Gérard Iooss
email: iooss.gerard@orange.fr

I U F Dieudonné

A ′′

A ′′′

→ +∞

Existence of orthogonal domain walls in Bénard-Rayleigh convection

Keywords: Reversible dynamical systems, Bifurcations, Heteroclinic connection, Domain walls in convection

.

Introduction

The Bénard-Rayleigh convection problem is a classical problem in fluid mechanics. It concerns the flow of a three-dimensional viscous fluid layer situated between two horizontal parallel plates and heated from below. Upon increasing the difference of temperature between the two plates, the simple conduction state looses stability at a critical value of the temperature difference corresponding to a critical value R c of the Rayleigh number. Beyond the instability threshold, a convective regime develops in which patterns are formed, such as convective rolls, hexagons, or squares [START_REF] Koschmieder | Bénard cells and Taylor vortices[END_REF]. Observed patterns are often accompanied by defects as for instance domain walls which occur between rolls with different orientations. We refer to the works [START_REF] Bodenschatz | Recent developments in Rayleigh-Bénard convection[END_REF][START_REF] Manneville | Rayleigh-Bénard convection, thirty years of experimental, theoretical, and modeling work[END_REF][START_REF] Manneville | A grain boundary in cellular structures near the onset of convection[END_REF], and the references therein, for experimental and analytical results, and detailed descriptions of these patterns and defects.

Mathematically, the governing equations are the Navier-Stokes equations coupled with an equation for the temperature, and completed by boundary conditions at the two plates. Observed patterns are then found as particular steady solutions of these equations. In [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF] and [START_REF] Haragus | Domain walls for the Bénard-Rayleigh convection problem with "rigid-free" boundary conditions[END_REF] Haragus and Iooss handled the full governing equations and proved, for various boundary conditions, the existence of symmetric domain walls in convection (however not yet observed experimentally).

The existence of orthogonal domain walls (effectively observed experimentally) has been studied formally by Manneville and Pomeau in [START_REF] Manneville | A grain boundary in cellular structures near the onset of convection[END_REF], and more recently by Buffoni et al [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF], where the authors handle the full governing equations, showing that the study leads to a small perturbation of the reduced system of amplitude equations in R 6 , the same system as the one predicted in [START_REF] Manneville | A grain boundary in cellular structures near the onset of convection[END_REF]:

A (4) = A(1 -A 2 -gB 2 )
(1)

B ′′ = ε 2 B(-1 + gA 2 + B 2 ).
By a variational argument Boris Buffoni et al [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF] prove the existence of an heteroclinic orbit, for any g > 1, and ε small enough, such that

A * (x), B * (x) > 0, (A * (x), B * (x)) → M -= (1, 0) as x → -∞ M + = (0, 1) as x → +∞ .
This orbit is expected to represent the connection between a set of convecting rolls parallel to the x direction, with a set of orthogonal rolls. Unfortunately, this type of elegant proof does not allow to prove the persistence of such heteroclinic curve under reversible perturbations of the vector field. Our purpose here is to use the analytic results of [START_REF] Iooss | Heteroclinic for a 6-dimensional reversible system related with orthogonal domain walls in convection[END_REF] for proving its persistence applied to orthogonal domain walls in Bénard-Rayleigh convection. It should be noticed that even though the present analysis looks similar to the one made in [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF] and [START_REF] Haragus | Domain walls for the Bénard-Rayleigh convection problem with "rigid-free" boundary conditions[END_REF], it really needs serious adaptation since, here we loose the symmetry of the wall defect, which plays an important role in [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF] and [START_REF] Haragus | Domain walls for the Bénard-Rayleigh convection problem with "rigid-free" boundary conditions[END_REF].

Starting from a formulation of the steady governing equations as an infinitedimensional dynamical system in which the horizontal coordinate x plays the role of evolutionary variable (spatial dynamics), and looking for solutions periodic in y, a center manifold reduction is performed, which leads to a 12dimensional reduced reversible dynamical system, reducing to 8-dimensional (R 4 × C 2 ), after restricting to solutions with reflection symmetry y → -y. A normal form up to cubic order for this reduced system is obtained in [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF], which, after some calculations and rescaling (see Appendix A.1) becomes

A (4) 0 = k -A ′′ 0 + A 0 (1 - k 2 - 4 -A 2 0 -g|B 0 | 2 ) + f , B ′′ 0 = ε 2 B 0 (-1 + gA 2 0 + |B 0 | 2 ) + g, (2)
with additional "cubic" terms of the form

f 0 = id 1 εA 0 (B 0 B 0 ′ -B 0 B ′ 0 ) + ε 2 [σ 0 k -A 3 0 + d 3 A ′′ 0 + d 4 A 2 0 A ′′ 0 + d 2 A 0 A ′2 0 + d 6 A 0 |B ′ 0 | 2 +d 7 A ′ 0 (B 0 B 0 ′ + B 0 B ′ 0 ) + d 5 A ′′ 0 |B 0 | 2 ] + id 8 ε 3 A ′′ 0 (B 0 B 0 ′ -B 0 B ′ 0 ) + O(ε 4 ), ( 3
)
g 0 = ε 3 [ic 0 B ′ 0 + ic 1 B ′ 0 |A 0 | 2 + ic 2 B ′ 0 |B 0 | 2 + ic 3 B 2 0 B 0 ′ + ic 9 B 0 A 0 A ′ 0 ] (4) +ε 4 [c 4 B ′ 0 (B 0 B 0 ′ -B 0 B ′ 0 ) + c 5 B 0 A 0 A ′′ 0 + c 6 B 0 A ′2 0 + c 7 B ′ 0 A 0 A ′ 0 ] +ε 5 [ic 8 B 0 A 0 A ′′′ 0 + ic 7 B ′ 0 A 0 A ′′ 0 + ic 10 B ′ 0 A ′2 0 + ic 11 B 0 A ′ 0 A ′′ 0 + O(ε 6 ),
where parameters are defined as (see Appendix A.1)

ε 4 ∼ R 1/2 -R 1/2 c , R Rayleigh number, k c (1 + ε 2 k -)
wave number in y direction, coefficients c j , d j are real and

f = f 0 + f 1 g = g 0 + g 1 , f 1 = ε 4 O[|X|(|X| 2 + |Y | 2 + ε 4 ) 2 ], g 1 = ε 6 O[(|X| 2 + |Y |)(|X| 2 + |Y | 2 + ε 4 ) 2 ], with X = (A 0 , A ′ 0 , A ′′ 0 , A ′′′ 0 ) t ∈ R 4 , Y = (B 0 , B ′ 0 ) t ∈ C 2 .
Moreover the system (2) commutes with the reversibility symmetry S 1 :

(A 0 , A ′ 0 , A ′′ 0 , A ′′′ 0 , B 0 , B ′ 0 ) → (A 0 , -A ′ 0 , A ′′ 0 , -A ′′′ 0 , B 0 , -B ′ 0 ), and
A (4) 0 is odd in X, B ′′ 0 is even in X,
which results from the equivariance of the original system under the shift by half of a wave length in the y direction (fixing the symmetry y → -y). The estimates for f 1 and g 1 result from the property of the normal form which does not contain terms of degree 4 in (X, Y ), and from the inequality

(a + b) 4 ≤ 4(a 2 + b 2 ) 2 for a, b ∈ R.
Remark 1 Notice that the above reduction is valid for the three classical boundary conditions for the Bénard-Rayleigh convection problem: rigid-rigid, free-free, free-rigid. However in the case of rigid-rigid or free-free boundary conditions, Y = 0 is an invariant subspace, which simplifies the estimate for g 1 .

Remark 2 Notice that the system (2) becomes just the system (1) for k -= f = g = 0, and B 0 real.

Remark 3 Notice also that the high order terms of size O(ε 4 ) for A (4) 0 and O(ε 6 ) for B ′′ 0 are functions of e ±i x 2ε . This is due to the fact that B 0 e i x 2ε is the original amplitude of the Y mode (see (59)).

Let us give here the results obtained in [START_REF] Iooss | Heteroclinic for a 6-dimensional reversible system related with orthogonal domain walls in convection[END_REF] for the system (1):

Theorem 4 Let us choose 0 < δ 0 < 1/3, then for δ 0 ≤ δ ≤ 1, η 0 such that for ε small enough with α = ε 1/3 , where 0 < α = [(1 + δ 2 )η 2 0 -1] 1/2
, the 3-dim unstable manifold of M -intersects transversally the 3-dim stable manifold of M + , except for a finite umber of values of δ. The connecting curve (A * , B * )(x) which is obtained is locally unique (it is the only curve for this intersection). Moreover its dependency in parameters (ε, δ) is analytic. In addition we have B * (x) and B ′ * (x) > 0 on (-∞, +∞), the principal part of B * (x) being given i) for x ∈ (-∞, 0], by

B * (x) = 1 (1 + δ 2 2 ) 1/2 cosh(x 0 -εδx) , cosh x 0 = 1 B 00 (1 + δ 2 2 ) 1/2 , B 00 = B * (0) = (1 -η 2 0 δ 2 ) 1/2 , ii) for x ∈ [0, +∞), by B * (x) = tanh(εx/ √ 2) + B 00 1 + B 00 tanh(εx/ √ 2) .
We intend to prove the following for the system (2):

Proposition 7 Except for a finite number of values of g = 1 + δ 2 and for ε such that Theorem 4 applies, the heteroclinic solution connecting an equilibrium at -∞ (representing convective rolls parallel to x -axis) and a periodic solution at +∞ (representing convective rolls orthogonal to the previous ones), persists as a one-parameter family of orthogonal domain walls, (see results in (55), (56)). The wave numbers of limiting periodic convective rolls are linked by a one parameter family of relationships depending on ε (the amplitude of rolls being of order ε 2 ) (see Remark 16).

Remark 8

The wave numbers of the sets of rolls at -∞ and at +∞ differ in general. This is a major difference with the symmetric case (of non orthogonal walls) treated in [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF] and [START_REF] Haragus | Domain walls for the Bénard-Rayleigh convection problem with "rigid-free" boundary conditions[END_REF].

Remark 9 In the result above, the size of the perturbed wave numbers depend on bounds proved for some coefficients, specially the term a 6 ε 5/2 in the last bifurcation equation (54). Notice that all such coefficients depend on the Prandtl number.

Remark 10 Values of δ such that 0.476 ≤ δ include values obtained for δ in the Bénard-Rayleigh convection problem where g is function of the Prandtl number P (see [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF]). With rigid-rigid, rigid-free, or free-free boundaries the minimum values of g are respectively (g min = 1.227, 1.332, 1.423) corresponding to δ min = 0.476, 0.576, 0.650. The restriction in Theorem 4 corresponds to 1 < g ≤ 2. The eligible values for the Prandtl number are respectively P > 0.5308, > 0.6222, > 0.8078. Since we leave now some freedom to the wave numbers, as well in the y direction, as in the x direction, the "end points" of the expected heteroclinic are no longer (1, 0) at -∞, and the circle (0, e iφ ) at +∞. In fact the classical study of steady convective rolls, shows that these should be respectively (A

(-∞) 0 (k -), B (-∞) 0
(k -)) and (0, B (+∞) 0

(ω, x)) (see [START_REF] Haragus | Local bifurcations, Center manifolds, and Normal forms in infinite-dimensional dynamical systems[END_REF] section 4.3.3, or [START_REF] Haragus | Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem[END_REF] sections 2 and 6.2, and Appendix A.2, A.3). We have indeed (A

(-∞) 0 ) 2 = 1 - k 2 - 4 + σ 0 ε 2 k -+ O(ε 2 |k -| 3 + ε 4 ), 1 -(A (-∞) 0 ) def = - ω 2 - 2 , ω 2 -= k 2 - 4 -σ 0 ε 2 k -+ O[k 4 -+ ε 2 |k -| 3 + ε 4 ], B (-∞) 0 = O(ε 6 ), e i x 2ε B (+∞) 0 (ω, x) = r 0 e iωx + O(ε 6 ), A (+∞) 0 = 0, ω def = 1 2ε + ε ω + = 1 + ε 2 k + 2ε + O(ε 7 ), B (+∞) 0 e -iε ω+x = C (+∞) 0 + iD (+∞) 0 r 2 0 = 1 - k 2 + 4 + O(ε 2 |k + | + ε 4 ) = 1 -O[( |ω + | + ε 2 ) 2 ], C (+∞) 0 = r 0 + O(ε 6 ), oscil. part(C (+∞) 0 ) = O(ε 6 ), D (+∞) 0 = O(ε 6 ).

