I − P^{+})Ψ(s)^{−1}y(s)ds  (15)

(ii) Assume n_{c}^{−} = 0. Then lim_{−∞} x = 0 if and only if

P^{−}x_{0} = ∫_{−∞}^{0} P^{−}Ψ(s)^{−1}y(s)ds  (16)

Proof First we show that

lim_{t→+∞} Ψ(t)P^{+}x_{0} + ∫_{0}^{t} Ψ(t)P^{+}Ψ(s)^{−1}y(s)ds = 0 for all x_{0} ∈ ℂ^{n}.  (17)

Applying (12) for P = P^{+} and s = 0 we obtain

|Ψ(t)P^{+}x_{0}| ≤ Ke^{−αt}|x_{0}| → 0 as t → +∞.

Now we prove the convergence of the second term. Let ε > 0 be an arbitrary positive number. Since lim_{+∞} y = 0, there exists t_{1} > 0 such that, for t > t_{1} we have |y(t)| < εα/2K. Let t_{2} > t_{1} be a number for which

\frac{K}{α} ∥y∥e^{α(t_{1}−t)} < \frac{ε}{2} for all t > t_{2}.

Then for t > t_{2} we have

|∫_{0}^{t} Ψ(t)P^{+}Ψ(s)^{−1}y(s)ds| ≤ ∫_{0}^{t_{1}} |Ψ(t)P^{+}Ψ(s)^{−1}y(s)|ds + ∫_{t_{1}}^{+∞}|Ψ(t)P^{+}Ψ(s)^{−1}y(s)|ds ≤

∥y∥Ke^{−αt} ∫_{0}^{t_{1}} e^{αs}ds + \frac{εα}{2K} Ke^{−αt} ∫_{t_{1}}^{+∞}e^{αs}ds < ε.

Thus (17) is verified. Similarly we can prove

lim_{t→−∞} (Ψ(t)(I − P^{−})x_{0} + ∫_{0}^{t} Ψ(t)(I − P^{−})Ψ(s)^{−1}y(s)ds) = 0 for all x_{0} ∈ ℂ^{n}.  (18)

Now we show that (15) is equivalent to

lim_{t→+∞} (Ψ(t)(I − P^{+})x_{0} + ∫_{0}^{t} Ψ(t)(I − P^{+})Ψ(s)^{−1}y(s)ds) = 0.  (19)

Applying (13) for P = P^{+} and t = 0 we can see that the integral in (15) is convergent, since y is a bounded function. Let us assume first that (15) holds. Then

Ψ(t)(I − P^{+})x_{0} + ∫_{0}^{t} Ψ(t)(I − P^{+})Ψ(s)^{−1}y(s)ds = − ∫_{t}^{+∞}Ψ(t)(I − P^{+})Ψ(s)^{−1}y(s)ds.

According to (13)

| ∫_{t}^{+∞}Ψ(t)(I − P^{+})Ψ(s)^{−1}y(s)ds| ≤ max_{(t,+∞)} |y(t)| ∫_{t}^{+∞}Le^{−β(s−t)}ds = max_{(t,+∞)} |y(t)|\frac{L}{β} → 0 as t → +∞.

Now let us assume that (19) holds. Let us introduce x^{∗} by

(I − P^{+})x^{∗} := ∫_{0}^{+∞}(I − P^{+})Ψ(s)^{−1}y(s)ds

Then

Ψ(t)(I −P^{+})x_{0} +∫_{0}^{t} Ψ(t)(I −P^{+})Ψ(s)^{−1}y(s)ds = Ψ(t)(I −P^{+})(x_{0} +x^{∗})− ∫_{t}^{+∞}Ψ(t)(I −P^{+})Ψ(s)^{−1}y(s)ds.

We have seen that the second term tends to zero as t → +∞, hence

lim_{t→+∞} Ψ(t)(I − P^{+})(x_{0} + x^{∗}) = 0

implying x_{0} + x^{∗} = 0 according to Proposition 2, hence x_{0} = −x^{∗}, yielding (15).

Similarly we can prove that (16) is equivalent to

lim_{t→−∞} (Ψ(t)P^{−}x_{0} + ∫_{0}^{t} Ψ(t)P^{−}Ψ(s)^{−1}y(s)ds) = 0.  (20)

Now the proof of statement (i) of the Lemma follows from the equation

x(t) = Ψ(t)P^{+}x_{0} + ∫_{0}^{t} Ψ(t)P^{+}Ψ(s)^{−1}y(s)ds + Ψ(t)(I − P^{+})x_{0} + ∫_{0}^{t} Ψ(t)(I − P^{+})Ψ(s)^{−1}y(s)ds.

According to (17) the first and second terms tend to zero as t → +∞. Hence lim_{+∞} x = 0 if and only if the sum of the third and fourth term tends to zero as t → +∞. According to (19) this sum tends to zero if and only if that (15) holds, what we had to prove.

The proof of statement (ii) of the Lemma follows similarly from (18) and (20). □

Lemma 5 Let us assume n_{c}^{+} = n_{c}^{−} = 0. The following two statements are equivalent.

(i) For any a ∈ E_{u}^{+} and any b ∈ E_{s}^{−} there exists a unique solution x_{0} of

(I − P^{+})x_{0} = a, P^{−}x_{0} = b.

