On the spectral stability of standing waves of the one-dimensional $M^5$-model

We consider the spectral stability problem for a family of standing pulse and wave front solutions to the one-dimensional version of the $M^5$-model formulated by Hillen [T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585--616], to describe the mesenchymal cell motion inside tissue. The stability analysis requires the definition of spectrum, which is divided into two disjoint sets: the point spectrum and the essential spectrum. Under this partition the eigenvalue zero belongs to the essential spectrum and not to the point spectrum. By excluding the eigenvalue zero we can bring the spectral problem into an equivalent scalar quadratic eigenvalue problem. This leads, naturally, to deduce the existence of a negative eigenvalue which also turns out to belong to the essential spectrum. Beyond this result, the scalar formulation enables us to use the integrated equation technique to establish, via energy methods, that the point spectrum is empty. Our main result is that the family of standing waves is spectrally stable. To prove it, we go back to the original scalar problem and show that the rest of the essential spectrum is a subset of the open left-half complex plane.


1.
Introduction. The ability of cancer cells to invade locally the host stroma and its eventual metastatic dissemination to distant organs has attracted considerable attention to the study of the process of cell migration. Friedl and Wolf [3] reported that mesenchymal-type movement is found in cells from connective tissue tumors. Mesenchymal motion is a class of individual cell migration that takes place inside the network of collagen fibres which form a supporting framework for tissues (the extracellular matrix, abbreviated as ECM). The interaction between cells and ECM plays an important role in the process of mesenchymal migration; during this process, directional information is provided by the orientation of the matrix fibres (a process known as contact guidance) and at the same time cells degrade these fibres through the secretion of matrix proteolytic enzymes to create a path to facilitate migration [21].
In [6], a transport model to describe the mesenchymal cell movement was put forward by Hillen. The M 5 -model introduced by the author is written in the form of two coupled integro-differential equations. The system consists of a transport equation for the changes in space and time of the moving cell population and a dynamic equation for the directional distribution of tissue fibres, in both equations cell-ECM interaction terms are included. In this model a distinction is made between undirected and directed tissue types. In undirected tissues the fibres are symmetric, so that the cells are unable to distinguish between two opposite directions of migration. In directed tissues the fibres are asymmetric and the two ends have a specific polarity.
Painter proposed in [11] an extended version of the M 5 -model for undirected tissues. Through numerical simulations in 2D, the author investigated the effect of contact guidance and the joint action of focussed proteolysis and new matrix assembly. The results obtained suggest that these processes are capable of producing stable steady nework patterns. These patterns are composed by a network of interconnected dense cell-chains sustained by highly aligned fibres parallel to the direction of cell movement, which is surrounded by zones of nearly unaligned ECM where the cell density is lower. The formation of these numerically stable networks offered Hillen et al. [7] a source of motivation to investigate analytically the steady states of the M 5 -model for the undirected case. Having constructed a suitable solution framework, Hillen and coworkers found that, in R n , cells and fibres uniformly distributed in space and directions (homogeneous tissue) constitute a steady state, in which they observed that in any steady state the cell orientation is completely given by the direction of the fibres. More interesting tissue configurations with regions of completely aligned fibres required the use of Dirac delta functions to represent fibres orientation. With the aim of designing network type steady states, Hillen et al. introduced the concepts of weak steady state and pointwise steady state. The first concept allowed them indentify the arrangements of strictly aligned matrix fibres (strictly aligned tissues) as steady states of the model. The second concept permits building patchy steady states and steady states of network type in R 2 . The former consists of homogeneous tissue disposed over disjoint open sets, which are divided by curves of finite lenght possibly closed but without intersections between them, the orientation of cells and fibres along the curves is determined by the direction of the tangent vector at each point. The latter is similar, differing in that intersections are admissible. At the end of [7] the question about the stability of the steady states is put on the table, this question also concerns if solutions can evolve to a traveling wave as t → ∞. As far as we know the question still remains without an answer.