Remark 11

The coefficient σ 0 introduced in the expression of (A (-∞) 0

) 2 depends on the Prandtl number.

Remark 12

We may notice that in case the system has the symmetry S 0 representing z → 1 -z (OK for rigid-rigid, or free-free boundary conditions), then B (-∞) 0 = 0, which simplifies computations.

Let us set

B 0 e -iε ω+x = C 0 + iD 0 , then

A (4) 0 = k -A ′′ 0 + A 0 [1 - k 2 - 4 -A 2 0 -g(C 2 0 + D 2 0 )] + f (2) becomes
C ′′ 0 = 2ε ω + D ′ 0 + ε 2 C 0 (-1 + ω 2 + + gA 2 0 + C 2 0 + D 2 0 ) + g r (5)
D ′′ 0 = -2ε ω + C ′ 0 + ε 2 D 0 (-1 + ω 2 + + gA 2 0 + C 2 0 + D 2 0 ) + g i with f = f , g r + ig i = ge -iε ω+x , (6)
and where the exponential factor disappears in the cubic part when we replace B 0 by (C 0 + iD 0 )e iε ω+x . Let us define

f = f 0 (ε, k -, X, Y, Y ) + f 1 (ωx, ε, k -, X, Y, Y ) g r = g r0 (ε, X, Y, Y ) + g r1 (ωx, ε, k -, X, Y, Y ) g i = g i0 (ε, X, Y, Y ) + g i1 (ωx, ε, k -, X, Y, Y ),
where f 0 , g r0 , g i0 come only from cubic terms of the normal form in (2), and where f 1 , g r1 , g i1 are 2π-periodic in ωx, smooth in their arguments, and satisfy estimates

|f 1 (ωx, ε, k -, X, Y, Y )| ≤ cε 4 |X|(|X| 2 + |Y | 2 ) 2 |g r1 (ωx, ε, k -, X, Y, Y )| + |g i1 (ωx, ε, k -, X, Y, Y )| ≤ cε 6 (|X| 2 + |Y |)(|X| 2 + |Y | 2 ) 2 , with X = (A 0 , A ′ 0 , A ′′ 0 , A ′′′ 0 ) Y = (C 0 + iD 0 , C ′ 0 + iD ′ 0 ).
Then we have from ( 3), ( 4):

f 0 = d 1 εA 0 (C 0 D ′ 0 -D 0 C ′ 0 ) + σ 0 ε 2 k -A 3 0 + d 2 ε 2 A 0 A ′2 0 + d 3 ε 2 A ′′ 0 (7) +d 4 ε 2 A 2 0 A ′′ 0 + d 5 ε 2 A ′′ 0 (C 2 0 + D 2 0 ) + d 6 ε 2 A 0 (C ′2 0 + D ′2 0 ) + +d 7 ε 2 A ′ 0 (C 0 C ′ 0 + D 0 D ′ 0 ) + d 8 ε 3 A ′′ 0 (C 0 D ′ 0 -D 0 C ′ 0 ) + O(ε 4 ), g r0 + ig i0 = iε 3 (C ′ 0 + iD ′ 0 )[c 0 + c 1 A 2 0 + c 2 (C 2 0 + D 2 0 )] ( 8
)
+ε 3 c 3 (C 0 + iD 0 )(C 0 D ′ 0 -D 0 C ′ 0 ) + iε 3 c 9 (C 0 + iD 0 )A 0 A ′ 0 +ε 4 c 4 (C ′ 0 + iD ′ 0 )(C 0 D ′ 0 -D 0 C ′ 0 ) + c 5 ε 4 A 0 A ′′ 0 (C 0 + iD 0 ) +ε 4 [c 6 A ′2 0 (C 0 + iD 0 ) + c 7 A 0 A ′ 0 (C ′ 0 + iD ′ 0 )] +iε 5 (C ′ 0 + iD ′ 0 )(c 7 A 0 A ′′ 0 + c 10 A ′2 0 ) +iε 5 (C 0 + iD 0 )(c 8 A 0 A ′′′ 0 + c 11 A ′ 0 A ′′ 0 ) + O(ε 6 ).

Now, let us set

A 0 = A * + A 0 C 0 = B * + C 0 D 0 = D 0
where we observe that we expect

A 0 → x=-∞ A (-∞) 0 -1 = - ω 2 - 2 , C 0 + iD 0 → x=-∞ C (-∞) 0 = B (-∞) 0 = O(ε 6 ), C 0 + i D 0 → x=+∞ C (+∞) 0 + iD (+∞) 0 -1 ∼ - ( ω + + O(ε 2 )) 2 2 .
Then (5,6) becomes the "perturbed system"

M g ( A 0 , C 0 ) = -k -(A ′′ * + A 0 ′′ ) + k 2 - 4 (A * + A 0 ) + φ 0 2 ω+ ε D 0 ′ + ω 2 + (B * + C 0 ) + ψ 0r , (9)
L g D 0 = - 2 ω + ε (B ′ * + C 0 ′ ) + ω 2 + D 0 + ψ 0i , (10)
where linear operators M g and L g are defined as

M g A C = -A (4) + (1 -3A 2 * -gB 2 * )A -2gA * B * C 1 ε 2 C ′′ + (1 -gA 2 * -3B 2 * )C -2gA * B * A , (11)
L g D = 1 ε 2 D ′′ + (1 -gA 2 * -B 2 * )D, (12)
and where φ 0 , ψ 0r , ψ 0i are smooth functions of (ωx, ε, k -, ω + , X, Y ) where

X = ( A 0 , A 0 ′ , A 0 ′′ , A 0 ′′′ ) Y = ( C 0 , D 0 , C 0 ′ , D 0 ′ ) φ 0 = φ 00 (ε, k -, X, Y ) + φ 01 (ωx, ε, k -, X, Y ) ψ 0r = ψ 0r0 (ε, k -, X, Y ) + ψ 0r1 (ωx, ε, k -, X, Y ) ψ 0i = ψ 0i0 (ε, k -, X, Y ) + ψ 0i1 (ωx, ε, k -, X, Y ) | φ 01 (ωx, ε, k -, X, Y )| ≤ cε 4 | ψ 0r1 (ωx, ε, k -, X, Y )| + | ψ 0i1 (ωx, ε, k -, X, Y )| ≤ cε 4 .
More precisely, we have

φ 00 (ε, k -, X, Y ) = 3A * A 0 2 + A 0 3 + 2gB * A 0 C 0 (13) +g(A * + A 0 )( C 0 2 + D 0 2 ) + f 00 , ψ 0r0 (ε, k -, X, Y ) = 2gA * A 0 C 0 + gB * A 0 2 + 2B * C 0 2 + g A 0 2 C 0 (14) +(B * + C 0 )( C 0 2 + D 0 2 ) + g 00r , ψ 0i0 (ε, k -, X, Y ) = 2gA * A 0 D 0 + 2B * C 0 D 0 + g A 0 2 D 0 + D 0 ( C 0 2 + D 0 2 ) + g 00i , (15)
and in using Theorem 4, Corollaries 5 and 6,

f 00 = σ 0 ε 2 k -A 3 * + O[ε 5/2 (e εδx χ (-∞,0) + e -δ ′ x √ 2 χ (0,∞) ) + ε 2 (| X| + | Y |) +ε| D 0 ′ |(χ (-∞,0) + e -δ ′ x √ 2 χ (0,∞) )], g 00r = O[ε 5/2 (e εδx χ (-∞,0) + e -δ ′ x √ 2 χ (0,∞) ) + ε 2 (| X| + | Y |) +ε(| C 0 ′ | + | D 0 ′ |)], g 00i = O[ε 3/2 (e εδx χ (-∞,0) + e -δ ′ x √ 2 χ (0,∞) ) + ε| X| +ε(| C 0 ′ | + | D 0 ′ | + ε| D 0 |)].
where f 00 and g 00r + ig 00i are smooth functions which come from the rest of the cubic normal form written in [START_REF] Iooss | Perturbed homoclinic solutions in reversible 1:1 resonance vector fields[END_REF][START_REF] Iooss | Heteroclinic for a 6-dimensional reversible system related with orthogonal domain walls in convection[END_REF]) and χ (-∞,0) and χ (0,∞) are the characteristic functions on the corresponding intervals.

Remark 13 We notice that the estimates for the main terms independent of X, Y come from for f 00 :

σ 0 ε 2 k -A 3 = - σ 0 ε 2 k - 4 + O(ε 5/2 ), for g 00r B ′ * dx ∼ ε 2 R c 5 A * A ′′ * B * B ′ * dx = O(ε 5/2 ), for g 00i B * dx = ε( c 0 2 + c 2 4 ) + ε(c 1 -c 9 ) R A 2 * B * B ′ * = O(ε),
where we notice

A ′′ * A ′
Before solving the system we need to change variables so that the variables and the right hand side of (9,10) tend towards 0 at infinity. Let us denote

X (-∞) = (A (-∞) 0 -1, 0, 0, 0) = (O( ω 2 -), 0, 0, 0) Y (-∞) = (C (-∞) 0
, 0, 0, 0) = (O(ε 6 ), 0, 0, 0),

X (+∞) = 0 Y (+∞) = (C (+∞) 0 -1, D (+∞) 0 , C (+∞)′ 0 , D (+∞)′ 0 ) = [O(( ω + + ε 2 ) 2 ), O(ε 6 ), O(ε 5 ), O(ε 5 )],
then, taking care, in [START_REF] Haragus | Domain walls for the Bénard-Rayleigh convection problem with "rigid-free" boundary conditions[END_REF][START_REF] Horn | Pattern formation at a fluid-ferrofluid interface[END_REF], of the forms of f , g r , g i , we notice that the limit terms in the right hand side of (9,10) as x → -∞ are

k 2 - 4 A (-∞) 0 + φ 0 (ωx, ε, k -, X (-∞) , Y (-∞)
) exp limit as e εδx (see f 00 ),

ω 2 + C (-∞) 0 + ψ 0r (ωx, ε, k -, X (-∞) , Y (-∞)
) exp limit as e εδx (see g 00r )

ψ 0i (ωx, ε, k -, X (-∞) , Y (-∞)
) exp limit as e εδx (as B ′ * and see g 00i ).

