Pierre Comon

Fellow, IEEE Lek-Heng Lim

Sparse Representations and Low-Rank Tensor Approximation

Keywords: Blind identification, under-determined linear mixtures, blind source separation, polyadic tensor decompositions, tensors, tensor rank, localization, best rank-r approximations, sparse representations, spark, compressed sensing, Kruskal's rank, coherence, multiarrays, multisensors

Approximating a tensor by another of lower rank is in general an ill posed problem. Yet, this kind of approximation is mandatory in the presence of measurement errors or noise. We show how tools recently developed in compressed sensing can be used to solve this problem. More precisely, a minimal angle between the columns of loading matrices allows to restore both existence and uniqueness of the best low rank approximation. We then show how these results can be applied to perform jointly localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors, in an entirely deterministic manner. The main interest in deterministic approaches is that they can be followed in the presence of strong channel nonstationarities.

I. INTRODUCTION

Tensor decomposition and approximation models arise naturally in multiarray multisensor signal processing, as already demonstrated in [START_REF] Pesquet | Cumulant based independence measures for linear mixtures[END_REF], [START_REF] Fernandes | Blind channel identification algorithms based on the Parafac decomposition of cumulant tensors: the single and multiuser cases[END_REF], [START_REF] Comon | Contrasts, independent component analysis, and blind deconvolution[END_REF], [START_REF] Cardoso | Asymptotic performance analysis of direction-finding algorithms based on fourth-order cumulants[END_REF], [START_REF] Shamsunder | Modeling of non-Gaussian array data using cumulants: DOA estimation of more sources with less sensors[END_REF] when high-order statistics are used, and in [START_REF] Sidiropoulos | Parallel factor analysis in sensor array processing[END_REF] when sensor arrays enjoy particular geometrical properties. However, the fact that approximating a tensor by another of lower rank is generally an ill-posed problem has not * Pierre Comon is with Lab. Informatique Signaux et Systèmes de Sophia-Antipolis (I3S), UMR6070 CNRS-UNS, 2000 route des Lucioles, BP.121, F06903 Sophia Antipolis cedex, France, and with INRIA, Galaad, 2004 route des Lucioles, BP.93, F06902 Sophia Antipolis cedex, France. ‡ Lek-Heng Lim is with the Department of Statistics, University of Chicago. February 21, 2011 DRAFT been taken into account in the latter works. This explains why numerical algorithms sometimes do not converge to the expected solution, and that they often converge quite slowly.

We explain in Section II why the problem is ill-posed, and what remedies have already been proposed to face it. Then we see in Section III how contributions borrowed from compressed sensing can be used to address the problem in a more convenient manner. The term compressed sensing should be understood in a broad sense, encompassing not only the ideas covered in [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF], [START_REF] Candès | Sparsity and incoherence in compressive sampling[END_REF], [START_REF] Cohen | Compressed sensing and best k-term approximation[END_REF], [START_REF] Donoho | Compressed sensing[END_REF], [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization[END_REF], [START_REF] Gribonval | Sparse representations in unions of bases[END_REF] but also in [START_REF] Candès | Exact matrix completion via convex optimization[END_REF], [START_REF] Candès | The power of convex relaxation: Near-optimal matrix completion[END_REF], [START_REF] Fazel | A rank minimization heuristic with application to minimum order system approximation[END_REF], [START_REF] Gross | Recovering low-rank matrices from few coefficients in any basis[END_REF]. Then, the usefulness of the proposed approach is demonstrated in Section IV, where several applications are pointed out, with an emphasis on the problem of joint localization and estimation of radiating sources with short data lengths, which can be solved deterministically.

II. PROBLEM POSITION AND FIRST REMEDIES

A tensor of order D is an object defined on a product of D linear spaces, S When a change of basis is operated in each linear space S d , defined by a matrix A (d) , the array of coordinates T must be modified accordingly into an array T ′ . We shall be concerned by the so-called contravariant tensors, which enjoy the multilinearity property below:

T ′ ij..k = i A (1) ip j A (2) jq • • • k A (D)
kr T pq..r which we shall denote compactly as

T ′ = (A (1) , A (2) , . . . , A (D) ) • T (1)

A. Canonical Polyadic decomposition

A tensor E of order D is said to be decomposable if it can be written as the tensor product of D vectors:

E = u (1) ⊗ ⊗ ⊗ u (2) ⊗ ⊗ ⊗ . . . ⊗ ⊗ ⊗ u (D)
February 21, 2011 DRAFT

In other words, its array of coordinates can be written as

E ij..k = u (1) i u (2) j .. u (1)
k . Any tensor T can always be decomposed in a sum of decomposable tensors [START_REF] Hitchcock | Multiple invariants and generalized rank of a p-way matrix or tensor[END_REF]:

T = R r=1 λ r E(r), (2)
E(r) = u (1) r ⊗ ⊗ ⊗ u (2) r ⊗ ⊗ ⊗ . . . ⊗ ⊗ ⊗ u (D) r
In addition, λ r can be imposed to be real nonnegative, and vectors u

(d) r
can be imposed to be of unit norm, for some suitably chosen norm.

Definition 1. The minimal number of decomposable terms necessary to meet the exact fit in equation ( 2) is referred to as the rank of tensor T:

rank{T} = min R T = R r=1 λ r E(r) (3)
In particular, decomposable tensors have a rank equal to one. When minimal, decomposition

(2) reveals the rank and is often called the Canonical Polyadic (CP) decomposition of T. Other terminologies have been used in various communities, including CanDecomp [START_REF] Carroll | Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition[END_REF] and Parafac [START_REF] Harshman | Foundations of the Parafac procedure: Models and conditions for an explanatory multimodal factor analysis[END_REF] in Psychometrics. The Linear Algebra community has taken the habit to use the acronym CP, standing for CanDecomp/Parafac, which fortunately coincides with the former.

Note that the CP decomposition can be written in compact form as T = (U (1) , U (2) , . . . ,

U (D) ) • Λ (4)
if Λ denotes the D-way diagonal array, whose sole nonzero entries are Λ rr..r = λ r , and matrices d) are each built with the R = rank{T} column vectors u

U (
(d) r , 1 ≤ d ≤ D, 1 ≤ r ≤ R.
At this stage, it is convenient to make a comparison with matrix decompositions, which are better known. Let a matrix M of rank R > 1. Then it can be decomposed in infinitely many ways into a sum of rank-1 terms as In [START_REF] Fernandes | Blind channel identification algorithms based on the Parafac decomposition of cumulant tensors: the single and multiuser cases[END_REF][START_REF] Comon | Contrasts, independent component analysis, and blind deconvolution[END_REF][START_REF] Cardoso | Asymptotic performance analysis of direction-finding algorithms based on fourth-order cumulants[END_REF], no orthogonality constraint is imposed. However, a sufficient condition for uniqueness has still been derived when D > 2 as now pointed out. We first need a definition Definition 2. The Kruskal's rank of a matrix M, or krank{M} in brief, is the maximal number κ such that any subset of κ columns of M are linearly independent.

M = R r=1 λ r u r v T
From this definition, originally introduced in [START_REF]Three-way arrays: Rank and uniqueness of trilinear decompositions[END_REF], it is clear that rank{M} ≥ krank{M}. Also note that the notion of spark introduced in compressed sensing [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization[END_REF], [START_REF] Gribonval | Sparse representations in unions of bases[END_REF] is related to Kruskal's rank, since spark{M} = krank{M}+1. Let's now turn to the following result, generally referred to as Kruskal's lemma [START_REF]Three-way arrays: Rank and uniqueness of trilinear decompositions[END_REF], [START_REF] Jiang | Kruskal's permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models[END_REF], [START_REF] Stegeman | On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition[END_REF]:

Lemma 3.
Let T be a tensor of order D. Then its CP decomposition ( 4) is unique if

2 rank{T} + 2 ≤ D d=1 krank{U (d) }. (5)
This condition has been proved to be sufficient but not proved to be necessary. We insist that uniqueness is here to be understood up to a scale factor. More precisely, there still remains a whole equivalence class of CP decompositions in the sense that one can replace each U (d) by d) , where matrices ∆ (d) are diagonal invertible, and satisfy the constraint D d=1 ∆ (d) = I R , the R × R identity matrix.

U (d) ∆ (

B. Ill-posedness of the best low-rank approximation

The best rank-R approximate is defined by the minimum of the objective Υ(U (1) r , ..,

U (D) r , Λ) = T -(U (1) , . . . , U (D) ) • Λ 2 F ( 6
)
where • F denotes the Frobenius norm defined by

T 2 F = ij..k |T ij..k | 2 , and matrices U (d) r
have R columns, R < rank{T}. It turns out that this best approximate may not exist for tensors of order D > 2. In fact, the set of tensors of rank at most β is not closed if β > 1, except when the latter set is the whole space, i.e. when β is maximal (but there is then no approximation).

This lack of closeness is now well known, and examples have been provided in the literature [START_REF] Silva | Tensor rank and the ill-posedness of the best low-rank approximation problem[END_REF], [START_REF] Comon | Tensor decompositions, alternating least squares and other tales[END_REF], which suffice to prove it.

T(n) = n (a + 1 n b) ⊗ ⊗ ⊗ 4 -a ⊗ ⊗ ⊗ 4
As n tends to infinity, this sequence converges towards

T ∞ = a ⊗ ⊗ ⊗ 3 ⊗ ⊗ ⊗ b + a ⊗ ⊗ ⊗ 2 ⊗ ⊗ ⊗ b ⊗ ⊗ ⊗ a + a ⊗ ⊗ ⊗ b ⊗ ⊗ ⊗ a ⊗ ⊗ ⊗ 2 + b ⊗ ⊗ ⊗ a ⊗ ⊗ ⊗ 3
which may be shown to be of rank 4. This demonstrates that the set of tensors of rank at most 2 is not closed.