Before the publication of [7], Wang et al. worked on the one-dimensional version of the M 5 -model [20], which corresponds to the case where the mesenchymal cell population moves within a tissue made up of totally aligned fibres. During their investigation they found that standing and traveling pulse and front solutions are admitted when the motion takes place in directed tissues, and that in undirected tissues these have no chance to exist. In this paper the spectral stability of the standing pulse and front solutions is investigated. Here we do not mean to answer the question put by Hillen et al., but we hope to make a path for future research into the asymptotic stability of the one-dimensional traveling pulse and front solutions or even of the steady states identified in [7].
The one-dimensional M 5 -model reads [6]: with (x, t) ∈ R × [0, ∞). The constants µ > 0 and κ > 0 denote the turning rate and the rate of matrix degradation, respectively. The function p ± (x, t) describes the density of cells moving to the right/left with constant speed ±s. The probability density for a cell to choose to move to the right/left is denoted by q ± (x, t).
In order to see what kind of movement patterns are used by cells in the process of tissue invasion, Wang et al. [20] sought traveling wave solutions for the system (1). They showed that q + + q − = 1 is an invariant manifold of the system (1). Using this fact they could rewrite the system (1) as a system of equations for the total cell population p = p + + p − , the population flux j = s(p + − p − ), and the probability of moving to the right q + . The equivalent model is composed of the following system By applying the traveling wave ansatz where c ≥ 0 is the wave speed, Wang et al. [20] obtained the traveling wave system: The solutions of interest are those that satisfy the boundary conditions where q + l and q + r are constans that satisfy 0 ≤ q + l , q + r ≤ 1 and q + l > q + r . These solutions correspond to pulses for the quantitiesp andj, and a wave front forq + .
Integration of the first equation in (3) together with the boundary conditions lead to the invariant of motionj = cp.
Substitution of (5) into the last two equations of system (1) reduces to Wang et al. [20] assume that c = s, since when c = s the problem (6) becomes singular with homogeneous steady solutionq + = 1, which does not satisfy the above boundary conditions. The system (6) finally takes the form The analysis of (7) performed by Wang and collaborators revealed that there exists a family of standing and traveling pulse solutions for the total cell populationp, and a family of standing and traveling front solutions for the probabilty densityq + . They noted that the system (7) has a continuum of steady states (p,q + ) = (0, θ) with 0 ≤ θ ≤ 1. They first proved that each fixed c on the segment 0 ≤ c < s, determines a constant θ * = c+s 2s with the property that the steady state (0, θ) is stable for all 0 ≤ θ < θ * , and unstable for all θ * < θ ≤ 1. Thereafter, the authors showed that for each left state q l with θ * < q + l < 1, the trajectory leaving the point (0, q + l ) finishes, as z → +∞, at some right steady state (0, q + r ) with 0 < q + r < θ * , giving rise to a nonnegative heteroclinic connection. The authors stress out the fact that neither a heteroclinic trajectory starting from (0, 1), nor a heteroclinic trajectory ending at (0, 0), can exist. For a detailed discussion we refer the reader to [20].
Here one may ask whether the model can maintain the pulses and wave fronts during cell spreading. This is a matter of orbital asymptotic stability. This issue refers to whether a solution initially close to a wave profile approach, as t → ∞, some translation thereof. In a notable study on stability of traveling waves in nonstrictly hyperbolic systems, Rottmann-Matthes [16,17] proves orbital stability due to the presence of spectral stability. In [17], the orbital stability problem is tackled by reformulating it as a partial differential algebraic equation. In general, spectral stability does not necessarily imply orbital stability, but in view of the results of Rottmann-Matthes, we believe that establishing spectral stability is a good advance towards our more ambitious goal of achieving orbital stability (see the Discussion section). As a first step in this direction, we focus on the familiy of standing pulse and wave front solutions, carrying out a careful analysis of its spectral stability. In this work we prove the following.
Theorem 1.1. The family of standing waves is spectrally stable.