The limit terms of the right hand side of (9,10) as x → +∞ is 0 exp limit as e

- √ δ √ 2 x (as A * ) 2 ω + ε (D (+∞) 0 ) ′ + ω 2 + C (+∞) 0 + ψ 0r (ωx, ε, k -, 0, Y (+∞) ) exp limit as e - √ δ 2 x (see g 00r ), - 2 ω + ε (C (+∞) 0 ) ′ + ω 2 + D (+∞) 0 + ψ 0i (ωx, ε, k -, 0, Y (+∞) ) exp limit as e -ε √ 2x (see g 00i ).
Let us change variables as

A 0 = α -χ -+ A 0 C 0 = β -χ -+ β + χ + + C 0 , D 0 = γ + χ + + D 0 ,
with (in using Appendix A.2 and (65) in Appendix A.3)

α -= (A (-∞) 0 -1) = -ω 2 -/2, β -= B (-∞) 0 , β + = (C (+∞) 0 (ωx) -1), γ + = D (+∞) 0 (ωx), const part of β + def = β (c) + = - ω 2 + 2 + σ 1 ε 2 ω + 2 + σ 2 ε 4 2 + O[(| ω + | + ε 2 ) 4 ], (16)
and where χ -and χ + are smooth functions, such that

χ -= 1 for x ∈ (-∞, -1), = 0 for x > 0 0 < χ -< 1 for x ∈ (-1, 0), χ + = 1 for x ∈ (1, ∞), = 0 for x < 0 0 < χ + < 1 for x ∈ (0, 1), such that ( A 0 , C 0 , D 0 ) → 0 as |x| → ∞.
2.1 Properties of linear operators M g and L g

Let us define the Hilbert spaces

L 2 η = {u; u(x)e η|x| ∈ L 2 (R)}, D 0 = {(A, C) ∈ H 4 η × H 2 η ; A ∈ H 4 η , C ∈ D 1 } D 1 = {C ∈ H 2 η ; ε -2 ||C ′′ || L 2 η + ε -1 ||C ′ || L 2 η + ||C|| L 2 η def = ||C|| D1 < ∞}
equiped with natural scalar products. Then we have the following result (proved in [START_REF] Iooss | Heteroclinic for a 6-dimensional reversible system related with orthogonal domain walls in convection[END_REF]):

Lemma 14 Except maybe for a set of isolated values of g, the kernel of M g in L 2 η is one dimensional, spanned by (A ′ * , B ′ * ), and its range has codimension 1, L 2 -orthogonal to (A ′ * , B ′ * ). M g has a pseudo-inverse acting from L 2 η to D 0 for any η > 0 small enough, with bound independent of ε.

The operator L g has a trivial kernel, and its range which has codimension 1, is L 2 -orthogonal to B * (B * / ∈ L 2 ). L g has a pseudo-inverse acting respectively from L 2 η to D 1 for η > 0 small enough, with bound independent of ε.

3 Calculation and estimates for M g ( A 0 , C 0 ) and

L g D 0 3.1 First component of M g ( A 0 , C 0 )
The first component is now the sum of small terms linear in ( A 0 , C 0 ) plus quadratic terms and terms independent of ( A 0 , C 0 ) which tend exponentially to 0 as e εδx for x → -∞ and e - √ 2εx for x → +∞ :

M g ( A 0 , C 0 )| 1 = -k -A 0 ′′ + k 2 - 4 A 0 + φ 0 + ϕ 1 (k -) (17)
with

ϕ 1 (k -) = -k -(A ′′ * + α -χ ′′ -) + k 2 - 4 (A * -χ -) + α -χ (4) - ( 18
) -3(1 -A 2 * )α -χ -+ gB 2 * α -χ -+ 2gA * B * (β -χ -+ β + χ + ), φ 0 = φ 0 (ωx, ε, k -, X, Y ) -χ -φ 0 (ωx, ε, k -, X (-∞) , Y (-∞) ).
More precisely we have, from (13)

φ 0 = 3[α 2 -(A * χ 2 --χ -) + 2α -A * χ -A 0 + A * A 0 2 ] + α 3 -(χ 3 --χ -) (19) +3α 2 -χ 2 -A 0 + 3α -χ -A 0 2 + A 0 3 + 2gB * [α -χ -C 0 + (β -χ -+ β + χ + ) A 0 + A 0 C 0 ] +g(A * + α -χ -+ A 0 )[(β -χ -+ β + χ + + C 0 ) 2 + (γ + χ + + D 0 ) 2 ] -χ -g(1 + α -)β 2 -+ f 00 , f 00 = σ 0 ε 2 k -(A 3 * -χ -) + O[ε 5/2 (e εδx χ (-∞,0) + e -δ ′ x √ 2 χ (0,∞) ) + ε(| D 0 ′ | + ε| D 0 |) + ε 2 | X|].
We notice that for η = εδ/2, and due to Corollary 6,

1 ε 2 β ′ + = O(ε 3 ), 1 ε 2 γ ′ + = O(ε 3 ), ||A ′ * || L 2 η = O(1), ||B ′ * || L 2 η = O(ε 1/2 ), ||A ′2 * || L 2 η = O(ε 1/2 ), ||B ′2 * || L 2 η = O(ε 3/2 ), ||A ′′ * || L 2 η = O(1), ||B ′′ * || L 2 η = O(ε 3/2
). Then we have the estimates (we use extensively 2|ab| ≤ a 2 + b 2 )

||ϕ 1 (k -)|| L 2 η ≤ c √ ε|k -| + k 2 -+ ε 4 √ ε + ω 2 + + ε 2 | ω + |) , (20)
R ϕ 1 (k -)A ′ * dx = O[(|k -| + | ω + | + ε 2 ) 2 ],
where we use η = εδ/2 (η < εδ is necessary), integration by parts and

R A ′ * A ′′ * dx = 0, R (A * -χ -)A ′ * dx = O(1) R (1 -A 2 * )A ′ * χ -dx = O(1).
Let us use the following little Lemma (adapted from a simple Sobolev inequality)

Lemma 15 For any u ∈ H 1 η and ε sufficiently small, we have

|u(x)| ≤ c(||u|| L 2 η + 1 ε ||u ′ || L 2 η )
where c is independent of ε.

Then in the next estimates we use extensively

| A 0 (m) (x)| ≤ c ε || A 0 || H 4 η , m = 0, 1, 2, 3 | C 0 (m) (x)| ≤ cε m || C 0 || D1 , m = 0, 1.
We obtain, for sufficiently small ε, k -, ω

+ , A 0 , C 0 , D 0 in R 3 × D 0 × D 1 || φ 0 || L 2 η ≤ c ε 2 + ε 3/2 |k -| + ω 4 - √ ε + ω 4 + + 1 ε ||( A 0 , C 0 )|| 2 D0 + 1 ε 2 ||( A 0 || 3 H 4 η + || D 0 || 2 D1 . (21) 3.2 Second component of M g ( A 0 , C 0 )
For the second component we have

M g ( A 0 , C 0 )| 2 = 2 ω + ε D 0 ′ + ω 2 + C 0 + ψ 0r + ϕ 2 (k -), (22)
with

ϕ 2 (k -) = ω 2 + (B * -χ + ) - 1 ε 2 β -χ ′′ -- 2 ε 2 β ′ + χ ′ + - 1 ε 2 β + χ ′′ + + 2 ω + ε γ + χ ′ + (23) -(3 -gA 2 * -3B 2 * )β + χ + + [1 -χ --g(A 2 * -χ -)]β -χ -+ 2gA * B * α -χ -, ψ 0r = ψ 0r (ωx, ε, k -, X, Y ) -χ + ψ 0r (ωx, ε, k -, 0, Y (+∞) ) -χ -ψ 0r (ωx, ε, k -, X (-∞) , Y (-∞) ),
where γ + = D (+∞) 0

. For ψ 0r we have

ψ 0r = 2gA * (α -χ -C 0 + (β -χ -+ β + χ + ) A 0 + A 0 C 0 ) (24) +g(B * + β + χ + + C 0 )(α 2 -χ 2 -+ 2α -χ -A 0 + A 0 2 ) +gβ -χ -[(α 2 -(χ 2 --1) + 2α -χ -A 0 + A 0 2 ] +[B * (β -χ -+ β + χ + ) 2 -χ + β 2 + ] + [B * (γ + χ + ) 2 -χ + γ 2 + ] +β + χ + (χ 2 + -1)(β 2 + + γ 2 + ) + β 3 -χ -(χ 2 --1) + C 0 [(β -χ -+ β + χ + + C 0 ) 2 + (γ + χ + + D 0 ) 2 ] +2(B * + β + χ + )(β -χ -+ β + χ + C 0 + γ + χ + D 0 ) +(B * + β -χ -+ β + χ + )( C 0 2 + D 0 2 ) + g 00r , g 00r = O(ε 5/2 (e εδx χ (-∞,0) + e -δ ′ x √ 2 χ (0,∞) ) + ε 2 (| X| + | Y |) + ε(| C 0 ′ | + | D 0 ′ |).
Now we obtain for sufficiently small ε, k -, ω

+ , A 0 , C 0 , D 0 in R 3 × D 0 × D 1 || ψ 0r || L 2 η ≤ c ε 2 + ω 4 -+ ω 4 + √ ε + 1 ε ||( A 0 , C 0 )|| 2 D0 + || D 0 || 2 D1 (25) +c ( ω 2 -+ ω 2 + )||( A 0 , C 0 )|| D0 + ε 2 || D 0 || D1 , ||ϕ 2 (k -)|| L 2 η ≤ c( ω 2 - √ ε + (| ω + | + ε 2 ) 2 ε 2 ), ( 26
) R ϕ 2 (k -)B ′ * dx = O[( ω 2 -+ ω 2 + + ε 4 )],
where the last estimates use

1 ε 2 1 0 β ′ + χ ′ + B ′ * dx = O(ε 4 ) 1 ε 2 1 0 β + χ ′′ + B ′ * dx = O(| ω + | + ε 2 ) 2
obtained, for the first integral in integrating by parts, and for the second one in separating the oscillating part of order ε 6 from the constant part β (c) + of β + , for which we make an integration by parts, in using

B ′′ * = O(ε 2 B * ). More precisely we have R ϕ 1 (k -)A ′ * dx + R ϕ 2 (k -)B ′ * dx = a 2 k 2 - 4 + a 3 σ 0 ε 2 k - (27) +O(|k 3 -| + ε 2 k 2 -+ ω 2 + + ε 4 ), with a 2 = R (A * -χ -)A ′ * dx -a 3 , a 3 = 0 -1 χ (4) -A ′ * -3 R (1 -A 2 * )A ′ * χ -dx + g R (A * B 2 * ) ′ χ -dx,
We observe that (see Corollay 5)

R (A * -χ -)A ′ * dx = 1 2 + O(ε 1/2 ) 0 -1 χ (4) -A ′ * dx = O(ε 1/2 ) g 0 -∞ (A * B 2 * ) ′ χ -dx = -g 0 -1 (A * B 2 * )χ ′ -dx = O(ε 1/3 ) -3 0 -∞ (1 -A 2 * )χ -A ′ * dx = 3 0 -1 (A * - A 3 * 3 - 2 3 )χ ′ -dx = 2 + O(ε 1/3 ) ε 6 b 0 = - 1 ε 2 0 -1 β -χ ′′ -B ′ * dx. ( 28
)
so that

a 2 = -3/2 + O(ε 1/3 ), ( 29
) a 3 = 2 + O(ε 1/3 ). ( 30
) 3.3 Component L g D 0
For the third component we obtain