The limit of tensors of rank R is said to be of border rank R. In general, the actual tensor rank is larger than the border rank, as in the above example. However, there are cases where they always coincide. In particular, this is the case of real tensors with nonnegative entries, if they are decomposed into a sum of real nonnegative decomposable tensors [START_REF] Lim | Nonnegative approximations of nonnegative tensors[END_REF]. But in the present framework, they differ, so that the lower rank approximation problem is ill-posed.

C. Searching a compact set via constrained optimization

The most natural way to face this problem is to change the set we are searching into a compact set. This can be done in several ways. In [START_REF] Comon | Independent Component Analysis[END_REF], loading marices U (d) are imposed to be orthogonal.

This solution is acceptable only in very restrictive conditions; in particular the rank of T must be smaller than its dimensions. In [START_REF] Kolda | Orthogonal tensor decomposition[END_REF], it has been proposed to impose orthogonality between the decomposable tensors; this constraint is less restrictive, but quite difficult to impose and still too restrictive. The first available practical technique was that proposed in [START_REF] Paatero | A weighted non-negative least squares algorithm for three-way Parafac factor analysis[END_REF], consisting of minimizing the objective Υ(U (1) , . . . ,

U (D) ) + β d U (d) 2
F , where β is an arbitrarily chosen regularization parameter. It can be seen that this is equivalent to constrain matrices U (d) to lie on a sphere D d=1 U (d) 2 F = ρ, ρ being determined a posteriori from β. The drawback of this efficient constraint is that ρ and β are arbitrary, and that they generally have no physical meaning.

We posed the problem slightly differently in [START_REF] Sidiropoulos | Parallel factor analysis in sensor array processing[END_REF], where each column vector is imposed to have a unit norm, which permits to define scale coefficients λ r properly. In other words, we minimize the Lagrangian Υ(U (1) , . . . , U (D) , Λ)

+ D d=1 R r=1 β d,r u (d) r 2 2
February 21, 2011 DRAFT where β d,r denote Lagrange multipliers, and • 2 the L 2 norm. However, contrary to [START_REF] Paatero | A weighted non-negative least squares algorithm for three-way Parafac factor analysis[END_REF], the set we are searching is not compact anymore, because of the presence of unbounded variables λ r (the entries of Λ) in the objective Υ. The goal of the next section is to define physically meaningful constraints, which will ensure the existence of a unique minimum, even if Λ is unbounded.

III. ANGULAR CONSTRAINT

A. Existence

The goal is to prevent the phenomenon we observed in Example 4 to occur, by imposing natural and weak constraints; we do not want to reduce the search to a compact set. It is clear that the objective (6) is not coercive, which explains why the minimum may not exist. But with an additional condition on the coherence, we shall be able to prove existence thanks to coercivity.

Definition 5.

Let H be a Hilbert space provided with scalar product • , and let V = {v 1 , .., v R } be a finite collection of vectors of unit norm. The coherence of the collection V is defined as

µ(V) def = max p =q | v p , v q |.
This notion has received different names in the literature: muual incoherence of two dictionaries [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization[END_REF], mutual coherence of two dictionaries [START_REF] Candès | Sparsity and incoherence in compressive sampling[END_REF], the coherence of a subspace projection [START_REF] Candès | The power of convex relaxation: Near-optimal matrix completion[END_REF], etc. The version here follows that of [START_REF] Gribonval | Sparse representations in unions of bases[END_REF]. We are interested in the case when H is finite dimensional, namely C N . Usually, dictionaries are finite or countable, but we have here a continuum of atoms. Clearly, 0 ≤ µ(V) ≤ 1, and µ(V) = 0 iff v 1 , . . . , v R are orthonormal, and µ(V) = 1 iff V contains at least a pair of collinear vectors.

The following shows that a solution to the bounded coherence best rank-R approxiamtion problem always exists: Proposition 6. Le T be a tensor of order D and dimensions N d , 1 ≤ d ≤ D, and define the sets of dictionaries of unit vectors of coherence not larger than µ d : Proof: Since all norms are equivalent on a finite dimensional space, we may assume the Frobenius norm • F . We have the following inequalities (U (1) , . . . ,

U (d) = {U (d) ∈ C | µ(U (d) ) ≤ µ d } (7)
If D d=1 µ d < 1 R , then η = inf T -(U (1) , . . . , U (D) ) • Λ λ ∈ C R , U (d) ∈ U (d) (8)
U (D) ) • Λ = R p,q=1 λ p λq D d=1 u (d) p , u (d) q ≥ R p=1 λ p λp D d=1 ||u (d) p || 2 - R p =q λ p λq D d=1 u (d) p , u (d) q ≥ R p=1 |λ p | 2 - D d=1 µ d p =q |λ p λq | ≥ ||λ|| 2 2 - D d=1 µ d ||λ|| 2 1 Now, use the fact that ||λ|| 2 1 ≤ √ R ||λ|| 2 2
for any vector λ of size R, to get eventually

(U (1) , . . . , U (D) ) • Λ ≥ 1 -R D d=1 µ d ||λ|| 2 2 (9)
Since by assumption R D d=1 µ d < 1, it is clear that the left hand side of (9) tends to infinity as ||λ|| 2 → ∞. And because ||T|| is fixed, T -(U (1) , . . . , U (D) ) • Λ also tends to infinity. This proves coercivity, and hence the proposition.

B. Uniqueness

In order to prove uniqueness, we shall call for Kruskal's lemma. For that purpose, the following lemma is needed.

Lemma 7. Let H be a Hilbert space and let

V = {v 1 , .., v R } be a finite collection of vectors of unit norm. Then krank{V} ≥ 1 µ(V) (10)
Proof: Let s = krank{V} + 1, the spark of µ(V). Then there exists a s-uple of distinct unit

vectors in V, {v 1 , . . . , v s } such that α 1 v 1 +• • •+α s v s = 0 with |α 1 | = max{|α 1 |, . . . , |α s |} > 0.
Taking inner product with v 1 we get We now characterize the uniqueness of the CP decomposition in terms of coherence introduced in Definition 5.

α 1 = -α 2 v 2 , v 1 -• • • -α s v s , v 1 and so |α 1 | ≤ (|α 2 | + • • • + |α s |)µ(V ). Dividing by |α 1 | then yields 1 ≤ (s -1)µ(V).

Proposition 9. Let T be a tensor of order D and a decomposition

T = R r=1 λ r E(r) into R decomposable tensors E(r) = u (1) r ⊗ ⊗ ⊗ u (2) r ⊗ ⊗ ⊗ . . . ⊗ ⊗ ⊗ u (D)
r , where u

1 2 D d=1 1 µ(U (d) r ) ≥ R + 1 (11)
then R = rank{T} and the decomposition is unique up to unimodulus scaling.

Proof: If inequality ( 11) is satisfied, then so is Kruskal's condition [START_REF] Shamsunder | Modeling of non-Gaussian array data using cumulants: DOA estimation of more sources with less sensors[END_REF] thanks to Lemma 7.

The results hence directly follows from Lemma 3.

Note that unlike the k-ranks in ( 5), the coherences in [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization[END_REF] are trivial to compute. In addition to uniqueness, an easy but important consequence of Proposition 9 is that it provides a readily checkable sufficient condition for tensor rank, which is NP-hard over any field [START_REF] Håstad | Tensor rank is np-complete[END_REF], [START_REF] Hillar | Most tensor problems are NP-hard[END_REF].

C. Existence and uniqueness

Now the following existence and uniqueness sufficient condition can be deduced from Propositions 6 and 9.

Corollary 10.

If D ≤ 5 and if coherences µ(U (d) ) satisfy D d=1 µ(U (d) ) 1/D ≤ D 2R + 2 (12)
then the bounded coherence best rank-R approximation problem has a unique solution up to unimodulus scaling.

Proof: The existence in the case R = 1 is ensured, because the set of tensors of rank 1 is closed (it is in fact a determinantal variety). Consider thus the case R ≥ 2. Since the function d) ) is smaller than 1/R, which permits to claim that the solution exists by calling for Proposition 6.

f (x) = 1 x -( D 2x+2 ) D is
Next in order to prove uniqueness, we use the inequality between harmonic and geometric means: if ( 12) is verified, then we also necessarily have

D [ D d=1 µ(U (d) ) -1 ] -1 ≤ D 2R+2 . Hence D d=1 µ(U (d)
) -1 ≥ 2R + 2 and we can apply Proposition 9.

IV. APPLICATIONS

The goal of this section is two-fold. First we want to show the usefulness of the CP decomposition in real world problems, amd second we want to know the meaning of the coherence conditions in terms of physical quantities.

A. Joint channel and source estimation

Consider a narrow band transmission problem in the far field. We assume here that we are in the context of wireless telecommunications, but the same principle could apply in other fields.

Let P signals impinge on an array, so that their mixture is recorded. It is wished to recover the original signals, and to estimate their directions of arrival and respective powers at the receiver.

If the channel is specular, some of these signals can correspond to different propagation paths of the same radiating source, and are hence correlated. In other words, P does not denote the number of sources, but the total number of distinct paths viewed from the receiver.

In the present framework, we assume that channels can be time-varying, but that they can be assumed constant over a sufficiently short observation length. The goal is hence to be able to work with extremely short samples.

In order to face this challenge, we assume that the sensor array is structured, as in [START_REF] Sidiropoulos | Parallel factor analysis in sensor array processing[END_REF]. More precisely, the sensor array is composed of a reference array containing I sensors, whose location is defined by a vector b i ∈ R 3 , and J -1 other subarrays, deduced from the reference array by a translation in space defined by a vector ∆ j ∈ R 3 , 1 < j ≤ J. The reference subarray is numbered with j = 1 in the remainder.