(ii) E_{s}^{+} ⊕ E_{u}^{−} = ℂ^{n}

Proof Let us assume that (i) holds. First we show that E_{s}^{+} ∩ E_{u}^{−} = {0}. Let z ∈ E_{s}^{+} ∩ E_{u}^{−}, then (I − P^{+})z = 0 and P^{−}z = 0, hence applying (i) with a = b = 0 we get z = 0. Now we show that for any z ∈ ℂ^{n} there exist z_{1} ∈ E_{s}^{+}, z_{2} ∈ E_{u}^{−}, such that z = z_{1} + z_{2}. Let z_{1} be the solution of (i) with a = 0, b = P^{−}z, that is

(I − P^{+})z_{1} = 0, P^{−}z_{1} = P^{−}z.

Let z_{2} be the solution of (i) with a = (I − P^{+})z, b = 0, that is

(I − P^{+})z_{2} = (I − P^{+})z, P^{−}z_{2} = 0.

Then (I − P^{+})z = (I − P^{+})(z_{1} + z_{2}) and P^{−}z = P^{−}(z_{1} + z_{2}), hence the uniqueness implies z = z_{1} + z_{2}.

Now let us assume that (ii) holds. First we show that the solution in (i) is unique. Let us assume that there are two solutions x′_{0} and x′′_{0}. Introducing x_{0} = x′_{0} − x′′_{0} we have P^{−}x_{0} = 0 and (I − P^{+})x_{0} = 0. Hence x_{0} ∈ E_{u}^{−} ∩ E_{s}^{+}, yielding x_{0} = 0, that is x′_{0} = x′′_{0}. Now we show that the solution x_{0} in (i) exists. Let z_{1} ∈ E_{s}^{+}, z_{2} ∈ E_{u}^{−}, such that a − b = z_{1} + z_{2} and let x_{0} = b + z_{2} = a − z_{1}. Then (I −P^{+})x_{0} = (I −P^{+})(a−z_{1}) = a, since P^{+}a = 0 and (I −P^{+})z_{1} = 0. Similarly P^{−}x_{0} = P^{−}(b+z_{2}) = b. □

The following lemma is proved in [16] Prop. 2.1 and in [5] p. 19, but in order to make the paper self-contained we present a short proof.

Lemma 6 Let us assume n_{c}^{+} = n_{c}^{−} = 0 and E_{s}^{+} ⊕ E_{u}^{−} = ℂ^{n}. Then system (11) possesses an exponential dichotomy in ℝ with a projection P^{∗}, for which Im P^{∗} = E_{s}^{+}, ker P^{∗} = E_{u}^{−}.

Proof Assumption E_{s}^{+} ⊕ E_{u}^{−} = ℂ^{n} implies that there exists a projection P^{∗}, for which Im P^{∗} = E_{s}^{+}, ker P^{∗} = E_{u}^{−}. Namely, for an arbitrary vector z ∈ ℂ^{n} we can define P^{∗}z as P^{∗}z = z_{1}, where z = z_{1} + z_{2} with z_{1} ∈ E_{s}^{+}, z_{2} ∈ E_{u}^{−}. Since Im P^{∗} = Im P^{+}, Lemma 2 and Lemma 3 imply that system (11) possesses an exponential dichotomy in [0, +∞) with the projection P^{∗}. On the other hand Im (I − P^{∗}) = Im (I −P^{−}), hence Lemma 2 and Lemma 3 imply that system (11) possesses an exponential dichotomy in (−∞, 0] with the projection P^{∗}. Since system (11) possesses an exponential dichotomy both in [0, +∞) and in (−∞, 0] with the projection P^{∗}, according to Lemma 2 it possesses an exponential dichotomy in ℝ. □

Theorem 2 Let us assume n_{c}^{+} = n_{c}^{−} = 0.

(i) If for any differentiable function y ∈ C_{0}, for which ẏ ∈ C_{0} there exists a unique solution x ∈ C_{0} of (9), then E_{s}^{+} ⊕ E_{u}^{−} = ℂ^{n}.

(ii) If E_{s}^{+} ⊕ E_{u}^{−} = ℂ^{n}, then system (9) has a unique solution x ∈ C_{0} for any y ∈ C_{0}, and there exists M > 0, such that for any y ∈ C_{0} and for the corresponding solution x ∈ C_{0} the inequality ∥x∥ ≤ M∥y∥ holds.

Proof (i) Let us assume that system (9) has a unique solution x ∈ C_{0} for any differentiable function y ∈ C_{0}, for which ẏ ∈ C_{0}. Let a ∈ E_{u}^{+} and b ∈ E_{s}^{−} be arbitrary vectors. We will show that there exists a differentiable function y ∈ C_{0}, with ẏ ∈ C_{0}, such that

a = − ∫_{0}^{+∞}(I − P^{+})Ψ(s)^{−1}y(s)ds,  b = ∫_{−∞}^{0} P^{−}Ψ(s)^{−1}y(s)ds.  (21)