L g D 0 = - 2 ω + ε C 0 ′ + ω 2 + D 0 + ψ 0i + ϕ 3 (k -), ( 31
)
ϕ 3 ( ω, k -, ωx) = - 2 ω + ε [B ′ * + β -χ ′ -+ β + χ ′ + ] - 2 ε 2 γ ′ + χ ′ + - 1 ε 2 γ + χ ′′ + -(1 -gA 2 * -B 2 * )γ + χ + ,
and for

ψ 0i = ψ 0i (ωx, ε, k -, X, Y ) -χ + ψ 0i (ωx, ε, k -, 0, Y (+∞) ) -χ -ψ 0i (ωx, ε, k -, X (-∞) , Y (-∞) ),
and for sufficiently small ε, k -, ω

+ , A 0 , C 0 , D 0 in R 3 × D 0 × D 1 , the estimate || ψ 0i || L 2 η ≤ c{ε + ( ω 2 -+ ω 2 + )|| D 0 || D1 + || A 0 D 0 || L 2 η +||( C 0 D 0 )|| L 2 η + || D 0 || 2 D1 }, ( 32
)
||ϕ 3 || L 2 η ≤ c(ε 3 + | ω + | √ ε + | ω 3 + | ε ), ( 33
)

Bifurcation equation

Let us use an adapted Lyapunov-Schmidt method. Since

M g (A ′ * , B ′ * ) = 0,
we now decompose ( A 0 , C 0 , D 0 ) as

A 0 = zA ′ * + u, (34)
C 0 = zB ′ * + v, D 0 = w, then equations (17,22) give (Q 0 is the projection in L 2 on the range of M g ) M g (u, v) = Q 0 -k -(zA ′ * + u) ′′ + k 2 - 4 (zA ′ * + u) + φ 0 + ϕ 1 (k -) 2 ω+ ε w ′ + ω 2 + (zB ′ * + v) + ψ 0r + ϕ 2 (k -) . ( 35
)

Resolution in ω + and w

We observe that (u, v) and w appear non symmetrically, so, we first solve equation (31), where the kernel of L g is empty, and its range of codimension 1 (see Lemma 14). This has the advantage to give w and ω + in function of (u, v, z, k -, ε). Since

1 0 1 ε 2 γ ′ + χ ′ + B * dx = - 1 0 1 ε 2 γ + (χ ′ + B * ) ′ dx = O(ε 4 ), we obtain R ϕ 3 B * dx = - ω + ε [1 + O(| ω + | + ε 2 ) 2 ] + O(ε 4 ), R ψ 0i B * dx = O[ε 1/2 + ( ω 2 -+ ω 2 + )|| D 0 || D1 + || D 0 || 2 D1 + || A 0 D 0 || L 2 η + || C 0 D 0 || L 2 η ] = O[ε 1/2 + ||(u, v)|| 2 D0 + ||w|| 2 D1 + ( ω 2 -+ ω 2 + )||w|| D1 )].
The compatibility condition for equation (31) leads to

2 ω + ε R B ′ * B * dx = R - 2 ω + ε (zB ′′ * + v ′ ) + ω 2 + w + ψ 0i + ϕ 3 B * dx, which gives ω + = R -2 ω + (zB ′′ * + v ′ ) + ε ω 2 + w B * dx +O[ε 3/2 + | ω + |(| ω + | + ε 2 ) 2 ] +εO(||(u, v)|| 2 D0 + ||w|| 2 D1 + ( ω 2 -+ ω 2 + )||w|| D1
). which is a smooth function of its arguments and may be solved with respect to ω + (or equivalently with respect to k + since ω + = k+ 2 + O(ε 6 )) by implicit function theorem in the neighborhood of 0 for

(u, v) ∈ D 0 , w ∈ D 1 , (ε, ω -, z) ∈ R 3 , ω + = k + (ε, ω -, z, (u, v), w) ∈ C 1 (R 3 × D 0 × D 1 ).
Indeed we have the estimate

|k + | ≤ c[ε 3/2 + ε ω 2 -||w|| D1 + ε(||(u, v)|| 2 D0 + ||w|| 2 D1 )]. ( 36
)
For solving equation ( 31) we now have

w = L -1 g [- 2k + ε (zB ′′ * + v ′ ) + k 2 + w + ϕ 3 + ψ 0i ]
which may be solved by implicit function theorem with respect to w in D 1 for

(ε, ω -, z, (u, v)) ∈ R 3 × D 0 .
We obtain, using (33), ( 32), (36) and

|| B ′′ * ε || L 2 η = O(ε 1/2 ), || v ′ ε || L 2 η ≤ ||v|| D1 , then w = w(ε, ω -, z, u, v) with ||w|| D1 ≤ c(ε + ε 1/2 ||(u, v)|| 2 D0 ), (37)
and we obtain

|k + | ≤ c(ε 3/2 + ε||(u, v)|| 2 D0 ). ( 38
)

Resolution in (u, v)

Now, we replace w and ω + by their expressions w and k + , and consider (35) which may be solved by implicit function theorem with respect to (u, v) in a neighborhood of 0 in D 0 for (ε, k -, z) close to 0 in R 3 . Indeed, the right hand side of (35) is smooth in its arguments and assuming

|k -| ≤ cε 3/4 , i.e. | ω -| ≤ c ′ ε 3/4 , (39)
|z| ≤ cε, (40)
||(u, v)|| D0 ≤ cε, (41)
using (34) and collecting results of (17,20,21) for the first component, and (22,26,25) for the second component, estimates in L 2 η of the right hand side are as follows

1st comp. = O( k 2 - √ ε + ε + z 2 ε + 1 ε ||(u, v)|| 2 D0 ) 2nd comp. = O(ε+ k 2 - √ ε + z 2 ε + 1 ε ||(u, v)|| 2 D0 ). Applying implicit function theorem for ( k- ε 3/4 , z ε ) satisfying (39,40) in R 2 , leads to (u, v) = (u, v)(ε, k -, z) ∈ D 0 with ||(u, v)|| D0 ≤ c(ε+ k 2 - √ ε + ε 1/2 |k -| + z 2 ε ), (42)
which satisfies the a priori estimate (41). Now using (37), (38), (39), ( 40) and ( 42)

||(u, v)|| D0 + ||w|| D1 ≤ cε, (43)
|k + | ≤ cε 3/2 , (44)
where (39) and (40) need to be checked at the end.

Final bifurcation equation

The orthogonality in L 2 of the right hand side of M g ( A 0 , C 0 ) with (A ′ * , B ′ * ) (see Lemma 14), gives one relationship, expressed as the cancelling of a function of (z, k -, ε), from which we extract the family of bifurcating solutions. It gives

0 = R [-k -(zA ′′′ * + u ′′ ) + k 2 - 4 (zA ′ * + u)]A ′ * dx + R ( φ 0 + ϕ 1 )A ′ * dx + R [ 2 ω + ε w ′ + ω 2 + (zB ′ * + v)]B ′ * dx + R ( ψ 0r + ϕ 2 )B ′ * dx. (45)
We define

a 1 = - R A ′′′ * A ′ * dx = R A ′′2 * dx > 0, (= O(1)) (46)
and we have, from (19), ( 24), ( 41), ( 43), ( 44), (39), (40) and Remark 13

R φ 0 A ′ * dx = z 2 [a ′ 0 + O(ε 3/2 )] + σ ′ 0 ε 2 k -+ O[ε 5/2 + |z|(ε 3/2 + k 2 -)],
with

a ′ 0 = R (3A * A ′3 * + 2gB * B ′ * A ′2 * + gA * A ′ * B ′2 * )dx = O(ε 1/2 ), σ ′ 0 = σ 0 R A ′ * (A 3 * -χ -)dx = σ 0 [ 3 4 + O(ε 1/2 )], (47)
where (for example) the estimated term in ε 3/2 |z| comes from

| R zA ′2 * udx| ≤ c|z|||A ′2 * || L 2 η ||u|| L 2 η ≤ c ′ ε 1/2 |z|ε, occuring in 3 R A * A ′ * (A ′ * z + u) 2 dx.
We also obtain

R ψ 0r B ′ * dx = z 2 a ′′ 0 + O(ε 5/2 + ε 2 |z|) with a ′′ 0 = R (gB * B ′ * A ′2 * + 2gA * A ′ * B ′2 * + B * B ′3 * )dx = O(ε). Hence R φ 0 A ′ * dx + R ψ 0r B ′ * dx = z 2 [a 0 + O(ε 3/2 )] + σ ′ 0 ε 2 k -+ O(ε 5/2 + ε 3/2 |z|), (48)
where we define

a 0 = 3 R (A * A ′3 * + gB * B ′ * A ′2 * + gA * A ′ * B ′2 * + (1/3)B * B ′3 * )dx = O(ε 1/2 ). ( 49
)
Using Corollaries 5 and 6, we notice that the main contribution of this integral is on (-∞, 0) and precisely

0 -∞ 3A * A ′3 * dx = O(ε 1/2 ), +∞ 0 3A * A ′3 * dx = O(ε 2 ).
Now collecting the expressions (46), ( 27), (48) in (45) we obtain the bifurcation equation

a 0 z 2 + a 1 k -z + a ′ 2 k 2 - 4 + a ′′ 3 εk -+ a 5 ε 3/2 z + a ′ 4 ε 5/2 = O(|k -| 3 + ε 3 ), ( 50
)
where

a 0 = a 0 + O(ε 3/2 ) = O(ε 1/2 ) a 1 = O(1) (51)
a ′ 2 = a 2 + O(ε 3/2 ) = -3/2 + O(ε 1/3 ) a ′′ 3 = a ′ 3 + ε(σ ′ 0 + a 3 σ 0 ) = O(1), a ′ 4 = a 4 + O(1), a 5 = O(1), with a ′ 3 εk -= -k - R A ′ * u ′′ dx
and (see Remark 13)

-a 2 ε 2 [c 5 R A * A ′′ * B * B ′ * dx + d 4 R A 2 * A ′′ * A ′ * dx] = a 4 ε 5/2 , ( 52
)
the term of order O(1) in a ′ 4 comes from the estimate of terms of order O(ε

5/2 ) in √ ε(||u||||v|| + ||u|| 2 + ||v|| 2 )
occuring in the estimate of

R φ 0 A ′ * dx.
The discriminant of the principal part of the quadratic form in (z, k -) of the left hand side of (50

) is ∆ = a 2 1 -a 0 a ′ 2 = O(1) (53)
which it is positive. The bifurcation equation (50) may be written as

a ′ 2 k - 2 + a 1 z + a ′′ 3 ε 2 -∆ z + a 1 a ′′ 3 ε ∆ - a ′ 2 a 5 ε 3/2 2∆ 2 + a 6 ε 5/2 (54) = O |k -| 3 + ε 3 , where a ′ 2 a ′ 4 ε 5/2 + a ′ 2 a ′′2 3 a 0 ε 2 -a 1 a ′ 2 a ′′ 3 a 5 ε 5/2 ∆ = a 6 ε 5/2 .
Then, using the implicit function theorem, we obtain a family of solutions such that i) if

a 6 > 0 z = - a 1 a ′′ 3 ε ∆ + -a ′ 2 a 6 ∆ ε 5/4 sinh φ + O(ε 3/2 ). a ′ 2 k - 2 = -a 1 z -a ′′ 3 ε + -a ′ 2 a 6 ε 5/4 cosh φ + O(ε 3/2 ). ( 55
) φ ∈ R; ii) if a 6 < 0 z = - a 1 a ′′ 3 ε ∆ + a ′ 2 a 6 ∆ ε 5/4 cosh φ + O(ε 3/2 ) a ′ 2 k - 2 = -a 1 z -a ′′ 3 ε + a ′ 2 a 6 ε 5/4 sinh φ + O(ε 3/2 ) (56) φ ∈ R.
Checking that the conditions (39), (40) are satisfied, Proposition 7 is proved.