Under these assumptions, the signal received at discrete time k on the ith sensor of the reference subarray can be written as:

s i,1 (k) = P p=1 σ p (k) exp(ψ i,p ) February 21, 2011 DRAFT with ψ i,p =  ω C (b T i d p )
where the dotless  denotes √ -1, vector d p is unit norm and denotes the direction of arrival of the pth path. Next, on the jth subarray, j > 1, we have

s i,j (k) = P p=1 σ p (k) exp(ψ i,j,p ) (13)
with

ψ i,j,p =  ω C (b T i d p + ∆ T j d p ).
If we let ∆ 1 be the null vector, then (13) also applies for the reference subarray. The interest of this structure is that variables i and j decouple in function exp(ψ i,j,p ), yielding a relation resembling the CP decomposition:

s i,j (k) = P p=1 λ p U (1) ip U (2)
jp U

(3) kp [START_REF] Candès | The power of convex relaxation: Near-optimal matrix completion[END_REF] where

U (1) ip = exp( ω C b T i d p ), U (2)
jp = exp( ω C ∆ T j d p ) and U (3)
kp = σ p (k)/||σ p ||, λ p = ||σ p ||.
Hence, by computing the CP decomposition of the I × J × K tensor S = s i,j (k) , it is possible to jointly estimate: (i) signal waveforms σ p (k), and (ii) the directions of arrival d p of each propagation path if b i or ∆ j are known.

However, the observation model ( 13) is not realistic, and an additional error term should be added in order to stand for modeling inaccuracies and background noise. It is customary (and realistic thanks to the central limit theorem) to assume that this additive error has a continuous probability distribution, so that tensor S has a generic rank. Yet, the generic rank takes values at least as large as ⌈IJK/(I + J + K -2)⌉, which is always larger than Kruskal's bound [START_REF] Comon | Tensor decompositions, alternating least squares and other tales[END_REF]. Therefore, we have to face the problem of approximating tensor S by another of rank P . And we have seen that the angular constraint imposed in Section III permits to deal with a well-posed problem. In order to see the physical meaning of this constraint, it is convenient to define first the tensor product between subarrays.

B. Towards the concept of tensor product between sensor subarrays

The sensor arrays we cope with are structured, in the sense that the whole array is generated by one subarray, defined by the collection of vector locations {b i ∈ R 3 , 1 ≤ i ≤ I}, and a collection of translations in space, {∆ j ∈ R 3 , 1 ≤ j ≤ J}. If we define vectors then this means that we may see all measurements as the superimposition of decomposable tensors:

u (1) p = exp( ω C b T i d p ) I i=1 / √ I u (2) p = exp( ω C ∆ T j d p ) J j=1 / √ J (15) u (3)
λ p u (1) p ⊗ ⊗ ⊗ u (2) p ⊗ ⊗ ⊗ u (3) p
The geometry of the sensor array is contained in u

(1)

p ⊗ ⊗ ⊗ u (2)
p , whereas λ p and u

p contain energy and time information on each path p, respectively. Note that the reference subarray and the set of translations play symmetric roles, in the sense that u

= ⊗ ⊗ ⊗ = ⊗ ⊗ ⊗
Another decomposition may be obtained as

= ⊗ ⊗ ⊗ = ⊗ ⊗ ⊗ In fact, = ⊗ ⊗ ⊗ and = ⊗ ⊗ ⊗
. However, it is important to stress that the various decompositions of the whole array into tensor products of subarrays are not equivalent from the point of view of performance. In particular, the Kruskal's bound can be different, as will be pointed out next. Let now have a look at the maximal number of sources P max that can be extracted from a tensor of size I × J × K. A sufficient condition is that the total number of paths, P , is smaller than Kruskal's bound [START_REF] Shamsunder | Modeling of non-Gaussian array data using cumulants: DOA estimation of more sources with less sensors[END_REF]. We shall simplify the bound by making two assumptions: (a) the loading matrices are generic, that is, they are full rank, and (b) the number of paths is larger than the sizes I and J of the two subarrays entering the array tensor product, and smaller than the number of time samples, K. Under these simplifying assumptions, Kruskal's bound becomes 2P ≤ I + J + P -2, or:

P max = I + J -2 (16)
The table below illustrates the fact that the choice of subarrays has an impact on this bound.

Array Subarray I J P max product

⊗ ⊗ ⊗ 3 2 3 ⊗ ⊗ ⊗ 4 2 4 ⊗ ⊗ ⊗ 2 3 3 ⊗ ⊗ ⊗ 3 3 4 ⊗ ⊗ ⊗ 6 2 6 ⊗ ⊗ ⊗ 4 4 6

C. Signification of the angular constraint

We are now in a position to interpret the meaning of angular constraints proposed in Section III. According to the notations given in [START_REF] Fazel | A rank minimization heuristic with application to minimum order system approximation[END_REF], the first coherence 1) q | corresponds to the angular separation viewed from the reference subarray. In fact, vectors b i and d p having a unit norm, as well as vectors u p , the quantity |u H p u q | may be seen as a measure of angular separation between d p and d q , as we shall now subsequently show in Proposition 15.

µ (1) = max p =q |u (1)H p u (

Definition 14. We shall say that a collection of vectors {b

i } 1≤i≤I is resolvent w.r.t. direction b k -b k0 if 0 < b k -b k0 < λ 2 (17)
where λ = 2πC ω denotes the wavelength.

Let b i , d p and u q be defined as in [START_REF] Fazel | A rank minimization heuristic with application to minimum order system approximation[END_REF], 1 ≤ i ≤ I, 1 ≤ p, q ≤ P . Proof: Assume |u H p u q | = 1. Then because they are unit norm, vectors u p and u q are collinear with a unit modulus proportionality factor. Hence from [START_REF] Fazel | A rank minimization heuristic with application to minimum order system approximation[END_REF], for all i, k, 1 ≤ i, k ≤ I, (b i -b k ) T (d p -d q ) ∈ λZ, where λ is defined in Definition 17. Since {b i } is resolvant, there exist (i, i0) such that 0 < b ib i0 < λ/2. Hence, because vectors d p are unit norm, d pd q ≤ 2 so that we necessarily have that (b ib i0 ) T (d pd q ) = 0. Vector (d pd q ) is consequently orthogonal to (b ib i0 ). The same reasoning can be carried out with two other independent vectors. Eventually, vector (d p -d q ) is null because it is orthogonal to three linearly independent vectors in R 3 . The converse is immediate, by the definition of u q .

Note that the condition of Definition 17 is not very restrictive, since sensor arrays usually contain sensors separated by half a wavelength or less.

From Section IV-B, one can claim that a similar interpretation can be put forward for the second coherence, which measures the minimal angular separation between paths, viewed from the subarray defining translations.

The third coherence is nothing else but the maximal correlation coefficient between signals received from various paths on the array:

µ (3) = max p =q |σ H p σ q | ||σ p || • ||σ q ||
As a conclusion, the tensor approximation exists and is unique if either signals propagating through various paths are not too much correlated, or if their direction of arrival are not too close.

By "not too" it should be understood that the product of coherencies need to satisfy inequality [START_REF] Gribonval | Sparse representations in unions of bases[END_REF] of Corollary 10. In other words, one can separate paths with high correlation provided they are sufficiently well separated in space.

Hence, the decomposition of an array into a tensor product of two (or more) subarrays should not only take into account Kruskal's bound, as elaborated in Section IV-B, but also the ability of the latter subarrays to separate two distinct directions of arrival (cf. Proposition 15).

  , 1 ≤ d ≤ D. Such a tensor may represent a map from S 1 ⊗ • • • ⊗ S γ onto S γ+1 ⊗ • • • ⊗ S D , for some γ, 0 < γ < D. If γ = D, one has a D-linear form defined on ⊗ D d=1 S d .Once the basis of each linear space S i is fixed, such a tensor can be represented by a D-way array of coordinates, T = T ij..k . The background field is assumed to be R or C.

r

  where U = {u r } and V = {v r } are collections of R unit-norm vectors. The Singular Value decomposition (SVD) of M yields one such decomposition, where vectors of U (resp. V) are orthogonal to each other.February 21, 2011 DRAFT

February 21, 2011 DRAFT Example 4 .

 4 Let's for instance consider two non collinear vectors, a and b, and define the sequence of rank-2 tensors:

February 21 ,

 21 2011 DRAFT is attained, where || • || denotes any norm on ⊗ D d=1 C N d . Above, vector λ contains the entries λ d of the diagonal tensor Λ.

February 21, 2011 DRAFT Definition 8 . 1 ,

 81 We shall say that the CP decomposition (4) is unique up to unimodulus scaling if each vector u (d) r can be multiplied by a scalar factor α (d) r of unit modulus, such that D d=1 α ∀r, 1 ≤ r ≤ R.

  strictly positive for x ≥ 2 and D ≤ 5, condition (12) implies that February 21, 2011 DRAFT D d=1 µ(U (

p=

  σ p /||σ p || February 21, 2011 DRAFT

p

  could be interchanged without changing the whole array. This will become clear with a few examples. When we are given a structured sensor array, there can be several ways of splitting it into a tensor product of two (or more) subarrays, as now shown by simple examples. Example 11. Define the matrix of sensor locations [b 1 , b 2 , b 3 ] = This subarray is depicted in Figure 1.b. By translating it according to the translation defined in Figure 1.c one obtains another subarray. The union of the two subarrays yields the array of Figure 1.a. The same array is obtained by interchanging the roles of the two subarrays, i.e. three subarrays of two sensors deduced from each other by two translations.

Fig. 1 .Example 12 .

 112 Fig. 1. Antenna array (a) is obtained as the tensor product between subarrays (b) and (c)

Fig. 2 .Example 13 .

 213 Fig. 2. Antenna array (a) is obtained as the tensor product between subarrays (b) and (c)

Proposition 15 .