In order to prove this result, we have organized the paper as follows. In Section 2 we state the main existence results of [20]. Then we prove that higher amplitude pulses travel with a slower speed than smaller amplitude pulses. We end by showing that the wave profiles converge exponentially to their end states. This last property will be of fundamental importance to the subsequent analysis. In Section 3 we formulate the spectral problem and define the resolvent set and the spectrum, as well as spectral stability of the wave profiles; the spectrum is defined to be the disjoint union of the point spectrum and the essential spectrum. We then prove that the eigenvalue zero belongs to the essential spectrum, thus remaining outside the point spectrum. The spectral problem is recast as an equivalent scalar quadratic eigenvalue problem in Section 4, where, we show that the minus value of the turning rate parameter, namely −µ, is an eigenvalue which is also an element of the essential spectrum. Section 5 is devoted to analyse in detail the spectrum of the scalar operator. The scalar spectral problem is reformulated as a integrated eigenvalue problem, and this approach helps us to show that the point spectrum of the original spectral problem is empty. The main result of this paper is stated in Section 6, this establishes that the family of standing wave profiles is spectrally stable. The proof consists in showing that the essential spectrum is a subset of the open left-half complex plane.
Notation. Throughout the paper, let L 2 denote the space of all square integrable functions on R with norm Existence and structure of standing and traveling wave profiles. We state the results obtained by Wang et al. [20], concerning the existence of standing and traveling wave solutions for the system (7).

Lemma 2.2. [20]
Given a speed c satisfying 0 ≤ c < s, the left and right equilibria (0, q + l ) and (0, q + r ) are related as Remark 1. It is worth noting that for each fixed c, the left state q + l is a free parameter whose variation generates a family of traveling wave solutions, and that, in turn, the wave speed c parametrizes a family of traveling waves for a given q + l . In [20], Wang et al. obtained an explicit expression for the (p,q + ) heteroclinic trajectory, namelyp Using this result they computed the maximal value ofp. They found that the maximump,p max , occurs atq + = θ * and is given by From (10), Wang et al. observed that p max is an increasing function of the left state q + l . This observation can be extended to the dependence of p max on the wave speed c. We have noticed that pulses of higher amplitude are slower than those of a lower amplitude. To make this idea precise, we establish the following result. Proposition 1. Let q + l be a given left state. Then the maximum value p max is a decreasing function of the wave speed.
A fundamental structural feature of the wave profiles is that the pulsep and the wave frontq + converge with an exponential rate to their end states. Lemma 2.3. Traveling wave solutionsp andq + satisfy for i = 0, 1, and some uniform C > 0.
Proof. We begin by obtaining an uncoupled differential equation forq + . Upon substituting formula (9) into the equation forq + in (7) we arrive at Computing the Taylor series of the right-hand side of this equation we find that, as and, thatq as z → −∞. Hence, from (12) we obtain as z → +∞, for some constant C 1 > 0. In addition, taking the absolute value of (12) and substituting (14), we can conclude as z → −∞, for some constant C 2 > 0. In view of this, we use (13) and (15) to get Proceeding analogously as before, it results from equation (9) that as z → +∞, and also that as z → −∞.
We substitute (14) into the absolute value of (16) to obtain as z → +∞. Similarly, from (15) and (17) we can deduce that, as z → −∞, Given thatp andq + converge exponentially fast to their stationary states, we can see that the dominant term in the right-hand side of (7) is the linear one. Thus, from (18) we infer that Finally, we let C be the upper bound of all constant terms that multiply the exponential functions.
3. Spectral problem. This section is concerned with the formulation of the spectral stability problem for the family of standing wave solutions of (2), whose existence and structure has been established by the results of Theorem 2.1, Lemma 2.2, and by formula (5). All members of this family consist of a function identically zero,j ≡ 0, and a profile (p,q + ) that satisfies system (7); for c = 0 the latter reads For a standing front with amplitude ε = q + l − q + r , 0 < ε < 1, one can infer from (8) that the connected left and right endpoints are Remark 2. If we attempt to directly solve the stationary version of system (2), it results from the boundary conditions (4) that j ≡ 0. As a consequence, any function q + solves the third stationary equation in (2). Thus the problem reduces to solving the scalar equation for a given function q + satisfying (4). Therefore, it is possible that q + is not a wave front or that it is a front that does not satisfy the second equation in (20). In this work we do not deal with these kind of solutions.