Namely, let h : ℝ → ℝ be a continuously differentiable function satisfying

h(0) = 0, ḣ(0) = 0, ∫_{0}^{+∞}h = −1, ∫_{−∞}^{0} h = 1, |h(t)| ≤ e^{−(q+1)|t|}, |ḣ(t)| ≤ re^{−(q+1)|t|} for |t| > 1,

with some r > 0, and where q ∈ ℝ is chosen to satisfy ∥A(t)∥ ≤ q for all t ∈ ℝ. It can be easily shown that there exists k > 0, such that ∥Ψ(t)∥ ≤ ke^{qt} for all t ∈ ℝ. Let

y(t) = \{ \begin{matrix}Ψ(t)h(t)a, & for t ≥ 0 \\ Ψ(t)h(t)b, & for t < 0 \end{matrix}

Then y and ẏ are continuous (also in zero, because their limits are zero from left and from right), and y, ẏ ∈ C_{0}, because for t > 1 we have

|y(t)| ≤ ∥Ψ(t)∥|h(t)||a| ≤ ke^{qt}e^{−(q+1)t}|a| = k|a|e^{−t},

and

|ẏ(t)| ≤ (∥Ψ̇(t)∥|h(t)| + ∥Ψ(t)∥|ḣ(t)|)|a| ≤ (q + r)k|a|e^{−t}.

Similar estimate can be derived for t < −1. Finally, we obtain

− ∫_{0}^{+∞}(I − P^{+})Ψ(s)^{−1}y(s)ds = − ∫_{0}^{+∞}(I − P^{+})Ψ(s)^{−1}Ψ(s)h(s)ads = −a ∫_{0}^{+∞}h = a,

∫_{−∞}^{0} P^{−}Ψ(s)^{−1}y(s)ds = ∫_{−∞}^{0} P^{−}Ψ(s)^{−1}Ψ(s)h(s)bds = b ∫_{−∞}^{0} h = b.

Let x ∈ C_{0} be the solution of (9) belonging to y. Then according to Lemma 4 for x_{0} = x(0) we have

a = (I − P^{+})x_{0},  b = P^{−}x_{0}.  (22)

Hence Lemma 5 implies E_{s}^{+} ⊕ E_{u}^{−} = ℂ^{n}.

(ii) Now let us assume E_{s}^{+} ⊕ E_{u}^{−} = ℂ^{n}. Let y ∈ C_{0} and a, b given by (21). According to Lemma 5 there exists a unique x_{0} ∈ ℂ^{n} satisfying (22), hence Lemma 4 implies x ∈ C_{0}. If x^{∗} ∈ C_{0} is another solution of (9), then x^{∗} − x ∈ C_{0} is a solution of (11). However, according to Corollary 1 x^{∗} − x ≡ 0, that is x^{∗} ≡ x.

Finally we prove the continuous dependence. According to Lemma 6 there exist a projection P^{∗}, and positive numbers K, L, α, β, for which

∥Ψ(t)P^{∗}Ψ(s)^{−1}∥ ≤ Ke^{−α(t−s)}  for t ≥ s, t, s ∈ ℝ  (23)
∥Ψ(t)(I − P^{∗})Ψ(s)^{−1}∥ ≤ Le^{−β(s−t)}  for s ≥ t, t, s ∈ ℝ  (24)

Repeating the proof of Lemma 4 replacing P^{+} and P^{−} with P^{∗} we get

(I − P^{∗})x_{0} = − ∫_{0}^{+∞}(I − P^{∗})Ψ(s)^{−1}y(s)ds, P^{∗}x_{0} = ∫_{−∞}^{0} P^{∗}Ψ(s)^{−1}y(s)ds.

Therefore from the variation of constant formula (10)

x(t) = Ψ(t)P^{∗}x_{0} + ∫_{0}^{t} Ψ(t)P^{∗}Ψ(s)^{−1}y(s)ds + Ψ(t)(I − P^{∗})x_{0} + ∫_{0}^{t} Ψ(t)(I − P^{∗})Ψ(s)^{−1}y(s)ds =

∫_{−∞}^{t} Ψ(t)P^{∗}Ψ(s)^{−1}y(s)ds + ∫_{t}^{+∞}Ψ(t)(I − P^{∗})Ψ(s)^{−1}y(s)ds.

From (23) and (24)

| ∫_{−∞}^{t} Ψ(t)P^{∗}Ψ(s)^{−1}y(s)ds| ≤ ∥y∥Ke^{−αt} ∫_{−∞}^{t} e^{αs}ds = ∥y∥\frac{K}{α} ,

| ∫_{t}^{+∞}Ψ(t)(I − P^{∗})Ψ(s)^{−1}y(s)ds| ≤ ∥y∥Le^{βt} ∫_{t}^{+∞}e^{−βs}ds = ∥y∥\frac{L}{β},

hence ∥x∥ ≤ (K/α + L/β)∥y∥. □

4 The spectrum of L

In Section 1 we have introduced the matrix functions A_{λ}, see (6). Since function Q tends to a limit at ±∞, the limits

A_{λ}^{±} = lim_{t→±∞} A_{λ}(t)

exist. We have seen in Section 2 that the dimension of the stable, unstable and central subspaces of the matrices A_{λ}^{±} play important role. Let us denote the number of eigenvalues (with multiplicity) of A_{λ}^{+} with positive, negative, zero real part by n_{u}^{+}(λ), n_{s}^{+}(λ), n_{c}^{+}(λ), respectively. We define n_{u}^{−}(λ), n_{s}^{−}(λ), n_{c}^{−}(λ) similarly using A_{λ}^{−}.