Remark 16 It should be noted that the one parameter family of solutions which are obtained, correspond to convective rolls at -∞ with wave numbers

k c (1 + ε 2 k -)
connected to convective rolls at +∞ with wave numbers

k c (1 + 2ε 2 ω + ).
The calculations made above, show that we obtain ω + and k -as functions of ε, φ where φ ∈ R. This is a one parameter family of relationships between wave numbers at each infinity, depending on the amplitude ε 2 of rolls.

A Appendix

A.1 Reduction of the normal form

We start with

X = (A 0 , A 1 , A 2 , A 3 ) t ∈ R 4 , Y = (B 0 , B 1 ) t ∈ C 2 ,
and the system under normal form (see [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF] )

dX dx = LX + N (X, Y, Y , µ, k) + F (X, Y, Y , µ, k), (57)
dY dx = L kc Y + M (X, Y, Y , µ) + G(X, Y, Y , µ), LX = (A 1 , A 2 , A 3 , 0) t , L kc Y = (ik c B 0 + B 1 , ik c B 1 ) t , |F (X, Y, Y , µ, k)| ≤ c|X|(|X| 2 + |Y | 2 + | k| + |µ|) 2 |G(X, Y, Y , µ)| ≤ c(|X| 2 + |Y |)(|X| 2 + |Y | 2 + |µ|) 2 , (58)
Then, the X part of the system (57) may be written as a 4th order real ODE, while the Y part becomes a 2nd order complex ODE as

A (4) 0 = A 0 [d 0 µ + (d ′′ 0 -b ′2 0 ) k 2 + d 1 A 2 0 + d ′ 1 kA 2 0 + d 5 B 0 B 0 + d ′ 1 kA 2 0 +id 6 ( B 0 B 0 ′ -B 0 B 0 ′ )] + (a 0 µ + 3b ′ 0 k)A ′′ 0 + a 1 A 2 0 A ′′ 0 + a 2 A 0 A ′2 0 +a 3 A 0 B 0 ′ B 0 ′ + a 4 A ′ 0 ( B 0 B 0 ′ + B 0 B 0 ′ ) + a 5 A ′′ 0 B 0 B 0 +3ib 6 A ′′ 0 ( B 0 B 0 ′ -B 0 B 0 ′ ) + a 6 A 0 A ′ 0 A ′′′ 0 + a 7 A 0 A ′′2 0 + a 8 A ′2 0 A ′′ 0 + O X (5), B 0 ′′ = B 0 [β 0 µ + β 1 A 2 0 + β 5 B 0 B 0 ] + ic 1 B 0 ′ A 2 0 + ic 2 B 0 ′ | B 0 | 2 + ic 3 B 0 ′ B 0 2 +2iα 0 µ B 0 ′ + ic 4 B 0 A 0 A ′ 0 -2α 6 B 0 ′ ( B 0 B 0 ′ -B 0 B 0 ′ ) +c 5 B 0 A 0 A ′′ 0 + c 6 B 0 A ′2 0 + c 7 B 0 ′ A 0 A ′ 0 + ic 8 B 0 A 0 A ′′′ 0 ic 9 B 0 ′ A 0 A ′′ 0 + ic 10 B 0 ′ A ′2 0 + ic 11 B 0 A ′ 0 A ′′ 0 + O Y (5), with real coefficients d j , d ′ 1 d ′′ 0 , a j , b j , b ′ 0 , c j , β j , α j and B 0 = B 0 e -ikcx , B 1 = B 1 e -ikcx , (59)
d 0 = -4k 2 c β 0 > 0, d 1 = -4k 2 c β 5 < 0, β 1 β 5 = d 5 d 1 := g > 0, b ′ 0 = 4k 2 c 3 , d ′′ 0 = - 20 9 k 4 c , O X (5) = O(|X|(|X| 2 + |Y | 2 + k 2 + |µ|) 2 ), O Y (5) = O[(|X| 2 + |Y |)(|X| 2 + |Y | 2 + |µ|) 2 ], X = (A 0 , A ′ 0 , A ′′ 0 , A ′′′ 0 ) t Y = ( B 0 , B 0 ′ ).
Notice that the high order rests O X (5) and O Y (5) are no longer autonomous, since they are functions of e ±ikcx . Now, as indicated in [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF] we make the following scaling

x = 1 2εk c x, µ = 4k 2 c -β 0 ε 4 , k = ε 2 k - (60) A 0 (x) = 2k c β 5 ε 2 A 0 ( x), B 0 (x) = 2k c β 5 ε 2 B 0 ( x),
so that the system above becomes, after suppressing the tildes,

A (4) 0 = k -A ′′ 0 + A 0 (1 - k 2 - 4 -A 2 0 -g|B 0 | 2 ) + f , B ′′ 0 = ε 2 B 0 (-1 + gA 2 0 + |B 0 | 2 ) + g, (61)
e ix 2ε in B 0 , so that the system is autonomous):

dB 0 dx = i 2ε B 0 + B 1 + iε 3 B 0 P + ε 7 g 0 (ε, Y, Y ) (62) dB 1 dx = i 2ε B 1 + ε 2 B 0 Q + iε 3 B 1 P + ε 6 g 1 (ε, Y, Y ), with Y = (B 0 , B 1 ) P = α + β|B 0 | 2 + εγK Q = -1 + |B 0 | 2 + εδK K = i 2 (B 0 B 1 -B 0 B 1 )
where we are looking for a periodic solution (B 0 , B 1 ), with wave number ω close to 1+ε 2 k+ 2ε .

A.3.1 Principal part

Let us first compute periodic solutions for g 0 = g 1 ≡ 0. Then these small terms will be perturbations treated by an adapted implicit function theorem. Without g 0 and g 1 , let us use polar coordinates (see [START_REF] Haragus | Local bifurcations, Center manifolds, and Normal forms in infinite-dimensional dynamical systems[END_REF] section 4.3.3)

B 0 = r 0 e iθ0 B 1 = ir 1 e iθ1 then K = r 0 r 1 cos(θ 0 -θ 1 ) = const dr 0 dx = r 1 sin(θ 0 -θ 1 ) dr 1 dx = ε 2 r 0 sin(θ 0 -θ 1 )Q(ε, r 2 0 , K) r 0 dθ 0 dx = r 0 2ε + r 1 cos(θ 0 -θ 1 ) + ε 3 r 0 P r 1 dθ 1 dx = r 1 2ε -ε 2 r 0 cos(θ 0 -θ 1 )Q(ε, r 2 0 , K) + ε 3 r 1 P.
The required periodic solutions correspond to r 0 and r 1 const

θ 0 = θ 1 , dθ 0 dx = 1 + ε 2 k + 2ε K = r 0 r 1 , hence εk + 2 = r 1 r 0 + ε 3 P (63) ( r 1 r 0 ) 2 = -ε 2 Q. ( 64
)
Solving (63) with respect to r 1 gives

r 1 = εr 0 k + -2ε 2 (α + βr 2 0 ) 2(1 + ε 4 γr 2 0 ) = εr 0 2 [k + -2ε 2 (α + βr 2 0 )](1 + O(ε 4 )),
and (64) leads to

1 4 [k + -2ε 2 (α + βr 2 0 )] 2 + ε 2 δr 2 0 2 [k + -2ε 2 (α + βr 2 0 )] = (1 -r 2 0 )(1 + γε 4 r 2 0 ) 2
which is solved with respect to r 2 0 , by implicit function theorem:

r 2 0 = 1 - k 2 + 4 + σ 1 ε 2 k + + σ 2 ε 4 + O[(|k + | + ε 2 ) 4 ], (65)
r 1 = εr 0 2 k + + O(ε 3 ),
where we notice that coefficients σ 1 and σ 2 are functions of the Prandtl number. We obtain a one-parameter family of periodic solutions (parameter k + ), with only the Fourier modes e ±is .

A.3.2 Estimates of higher order terms

The proof below is new and self contained. There is a geometrical proof without estimates in Iooss-Pérouème [START_REF] Iooss | Perturbed homoclinic solutions in reversible 1:1 resonance vector fields[END_REF], and a more precise proof by Horn in [6] section 3.5. Let us define by ω the frequency of periodic solutions, where ω is close to

ω 0 = 1 + ε 2 k + 2ε ,
ε 2 Q 0 r 0 + [(ω 0 - 1 2ε
-ε 3 P 0 ]r 1 = 0, with P 0 = α + βr 2 0 + εγr 0 r 1 , Q 0 = -1 + r 2 0 + εδr 0 r 1 . Then the system (62), to be completed by its complex conjugate, becomes:

ωV 0 e is + L 0 B 0 B 1 = i ω d ds B 0 B 1 + ε 3 r 0 P lin -ε 2 r 0 Q lin + ε 3 r 1 P lin + R 0 ( Y , Y ) R 1 ( Y , Y ) , (66)
where R 0 ( Y , Y ) = ε 3 r 0 e is P quad + ε 3 B 0 (e -is P lin + P quad ) -iε 7 g 0 , R 1 ( Y , Y ) = -ε 2 r 0 e is Q quad -ε 2 B 0 (e -is Q lin + Q quad ) +ε 3 r 1 e is P quad + ε 3 B 1 (e -is P lin + P quad ) -ε 6 g 1 , with

P
Q quad = B 0 B 0 + εδ 2 ( B 0 B 1 + B 1 B 0 ) P quad = β B 0 B 0 + εγ 2 ( B 0 B 1 + B 1 B 0 ). Let us decompose B 0 B 1 = y r 1 e is -r 0 e is + B 0 B 1
where B 0 and B 1 have no Fourier component in e is , and we take the component in e is orthogonal to V 0 e is , since adding a component proportional to (r 0 , r 1 ) is equivalent to adapt (r 0 , r 1 ). We first solve (66) with respect to ( B 0 , B 1 ) in using the implicit function theorem, since we observe (notice the term nω 0 = n 2ε (1 + ε 2 k + ) in the operator for a Fourier component e nis ), that the pseudo-inverse of L 0 is bounded from H 1 (R/2πZ) × L 2 (R/2πZ) to H 2 (R/2πZ) × H 1 (R/2πZ). Let us notice that the difference with the classical Hopf bifurcation proof is that, norms in these spaces are chosen as, for example

||u|| H 2 = 1 ε 2 ||u ′′ || L 2 + 1 ε ||u ′ || L 2 + ||u|| L 2 ,
and notice that H 1 (R/2πZ) is an algebra. It results that we obtain an estimate such that ||( B 0 , B 1 )|| H 2 ×H 1 ≤ c(ε 2 | y| + ε 6 ).