In order to investigate the spectral stability of the family of standing waves, we consider solutions to system (2) of the form Substituting (22) into (2) and neglecting nonlinear terms in the perturbations, we obtain the linearized system about the waves: Assuming that perturbations are of the form e λt p(x), e λt j(x) and e λt q + (x) with λ ∈ C, substitution yields the spectral problem We are interested in solutions of problem (23) in the space H 1 (R; C 3 ). We start our study by giving some definitions used in the stability theory [10].
Definition 3.1. Let X be a Banach space and let L : D(L) → X be a linear operator with dense domain D(L) ⊂ X . The resolvent set of L is the set of all numbers λ ∈ C such that the operator L − λI has a bounded inverse. The complement of the resolvent is called the spectrum σ(L). We say that λ ∈ σ(L) is an eigenvalue of L if L − λI has a nontrivial kernel.
Definition 3.2. Let L : D(L) → X be a linear, closed, densely defined operator. Its spectrum is divided into the point spectrum σ pt (L), which is composed of those eigenvalues λ such that L − λI is Fredholm with index zero, and the essential spectrum σ ess (L) which is the remaining part; σ ess (L) = σ(L)\σ pt (L).
Recall that a linear operator L : X → Y is said to be a Fredholm operator whenever its range R(L) is closed, and dim[ker(L)] and codim[R(L)] are both finite.
Remark 3. In general, the point spectrum does not represent the entire set of eigenvalues since some eigenvalues may belong to the essential spectrum. Indeed, we show in Lemma 3.4 immediately below that λ = 0 is an eigenvalue, but nevertheless it does not lie in the point spectrum; it belongs to the essential spectrum instead, because the operator fails to be Fredholm. The same situation occurs in combustion fronts [4], KdV solitons [14] and fronts in isothermal autocatalytic chemical reactions [22], just to mention a few.
The spectral problem (23) can be recast as where the operator L : Definition 3.3. We say that the standing wavesp,q + and the identically zero functionj are spectrally stable if Lemma 3.4. λ = 0 is an eigenvalue of L embedded in the essential spectrum with an infinite dimensional eigenspace.
Proof. Take λ = 0 in the spectral system (23); it immediately follows that j ≡ 0. Therefore, it all comes down to solve the differential equation The problem consists in finding nontrivial solutions (p, q + ) in the space H 1 (R; C 2 ). The existence of one of the desired solutions follows from the first equation in (20). If we differentiate the first equation in (20), multiply by s 2 and move all terms to the right-hand side, we can infer that (p x ,q + x ) satisfies (25). In fact, it turns out that for q + =q + x equation (25) has an infinite number of linearly independent solutions p belonging to H 1 (R; C). In other words, {(p,q + x )} is an infinite collection of linearly independent solutions in the prescribed space H 1 (R; C 2 ) provided p is a solution of (25). In order to obtain such solutions, we first show thatq + x ∈ H 1 (R; C). Thereafter we solve (25) and verify that the solutions p belong to H 1 (R; C) for given q + =q + x . Sinceq + x is continuous and decays exponentially to zero as |x| → +∞ (see Lemma 2.3), thenq + x ∈ L 2 (R; C). Differentiating the second equation in (20), we find We apply the triangle inequality to the right-hand side and take squares. We then integrate over R to obtain We want to prove that the right-hand side of this inequality is uniformly bounded. From (21) and given that q + r ≤q + ≤ q + l , we get Applying Hölder's inequality to (26) and using (27), we obtain We can conclude thatq + xx ∈ L 2 (R; C) because of continuity ofp x and Lemma 2.3. It is thus proven thatq + x ∈ H 1 (R; C). Now we solve equation (25) for the variable p. From the equation forp in (20) we havep let q + =q + x , substitution of (28) into (25) yields Solving for p we get We readily note that there is a value of C that allows us to recover the solution p =p x . Next, we show that p is an element of H 1 (R; C). To this end, we make estimates for p and p x . Thus hence the right-hand side of the above equation is bounded above by Thatp ∈ L 2 (R; C) is a consequence of Lemma 2.3. Therefore, the above estimate implies that p ∈ L 2 (R; C). Now, from (29) and the triangle inequality we have that Hölder's inequality applied to (31) and the absolute value of (28) combined with (27) give This shows that p ∈ H 1 (R; C). The infinite dimension of the eigenspace is a direct consequence of the linear independence of the solutions given by formula (30): let p 1 and p 2 be two solutions corresponding to two different values of C, say C 1 and C 2 ; it is not difficult to check that the Wronskian of these solutions is W (p 1 , p 2 )(x) = 2µ sp 2q+ x (C 1 − C 2 ). Sincē p > 0 and 0 <q + < 1 on the whole real line, we see from the equation forq + in (20) thatq + x < 0 for all x ∈ R. Then it holds that W (p 1 , p 2 )(x) = 0 for all x ∈ R. Therefore, we conclude that equation (25) has infinitely many linearly independent solutions.