Theorem 3 Let us assume that at least one of the following two conditions holds:

(a) n_{c}^{+}(λ) > 0 and ∫_{0}^{+∞} |A_{λ}(t) − A_{λ}^{+}| < ∞

(b) n_{c}^{−}(λ) > 0 and ∫_{−∞}^{0} |A_{λ}(t) − A_{λ}^{−}| < ∞.

Then λ ∈ σ(L).

Proof Let us assume the contrary, i.e. that λ is a regular value of L. Then according to (ii) of Lemma 1 for any differentiable function y ∈ C_{0}(ℝ, ℂ^{2m}), for which ẏ ∈ C_{0}(ℝ, ℂ^{2m}), there exists a unique solution x ∈ C_{0}(ℝ, ℂ^{2m}) of system (5); and there exists M > 0 such that for any y the inequality ∥x∥ ≤ M(∥y∥+∥ẏ∥) holds. However, Theorem 1 yields that this M cannot exist, which is a contradiction. □

In the further considerations we deal with the case n_{c}^{+}(λ) = 0 = n_{c}^{−}(λ). In this case we introduce the subspaces

E_{s}^{+}(λ), E_{u}^{+}(λ), E_{s}^{−}(λ), E_{u}^{−}(λ)

in the same way as in Section 2. (Now they depend also on λ, because A depends on λ.)

Theorem 4 Let us assume n_{c}^{+}(λ) = 0 = n_{c}^{−}(λ).

(i) λ is an eigenvalue of L if and only if dim(E_{s}^{+}(λ) ∩ E_{u}^{−}(λ)) > 0.

(ii) λ is a regular value of L if and only if E_{s}^{+}(λ) ⊕ E_{u}^{−}(λ) = ℂ^{2m}.

Proof

(i) If λ is an eigenvalue of L, then there exists V ∈ C_{0}(ℝ, ℂ^{m}), V ≠ 0, for which LV = λV. According to Proposition 1 V′ ∈ C_{0}(ℝ, ℂ^{m}), hence for x = (V, V′)^{T} we have x ∈ C_{0}(ℝ, ℂ^{2m}) and it is a nonzero solution of ẋ(t) = A_{λ}(t)x(t). Therefore the statement follows from Corollary 1.

If dim(E_{s}^{+}(λ) ∩ E_{u}^{−}(λ)) > 0, then according to Corollary 1 there is a nonzero solution x ∈ C_{0}(ℝ, ℂ^{2m}) of ẋ(t) = A_{λ}(t)x(t). Let V = (x_{1}, …, x_{m})^{T}, U = (x_{m+1}, …, x_{2m})^{T}, then U = V′, hence V is twice differentiable, V ≠ 0 (otherwise U ≡ 0 and x ≡ 0), and LV = λV. (ii) If λ is a regular value of L, then according to (ii) of Lemma 1 for any differentiable y ∈ C_{0}(ℝ, ℂ^{2m}) for which ẏ ∈ C_{0}(ℝ, ℂ^{2m}) there exists a unique solution x ∈ C_{0}(ℝ, ℂ^{2m}) of (5). Then Theorem 2 implies E_{s}^{+}(λ) ⊕ E_{u}^{−}(λ) = ℂ^{2m}.

If E_{s}^{+}(λ) ⊕ E_{u}^{−}(λ) = ℂ^{2m}, then according to Theorem 2 for any y ∈ C_{0}(ℝ, ℂ^{2m}) there exists a unique solution x ∈ C_{0}(ℝ, ℂ^{2m}) of (5) and it depends continuously on y. Hence by (i) of Lemma 1 λ is a regular value of L. □

Using that E_{s}^{+}(λ) ⊕ E_{u}^{−}(λ) = ℂ^{2m} is equivalent to dim E_{s}^{+}(λ) + dim E_{u}^{−}(λ) = 2m and dim(E_{s}^{+}(λ) ∩ E_{u}^{−}(λ)) = 0, the following statements are obvious consequences of the theorem above.

Corollary 2 Let us assume n_{c}^{+}(λ) = 0 = n_{c}^{−}(λ).

1. If dim E_{s}^{+}(λ) + dim E_{u}^{−}(λ) > 2m, then λ is an eigenvalue of L.

2. If dim E_{s}^{+}(λ) + dim E_{u}^{−}(λ) < 2m, then λ ∈ σ(L).

3. If dim E_{s}^{+}(λ) + dim E_{u}^{−}(λ) = 2m and dim(E_{s}^{+}(λ) ∩ E_{u}^{−}(λ)) = 0, then λ is a regular value of L.

4. If dim E_{s}^{+}(λ) + dim E_{u}^{−}(λ) = 2m and dim(E_{s}^{+}(λ) ∩ E_{u}^{−}(λ)) > 0, then λ is an eigenvalue of L.

Remark 1 If n_{c}^{+}(λ) = 0 = n_{c}^{−}(λ), then the operator L − λI is Fredholm, and its Fredholm index is α(L − λI) = dim E_{s}^{+}(λ) + dim E_{u}^{−}(λ) − 2m, see [14, 16].