It then remains to solve the 2-dimensional system in ( ω, y) which is a real system, due to the reversibility symmetry:

Figure 1 :

 1 Figure 1: Orthogonal domain wall

s

  = ωx, ω = ω 0 + ω B 0 (s) = r 0 e is + B 0 B 1 (s) = ir 1 e is + i B 1 ,where B 0 and B 1 are 2π-periodic in s, and r 0 , r 1 are solution of (63,64). Let us introduce the linear operatorL 0 = -(iω 0 d ds + 1 2ε + ε 3 P 0 ) -1 ε 2 Q 0 -(iω 0 d ds + 1 2ε + ε 3 P 0 ) , acting in the function space H 1 (R/2πZ) × L 2 (R/2πZ). It appears that L 0 has a one-dimensional kernel (r 0 e is , r 1 e is ) def = V 0 e is since (63,64) implies [(ω 0 -1 2ε -ε 3 P 0 ]r 0 -r 1 = 0

  lin = e 2is [βr 0 B 0 + εγ 2 (r 0 B 1 + r 1 B 0 )] +[βr 0 B 0 + εγ 2 (r 0 B 1 + r 1 B 0 )] Q lin = e 2is [-r 0 B 0 + εδ 2 (r 0 B 1 + r 1 B 0 )] +[-r 0 B 0 + εδ 2 (r 0 B 1 + r 1 B 0 )],

ωr 0 + yr 1 =

 1 -ω yr 1 + O(ε 4 | y| + ε 3 | y| + ε 7 ) ωr 1 -yr 0 = ω yr 0 + O(ε 3 | y| + ε 2 | y| + ε 6 ), which gives ω = O(ε 7 ) y = O(ε 6 ).It results finally that the family of periodic solutions at M + are such thatB 0 = r 0 e iωx + O(ε 6 ), B 1 = ir 1 e iωx + O(ε 6 ),

N (X, Y, Y , µ) =     0 A 0 P 1 A 1 P 1 + c 8 u 8 + c 13 u 13 A 2 P 1 + A 0 P 3 + +c 8 v 8 + c 13 v 13 + d 14 u 14

where

.

The (reversible) system anticomutes with the symmetry S 1 (represents the reflection x → -x). and commutes with τ π (shift by half of one period in y direction):

Remark 17 In the case of rigid-free boundary conditions, we have no symmetry z → 1-z. We don't use this symmetry here (valid only in rigid-rigid or free-free boundaries). The symmetry τ π implies that F is odd in X and G even in X. It can be shown that there is no term of degree 4 in X, Y, Y in the normal form.

with additional cubic terms of the form

A.2 Equilibrium solution at x = -∞

Let us look for equilibria of ( 2), which should correspond to the convective rolls at x = -∞ parallel to x -axis. Cancelling all derivatives with respect to x, we obtain a system commuting with the symmetry (A 0 , B 0 ) → (A 0 , B 0 ). It then results a system of 2 real equations for A 0 , B 0 :

where we may observe that the terms O(ε 4 ) in the second equation contain at least terms of degree 1 in B 0 , since they come from terms of order 5 in (A 0 , B 0 , B 0 ). The first terms not containing B 0 may be found at order 6 in A 0 , which makes order ε 6 after the scaling (60) in the rest (12-6=6).

It then results that the equilibrium that we are looking for satisfies (by implicit function theorem)

Remark 18 In the cases where symmetry z → 1-z applies, the additional symmetry S 0 changes the signs of A 0 and B 0 , implying that Y = 0 is an invariant subspace, so that in such cases B 0 = 0 for equilibrium at -∞.

A.3 Periodic solution in M +

Let us consider the 4-dimensional reversible vector field corresponding to the system (57) with X = 0 and rescaled. We intend to give precise estimates on the family of periodic bifurcating solutions B (+∞) 0

(k + , x), here corresponding to the periodic convection rolls at infinity in M + with wave numbers close to k c (becomes 1/2ε after the scaling (60)).

Since we use the normal form up to cubic order, and since there is no term of order 4, it takes the form (after the scaling used in [START_REF] Buffoni | Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls[END_REF], but before we incorporate


Jérôme Boursier
email: jeboursier@chu-angers.fr

Victor De Ledinghen

Vincent Leroy

Rodolphe Anty

Sven Francque

Dominique Salmon

Adrien Lannes

Sandrine Bertrais

Frederic Oberti

Isabelle Fouchard-Hubert

Paul Calès

Jérôme Boursier

Dr Jérôme Boursier

Jerome Boursier

A stepwise algorithm using a at-a-glance first-line test for the non-invasive diagnosis of advanced liver fibrosis and cirrhosis

Keywords: liver fibrosis, cirrhosis, non-invasive tests, algorithm Word count (including abstract, references, tables and figure legends): 5294, Tables: 4

and interpretation of data, drafting of the manuscript, obtained funding, study supervision; Victor de Ledinghen: acquisition of data, critical revision of the manuscript for important intellectual content; Vincent Leroy: acquisition of data, critical revision of the manuscript for important intellectual content; Rodolphe Anty: acquisition of data, critical revision of the manuscript for important intellectual content; Sven Francque: acquisition of data, critical revision of the manuscript for important intellectual content; Dominique Salmon: acquisition of data; Adrien Lannes: acquisition of data; Sandrine Bertrais: statistical analysis; Frederic Oberti: acquisition of data; Isabelle Fouchard-Hubert: acquisition of data; Paul Calès: acquisition of data, critical revision of the manuscript for important intellectual content, obtained funding Abbreviations: CLD, Chronic Liver Disease; NAFLD: Non-Alcoholic Fatty Liver Disease; HCC: HepatoCellular Carcinoma; LSM: Liver Stiffness Measurement; AUROC: Area Under the

Receiving Operating Characteristics

Clinical trial registration: no registration (analysis of pooled data from previously published diagnostic studies) ABSTRACT Background & Aims. Chronic liver diseases (CLD) are highly frequent and thus mainly managed by non-hepatologists. Because they lack access to the best non-invasive tests of liver fibrosis, these physicians cannot accurately determine the disease severity and the need for referral to a hepatologist. We aimed to implement an algorithm, comprising a new first-line test usable by all physicians, for the detection of advanced liver fibrosis in all CLD patients.

Methods. Diagnostic study: 3,754 CLD patients with liver biopsy were 2:1 randomized into derivation and validation sets. Prognostic study: longitudinal follow-up of 1,275 CLD patients with baseline fibrosis tests.

Results: Diagnostic study: the easy LIver Fibrosis Test (eLIFT), an "at-a-glance" sum of points attributed to age, gender, gamma-GT, AST, platelets and prothrombin time, was developed for the diagnosis of advanced fibrosis. In the validation set, eLIFT and FIB4 had the same sensitivity (78.0% vs 76.6%, p=0.470) but eLIFT gave less false-positive results, especially in patients ≥60 years old (53.8% vs 82.0%, p<0.001), and was thus more suitable as screening test. FibroMeter VCTE was the most accurate among the eight fibrosis tests evaluated. The sensitivity of the eLIFT-FM VCTE algorithm (first-line eLIFT, second-line FibroMeter VCTE ) was 76.1% for advanced fibrosis and 92.1% for cirrhosis. Prognostic study: patients diagnosed as having "no/mild fibrosis" by the algorithm had excellent liver-related prognosis with thus no need for referral to a hepatologist.

Conclusion:

The eLIFT-FM VCTE algorithm extends the detection of advanced liver fibrosis to all CLD patients and reduces unnecessary referrals of patients without significant CLD to hepatologists.

LAY SUMMARY

Blood fibrosis tests and transient elastography accurately diagnose advanced liver fibrosis in

the large population of patients having chronic liver disease, but these non-invasive tests are currently available only in specialized centers. We have developed an algorithm including as first-line procedure the easy LIver Fibrosis Test (eLIFT), a new simple and widely available blood test, that selects at-risk patients who need further evaluation with the FibroMeter VCTE , an accurate fibrosis test combining blood markers and transient elastography result. This new algorithm, called the eLIFT-FM VCTE , accurately identifies the patients with advanced chronic liver disease who need referral to a specialist, and those with no or mild liver lesions who can remain under the care of their usual physician.

INTRODUCTION

Chronic liver diseases (CLD) are very common: worldwide, an estimated 160 million people have chronic hepatitis C (1), 240 million have chronic hepatitis B (2), and 25% of the general population has non-alcoholic fatty liver disease (NAFLD) [START_REF] Younossi | Global epidemiology of nonalcoholic fatty liver disease-Meta-analytic assessment of prevalence, incidence, and outcomes[END_REF]. CLD can lead to a progressive accumulation of fibrosis in the liver which progressively evolves to cirrhosis and its lifethreatening complications such as hepatocellular carcinoma (HCC), liver failure, variceal bleeding, or renal insufficiency. In 2012, driven by the growing worldwide burden of CLD, cirrhosis was responsible for more than 35 million years of lost life and thus became the eleventh leading cause of mortality among non-communicable diseases (4). Additionally, HCC has become the sixth leading incident cancer and the second leading cause of cancerrelated death worldwide (4).

Both the prognosis and the management of CLD patients are closely linked to the level of liver fibrosis. Treatment of the cause of CLD is mandatory in patients who develop advanced septal fibrosis to prevent further progression to cirrhosis and its complications [START_REF] Association For Study Of | EASL Recommendations on Treatment of Hepatitis C 2015[END_REF][START_REF] Terrault | AASLD guidelines for treatment of chronic hepatitis B[END_REF][START_REF]EASL-EASD-EASO Clinical Practice Guidelines for the management of non-alcoholic fatty liver disease[END_REF]. In cirrhotic patients, screening procedures are required for early detection of HCC and identification of large esophageal varices. Liver biopsy is the reference procedure for liver fibrosis evaluation but its invasive nature makes it unsuitable as first-line procedure in the large number of CLD patients. Blood tests and liver stiffness measurement (LSM) by elastography have been recently developed for the non-invasive evaluation of liver fibrosis and provide an exciting alternative to biopsy [START_REF]European Association for Study of the Liver, Asociacion Latinoamericana para el Estudio del Higado. EASL-ALEH Clinical Practice Guidelines: Non-invasive tests for evaluation of liver disease severity and prognosis[END_REF]. However, the high cost of the most accurate blood fibrosis tests limits their widespread use, and liver elastometry is only accessible in specialized centers.

CLD patients are numerous and thus not all of them can be referred to the few specialized hepatology clinics. In practice, most CLD patients are managed by non-hepatologists who encounter challenges in the evaluation of the liver disease that remains silent for many years with normal physical examination and normal routine diagnostic tests. In addition, nonhepatologists have very limited access to the best non-invasive liver fibrosis tests.

Resultantly, liver fibrosis is unevaluated in many CLD patients with progressive fibrosis. These patients are finally diagnosed too late when they have reached the stage of cirrhosis complications with an impaired short-term prognosis.

Therefore, in the present work, we aimed to develop and validate a stepwise algorithm that can be easily instigated by all physicians to facilitate the widespread detection of advanced liver fibrosis in all CLD patients. Such an algorithm should prove very helpful in the regulation of the large flow of CLD patients between primary care and specialized centers, and especially in the identification of CLD patients who needs referral to specialized hepatologists and those who do not.

PATIENTS AND METHODS

The study protocol conformed to the ethical guidelines of the current Declaration of Helsinki. All patients included in the cross-sectional population and the longitudinal cohort gave informed written consent for their participation.