To finish the proof we argue as follows. Previously, we have found that the problem L 0 p := −s 2 p x + µs 2q + − 1 p + 2pq + = 0 has an infinite number of solutions, then according to definition of Fredholm operator, this means that the operator L 0 is not Fredholm. Thus, it follows that λ = 0 is an eigenvalue that belongs to the essential spectrum; see Definition 3.2. This completes the proof. 4. The quadratic eigenvalue problem. With the purpose of characterizing the whole spectrum, we assume λ = 0 in (23). We then multiply the second equation in (23) to obtain from the first and third equations in (23), we have by substituting the equation forq + in (20) into (32), we get the quadratic eigenvalue problem Lemma 4.1. −µ is an eigenvalue of L embedded in the essential spectrum associated with the one-dimensional eigenspace spanned byp .
Proof. We begin by noting that (33) can be written in the form using (28) to substitute 2q + − 1 = sp x /µp into (34) gives us Letting λ = −µ, leads us to the equation By intregrating this equation, we have for an arbitrary constant C.
Recalling (21), we deduce from (28) thatp x /p → ∓εµ/s as x → ±∞. From this, and the requirement j, j x → 0 as x → ±∞, we infer that the left-hand side of (35) tends to 0 as x → ±∞, which implies that C = 0. We multiply (35) by 1/p to arrive at Thus, the solution is j = C 0p , for some constant C 0 . Now, substituting λ = −µ and j =p into the first and third equation in (23) we obtain that p =p x /µ and q + =q + x /µ. To obtain q + it is necessary to use the second equation of (20). We conclude therefore that −µ is an eigenvalue of (23) associated with the eigenspace spanned by (p x /µ,p,q x /µ) t .
The fact that −µ is an element of the essential spectrum is because the differential operator defined by (33) is not Fredholm when λ = −µ. For convenience, we provide this result at the end of Section 6 below. 5. The spectrum. To treat the problem (33), it is useful to introduce the parameterλ := λ 2 + µλ. Thus, the eigenvalue problem now reads so that the eigenvalues of (33) and consequently, those of (23), are given by solutions of the equation 5.1. The essential spectrum. In order to analyze the spectrum of L, we proceed as Alexander et al. [1] by rewriting (36) as the first-order system The aim of the reformulation is to use the exponential dichotomies enjoyed by (37). Hence, we introduce the definition of exponential dichotomy and Morse index given by Sandstede and Scheel in [19].
Definition 5.1. Consider the differential equation Let I = R + , R − or R, and fix λ * ∈ C. We say that (39), with λ = λ * fixed, has an exponential dichotomy on I if there exist positive constants K, k s and k u and a family of projections P (x) defined and continuous for x ∈ I such that the following is true. 1. For any fixed y ∈ I and u 0 ∈ C n , there exists a solution ϕ s (x, y)u 0 of (39) with initial value ϕ s (y, y)u 0 = P (y)u 0 for x = y, and |ϕ s (x, y)| ≤ Ke −k s |x−y| , for all x ≥ y, x, y ∈ I.
2. For any fixed y ∈ I and u 0 ∈ C n , there exists a solution ϕ u (x, y)u 0 of (39) with initial value ϕ u (y, y)u 0 = (I − P (y))u 0 for x = y, and 3. The solutions ϕ s (x, y)u 0 and ϕ u (x, y)u 0 satisfy ϕ s (x, y)u 0 ∈ R(P (x)) for all x ≥ y, x, y ∈ I, ϕ u (x, y)u 0 ∈ N(P (x)) for all x ≤ y, x, y ∈ I.