 15 If {b i } 1≤i≤I is resolvent w.r.t. three linearly independent directions, then|u H p u q | = 1 ⇔ d p = d q(18)February 21, 2011 DRAFT

February 21, 2011 DRAFT


Stéphane Labbé
email: stephane.labbe@imag.fr

Yannick Privat
email: yannick.privat@bretagne.ens-cachan.fr

Emmanuel Trélat
email: emmanuel.trelat@upmc.fr

Stability properties of steady-states for a network of ferromagnetic nanowires

Keywords: Landau-Lifshitz equation, steady-states, elliptic functions, spectral theory, stability

We investigate the problem of describing the possible stationary configurations of the magnetic moment in a network of ferromagnetic nanowires with length L connected by semiconductor devices, or equivalently, of its possible L-periodic stationary configurations in an infinite nanowire. The dynamical model that we use is based on the one-dimensional Landau-Lifshitz equation of micromagnetism. We compute all L-periodic steady-states of that system, define an associated energy functional, and these steady-states share a quantification property in the sense that their energy can only take some precise discrete values. Then, based on a precise spectral study of the linearized system, we investigate the stability properties of the steady-states.

Introduction

Ferromagnetic materials are nowadays in the heart of innovating technological applications. A concrete example of current use concerns magnetic storage for hard disks, magnetic memories MRAMs or mobile phones. In particular, the ferromagnetic nanowires are objects that establish themselves in the domain of nanoelectronics and in the conception of the memories of the future. Indeed, the storage of magnetic bits all along nanowires seems to be a promising option not only in terms of footprint but also in terms of speed access to the informations (see [START_REF] Parkin | Magnetic domain-wall racetrack memory[END_REF][START_REF] Parkin | Magnetically engineered spintronic sensors and memory[END_REF]). The conception of three dimensional memories based on the use of spin injection permits to hope access millions times shorter than the one observed nowadays in hard disks. In view of such potential application issues to rapid magnetic recording, it is of interest to be able to describes all possible stationary configurations of the magnetic moment and to investigate their natural stability properties; this is also a first step towards potential control issues, where the control may be for instance 1 an external magnetic field, or an electric current crossing the magnetic domain, in order to act on the configuration of the magnetic moment.

The most common model used to describe the static behavior of ferromagnetic materials was introduced by W.-F. Brown in the 60's (see [START_REF] Brown | Micromagnetics[END_REF]). From this point of view, the equilibrium states of the magnetization are seen as the minimizers of a given functional energy, consisting of several components. When we consider a ferromagnetic material occupying a domain Ω ⊂ R 3 , characterized by the presence of a spontaneous magnetization m almost everywhere, of norm 1 in Ω, the associated energy E(m) takes the form (see [START_REF] Hubert | Magnetic domains: the analysis of magnetic microstructures[END_REF])

E(m) = A Ω |∇m| 2 dx - Ω H a • m dx + 1 2 R 3 |H d (m)| 2 dx, (1)
and other relevant terms can be added for a more accurate physical model (e.g. anisotropic behavior of the crystal composing the ferromagnetic material) but these terms already explain a wide variety of phenomena. The first term is usually called "exchange term", and A > 0 is the exchange constant. The second term is the external energy, resulting from the possible presence of an external magnetic field H a and the last term is the socalled "demagnetizing-field", which reflects the energy of the stray-field H d (m) induced by the distribution m and is obtained by solving

div(H d + m) = 0 in D ′ (R 3 ), curl(H d ) = 0 in D ′ (R 3 ), ( 2
)
where m is extended to R 3 by 0 outside Ω, and D ′ (R 3 ) denotes the space of distributions on R 3 . The dynamical aspects of micromagnetism are usually described by the Landau-Lifshitz equation introduced in the 30's in [START_REF] Landau | Electrodynamics of continuous media[END_REF], written as

∂m ∂t = -m ∧ H e (m) -m ∧ (m ∧ H e (m)), (3)
where m(t, x) is the magnetic moment of the ferromagnetic material at time t, and H e = 2A∆u + H d (u) + H a is called the effective field. The existence of global weak solutions of that equation has been studied in [START_REF] Alouges | On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness[END_REF][START_REF] Visintin | On Landau-Lifshitz equations for ferromagnetism[END_REF]. Results on strong solutions locally in time and initial data have been derived in [START_REF] Carbou | Regular solutions for Landau-Lifshitz equation in R 3[END_REF]. For more details about modelization, stability and homogenization properties, we refer the reader to [START_REF] Simone | 2-d stability of the Néel wall[END_REF][START_REF] Simone | Two-dimensional modelling of soft ferromagnetic films[END_REF][START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF][START_REF] Hubert | Magnetic domains: the analysis of magnetic microstructures[END_REF][START_REF] Ignat | Lower bound for the energy of Bloch Walls in micromagnetics[END_REF][START_REF] Joly | Global solutions to Maxwell equations in a ferromagnetic medium[END_REF][START_REF] Sanchez | Behaviour of the Landau-Lifshitz equation in a periodic thin layer[END_REF][START_REF] Santugini-Repiquet | Modelization of a split in a ferromagnetic body by an equivalent boundary condition[END_REF][START_REF] Visintin | On Landau-Lifshitz equations for ferromagnetism[END_REF] and references therein. Numerical aspects have been investigated e.g. in [START_REF] Alouges | A new finite element scheme for Landau-Lifshitz equations[END_REF][START_REF] Simone | Two-dimensional modelling of soft ferromagnetic films[END_REF][START_REF] Labbé | Fast computation for large magnetostatic systems adapted for micromagnetism[END_REF], and control issues using such models have been addresses in [START_REF] Alouges | Magnetization switching on small ferromagnetic ellipsoidal samples[END_REF][START_REF] Carbou | Control of travelling walls in a ferromagnetic nanowire[END_REF][START_REF] Carbou | Smooth control of nanowires by means of a magnetic field[END_REF] for particular magnetic domains.

Notice that, given a solution m of (3), there holds

d dt (E(m(t, •)) = - Ω H e (m(t, x)) -H e (m(t, x)), m(t, x) m(t, x) 2 dx,
and thus this energy functional is naturally nonincreasing along a solution of (3). Every steady-state of (3) must satisfy m ∧ H e (m) = 0 since both terms appearing in the righthand side of (3) are orthogonal, and as expected the set of steady-states coincides with extremal points of the energy functional [START_REF] Alouges | A new finite element scheme for Landau-Lifshitz equations[END_REF].

In this article, we consider a one-dimensional model of a ferromagnetic nanowire, for which Γ convergence arguments permit to derive the one-dimensional version of the Landau-Lifshitz equation

∂u ∂t = -u ∧ h(u) -u ∧ (u ∧ h(u)), (4)
(see [START_REF] Sanchez | Behaviour of the Landau-Lifshitz equation in a periodic thin layer[END_REF], see also [START_REF] Carbou | Stabilization of walls for nanowires of finite length[END_REF] for arguments concerning a finite length nanowire) where u(t, x) ∈ R 3 denotes the magnetization vector, for every x ∈ R and every time t (recall that it is a unit vector), and where h(u) = ∂ 2 u ∂x 2 -u 2 e 2 -u 3 e 3 (assuming without loss of generality A = 1/2). Here, (e 1 , e 2 , e 3 ) denotes the canonical basis of R 3 and the nanowire coincides with the real axis Re 1 .

Given a positive real number L, our aim is to obtain a complete description of the Lperiodic steady-states of (4) and to investigate their stability properties. The motivation of this question is double. First, the equation above, combined with L-periodic conditions on u and ∂u ∂x , is the limit model for a straightline network of ferromagnetic nanowires of length L, connected by semiconductor devices. In that case, the period L is imposed by the physical setting. Second, our study will provide a description of all possible periodic steady-states of an infinite length one-dimensional ferromagnetic nanowire, which can be seen as the limit case of L-periodic steady-states in a finite length nanowire where L is very small compared with the length of the nanowire. Note that the authors of [START_REF] Carbou | Stability for walls in ferromagnetic nanowire[END_REF] have studied particular steady-states called travelling walls for straight ferromagnetic nanowires of infinite length. In [START_REF] Carbou | Stabilization of walls for nanowires of finite length[END_REF], the stability of one particular steady-state is investigated in a finite length nanowire with Neumann boundary conditions.

The article is organized as follows. We compute all possible L-periodic steady-states of (4) in Section 2 and prove that they share an energy quantification property, in the sense that their energy can only take isolated values. The stability properties of these steady-states are investigated in details in Section 3, based on a spectral study of the linearized system. Section 4 is devoted to the proof of our main result on quantification.

Computation of all periodic steady states

In what follows, the prime stands for the derivation with respect to the space variable x, and S 2 denotes the unit sphere of R 3 centered at the origin.

Definition 1. A L-periodic steady-state of (4) is a function u ∈ C 2 (R, S 2 ) such that u ∧ h(u) = 0 on (0, L), u(0) = u(L), u ′ (0) = u ′ (L). ( 5
)
Denoting as previously (e 1 , e 2 , e 3 ) the canonical basis of R 3 , with the agreement that the nanowire coincides with the axis Re 1 , every steady-state can be written as u = u 1 e 1 + u 2 e 2 + u 3 e 3 , and (5) yields

u 1 u ′′ 3 -u ′′ 1 u 3 -u 1 u 3 = 0 on (0, L), u 2 u ′′ 3 -u 3 u ′′ 2 = 0 on (0, L), u 1 u ′′ 2 -u ′′ 1 u 2 -u 1 u 2 = 0 on (0, L), u 2 1 + u 2 2 + u 2 3 = 1 on (0, L), u(0) = u(L), u ′ (0) = u ′ (L). (6)
The integration of the second equation of (6) yields the existence of a real number α such that u 2 u ′ 3 -u ′ 2 u 3 = α on [0, L]. Moreover, since u takes its values in S 2 , we set

u 1 (x) = cos θ α (x), u 2 (x) = cos ω α (x) sin θ α (x), u 3 (x) = sin ω α (x) sin θ α (x), (7)
for every x ∈ R. Then, we infer from (6) that