The dimension of E_{s}^{+}(λ) and E_{u}^{−}(λ) can be determined explicitly, because only the eigenvalues of the matrices A_{λ}^{±} have to be determined to get these dimensions. However, to get dim(E_{s}^{+}(λ) ∩ E_{u}^{−}(λ)) the time dependent system must be solved numerically. This leads to the definition of the Evans function. Let

Ω = {λ ∈ ℂ : n_{c}^{+}(λ) = 0 = n_{c}^{−}(λ), n_{s}^{+}(λ) ≠ 0 ≠ n_{u}^{−}(λ), and n_{s}^{+}(λ) + n_{u}^{−}(λ) = 2m}.

For λ ∈ Ω let us denote the base of the subspace E_{s}^{+}(λ) by v_{1}^{+}, …, v_{n_{s}^{+}}^{+}, and the base of the subspace E_{u}^{−}(λ) by v_{1}^{−}, …, v_{n_{u}^{−}}^{−}. The assumption dim(E_{s}^{+}(λ) ∩ E_{u}^{−}(λ)) > 0 means that the two bases together give a linearly dependent system of vectors. The Evans function is defined as the determinant formed by these 2m vectors. That is the determinant is zero if λ is an eigenvalue.

Definition 2 The Evans function belonging to the operator L is D : Ω → ℂ

D(λ) = det (v_{1}^{+} … v_{n_{s}^{+}}^{+}v_{1}^{−} … v_{n_{u}^{−}}^{−})

We have proved that the eigenvalues are the zeros of the Evans function. It can be also shown that the multiplicity of an eigenvalue is equal to the multiplicity of the zero of the Evans function, and that the Evans function is an analytic function on the domain Ω [1]. Hence the zeros of D are isolated, that is in the domain where dim E_{s}^{+}(λ) + dim E_{u}^{−}(λ) = 2m there can be only isolated eigenvalues. This statement together with Corollary 2 enables us to determine the essential spectrum explicitly.

Corollary 3 The essential spectrum of L is

σ_{e}(L) = {λ ∈ ℂ : n_{c}^{+}(λ) > 0 or n_{c}^{−}(λ) > 0 or n_{s}^{+}(λ) + n_{u}^{−}(λ) ≠ 2m}.

The bases of the stable and unstable subspaces can be determined numerically in the following way. We calculate the eigenvalues of A_{λ}^{+} with negative real part, and its corresponding eigenvectors. Let us denote these eigenvalues by μ_{1}, …, μ_{k}, and the eigenvectors by u_{1}, …, u_{k} (for short we used the notation k = n_{s}^{+}(λ)). Similarly, let us denote the eigenvalues of A_{λ}^{−} with positive real part by ν_{1}, …, ν_{l}, and the corresponding eigenvectors by v_{1}, …, v_{l} (for short we used the notation l = n_{u}^{−}(λ)). Then choosing a sufficiently large number ℓ we solve the homogeneous equation ẋ(t) = A_{λ}(t)x(t) in [0, ℓ] starting from the right end point with initial condition x(ℓ) = u_{i}e^{μ_{i}ℓ} for i = 1, …, k. Hence we get k = n_{s}^{+}(λ) linearly independent (approximating) solutions of the differential equations, therefore their values at 0 give a base of E_{s}^{+}(λ). Similarly, solving the differential equation in [−ℓ, 0] we get a base of E_{u}^{−}(λ), and the determinant defining the Evans function can be computed. We note that if ℓ is very large and there is a significant difference between the real parts of the eigenvalues μ_{1}, …, μ_{k}, then the solution belonging to the eigenvalue with largest real part will dominate and the solutions starting from linearly independent initial conditions will be practically linearly dependent at zero. (Similar case can occur in [−ℓ, 0] as well.) To overcome this difficulty the problem can be extended to a wedge product space of higher dimension [3].

Now we show a method to determine the eigenvalues and eigenvectors of A_{λ}^{±}, which determine the dimensions of E_{s}^{+}(λ) and E_{u}^{−}(λ). We will deal with the two cases together, therefore for short we introduce

A_{λ} = ( \begin{matrix}0 & I \\ D^{−1}(λI − Q) & −cD^{−1} \end{matrix} ) ,  (25)

where Q can be Q^{+} or Q^{−}. Let us denote an eigenvalue of A_{λ} by μ and an eigenvector by u = (u_{1}, u_{2})^{T}. That is

( \begin{matrix}0 & I \\ D^{−1}(λI − Q) & −cD^{−1} \end{matrix} ) ( \begin{matrix}u_{1} \\ u_{2} \end{matrix} ) = μ ( \begin{matrix}u_{1} \\ u_{2} \end{matrix} ) .

Then u_{2} = μu_{1} and D^{−1}(λI − Q)u_{1} − cD^{−1}u_{2} = μu_{2}, hence

D^{−1}(λI − Q)u_{1} − cD^{−1}μu_{1} = μ^{2}u_{1},

yielding

(μ^{2}D + μcI + Q − λI)u_{1} = 0.

Thus we have proved the following proposition.

Proposition 3 The number μ is an eigenvalue and u = (u_{1}, u_{2})^{T} is an eigenvector of A_{λ} if and only if

det(μ^{2}D + μcI + Q − λI) = 0  (26)

and

u_{1} ∈ ker(μ^{2}D + μcI + Q − λI), u_{2} = μu_{1}.