The x-independent dimension N(P (x)) is referred to as the Morse index i(λ * ) of the exponential dichotomy on I. If (39) has exponential dichotomies on R + and on R − , the associated Morse indices are denoted by i + (λ * ) and i − (λ * ), respectively.
Following the ideas of Flores and Plaza [2], and Sandstede [18], we consider the family of operators forλ ∈ C.

By Lemma 2.3, A(x,λ) → A ± (λ) exponentially fast as x → ±∞, with
The operators L −λ and T (λ) are linked by their Fredholm properties. In this sense, if one operator is Fredholm so is the other, in addition to having the same Fredholm index (see [18] and its references). In turn, T (λ) is Fredholm if and only if (37) has an exponential dichotomy on both half-lines R + = [0, ∞) and R − = (−∞, 0], [12,13]. In such case the Fredholm index is computed by Here is where the asymptotic matrices A ± (λ) come into play. An exponential dichotomy on R + exists if and only if A + (λ) is hyperbolic, in which case, the Morse index i + (λ) is equal to the dimension of the unstable eigenspace of A + (λ). Likewise, the hyperbolicity of A − (λ) determines the existence of an exponential dichotomy on R − and i − (λ) is given by the dimension of the unstable eigenspace of A + (λ) (cf. [18]).
In the light of all this, the essential spectrum of L comprises allλ for which (37) has exponential dichotomies on both R + and R − with distinct Morse indices, that is ind T (λ) = 0, and thoseλ such that (37) has no an exponential dichotomy on at least one half-line.
Note that the curvesλ ± (a) describe one single parabola. Along the parabola the limiting matrices A ± (λ) have at least one purely imaginary eigenvalue, outside, the matrices are hyperbolic. Denote Ω to be the open set in the complex plane bounded on the left by the parabola (43), and let Θ denote the complemet of the closure of Ω. We further denote by E s ± (λ) and E u ± (λ) the stable and unstable eigenspaces of A ± (λ), respectively. Proposition 2. The following statements are true.
Proof. Suppose that the matrices A ± (λ) are hyperbolic. By Theorem 3.3 in [18], this leads to the existence of exponential dichotomies for the equation (37) on R + and R − . Additionally, Theorem 3.3 tells us that the Morse indices i ± (λ) are equal to dim[E u ± (λ)]. According to Lemma 4.2 of [12], this means that Statement (i) of Proposition 2 yields that ind T (λ) = 0 for allλ ∈ Ω, which implies that σ ess (L) ⊂ C \ Ω. To show that σ ess (L) covers C \ Ω we argue as follows. From statement (ii) of Proposition 2 we obtain that ind T (λ) = 2 for allλ ∈ Θ, therefore Θ ⊂ σ ess (L). In accordance with Theorem 3.3, the lack of hyperbolicity A ± (λ) on the set (43) results in the absence of exponential dichotomies of (37). By Palmer's Theorem in [13], this entails that T (λ) is not Fredholm, thereby the parabola described by (43) is a subset of σ ess (L). Since we deduce that C \ Ω ⊂ σ ess (L), and hence that σ ess (L) = C \ Ω.
From the above lemma we observe that the point spectrum of (40) must only contain complex numbers that belong to the region Ω. Accordingly,λ = 0 is an eigenvalue that does not belong to the point spectrum, σ pt (L).
We show below that any eigenfunction of the operator T (λ) must have exponential decay. To this end, as in [10,14], we rewrite equation (37) in the following form for allλ ∈ Ω.
According to the Gap Lemma [24], if V ± j (λ) are eigenvectors of A ± (λ) associated with the eigenvalues η ± j (λ), j = 1, 2, the decay estimate (47) implies that for all α < µε s , the system (37) has a set of solutions Y ± j (x,λ), j = 1, 2, that satisfy for anyλ ∈ Ω. The importance of these relations stems from the fact that they allow us to characterize the asymptotic behaviour of the eigenfunctions of T (λ). Indeed, we have found previously that Reη ± 1 (λ) > 0 and Reη ± 2 (λ) < 0 provided thatλ ∈ Ω, since we are interested in solutions to (37) in H 1 (R; C 2 ), we observe from (48) that one can construct such solutions only if they decay exponentially to zero as |x| → +∞. We summarize this result as follows.