2θ ′′ α sin ω α + ω ′′ α cos ω α sin(2θ α ) -(ω ′2 α + 1) sin ω α sin(2θ α ) + 4ω ′ α θ ′ α cos ω α cos 2 θ α = 0, 2θ ′′ α cos ω α -ω ′′ α sin ω α sin(2θ α ) -(ω ′2 α + 1) cos ω α sin(2θ α ) -4ω ′ α θ ′ α sin ω α cos 2 θ α = 0, ω ′ α sin 2 θ α = α, θ α (0) = θ α (L) mod 2π, θ ′ α (0) = θ ′ α (L), ω α (0) = ω α (L) mod 2π, ω ′ α (0) = ω ′ α (L). (8)
Multiplying the first equation by sin ω α , the second one by cos ω α and adding these two equalities, it follows that (θ α , ω α ) is solution of

ω ′ α sin 2 θ α = α, -θ ′′ α + 1 2 ω ′2 α + 1 sin(2θ α ) = 0, θ α (0) = θ α (L) mod 2π, θ ′ α (0) = θ ′ α (L), ω α (0) = ω α (L) mod 2π, ω ′ α (0) = ω ′ α (L). (9)
At this step, the parameter α plays a particular role. First of all, observe that, if there exists x 0 ∈ [0, L] such that sin 2 θ α (x 0 ) = 0, then there must hold α = 0. In that case, ω 0 is constant, and θ 0 satisfies the pendulum equation

θ ′′ 0 - 1 2 sin(2θ 0 ) = 0, (10)
with periodic boundary conditions

θ 0 (0) = θ 0 (L) mod 2π, θ ′ 0 (0) = θ ′ 0 (L). ( 11
)
The case α = 0 can only occur provided sin 2 θ α (x) > 0, for every x ∈ [0, L]. In that case, we infer from (9) that θ α satisfies the equation

θ ′′ α - 1 2 α 2 sin 4 θ α + 1 sin(2θ α ) = 0, (12)
with periodic boundary conditions

θ α (0) = θ α (L) mod 2π, θ ′ α (0) = θ ′ α (L). ( 13
)
Remark 1. Note that, for every solution θ α of (12), the function

x → θ ′ α (x) 2 + α 2 sin 2 θ α (x) + cos 2 θ α (x)
is constant, and we define the functional

E α (θ α ) = θ ′2 α + α 2 sin 2 θ α + cos 2 θ α . (14)
It is related to the energy defined by (1) in the following way. Let u be a steady-state, associated with (θ α , ω α ) by the formula [START_REF] Carbou | Control of travelling walls in a ferromagnetic nanowire[END_REF], where θ α and ω α are solutions of [START_REF] Carbou | Regular solutions for Landau-Lifshitz equation in R 3[END_REF]. Then the energy E(u) defined by ( 1) is given by

E(u) = 1 2 L 0 θ ′ α (x) 2 + α 2 sin 2 θ α (x) + cos 2 θ α (x) dx = L 2 E α (θ α ). ( 15
)
In Section 4, we prove the following result.

Theorem 1. The set of real numbers α for which there exists a steady-state (θ α , ω α ) consists of isolated values, and contains in particular α = 0. Furthermore, if α denotes any of these isolated values, there exists a family

(E n ) n∈N * such that E α (θ α ) ∈ {E n } n∈N * .
The proof of that result is quite long and technical, and is postponed to Section 4. Notice that, using Remark 1, the energy of any steady-state u α with α = 0 is greater than the energy of any steady-state u 0 with α = 0, that is,

E(u α ) > E(u 0 ).
This property makes steady-states with α = 0 of particular interest, and in the sequel we focus on them. We next provide a precise description of all steady-states with α = 0. In that case, θ 0 is solution of the pendulum equation [START_REF] Simone | 2-d stability of the Néel wall[END_REF], the solutions of which are well known in terms of elliptic functions (see [START_REF] Lawden | Elliptic functions and applications[END_REF]), as recalled next.

First of all, recall that, for every solution θ 0 of (10), the function x → θ ′ 0 (x) 2 +cos 2 θ 0 (x) is constant, and the value of the constant is E 0 (θ 0 ).

Recall that, given k ∈ (0, 1), k = √ 1 -k 2 and η ∈ [0, 1], the Jacobi elliptic functions cn, sn and dn are defined from their inverse functions with respect to the first variable,

cn -1 : (η, k) -→ 1 η dt (1 -t 2 )( k2 + k 2 t 2 ) sn -1 : (η, k) -→ η 0 dt (1 -t 2 )(1 -k 2 t 2 ) dn -1 : (η, k) -→ 1 η dt (1 -t 2 )(t 2 + k 2 -1) (η 1 -k 2 in that case)
and the complete integral of the first kind is defined by

K(k) = π/2 0 dθ 1 -k 2 sin 2 θ .
The functions cn and sn are periodic with period 4K(k) while dn is periodic with period 2K(k). Using these elliptic functions, solutions of ( 10) can be integrated as follows, depending on the value of the energy E 0 (θ 0 ).

If E 0 (θ 0 ) = 0, then θ 0 (x) = π 2 for every x ∈ [0, L]. If 0 < E 0 (θ 0 ) < 1, then θ ′ 0 (x) = k cn x + sn -1 1 k cos θ(0), k , k , (16)
cos θ 0 (x) = k sn x + sn -1 1 k cos θ(0), k , k , (17)
for every

x ∈ [0, L], with E 0 (θ 0 ) = k 2 . The period of θ 0 is T = 4K(k) = 4K( E 0 (θ 0 )).
This case corresponds to the closed curves of Figure 1.

If E 0 (θ 0 ) = 1, then θ ′ 0 (x) = 1/cosh x + argth -1 (cos θ(0)) , (18)
cos θ 0 (x) = tanh x + argth -1 (cos θ(0)) . (19)
This case corresponds to the separatrices (in bold) of the phase portrait drawn on Figure 1.

If E 0 (θ 0 ) > 1, then θ ′ 0 (x) = 1 k dn x k + sn -1 (cos θ(0), k) , k , (20)
cos θ 0 (x) = sn x k + sn -1 (cos θ(0), k) , k , (21)
for every

x ∈ [0, L], with E 0 (θ 0 ) = 1/k 2 . Moreover, θ 0 (x + T ) = θ 0 (x) + 2π for every x ∈ [0, L] with T = 2kK(k) = 2K(1/ E 0 (θ 0 ))/ E 0 (θ 0 )
. This case corresponds to the curves located above and under the separatrices of Figure 1. Every steady-state must moreover satisfy the boundary conditions [START_REF] Simone | Two-dimensional modelling of soft ferromagnetic films[END_REF], with the period L. These boundary conditions appear as an additional constraint to be satisfied by the solutions above, which turns into a quantification property, as explained in the next result, that makes the conclusion of Theorem 1 more precise.

Theorem 2. (Case α = 0) Set N 0 = L 2π
, where the bracket notation stands for the integer part. Then, there exists a family (E n ) 1 n N 0 of elements of (0, 1) and a countable family ( E n ) n∈N * of elements of (1, +∞) such that, for every steady-state,

• if 0 E 0 (θ 0 ) < 1, then E 0 (θ 0 ) ∈ {E 1 , . . . , E N 0 }; K1 0 1 2 3 4 K2 K1 1 2 Figure 1: Phase portrait of (10) (in the plane (θ, θ ′ )) • if E 0 (θ 0 ) > 1, then E 0 (θ 0 ) ∈ { Ẽn | n ∈ N * }.
Furthermore, there are steady states corresponding to the energy level E 0 (θ 0 ) = 1.

Remark 2. Note that, if L < 2π, there is no solution satisfying E 0 (θ 0 ) < 1.

Remark 3. Using [START_REF] Joly | Global solutions to Maxwell equations in a ferromagnetic medium[END_REF], this theorem turns into a quantification property of the physical energies of steady-states.

Proof. To take into account the boundary conditions [START_REF] Simone | Two-dimensional modelling of soft ferromagnetic films[END_REF], we have to impose that L is equal to an integer multiple of the period T of θ 0 . The expression of T using the elliptic function K has been given previously, depending on the energy E 0 (θ 0 ). Recall that K is an increasing function from [0, 1) into [π/2, +∞). The graph of the period T as a function of E 0 (θ 0 ) is given on Figure 2. The conclusion follows easily.

Remark 4. If L tends to +∞ then the steady-state tends to one of the separatrices of Figure [START_REF] Alouges | A new finite element scheme for Landau-Lifshitz equations[END_REF]. Analytically, this means that θ tends to the solution of ( 18)- [START_REF] Kong | Dependence of the nth Sturm-Liouville eigenvalue on the problem[END_REF]. This corresponds to the case of an infinite length nanowire and to the steady-state studied in [START_REF] Carbou | Stability for walls in ferromagnetic nanowire[END_REF][START_REF] Carbou | Control of travelling walls in a ferromagnetic nanowire[END_REF].

3 Stability properties of the steady-states with α = 0

In order to investigate the stability properties of the steady-states such that α = 0, we compute the linearized system around a given steady-state and study its spectral properties. In what follows, define the spaces Endowed respectively with the usual H 1 and H 2 inner product, these are Hilbertian spaces. Let M 0 be a steady-state with α = 0. The results of the previous section show that, in the spherical coordinates (θ, ω) that have been used, the component ω is constant. Clearly, the equation ( 4) is invariant with respect to rotations around the axis Re 1 . Then, up to a rotation of angle ω around the axis Re 1 , we assume that

H 1 per (0, L; R 3 ) = {u ∈ H 1 (0, L; R 3 ) | u(0) = u(L)}, H 2 per (0, L; R 3 ) = {u ∈ H 2 (0, L; R 3 ) | u(0) = u(L) and u ′ (0) = u ′ (L)}.
M 0 (x) =   cos θ(x) sin θ(x) 0   ,
where θ is solution of ( 10), [START_REF] Simone | Two-dimensional modelling of soft ferromagnetic films[END_REF] as described in Section 2. In Section 3.1, we compute the linearized system around this steady-state. The operator underlying this linearized system is a matrix of one-dimensional operators, one of which, denoted A, plays an important role.

We study in details the spectral properties of A in Section 3.2. Based on this preliminary study, we investigate in Section 3.3 the stability properties of the steady-state M 0 . Notice that the linearized system is as well invariant with respect to rotations around the axis Re 1 , and hence these results hold for every L-periodic steady-state. Finally, Section ?? is devoted to prove that the eigenvalues of the linearized system are simple except for certain discrete values of L.