Thus the eigenvalues of A_{λ}^{±} are determined by equation (26) of degree 2m. In the special case when Q is an upper or lower triangular matrix the l.h.s. of the equation is a product of m second degree polynomials, hence the solutions can be computed explicitly [20].

5 Applications

5.1 Case of a single equation, m = 1

Now let U : ℝ → ℝ be a solution of

U′′ + cU′ + f(U) = 0  (27)
U(−∞) = U_{−}, U(+∞) = U_{+},  (28)

where f : ℝ → ℝ is a continuously differentiable function, U_{−}, U_{+} ∈ ℝ and it is assumed that c ≥ 0. The stability of U is determined by the spectrum of the operator

L(V) = V′′ + cV′ + f′(U)V.  (29)

The function q(t) = f′(U(t)) is continuous and has limits at ±∞,

q^{+} = f′(U_{+}),  q^{−} = f′(U_{−}).  (30)

If U tends to the limits U_{+} and U_{−} exponentially, then the integrals in the assumptions of Theorem 3 are convergent. Now A_{λ}^{±} are 2-by-2 matrices and according to Proposition 3 their eigenvalues (μ_{1,2}) are determined by the equation

μ^{2} + cμ + q^{±} − λ = 0.  (31)

The essential spectrum can be determined by calculating the dimensions of E_{s}^{+}(λ) and E_{u}^{−}(λ). These dimensions can be easily determined from the sets where n_{c}^{+}(λ) ≥ 1 and n_{c}^{−}(λ) ≥ 1. These sets are formed by those values of λ to which μ = iω is a solution of (31). Hence the set of λ values where n_{c}^{+}(λ) ≥ 1 is the parabola

P^{+} = \{λ_{1} + iλ_{2} ∈ ℂ | λ_{1} = q^{+} − (\frac{λ_{2}}{c} )^{2}\}.  (32)

It is easy to show that on the left hand side of the parabola dim E_{s}^{+}(λ) = 2, and on the right hand side dim E_{s}^{+}(λ) = 1, see Figure 1. Similarly, the set of λ values where n_{c}^{−}(λ) ≥ 1 is the parabola

P^{−} = \{λ_{1} + iλ_{2} ∈ ℂ | λ_{1} = q^{−} − (\frac{λ_{2}}{c} )^{2}\}.  (33)

It is easy to show that on the left hand side of the parabola dim E_{u}^{−}(λ) = 0, and on the right hand side dim E_{u}^{−}(λ) = 1, see Figure 1.

Figure 1. The parabolas given by (32) and (33) for q_{+} > q_{−} in (a) and for q_{−} > q_{+} in (b). The dimensions of the subspaces E_{s}^{+}(λ) and E_{u}^{−}(λ) are shown, the upper numbers correspond to the former and the lower numbers correspond to the latter.

Now applying Theorem 3 and Corollary 2 we have the following results concerning the spectrum of L.

• Both parabolas belong to the essential spectrum of L.

• The domain lying on the left hand side of both parabolas consists of regular values of L.

• The domain lying on the right hand side of both parabolas contains all the isolated eigenvalues of L the remaining points of this domain (which are not isolated eigenvalues) are regular values of L.

• If q^{+} > q^{−}, then the domain between the two parabolas is filled with eigenvalues.

• If q^{+} < q^{−}, then the domain between the two parabolas is filled with points belonging to the essential spectrum, but they are not eigenvalues.

In this special case of m = 1 the location of the isolated eigenvalues with respect to the imaginary axis can also be determined. It can be shown that zero is a simple eigenvalue and all other isolated eigenvalues of L are negative (real) if and only if U is strictly monotone and f′(U_{−}) < 0, f′(U_{+}) < 0.

5.2 Flame propagation in a three variable model

Let us consider the travelling wave solutions of the problem

∂_{τ} a = L_{A}^{−1}∂_{x}^{2}a − af_{1}(b)
∂_{τ} w = L_{W}^{−1}∂_{x}^{2}w − βwf_{2}(b)
∂_{τ} b = ∂_{x}^{2}b + af_{1}(b) − αwf_{2}(b)

where L_{A}, L_{W} , α, β are positive constants (L_{A}, L_{W} are the Lewis numbers) and

f_{1}(b) = e^{(b−1)/εb}, f_{2}(b) = e^{μ(b−1)/εb},

with some positive ε and μ. A travelling wave solution, propagating with velocity c, will also be denoted by (a, w, b), and satisfies the boundary conditions

a(y) → 1, w(y) → 1, b(y) → 0  as y → −∞,  (34)
a′(y) → 0, w′(y) → 0, b′(y) → 0  as y → +∞.  (35)

Here a is the concentration of the fuel, w is the concentration of an inhibitor species and b is scaled temperature. The travelling wave solution describes a flame propagating with velocity c. The number and stability of travelling waves of this system was investigated in [19]. We proved that the solutions a, w and b have limits at +∞, that are denoted by a_{+}, w_{+} and b_{+}. If b_{+} = 0, then we refer to the travelling wave solution as a pulse solution. If b_{+} > 0 then we call it a front solution. In the latter case a_{+} = 0 = w_{+}. It was also shown that a saddle-node bifurcation may occur and there can be 1, 2 or 3 travelling wave solutions. The stability of these solutions can also change through Hopf bifurcation. The saddle-node and Hopf bifurcation curves were determined numerically. Here we only show how the method described in the previous Section works for this system to determine the essential spectrum of the corresponding linearizdimen ed operator. The results obtained by the Evans function method will be only cited from [19].