Proposition 3. Letλ be an element of the point spectrum of T (λ) and assume that Y(x,λ) is the associated eigenfunction. Then Y(x,λ) decays exponentially fast as |x| → +∞, satisfying where V − 1 (λ) and V + 2 (λ) are eigenvectors associated to the unstable and stable eigenvalues η − 1 (λ) and η + 2 (λ), respectively. 5.2. Integrated equation. Suppose thatλ ∈ σ pt (L) is an eigenvalue with a corresponding eigenfunction j ∈ H 2 (R; C). Let us rewrite equation (36) in the form from which we obtain Applying the technique conceived by Goodman [5], we introduce the integrated variable We integrate (49) from −∞ to x and obtaiñ then, we substitute j for w in (51) in order to obtain the integrated eigenvalue equation The significance of the above equation arises from the fact that the point spectrum of L and L is the same (see Proposition 4 below). This result will prove very useful for characterizing the point spectrum of L as the integrated eigenvalue problem (52) will provide the required information. The same approach has been carried out by Zumbrun [23] and Humpherys [9] in the context of viscous conservation laws.
The family of operators associated with (52) is given by where W = (w, w x ) t and Remark 5. We point out that A I (x,λ) has the same asymptotic limits as A(x,λ).
Thus the essential spectrum of L and L coincide, and therefore the point spectrum of L is also contained in the set Ω. We may use arguments similar to those that led to Proposition 3, to conclude that for a givenλ ∈ σ pt (L ), the corresponding eigenfunction W(x,λ) has the asymptotic behavior Proof. We begin by proving that σ pt (L) ⊂ σ pt (L ). Observe that the existence of the eigenpair (j,λ) of (36) gives rise to a solution (w,λ) of equation (52), then the problem consists in checking that w belongs to H 2 (R; C). Since j ∈ H 2 (R; C), it is clear that w x , w xx ∈ L 2 (R; C). Thus, we only need to show that w ∈ L 2 (R; C). By Plancherel's theorem, it suffices to show thatŵ ∈ L 2 (R; C). For this purpose, we differentiate (50) and take the Fourier transform to obtain ikŵ(k) =ĵ(k). So we have that The above integral may be split into three parts with a > 1. The first and the last integral converge because are both bounded above by ĵ 2 L 2 . Given thatĵ(k) is a continuous function, then to establish the convergence of the second integral we only need to show thatĵ(k)/k tends to a finite limit as k → 0. First note that On the other hand, sinceλ = 0 and, j and j x decay to zero at x = ±∞, it follows from (51) that w approaches zero as x → +∞. Which implies thatĵ(0) = 0. Hence, using L'Hospital's rule, we get The fact that j decays exponentially fast to zero as |y| → ∞ ensures the convergence of the integral. Next we show that σ pt (L ) ⊂ σ pt (L). Let w ∈ H 2 (R; C) be an eigenfunction of (52) forλ ∈ σ pt (L ). Setting j = w x , it is readily seen that Substituting w x = j and (53) into (52), and differentiating, we obtain Thus, if we show that j belongs to the desired space H 2 (R; C), then it would follow that σ pt (L ) ⊂ σ pt (L). Clearly j ∈ H 1 (R; C). Then it is only necessary to check that j xx ∈ L 2 (R; C). This is achieved by showing that w xxx ∈ L 2 (R; C). Hence, after differentiation of (52) and upon substitution of w xx from (52), we arrive at Therefore, we use the exponential decay of w and w x together with the boundedness ofq + andq + x , to conclude that w xxx ∈ L 2 (R; C).

Energy estimates.