Linearization of (4) around a steady-state

Let u be a solution of (4). As in [START_REF] Carbou | Stability for walls in ferromagnetic nanowire[END_REF], we complete M 0 into the mobile frame (M 0 (x), M 1 (x), M 2 ), where M 1 and M 2 are defined by

M 1 (x) =   -sin θ(x) cos θ(x) 0   , M 2 =   0 0 1   .
Considering u as a perturbation of the steady-state M 0 , since |u(t, x)| = 1 pointwisely, we decompose u : R + × R -→ S 2 ⊂ R 3 in the mobile frame as

u(t, x) = 1 -r 2 1 (t, x) -r 2 2 (t, x)M 0 (x) + r 1 (t, x)M 1 (x) + r 2 (t, x)M 2 . ( 22
)
Easy but lengthy computations show that u is solution of (4) if and only

if r = r 1 r 2 satisfies ∂r ∂t = Lr + R(x, r, r x , r xx ), (23)
where

R(x, r, r x , r xx ) = G(r)r xx + H 1 (x, r)r x + H 2 (r)(r x , r x ),
and

• L = A + Id A + E 0 (θ)Id -(A + Id) A + E 0 (θ)Id with A = ∂ 2 xx -2 cos 2 θ Id defined on the domain D(A) = H 2 per (0, L),
• G(r) is the matrix defined by

G(r) =   r 1 r 2 √ 1-|r| 2 r 2 2 √ 1-|r| 2 + 1 -|r| 2 -1 - r 2 1 √ 1-|r| 2 -1 -|r| 2 + 1 -r 1 r 2 √ 1-|r| 2   ,
• H 1 (x, r) is the matrix defined by

H 1 (x, r) = 2θ ′ (x) 1 -|r| 2 r 2 1 -|r| 2 -r 1 r 2 2 -r 2 (1 -r 2 1 ) r 2 (1 -r 2 2 ) 1 -|r| 2 r 2 + r 1 r 2 2 ,
• H 2 (r) is the quadratic form on R 2 defined by

H 2 (r)(X, X) = (1 -|r| 2 )X ⊤ X + (r ⊤ X) 2 (1 -|r|) 3/2 1 -|r| 2 r 1 + r 2 1 -|r| 2 r 2 -r 1 ,
with the estimates

G(r) = O(|r| 2 ), H 1 (r) = O(|r|), H 2 (r) = O(|r|).
It is not difficult to prove that there exists a constant C > 0 such that, if |r| 2 1 2 , then, there holds for every x ∈ R, for every (p,

q) ∈ (R 2 ) 2 , |R(x, r, p, q)| C(|r| 2 |q| + |r||p| + |r||p| 2 ).
This a priori estimate shows that R(x, r, r x , r xx ) is a remainder term in [START_REF] Parkin | Magnetic domain-wall racetrack memory[END_REF].

Spectral study of the operator

A = ∂ 2 xx -2 cos 2 θ Id
In this section, we derive spectral properties of the operator A appearing in the expression of the linearized operator L, which will be useful for the stability analysis of Section 3.3. The domain of A is H 2 per (0, L; R 3 ), but of course it is equivalent to study A on the domain

D(A) = H 2 per (0, L; R) (denoted shortly H 2 per (0, L)). Every eigenpair (λ, u) of A must satisfy u ′′ -2 cos 2 θ u = λu, u(0) = u(L), u ′ (0) = u ′ (L).
This is a particular case of Sturm-Liouville type problems with real coupled self-adjoint boundary conditions (see [START_REF] Kong | Dependence of eigenvalues of Sturm-Liouville problems on the boundary[END_REF][START_REF] Kong | Eigenvalues of regular Sturm-Liouville problems[END_REF][START_REF] Kong | Dependence of the nth Sturm-Liouville eigenvalue on the problem[END_REF]). The following result provides some spectral properties of A.

Proposition 1. The operator A, defined on D(A) = H 2 per (0, L), is selfadjoint in L 2 (0, L) and there exists a hilbertian basis (e k ) k∈N of L 2 (0, L), consisting of eigenfunctions of A, associated with real eigenvalues λ k that are at most double, with

-∞ < • • • λ k • • • λ 1 λ 0 , (24)
and λ k → -∞ as k → +∞. Moreover,

• the eigenvalue λ 0 is simple, and its associated eigenfunction e 0 vanishes 0 or 1 time on [0, L];

• the eigenfunction e k vanishes k -1 or k or k + 1 times on [0, L].

Remark 5. A simple computation shows that

A sin θ = -E 0 (θ) sin θ,

Aθ ′ = -θ ′ , A cos θ = -(1 + E 0 (θ)) cos θ.
Hence, sin θ, θ ′ and cos θ are eigenfunctions of A associated respectively with the eigenvalues -E 0 (θ), -1, -(1 + E 0 (θ)). We are not able to exhibit nor compute explicitly some other eigenelements of A.

Note that, if the steady-state under consideration satisfies E 0 (θ) > 1 (that is, the corresponding trajectory on the phase portrait of Figure 1 is outside the separatrices), then the function θ ′ does not vanish, and it follows from Proposition 1 that λ 0 = -1, that is, -1 is the largest eigenvalue of A, and e 0 = θ ′ . Indeed, according to Proposition 1, the function e 1 could vanish 0, 1 or 2 times. Nevertheless, this is not the case since the inner product between e 0 and e 1 must be zero, which indicates that e 1 vanishes at least one time.

If the steady-state under consideration satisfies E 0 (θ) < 1 (that is, the corresponding trajectory on the phase portrait of Figure 1 is inside the separatrices), then the function sin θ does not vanish, and it follows from Proposition 1 that λ 0 = -E 0 (θ), that is, -E 0 (θ) is the largest eigenvalue of A, and e 0 = sin θ.

In the particular case θ = π/2 (corresponding to E 0 (θ) = 0), one has θ ′ = 0 and cos θ = 0 and thus they are not eigenfunctions. In that case, λ 0 = 0, and e 0 = 1. By the way, all eigenvalues can be easily computed as λ k = -2kπ L 2 , and they are all double except for k = 0.

Proof. The proof follows standard arguments. However, we include it from the convenience of the reader. We first prove that the operator A is diagonalisable. Consider the ordinary differential equation with boundary conditions

-u ′′ + (2 cos 2 θ + 1)u = f, u(0) = u(L), u ′ (0) = u ′ (L). ( 25
)
This problem is equivalent to the problem of determining u ∈ H 2 per (0, L) such that b(u, v) = g(v) for every v ∈ H 1 per (0, L), where the bilinear form b and the linear form g are defined by

b(u, v) = L 0 u ′ (x)v ′ (x)dx + L 0 (2 cos 2 θ(x) + 1)u(x)v(x)dx, g(v) = L 0 f (x)v(x)dx. Moreover, it is clear that u 2 H 1 (0,L) b(u, u), |b(u, v)| 4 u H 1 (0,L) v H 1 (0,L) , |g(v)| f L 2 (0,L) v H 1 (0,L) ,
for all u, v ∈ H 1 per (0, L). This implies that b is continuous and coercive, and g is continuous. Lax-Milgram's Theorem then implies the existence of a unique weak solution in H 1 per (0, L), and it is easy to prove that this solution is strong and belongs to H 2 per (0, L), using a standard bootstrap argument. It is then possible to define the linear operator

F : L 2 (0, L) -→ L 2 (0, L) f -→ u
where u is the unique solution of [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF]. The operator F is compact. Indeed, let u = F f , for f ∈ L 2 (0, L). Then,

u 2 H 1 (0,L) b(u, u) f L 2 (0,L) u H 1 (0,L) ,
and hence u H 1 (0,L) = F f H 1 (0,L) f L 2 (0,L) . Since the imbedding of H 1 (0, L) into L 2 (0, L) is compact, it follows that the operator F is compact. For f 1 , f 2 ∈ L 2 (0, L), denoting u 1 = F f 1 and u 2 = F f 2 , one has F f 1 , f 2 L 2 (0,L) = u 1 , f 2 L 2 (0,L) = b(u 1 , u 2 ) = f 1 , F f 2 L 2 (0,L) ,
and hence, since F is bounded on L 2 (0, L), F is selfadjoint. Since F is compact and selfadjoint, it follows that the operator A is diagonalisable with real eigenvalues satisfying [START_REF] Parkin | Magnetically engineered spintronic sensors and memory[END_REF]. The eigenvalues λ k are at most double because the associated eigenfunctions are solutions of a linear ordinary differential equation of order two. There cannot be two successive equalities in [START_REF] Parkin | Magnetically engineered spintronic sensors and memory[END_REF] because the eigenproblem associated to λ n has exactly two linearly independent solutions. The assertions concerning the zero properties of the eigenfunctions follow from [START_REF] Kong | Dependence of the nth Sturm-Liouville eigenvalue on the problem[END_REF].

Stability properties of the steady-states

Consider the linear system

∂z ∂t = Lz z(t, 0) = z(t, L), z ′ (t, 0) = z ′ (t, L), (26)
obtained in Section 3.1 by linearizing the Landau-Lifshitz equation ( 4) around the steady state M 0 . As stated in Lemma 1, since (e k ) k 0 is a hilbertian basis of L 2 (0, L) whose elements are eigenfunctions of the operator A, we can write

z(t, x) = z 1 (t, x) z 2 (t, x)
for almost every (t, x) ∈ R + × (0, L), where

z i (t, x) = +∞ k=0 z i k (t)e k (x) for i = 1, 2, with z i k (t) = z i (t, •), e k L 2 (0,L) for every k ∈ N.
Then, it is easy to see that ( 26) is equivalent to the series of 2 × 2 linear systems

∂z k ∂t = L k z, z k (0) = z k (L), z ′ k (0) = z ′ k (L), for every k ∈ N, where L k = λ k + 1 λ k + E 0 (θ) -(λ k + 1) λ k + E 0 (θ)
.