The operator corresponding to a travelling wave solution (a, w, b) of the above system takes the form

LV = ( \begin{matrix}L_{A}^{−1}V′′_{1} − cV′_{1} − f_{1}(b)V_{1} − af′_{1}(b)V_{3} \\ L_{W}^{−1}V′′_{2} − cV′_{2} − βf_{2}(b)V_{2} − βwf′_{2}(b)V_{3} \\ V′′_{3} − cV′_{3} + f_{1}(b)V_{1} − αf_{2}(b)V_{2} + (af′_{1}(b) − αwf′_{2}(b))V_{3} \end{matrix} )  (36)

We consider L as an operator defined for the C^{2} functions in the space

C_{0} = {V : ℝ → ℂ^{3} | V is continuous, lim_{t→±∞} V (t) = 0}

endowed with the supremum norm.

Now A_{λ}^{±} are the following 6-by-6 matrices

A_{λ}^{±} = ( \begin{matrix}0 & I \\ D^{−1}(λI − Q^{±}) & cD^{−1} \end{matrix} )

where Q^{−} is a 3 × 3 zero matrix,

Q^{+} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) for pulses, Q^{+} = ( \begin{matrix}−q_{1} & 0 & 0 \\ 0 & −βq_{2} & 0 \\ q_{1} & −αq_{2} & 0 \end{matrix} ) for fronts,

and q_{1} = f_{1}(1 − α/β), q_{2} = f_{2}(1 − α/β). According to Proposition 3 the eigenvalues (μ_{1}, …, μ_{6}) of A_{λ}^{±} are determined by the equation

(L_{A}^{−1}μ^{2} − cμ + Q_{11}^{±} − λ)(L_{W}^{−1}μ^{2} − cμ + Q_{22}^{±} − λ)(μ^{2} − cμ + Q_{33}^{±} − λ) = 0,  (37)

since Q^{±} are lower triangular matrices. Therefore the set of those λ values for which n_{c}^{+}(λ) ≥ 1, that is μ = iω (for some ω ∈ ℝ) is a solution of equation (37), consists of three parabolas, denoted by P_{1}^{+}, P_{2}^{+}, P_{3}^{+}, see Figure 2 (a) and (b). In Figure 2 (a) the parabolas are shown in the case of a pulse solution, in Figure 2 (b) the parabolas corresponding to a front solution are shown. We define the parabolas P_{1}^{−}, P_{2}^{−}, P_{3}^{−} similarly as the loci of those λ values for which n_{c}^{−}(λ) ≥ 1, they are shown in Figure 2 (c). The parabolas are given explicitly by

P_{1}^{+} = \{λ ∈ ℂ : Reλ = −q_{1} − \frac{(Imλ)^{2}}{L_{A}c^{2}} \}, P_{1}^{−} = \{λ ∈ ℂ : Reλ = −\frac{(Imλ)^{2}}{L_{A}c^{2}} \},

P_{2}^{+} = \{λ ∈ ℂ : Reλ = −βq_{2} − \frac{(Imλ)^{2}}{L_{W} c^{2}} \}, P_{2}^{−} = \{λ ∈ ℂ : Reλ = −\frac{(Imλ)^{2}}{L_{W} c^{2}} \},

P_{3}^{+} = \{λ ∈ ℂ : Reλ = −\frac{(Imλ)^{2}}{c^{2}} \}, P_{3}^{−} = \{λ ∈ ℂ : Reλ = −\frac{(Imλ)^{2}}{c^{2}} \}.

Figure 2. The parabolas determining the essential spectrum of the operator (36). The dimension of the subspace E_{s}^{+}(λ) is shown in part (a) and (b). The dimension of the subspace E_{u}^{−}(λ) is shown in part (c). (a) corresponds to pulses, (b) corresponds to fronts.

It can easily be seen that if λ is to the left of P_{1}^{+}, then both solutions for μ of equation L_{A}^{−1}μ^{2} − cμ + Q_{11}^{±} − λ = 0 have positive real part, and if λ is to the right of P_{1}^{+}, then one of the solutions has positive real part, the other one has negative real part. The same is true for the other parabolas. Therefore we can determine, for any λ, the value of n_{s}^{+}(λ), i.e. the dimension of E_{s}^{+}(λ), and the value of n_{u}^{−}(λ), i.e. the dimension of E_{u}^{−}(λ). The values of these numbers are shown in Figure 2 in the different domains determined by the parabolas. Now applying Theorem 3 and Corollary 2 we have the following results concerning the spectrum of L.

Proposition 4 1. All parabolas belong to the essential spectrum of L.

2. The domain lying to the left of all parabolas consists of regular values.

3. The Evans function can be defined in the domain lying to the right of all parabolas.

4. In the case of fronts there are open domains filled with eigenvalues. In the case of pulses this kind of domain does not exist.

Proof We will use the dimensions of the spaces E_{s}^{+}(λ), E_{u}^{−}(λ) as they are given in Figure 2.