Cross-sectional population

The cross-sectional population was used to develop and validate the new stepwise algorithm. This population was obtained by pooling the data of seven published studies that evaluated non-invasive liver fibrosis tests using liver biopsy as the reference [START_REF] Boursier | Diagnostic accuracy and prognostic significance of blood fibrosis tests and liver stiffness measurement by FibroScan in nonalcoholic fatty liver disease[END_REF][START_REF] Anty | A new composite model including metabolic syndrome, alanine aminotransferase and cytokeratin-18 for the diagnosis of non-alcoholic steatohepatitis in morbidly obese patients[END_REF][START_REF] Boursier | Comparison of 8 diagnostic algorithms for liver fibrosis in hepatitis C: new algorithms are more precise and entirely non-invasive[END_REF][START_REF] Boursier | The combination of a blood test and Fibroscan improves the non-invasive diagnosis of liver fibrosis[END_REF][START_REF] Castera | Comparison of transient elastography (FibroScan), FibroTest, APRI and two algorithms combining these non-invasive tests for liver fibrosis staging in HIV/HCV coinfected patients: ANRS CO13 HEPAVIH and FIBROSTIC collaboration[END_REF][START_REF] Francque | Noninvasive assessment of nonalcoholic Fatty liver disease in obese or overweight patients[END_REF][START_REF] Lignon | Screening for significant chronic liver disease by using three simple ultrasound parameters[END_REF]. We provide here the main inclusion/exclusion criteria and methods of these seven studies.

Patients

All included patients had CLD without decompensated cirrhosis or HCC. Patient duplication between studies was corrected so as to ensure that all patients were included only once in the statistical analysis for the present work.

Liver biopsy

All patients had a liver biopsy taken and used as the reference for liver fibrosis evaluation.

Pathological examinations were performed in each center by senior experts specialized in hepatology and blinded for patient data. Liver fibrosis was evaluated according to NASH CRN staging in patients with NAFLD, and METAVIR staging in patients with other causes of CLD.

Although the two semi-quantitative scoring systems comprise stages from F0 to F4, they do not completely correspond (Supplementary Table s1). For the present study, we defined "no/mild fibrosis" as NASH CRN F0-2 or METAVIR F0-1, "septal fibrosis" as NASH CRN F3 or Metavir F2-3, and "cirrhosis" as NASH CRN F4 or METAVIR F4. Advanced fibrosis, which was defined as NASH CRN F≥3 or Metavir F≥2 (Table s1c), was the primary diagnostic target of the study.

Blood fibrosis tests

Fasting blood samples were taken the day of or within the three months before or after liver biopsy. The data available from the seven studies enabled the calculation of six blood fibrosis tests according to published or patented formulas: APRI ( 14), FIB4 [START_REF] Sterling | Development of a simple noninvasive index to predict significant fibrosis in patients with HIV/HCV coinfection[END_REF], Hepascore [START_REF] Adams | Hepascore: an accurate validated predictor of liver fibrosis in chronic hepatitis C infection[END_REF], FibroMeter V2G (17), FibroMeter V3G [START_REF] Cales | Optimization and robustness of blood tests for liver fibrosis and cirrhosis[END_REF], and FibroMeter VCTE [START_REF] Cales | Improved fibrosis staging by elastometry and blood test in chronic hepatitis C[END_REF]. FibroMeter V3G is the same blood fibrosis test than FibroMeter V2G but hyaluronate, a costly and difficult-to-obtain marker, has been replaced by the gammaGT. We have previously shown that FibroMeter V2G and FibroMeter V3G have comparable diagnostic accuracy in chronic hepatitis C [START_REF] Cales | Optimization and robustness of blood tests for liver fibrosis and cirrhosis[END_REF].

FibroMeter VCTE is a fibrosis test that combines in a single formula both the blood markers of the FibroMeter V3G with the Fibroscan result. We have previously shown that FibroMeter VCTE was significantly more accurate than FibroMeter V2G and Fibroscan in chronic hepatitis C [START_REF] Cales | Improved fibrosis staging by elastometry and blood test in chronic hepatitis C[END_REF].

In the present study, APRI and FIB4 were considered as "simple fibrosis tests" since they use common, inexpensive variables and easy-to-calculate formulas. The other fibrosis tests include more expensive parameters in complex equations that require computerized calculation.

Fibroscan

LSM with Fibroscan was performed in each center by an experienced operator blinded for patient data using the standard M probe. LSM was performed in fasting conditions, the day of or within the three months before or after liver biopsy. Examination conditions were those recommended by the manufacturer [START_REF] Castera | Non-invasive evaluation of liver fibrosis using transient elastography[END_REF]. LSM result (kilo Pascal: kPa) corresponded to the median value of the 10 valid measurements recorded.

Longitudinal cohort

The prognostic longitudinal cohort was used to validate the clinical significance of the new stepwise algorithm developed in the cross-sectional population. We used a previouslyestablished local database that retrospectively included all consecutive CLD patients seen in the Hepatology Department of the Angers University Hospital for a non-invasive evaluation of liver fibrosis between January 2005 and December 2009 [START_REF] Cales | Improved fibrosis staging by elastometry and blood test in chronic hepatitis C[END_REF]. Exclusion criteria for the present study were: prothrombin time <70% or serum bilirubin ≥30 µmol/L (i.e., no need for a fibrosis test to diagnose advanced fibrosis), missing LSM or blood test results, and an interval between blood fibrosis tests and LSM >6 months. Follow-up started the day of the non-invasive evaluation of liver fibrosis and ended January 1 st , 2011. Dates and causes of death were obtained from the computerized National Registry of Individuals (CepiDC-Inserm, France). For those patients who could not be matched individually within the national registry, mortality data were obtained from the hospital database, or from the concerned general practitioner.

Statistics

In the cross-sectional population, the diagnostic accuracies of the fibrosis tests were mainly expressed as the area under the receiver operating characteristics (AUROC) and compared using the Delong test [START_REF] Delong | Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach[END_REF]. In the longitudinal cohort, the prognostic accuracies of fibrosis tests were evaluated using the Harrell C-index, as previously described [START_REF] Boursier | Combination of blood tests for significant fibrosis and cirrhosis improves the assessment of liver-prognosis in chronic hepatitis C[END_REF]. Briefly, the Harrell C-index is an extension of the AUROC for time-to-event (survival) data; it evaluates the concordance between the predicted risk of the event and the observed survival time (discriminative ability). Its results vary from 0 to 1, with a value of 1 indicating a perfect concordance. Survival curves were determined using the Kaplan-Meier method and compared with the log rank test. Statistical analyses were performed using SPSS version 18.0 software (IBM, Armonk, NY, USA) and SAS 9.1 (SAS Institute Inc., Cary, NC, USA). This study was reported in accordance with the recently published LiverFibroSTARD statements [START_REF] Boursier | An extension of STARD statements for reporting diagnostic accuracy studies on liver fibrosis tests: the Liver-FibroSTARD standards[END_REF].

RESULTS

Cross-sectional population: derivation and validation of the new algorithm

Patient characteristics

The cross-sectional population included 3,754 patients with CLD. Patient characteristics at inclusion are detailed in Table 1. 56.6% of patients were male and mean age was 49.5±13.1 years. The main causes of CLD were chronic hepatitis C (45.5%) and NAFLD (34.2%). Mean biopsy length was 25±10 mm (median: 23mm; 1 st quartile: 17mm; 3 rd quartile: 30mm). The prevalence of advanced fibrosis was 46.0% and cirrhosis 13.2%. The 3,754 patients were 2:1 randomly divided into derivation and validation sets (Supplementary Figure s1), the characteristics of which were not significantly different (Table 1).

Determination of the first-line "simple" fibrosis test

Evaluation of APRI and FIB4

Because they are simple fibrosis tests available to all physicians, we evaluated APRI and FIB4 for the first-line test of the new algorithm. FIB4 had significantly higher AUROCs than APRI for the diagnosis of advanced fibrosis or cirrhosis (Table s2). As calculated in the derivation set, the 80% sensitivity thresholds of APRI and FIB4 for advanced fibrosis were respectively 0.40 and 1.20. Using these thresholds in the validation set, FIB4 and APRI had similar sensitivity for advanced fibrosis (respectively 76.6% vs 75.8%, p=0.688), but FIB4 had significantly better sensitivity for cirrhosis (93.1% vs 86.7%, p=0.035). The rate of patients with FIB4 <1.20 was significantly higher than the rate of patients with APRI <0.40 (44.7% vs 41.3%, p=0.010). Taken together, these results suggested that FIB4 should be better than APRI as a first-line test in the new algorithm. However, the FIB4 equation includes age and Figure 1 shows that the rate of false positives (i.e., the rate of FIB4 ≥1.20 among patients with no/mild fibrosis) dramatically rose with age. In patients ≥60 years old, the false positive rate was 82.0%, which limited the interest of FIB4 as a first-line screening test.

Development of a new first-line "simple" fibrosis test

We decided to develop a new fibrosis test to circumvent the previously identified limitations while improving ease-of-use and availability for all physicians. Toward this, we chose the clinical and blood parameters commonly assessed in CLD patients: age, gender, gamma-GT, ALT, AST, platelet count, and prothrombin time. Continuous variables were transformed into qualitative and ordinal variables according to the thresholds corresponding to 33% and 66% prevalence for advanced fibrosis (see Figure s2 for more details). By multivariate analysis in the derivation set, all variables except ALT were independent predictors of advanced fibrosis (Table s3). The β coefficients of the multivariate analysis (β coefficients x3, rounded) were used to determine a new, simple and user-friendly liver fibrosis score: the easy LIver Fibrosis Test (eLIFT, Table 2). To keep AST from weighing too heavily in the score, it was capped at four points, as were platelets and prothrombin time. In the derivation set, the 80% sensitivity threshold of eLIFT was calculated at eight. This new eLIFT was then evaluated in the validation set versus FIB4. Significant differences between the AUROCs of eLIFT and FIB4 were observed neither for advanced fibrosis (respectively, 0.781±0.013 vs 0.789±0.013, p=0.421) nor for cirrhosis (0.853±0.015 vs 0.844±0.015, p=0.424). There was no significant difference in accuracy between eLIFT ≥8 and FIB4 ≥1.20: their sensitivities for advanced fibrosis were respectively 78.0% and 76.6% (p=0.470), for cirrhosis 94.2% and 93.1% (p=0.791), and the rates of patients with negative results were 43.5% and 44.7% (p=0.383). However, eLIFT was less influenced by age (Figure s3), providing a significantly lower rate of false positives in patients ≥60 years old: 53.8% vs 82.0% with FIB4 (p<0.001). This latter result, in addition to the test's "at-a-glance" ease of calculation, placed eLIFT as a better choice than FIB4 for the first-line test in our algorithm.

Determination of the second-line "diagnostic" fibrosis test

Blood fibrosis tests (APRI, FIB4, eLIFT, Hepascore, FibroMeter V2G , FibroMeter V3G ), LSM by Fibroscan, and FibroMeter VCTE were all available in a core group of 1,946 patients. This core group had more severe liver disease than the other 1,808 patients (Table s4). The core group comprised 1,282 patients from the derivation set and 664 from the validation set (Figure s1), with no significant differences between these two subsets (Table s4).

FibroMeter VCTE had significantly highest AUROC for the diagnosis of advanced fibrosis compared to the seven other fibrosis tests evaluated in the core group (Table s5). It also had significantly highest AUROC for the diagnosis of cirrhosis compared to the six blood fibrosis tests (Table s6). Consequently, we chose FibroMeter VCTE as the second-line test for our algorithm. Two FibroMeter VCTE thresholds for 90% sensitivity (0.384) and 90% specificity (0.715) for advanced fibrosis were calculated in the 1,282 patients from the derivation set. These two thresholds delineated three patient subgroups with the following diagnoses: <0.384: no/mild fibrosis, ≥0.384 and <0.715: undetermined (grey zone), ≥0.715: advanced fibrosis. Using liver biopsy in the grey zone, this patient classification had 89.5% diagnostic accuracy in the derivation set vs 88.3% in the 664 patients from the validation set (p=0.399, Figure s4).