In what follows we use energy methods [2,4,8] to prove that the point spectrum of the operator L is the empty set. Proof. We use (28) to substitute 2q + − 1 = sp x /µp into (52). This gives By multiplying (54) by the integrating factor 1/p, we obtain that w satisfies We now multiply (55) by the complex conjugate w * and integrate over R to obtain Claim. w/ √p , w x / √p → 0 exponentially as |x| → ∞.
Indeed, from Lemma 2.3 and Remark 5 we have that as x → +∞.
Recall that W = (w, w x ) t . Thus, noting that Re µ 2 s 2 + 4λ = |µ 2 s 2 + 4λ| 1 2 cos(arg(µ 2 s 2 + 4λ)/2) is positive for allλ ∈ σ pt (L ) ⊂ Ω, we obtain the exponential convergence to zero. In view of the claim, we can integrate the left-hand side by parts. We infer that This shows thatλ < 0. But this contradictsλ ∈ σ pt (L ) ⊂ Ω, because of the fact that the negative half-real line is a subset of σ ess (L ) = C \ Ω (see Lemma 5.2). Therefore, there is no point spectrum for L . 6. Spectral stability. We begin the section with our main result, previously stated in the Introduction.
Theorem 1.1. The family of standing waves is spectrally stable.
In view of Theorem 5.4, the point spectrum of L is empty, meaning that the spectrum is made up completely of essential spectrum. This being so, we must show that the rest of the essential spectrum, namely the non-zero elements, is composed of complex numbers with negative real part. In this fashion, we finish the proof of the main Theorem. For this purpose, we will show that the essential spectrum of the equivalent problem (33) is a subset of the stable half plane. As before, we may associate (33) with the family of operators where D(a) = µ 2 − 4(a 2 s 2 ∓ aµsεi).
6.1. End of the proof of Lemma 4.1. Observe that (λ, a) = (−µ, 0) is a solution of the characteristic polynomial in (57), this means that 0 is an eigenvalue of the asymptotic matrices A ± (0), or in other words, that A ± (0) are nonhyperbolic. As consequence of the Theorem 3.3 of [18] and Palmer's Theorem in [13], the operator T (0) is not Freholm, therefore −µ belongs to the essential spectrum of L.
7. Discussion. Through our investigation we have shown spectral stability for all members of the family of standing wave solutions. It was found that the spectrum of the linearized perturbation problem (24) consists of pure essential spectrum. The latter belongs to the left-half complex plane and intersects the imaginary axis only at zero. We proved that λ = 0 is an eigenvalue within the essential spectrum whose eigenspace has an infinite dimension, which means that the operator L is not Fredholm. As we mentioned in the introduction, the work of Rottmann-Matthes shows that spectral stability implies orbital stability for a large class of hyperbolic systems. In our case, we cannot apply Rottmann-Matthes's theory because one of the main assumptions in [16,17] is that there must exist a spectral gap, that is a separation between the boundary of the essential spectrum and the imaginary axis. As we have showed, this hypotesis is not satisfied here. A standard technique to circumvent this difficulty is the use of exponential weighted spaces. Nevertheless, in this case, zero remains in the boundary of the essential spectrum, because it keeps an eigenspace with infinite dimension, independently of the weight function that is used (calculations not presented here). To show orbital stability we must rely on more sophisticated techniques (see e.g. [23,24] for viscous shocks), for which spectral stability is a fundamental starting point. Another remaining open problem is the stability of traveling pulses and fronts with c > 0. In a work in preparation, we apply Evans function techniques [15] to analize the case of small wave speeds 0 < c < < 1, using perturbation arguments to show that the point spectrum at c = 0 is preserved on a neighborhood thereof; the Evans function is an analytic function whose zeroes away from the essential spectrum correspond to the point spectrum of the linearized operator. In the traveling case, the same problem of the absence of a gap between the eigenvalue zero and the boundary of the essential spectrum is present. However, the situation there is very different, as it is possible to build an appropiate weighted space for which the essential spectrum lies in the open left-half complex plane with a gap between its boundary and the imaginary axis.
There is still a long way to go before drawing conclusions about the persistence of a propagating pulse of mesenchymal cells and its respective wave front of aligned fibres. Properties like the wave speed and the amplitude of the wave front could be fundamental for the orbital stability.