1. This statement is a direct consequence of Theorem 3.

2. If λ is in the domain lying to the left of all parabolas, then dim(E_{s}^{+}(λ)) = 0, dim(E_{u}^{−}(λ) = 6, hence conditions of statement 3 of Corollary 2 are fulfilled, thus λ is a regular value.

3. If λ is in the domain lying to the right of all parabolas, then dim(E_{s}^{+}(λ)) = 3, dim(E_{u}^{−}(λ)) = 3, hence λ ∈ Ω, which is the domain of the Evans function.

4. In the case of fronts in the domain lying between P_{1}^{+} and P_{3}^{+} we have dim(E_{s}^{+}(λ)) = 2 (see Figure 2b), dim(E_{u}^{−}(λ)) = 6, hence conditions of Corollary 2 1. are fulfilled, thus any point λ of this domain is an eigenvalue, i.e. this domain is filled with eigenvalues. (We can find other domains where conditions of Corollary 2 1. are fulfilled.) In the case of pulses P_{i}^{+} = P_{i}^{−} for i = 1, 2, 3, hence from Figure 2a and 2c we can see that dim(E_{s}^{+}(λ)) + dim(E_{u}^{−}(λ)) = 6 holds for any λ ∈ ℂ except on the parabolas. Hence according to statements 3. and 4. of Corollary 2 any λ which is not on the parabolas is a regular value or an isolated eigenvalue.

We can decide whether there are eigenvalues with positive real part by computing the image of a half circle centred at the origin and lying in the right half plane under the Evans function D. If the image winds around the origin, then by the argument principle there is (at least one) zero of D in the half circle. Choosing a sufficiently large half circle all the eigenvalues with positive real part are inside the half circle, because an estimate can be derived for the eigenvalues with positive real part. In [19] we computed the Evans function numerically and showed that Hopf bifurcation occurs for some value of L_{A} between 3 and 4. (The bifurcation value was determined more exactly.) For L_{A} = 3 the image of the half circle does not wind around the origin, hence there is no zero of D in the half circle. This value is below the Hopf bifurcation value. For L_{A} = 4 the image of the half circle winds twice around the origin, hence there are two zeros of D in the half circle. This value is above the Hopf bifurcation value. This shows that the Hopf bifurcation value of L_{A} is between 3 and 4.

5.3 The generalized KdV equation

Let us consider the travelling wave solutions of the problem

∂_{τ} u + ∂_{x}f(u) + ∂_{x}^{3}u = 0  (38)

where f is a twice differentiable convex function with f(0) = 0 = f′(0) and f(u)/u increasing. The motivating example is f(u) = u^{p+1}/(p + 1). This equation has a solitary (travelling) wave solution u(τ, x) = U(x − cτ) for any c > 0, satisfying the boundary conditions U(z) → 0 as |z| → ∞, see e.g. [17]. (This solution can be given explicitly if f(u) = u^{p+1}/(p + 1)).

The operator determining the stability of a travelling wave solution U takes the form

LV = V′′′ − cV′ + (f′(U)V)′  (39)

The first order system corresponding to the third order equation LV − λV = W can be written in the form (5), where x = (V, V′, V′′)^{T}, y = (0, 0, W)^{T} and

A_{λ}(t) = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ λ − ġ(t) & c − g(t) & 0 \end{matrix} )

Here we used the notation g(t) = f′(U(t)).

Now A_{λ}^{±} are the same matrices

A_{λ}^{±} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ λ & c & 0 \end{matrix} )

The eigenvalues of A_{λ}^{±} are determined by the characteristic polynomial

μ^{3} − μc − λ = 0.

Substituting μ = iω into the characteristic equation we obtain Reλ = 0, therefore the set of those λ values for which n_{c}^{±}(λ) ≥ 1 is the imaginary axis.

Since A_{λ}^{+} = A_{λ}^{−}, we have E_{s}^{+}(λ) = E_{s}^{−}(λ) and E_{u}^{+}(λ) = E_{u}^{−}(λ). Hence in the case n_{c}^{±}(λ) = 0 we have dim(E_{s}^{+}(λ)) + dim(E_{u}^{−}(λ)) = 3. In fact, as an elementary computation shows, for Reλ > 0 we have dim(E_{s}^{+}(λ)) = 2, dim(E_{u}^{−}(λ)) = 1. For Reλ < 0 we have dim(E_{s}^{+}(λ)) = 1, dim(E_{u}^{−}(λ)) = 2. Now applying Corollary 2 and Corollary 3 we have the following results concerning the spectrum of L.

Proposition 5 1. The essential spectrum of L is the imaginary axis.

2. If λ is not purely imaginary, then it is either a regular value or an isolated eigenvalue.

3. The Evans function can be defined either in the open right half complex plane or in the open left half plane.

In [17] a method is developped for the investigation of the behaviour of the Evans function on the positive half of the real line. The essence of the method is to compute the derivatives of the Evans function at zero and its limit at infinity. It is shown in [17] that D(λ) → 1 as λ → +∞ (and λ is real). It is also shown that D(0) = 0 = D′(0) and that D′′(0) < 0 if p > 4 and f(u) = u^{p+}/(p + 1). Hence D(λ) < 0 for small values of λ, therefore D has a positive real root, that is the solitary wave solution is unstable when p > 4.

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