Recall that a matrix is said Hurwitzian whenever all its eigenvalues have their real part lower than 0. One has the following result.

Lemma 1. For every k ∈ N, the matrix L k is Hurwitzian if and only if λ k < min(-1, -E 0 (θ)).

Proof. Set m = min(-1, -E 0 (θ)) and M = max(-1, -E 0 (θ)). The matrix L k is Hurwitzian if and only if its determinant is positive and its trace is negative, that is, if and only if (λ k + 1)(λ k + E 0 (θ)) > 0 and 2λ k + 1 + E 0 (θ) < 0. The trace condition yields λ k < m+M 2 , and the determinant condition yields λ k < m or λ k > M . The conclusion follows.

To establish spectral properties of the steady-states, we distinguish between four cases, depending on value of the energy E 0 (θ) of the steady-state under consideration.

Case E 0 (θ) = 0

In this case, there holds θ = π/2 and θ ′ = 0. Hence, A = ∂ 2 xx , and in that case all eigenvalues of A are explicitly computed as λ k = -2kπ L 2 , for k ∈ N. Unstable modes correspond to the eigenvalues λ k satisfying λ k > -1, and hence there are exactly L 2π + 1 unstable modes whenever L 2π is not integer, and L 2π whenever it is an integer. In particular, there is always at least one unstable mode, corresponding to the eigenvalue 0 and the eigenfunction 1.

3.3.2 Case E 0 (θ) ∈ (0, 1)
This case corresponds to periodic trajectories of the pendulum phase portrait (see Figure 1) that are inside the separatrices.

Lemma 2. The operator A + E 0 (θ)Id admits the factorization

A + E 0 (θ)Id = -ℓ * ℓ,
where the operator ℓ is defined by ℓ = ∂ x -θ ′ cotanθ Id on the domain D(ℓ) = H 1 per (0, L). As a consequence, the largest eigenvalue of A is λ 0 = -E 0 (θ).

Proof. First of all, note that sin θ(x) = 0 for every x ∈ [0, L]. Indeed, the identity θ ′2 (x) + cos 2 θ(x) = E 0 (θ) < 1 yields cos 2 θ(x) < 1 for every x ∈ [0, L] and hence sin θ(x) = 0. Defining ℓ as in the statement of Lemma 2, there holds

ℓ * = -∂ x -θ ′ cotan θ Id, with D(ℓ * ) = D(ℓ) = H 1 per (0, L). One has H 2 per (0, L) = D(A + E 0 (θ)Id) ⊂ D(ℓ) and ℓ(D(A + E 0 (θ)Id)) ⊂ D(ℓ * ),

and one computes

-ℓ * ℓ = -(-∂ x -θ ′ cotanθ Id) • (∂ x -θ ′ cotan θ Id) = ∂ 2 xx -θ ′′ cotanθ Id + θ ′2 sin 2 θ Id -θ ′ cotanθ ∂ x +θ ′ cotanθ ∂ x + θ ′2 cotan 2 θ Id = ∂ 2 xx + (E 0 (θ) -2 cos 2 θ)Id, since θ ′2 = E 0 (θ) -cos 2 θ.
It follows from this factorization that the operator A + E 0 (θ)Id is nonpositive, and hence, since -E 0 (θ) is an eigenvalue of A, λ 0 = -E 0 (θ).

From Lemma 1, the matrix L k is Hurwitzian if and only if λ k < -1. Then, there is always a finite number of unstable modes, corresponding to the eigenvalues λ k such that -1 < λ k -E 0 (θ). In particular, using Remark 5, e 0 = sin θ is an unstable mode associated with λ 0 = -E 0 (θ). Moreover, if L is large, then, when solving T = L/n as in the proof of Theorem 2, the steady-state may be such that the integer n may be large (note that n ∈ {1, . . . , N 0 } with N 0 = [ L 2π ]). On the phase portrait of the pendulum (Figure 1), this means that, for this situation, the corresponding trajectory turns n times around the center point θ = π/2, θ ′ = 0 on the interval [0, L], and hence θ ′ vanishes 2n times; it then follows from Proposition 1 that θ ′ is the k th eigenfunction, with k ∈ {2n -1, 2n, 2n + 1}. Therefore, in that situation, since the eigenvalue -1 is at most double, there exist at least 2n -1 and at most 2n + 1 unstable modes.

The eigenvalue -1 (associated at least with the eigenfunction θ ′ , from Remark 5), corresponds to a central manifold for the nonlinear system (4) around the steady-state M 0 .

All other eigenvalues λ k , such that λ k < -1, correspond to stable modes (in infinite number).

Notice that, since n N 0 , for every L-periodic steady-state such that E 0 (θ) ∈ (0, 1), there are at most 2[ L 2π ] + 1 unstable modes.

3.3.3 Case E 0 (θ) = 1
In this case, there must hold either θ = θ ′ = 0, or θ = π and θ ′ = 0. Hence, cos θ is constant, equal to 1 or -1. Since it does not vanish, it follows from Proposition 1 and Remark 5 that λ 0 = -2. Actually, in that case, one has A = ∂ xx -2Id, and all eigenvalues can be easily computed. The corresponding steady-state is M 0 = (1, 0, 0) T , or M 0 = (-1, 0, 0) T (the resulting magnetic field is constant, tangent to the nanowire). It is locally asymptotically stable for the system (4).

3.3.4 Case E 0 (θ) > 1
This case corresponds to periodic trajectories of the pendulum phase portrait (see Figure 1) that are outside the separatrices. Note that, in that case, the factorization of Lemma 2 does not hold. This is due to the fact that sin θ vanishes.

From Lemma 1, the matrix L k is Hurwitzian if and only if λ k < -E 0 (θ). The situation is similar to the case E 0 (θ) ∈ (0, 1), except that the roles of -1 and -E 0 (θ) are exchanged. More precisely, there is always a finite number of unstable modes, corresponding to the eigenvalues λ k such that -E 0 (θ) < λ k -1. In particular, using Remark 5, e 0 = θ ′ is an unstable mode associated with λ 0 = -1. Moreover, as previously, when solving T = L/n as in the proof of Theorem 2, the steady-state may be such that the integer n may be large (and contrarily to the case E 0 (θ) ∈ (0, 1), there exist steady-states such that n is arbitrarily large). This means that, for this situation, sin θ vanishes a 2n times; it then follows from Proposition 1 that sin θ is the k th eigenfunction, with k ∈ {2n -1, 2n, 2n + 1}. Therefore, in that situation, since -E 0 (θ) is at most double, there exist at least 2n -1 and at most 2n + 1 unstable modes.

Notice that, for every integer p, there exists a L-periodic steady-state for which E 0 (θ) > 1, such that the corresponding operator A admits at least p unstable modes.

Proof of Theorem 1

Consider a L-periodic steady-state in the case α = 0. Recall that

E α (θ α ) = θ ′2 α + α 2 sin 2 θ α + cos 2 θ α
is a constant, and that, since α = 0, there must hold sin θ α (x) = 0, for every x ∈ [0, L], and hence θ α (x) ∈ (pπ, (p + 1)π), for some p ∈ Z. The phase portrait of ( 12), drawn on Figure 3 is then very different of the one of the pendulum studied previously. The vertical lines θ = 0 mod π are made of singular points. The region of the phase portrait of the pendulum (Figure 1) inside the separatrices can be seen as a sort of compactification process in which both vertical lines θ = 0 and θ = π would join to form the separatrices. The trajectories that are outside the separatrices of the phase portrait of the pendulum do not exist in the case α = 0.

K1 0 1 2 3 4 K2 K1 1 2 Figure 3: Phase portrait of (12) (case α = 0)
First of all, note that if E α (θ α ) = α 2 then necessarily θ α is constant, equal to π 2 mod π; this corresponds to the singular points θ = π 2 mod π, θ = 0, of Figure 3. For any other solution there must hold necessarily E α (θ α ) > α 2 . Lemma 3. Every solution θ α of [START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF] such that E α (θ α ) > α 2 is periodic, with period

T α = 4 √ 2 √ d α K 2 E α (θ α ) -α 2 d α , (27)
where

d α = E α (θ α ) + 1 + (1 -E α (θ α )) 2 + 4α 2 .
Proof. It can be easily seen that every such solution of [START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF] such that E α (θ α ) > α 2 is periodic. We assume that θ α (x) ∈ (0, π). Denote by θ - α and θ + α the extremal values of θ α (x). They are computed by solving the equation

sin 4 θ + (E α (θ α ) -1) sin 2 θ -α 2 = 0.
This leads to

θ - α = arcsin 1 -E α (θ α ) + (E α (θ α ) -1) 2 + 4α 2 2 , θ + α = π -θ - α .
Notice that the function x → θ(x) is monotone between two such successive extremal values. Then,

T α = 2 Tα/2 0 dt = 2 θ + α θ - α dθ E α (θ α ) -cos 2 θ -α 2 sin 2 θ = 4 θ + α π/2 dθ E α (θ α ) -cos 2 θ -α 2 sin 2 θ = 4 θ + α π/2
sin θdθ

sin 4 θ + (E α (θ α ) -1) sin 2 θ -α 2 = 4 θ + α π/2
sin θdθ

cos 4 θ -(E α (θ α ) + 1) cos 2 θ + E α (θ α ) -α 2 = 4 cos θ - α 0 du u 4 -(E α (θ α ) + 1)u 2 + E α (θ α ) -α 2 . Note that (1 -E α (θ α )) 2 + 4α 2 = (1 + E α (θ α )) 2 -4(E α (θ α ) -α 2 ),
and

cos θ - α = 1 + E α (θ α ) -(1 -E α (θ α )) 2 + 4α 2 2 . (28)
Setting

δ α = 1 4 (1 -E α (θ α )) 2 + 4α 2 and β α = Eα(θα)+1 2 √
δα , one ends up with

T α = 4 √ δ α cos θ - α 0 du u 2 √ δα -Eα(θα)+1 2 √ δα 2 -1 = 4 δ 1/4 α cos θ - α δ 1/4 α 0 dw (w 2 -β α ) 2 -1 .
It is known (see [START_REF] Lawden | Elliptic functions and applications[END_REF]) that dw

(w 2 -β) 2 -1 = 1 √ β + 1 F w √ β -1 , β -1 β + 1 ,
where

F (sin φ, k) = φ 0 dθ 1 -k 2 sin 2 θ
is the uncomplete elliptic integral of the first kind. Noticing that cos θ - α = δ

1/4 α √ β α -1 and that βα-1 βα+1 = 2 √ Eα(θα)-α 2 dα
, we get

T α = 4 √ 2 √ d α F 1, 2 E α (θ α ) -α 2 d α , with d α = E α (θ α ) + 1 + (E α (θ α ) -1) 2 + 4α 2
, which is the expected result.

Remark 6. For α = 0, we recover the period obtained in the previous section for trajectories that are inside the separatrices. Indeed, taking α = 0 in ( 27) leads to

T 0 = 4 √ 2 E 0 (θ 0 ) + 1 + |E 0 (θ 0 ) -1| K 2 E 0 (θ 0 ) E 0 (θ 0 ) + 1 + |E 0 (θ 0 ) -1| , (29)
and hence

T 0 = 4K( E 0 (θ 0 )) if 0 E 0 (θ 0 ) < 1, +∞ if E 0 (θ 0 ) = 1.
The function T 0 defined by ( 29) is also defined for E 0 (θ 0 ) > 1, however it differs from the period of trajectories of the pendulum phase portrait (see previous section) that are outside the separatrices. This is not surprising, since these trajectories do not exist in the case α = 0, as explained previously.

For every η > 0, define

f 1 (η) = η + α 2 + 1 + (η + α 2 -1) 2 + 4α 2 and T α (η) = 4 √ 2 f 1 (η) K 2 √ η f 1 (η) . ( 30
)
The function T α is smooth on (0, +∞), and according to Lemma 3 the period of every solution θ α of [START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF] such that

E α (θ α ) > α 2 is T α = T α (E α (θ α ) -α 2
). Note that the function T α can be extended as a continuous function on [0, +∞), with

T α (0) = 2π √ α 2 + 1 . (31)
A lengthy computation shows that

T ′ α (η) = 4 √ 2 (f 1 (η)) 3/2 f ′ 1 (η) 2 K 2 √ η f 1 (η) + 1 √ η K ′ 2 √ η f 1 (η) -f ′ 1 (η) K 2 √ η f 1 (η) ,
for every η > 0, where

K ′ (k) = -1 k K(k) + 1 k K(k), and
K(k) = π/2 0 dθ (1 -k 2 sin 2 θ) 3/2 .
Moreover,

T ′ α (η) ∼ η→0 π(1 -2α 2 ) 2(α 2 + 1) 5/2 , (32)
and

T ′ α (η) ∼ η→+∞ - π η 3/2 . ( 33
)
A tedious but straightforward study leads to the following result, describing some monotonicity properties of that function.

Lemma 4.

• For every α ∈ (0, √

2 ), there exists η * α ∈ (0, 1) such that the function T α is increasing on (0, η * α ) and decreasing on (η * α , +∞). Moreover, T α (η * α ) → +∞ and η * α → 1 whenever α → 0. For every α √ 2 2 , the function T α is decreasing on (0, +∞). • For every α > 0, T α (η) → 0 whenever η + ∞.

The graph of the function T α is given on Figure 4 for different values of α. Every steady-state must moreover satisfy the boundary conditions [START_REF] Hubert | Magnetic domains: the analysis of magnetic microstructures[END_REF]. As in the previous section, since L is fixed, using Lemma 4, this constraint leads to a quantification property of the energy E α (θ α ). There is however one additional constraint coming from the periodicity of ω α (see first and last lines of ( 9)), that results into the constraint

α L 0 dx sin 2 θ α (x)
= 0 mod 2π.

Since α = 0, this implies the existence of a nonzero integer k α such that

α L 0 dx sin 2 θ α (x) = 2k α π. (34)
This new constraint did not exist in the case α = 0 studied in the previous section. Here, for α = 0, (34) appears as an additional constraint driving to an overdetermined system.

This will imply that such steady-states can only exist for exceptional values of L, as proved below. Indeed, assume that there exists a steady-state θ α 0 , for α 0 = 0, satisfying this additional constraint (34). It is not restrictive to assume α 0 > 0. The positive real number L must be an integer multiple of the period, hence there exists n ∈ N * such that L = nT α 0 = nT α 0 (E α 0 (θ α 0 ) -α 2 0 ). We will vary α and follow a path of solutions θ α satisfying ( 12) and ( 13), such that θ α = θ α 0 for α = α 0 , having the same period T α = T α 0 , and then use analytic arguments. We stress that, the period T α of θ α is kept constant along this homotopy procedure. The existence of such a homotopic path of solutions θ α with fixed period T α 0 = L/n, for α close to α 0 , follows from the following arguments. It suffices to find a path of initial conditions θ 0 (α) (with θ ′ 0 (α) = 0), with θ 0 (α 0 ) = θ α 0 (0), for which the corresponding period is exactly L/n. To justify this fact, denote by Υ α (θ 0 ) the period of the solution of ( 12) with θ α (0) = θ 0 and θ ′ α (0) = 0, and one has to solve the equation Υ α (θ 0 (α)) = L/n in a neighborhood of α 0 . This follows immediately from an implicit function argument, noticing that Υ α (θ 0 (α)) = T α (E α (θ 0 ) -α 2 ), provided that the function T α is strictly monotonous at this point and that θ 0 = π/2 (since then the gradient of the energy is non zero along the corresponding level set).

We argue by contradiction, and assume that α 0 is not an isolated point of the set of real numbers α such that there exists a steady-state (θ α , ω α ). According to the above arguments, there exists locally around α 0 a path of solutions θ α satisfying ( 12) and ( 13), such that θ α = θ α 0 for α = α 0 , whose period is exactly L/n, and such that the additional constraint (34) is satisfied. We distinguish between two cases.

Case 0 < α 0 < √ 2/2. In the above construction, we decrease α (at least in a neighborhood of α 0 ) and follow a path of solutions θ α satisfying ( 12) and ( 13), such that θ α = θ α 0 for α = α 0 . Using Lemma 4 and in particular the fact that the maximum T α (η * α ) tends to +∞, it is clear that it is possible to make α decrease down to 0 and to follow a path such that E α (θ α ) < 1. Moreover, combining the expression of T 0 and the formula [START_REF] Visintin | On Landau-Lifshitz equations for ferromagnetism[END_REF], it is clear that this path shares the following crucial property: there exists ε > 0 such that, for every α ∈ (0, α 0 ), there holds ε θ α (x) π -ε. This implies that there exists M > 0 such that, for every α ∈ (0, α 0 ),

L 0 dx sin 2 θ α (x) M. (35)
According to (34), and since the function α → α L 0 dx sin 2 θα(x) is analytic, this function must be constant on (0, α 0 ), which raises a contradiction with (35).

Case α 0 > √ 2/2. In the above construction, we increase α (at least in a neighborhood of α 0 ) and follow a path of solutions θ α satisfying ( 12) and ( 13), such that θ α = θ α 0 for α = α 0 . According to the properties of the function T α settled in Lemma 4, it is possible to increase α up to a value α 1 satisfying T α 1 (0) = L/n, that is, 2π

α 2 1 + 1 = L n .
On the phase portrait of Figure 3, this corresponds to tracking trajectories shrinking to the center point θ = π/2, θ ′ = 0. For this limit case, passing to the limit in (34) as α tends to α 1 leads to the additional relation Lα 1 = 2k α 1 π. It is clear that these two relations are exclusive except for some isolated values of L. It is however possible to refine our argument for these exceptional values, considering first-order expansions, as follows.

As in the previous case, it is clear that there exists ε > 0 such that ε θ α (x) π -x for every α ∈ [α 0 , α 1 ]. Furthermore, this construction implies the existence of a positive integer k such that In what follows, we are going to expand asymptotically the θ α as α → α 1 and express, at the first order, the constraint (34). Using [START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF] and ( 14), we get,

θ(x) - π 2 ∼ α→α 1 η α 1 + α 2 1 sin( 1 + α 2 1 x).
Using [START_REF] Haddar | Stability of thin layer approximation of electromagnetic waves scattering by linear and nonlinear coatings[END_REF], we get lim

α→α 1     α 1 L 0 dx cos 2 ηα 1+α 2 1 sin 1 + α 2 1 x     = 2kπ. (36)
A simple asymptotic computation of the integral term of (36) leads to α 1 L + α 1 η α 4(1 + α 2 1 ) 3/2 (X 1 -sin X 1 ) + o α→α 1 (η α ) = 2kπ, with X 1 = 2 1 + α 2 1 L. Letting α tend to α 1 yields α 1 L = 2kπ (as noticed above) and dividing then this equality by η α and letting α tend to α 1 yields α 1 4(1 + α 2 1 ) 3/2 (X 1 -sin X 1 ) = 0, which is a contradiction.

Finally, the second conclusion of Theorem 1 is a direct consequence of Lemma 4 and of the fact that any energy level E α (θ α ) must satisfy

∃n ∈ N * | T α (E α (θ α )) = L n .
The proof of Theorem 1 is complete.

Figure 2 :

 2 Figure 2: Graph of the period T in function of E 0 (θ 0 ) (case α = 0)

Figure 4 :

 4 Figure 4: Graph of T α for α ∈ {0, 0.1, 0.5, √ 2 2 , 1, 3}

0 dx sin 2

 02 neighborhood of α 0 , and by analyticity of α → αL θα(x) on the whole interval [α 0 , α 1 ]. For α < α 1 , α close to α 1 , set E α (θ α ) = α 2 + η α , with η α > 0 small. By construction, η α = o α→α 1 (α -α 1 ).