New algorithm

The first-line "simple" eLIFT was combined with the second-line "diagnostic" FibroMeter VCTE into a new stepwise algorithm called eLIFT-FM VCTE (Figure 2). When applied to the validation set (n=664 patients), eLIFT-FM VCTE diagnosed 32.7% of the patients as having no/mild fibrosis at the first step (negative eLIFT <8) and 13.7% at the second (positive eLIFT ≥8 but negative FibroMeter VCTE <0.384). Thus, there was no need for referral to a hepatologist for 46.4% of the patients. Of the remaining patients, 19.4% were included in the grey zone and 34.2% were diagnosed as having advanced fibrosis. Using liver biopsy in the grey zone, the diagnostic accuracy of eLIFT-FM VCTE was 83.3%, sensitivity and specificity for advanced fibrosis were respectively 76.1% and 92.2%, and sensitivity and specificity for cirrhosis were respectively 92.1% and 76.2% (Table 3, see Table s7 for detailed results).

Compared to the other causes of chronic liver disease, eLIFT ≥8 had higher sensitivity and lower specificity in the alcohol subgroup (Figure s5). Interestingly, these differences were erased after the use of the FibroMeter VCTE : sensitivity and specificity for advanced fibrosis as well as the rate of well-classified patients were not significantly different between etiologies by using the eLIFT-FM VCTE algorithm. By multivariate analysis performed in the whole population, we evaluated the influence of age, sex, CLD cause, derivation/validation set, biopsy length and LSM IQR/median ratio on the diagnostic accuracy of the eLIFT-FM VCTE .

None of these parameters were independently associated with the rate of well-classified patients by the algorithm.

We finally compared the eLIFT-FM VCTE to strategies using FIB4 or FibroMeter V2G alone. We chose FIB4 as it was the best "simple" test compared to APRI for the diagnosis of advanced fibrosis, and FibroMeter V2G because it was the most accurate among all blood tests evaluated (Table s5). Results in the validation set showed that the eLIFT-FM VCTE had the same diagnostic accuracy than blood tests alone, but it had the strong advantage to require significantly less liver biopsies (Table s8).

Longitudinal cohort: evaluation of the prognostic significance of the new algorithm

The aim of the prognostic study was to evaluate the prognostic significance of the algorithm, especially in the subgroup of patients diagnosed as "no/mild fibrosis" as it can include false negative results.

Patient characteristics

The longitudinal cohort included 1,275 patients. Their characteristics at baseline are detailed in the Table s9. Mean age was 52.8±14.7 years and 66.8% were male. One hundred and twenty-eight patients died during the median follow-up period of 2.9 years (interquartile range: 1.9-4.2 years; 3,807 person-years). Death was related to liver complications in 43 patients. According to the eLIFT-FM VCTE algorithm, 60.1% of patients were diagnosed as having no/mild fibrosis (46.8% with eLIFT <8, and 13.3% with eLIFT ≥8 but FibroMeter VCTE <0.384), 12.8% were included in the grey zone, and 27.1% were diagnosed as having advanced fibrosis.

Prognostic accuracy of fibrosis tests

The Harrell C-indexes of eLIFT and FIB4 for the prediction of all-cause mortality were not significantly different (Table 4). However, the discriminative ability of eLIFT was significantly better than that of FIB4 for the prediction of death related to liver complications, thus reinforcing our choice to use it rather than FIB4 as the first-line test in our new algorithm.

FibroMeter VCTE had the significantly highest Harrell C-index for the prediction of death related to liver complications compared to the seven other fibrosis tests evaluated, confirming its use as the second-line test in the algorithm.

Prognostic accuracy of the new algorithm

Figure 3a shows overall survival and Figure 3b survival free of death related to liver complications as a function of the four patient subgroups defined by the eLIFT-FM VCTE algorithm. Interestingly, there were no significant differences in prognosis between patients diagnosed as having no/mild fibrosis with a negative eLIFT (<8) and those diagnosed as no/mild fibrosis with a positive eLIFT (≥8) but a negative FibroMeter VCTE (<0.384). Patients diagnosed as having no/mild fibrosis by the eLIFT-FM VCTE had an excellent liver-related prognosis with only two of them who died from liver-related complication during the followup. Finally, as expected, patients diagnosed as having advanced fibrosis according to eLIFT-FM VCTE had the worse prognosis among the four subgroups.

DISCUSSION

Despite the recommendation of early management for patients with CLD by all international guidelines, it is estimate that two thirds of cirrhosis cases are diagnosed belatedly when liver-related complications appear and the short-term prognosis is already impaired [START_REF] D'amico | Natural history and prognostic indicators of survival in cirrhosis: a systematic review of 118 studies[END_REF][START_REF] Conn | Cirrhosis[END_REF]. This demonstrates a crucial need for simple and accurate tools able of identifying the subset of CLD patients who require specific management by specialized physicians. In this setting, the non-invasive diagnosis of liver fibrosis is a rapidly growing field of research; blood fibrosis tests, liver elastometry, and their combinations are improving continuously and thus supplying constant increases in diagnostic accuracy [START_REF]European Association for Study of the Liver, Asociacion Latinoamericana para el Estudio del Higado. EASL-ALEH Clinical Practice Guidelines: Non-invasive tests for evaluation of liver disease severity and prognosis[END_REF]. Recently, a new three-dimensional magnetic resonance elastography technique showed an almost perfect AUROCs of 0.971 and 0.979 for the diagnoses of advanced liver fibrosis and cirrhosis, respectively [START_REF] Shi | Magnetic Resonance Elastography for the Evaluation of Liver Fibrosis in Chronic Hepatitis B and C by Using Both Gradient-Recalled Echo and Spin-Echo Echo Planar Imaging: A Prospective Study[END_REF]. However, the best means for the non-invasive evaluation of liver fibrosis are currently available only in specialized tertiary centers, which creates a dilemma. First, in chronic liver diseases, where NAFLD patients alone represent at least 25% of the general population (3), there are simply not enough of these centers to screen all CLD patients for advanced fibrosis. Second, even if we could do so, this would result in an incredibly high rate of unnecessary evaluations; indeed, a large proportion of CLD patients have alcoholic liver disease or NAFLD with no or mild liver fibrosis and thus do not require the "firepower" of a tertiary center or specific management by hepatologists.

FIB4 is a simple and widely available blood fibrosis test whose published diagnostic cut-offs are 1.45 and 3.25 [START_REF] Sterling | Development of a simple noninvasive index to predict significant fibrosis in patients with HIV/HCV coinfection[END_REF]. The low cut-off has been calculated to exclude severe fibrosis (Ishak ≥4) with 90% negative predictive value in the princeps study conducted in HIV-HCV coinfected patients [START_REF] Sterling | Development of a simple noninvasive index to predict significant fibrosis in patients with HIV/HCV coinfection[END_REF]. In our work, FIB4 >1.45 had only 67.2% sensitivity for the diagnosis of advanced fibrosis. As we aimed 80% sensitivity for the first-line test, we determined the corresponding FIB4 threshold that was calculated at 1.20. Using this cut-off, 90% of all the patients who were ≥60 years old had a positive FIB4 leading unsuitable this test for the screening of advanced fibrosis in elderly patients. We acknowledge that a lower cut-off has amplified the deleterious effect of age on FIB4 accuracy already observed by others teams [START_REF] Mcpherson | Age as a Confounding Factor for the Accurate Non-Invasive Diagnosis of Advanced NAFLD Fibrosis[END_REF]. However, this finally demonstrates the limits of FIB4 for the screening of advanced fibrosis: either the FIB4 threshold is low to ensure a good sensitivity but then increasing age induces rapidly a high rate of false positive results; or the FIB4 threshold is higher with less influence of age but then the sensitivity is low and insufficient.

We demonstrated here that it is possible to organize an accurate and generalized detection of advanced liver fibrosis and cirrhosis in all CLD patients. Our process starts with eLIFT, a new test designed for use by all physicians, as it is based on parameters commonly assessed in CLD. Compared to FIB4, the new eLIFT has two main advantages. First, while the FIB4 needs a computer for calculation, eLIFT is very easily calculated at-a-glance and in one's head. This makes eLIFT easier and faster to use in clinical practice than FIB4. More importantly, because the age is capped at 40 years, the rate of false positive results with eLIFT is much lower in elderly patients. In clinical practice, the purpose of eLIFT is to be used by any physician to identify patients at risk of advanced fibrosis who require further evaluation with a more accurate fibrosis test, i.e. FibroMeter VCTE . Together, they compose the eLIFT-FM VCTE algorithm. We showed here that eLIFT-FM VCTE had excellent sensitivity for detecting cirrhotic patients and good sensitivity for detecting patients with advanced fibrosis. The strength of our cross-sectional diagnostic study was the large size of the population where eLIFT-FM VCTE was derived and validated. eLIFT ≥8 was more sensitive and less specific for advanced fibrosis in the subgroup of patients with alcoholic disease (Figure s5). False positive results linked to increased AST and GGT in the context of chronic excessive alcohol consumption could have participated to this lack of specificity. Interestingly, the differences observed with eLIFT between etiologies were erased after the use of FibroMeter VCTE as second line test in the eLIFT-FM VCTE algorithm. This suggested that the eLIFT-FM VCTE could be used for the screening of advanced fibrosis in alcoholic patients but at the cost of a higher rate of patients requiring a second-line test. Due to the small rate of alcoholic patients included in our work, further studies evaluating the eLIFT-FM VCTE specifically in this population are required.

An important strength of our work is the use of a longitudinal cohort "from the real life" to validate the prognostic significance of eLIFT-FM VCTE . Indeed, following the eLIFT-FM VCTE algorithm, patients diagnosed as having no/mild fibrosis are not referred to a specialized hepatologist (Figure 2). The results obtained in the longitudinal cohort validated this management approach: the no/mild fibrosis patients had an excellent liver-related prognosis with only two deaths related to liver complication during the follow-up. Additionally, we have previously shown in chronic hepatitis C that the patient prognosis is linked to the evolution of the results of the non-invasive liver fibrosis tests [START_REF] Vergniol | Evolution of noninvasive tests of liver fibrosis is associated with prognosis in patients with chronic hepatitis C[END_REF]. Thus, in patients diagnosed as having no/mild fibrosis, the non-invasive tests can be repeated in the follow-up and, should the results change, spark the second-line test or a referral to a hepatologist.

Finally, as shown in Figure 2, eLIFT-FM VCTE clearly defines a pathway (algorithm) and the actors in that pathway (first-line non-hepatologist physicians, second-line platforms for FibroMeter VCTE realization, and finally specialized hepatologists for patients needing them).

In this way, it will minimize unnecessary referrals of CLD patients with no/mild fibrosis to specialized centers, and especially increase the detection of CLD patients with advanced fibrosis who need specialized management. A recent study showed that CLD patients with access to ambulatory gastrointestinal subspecialty care had improved survival [START_REF] Mellinger | Access to Subspecialty Care And Survival Among Patients With Liver Disease[END_REF]. In this setting, thanks to an increase in the detection of CLD patients with advanced fibrosis, we expect that the eLIFT-FM VCTE algorithm can improve the global CLD patient care.

Our algorithm is limited by the presence of a subgroup with undetermined diagnosis. Further works will be needed to see if a more accurate non-invasive fibrosis tests, such as magnetic resonance elastography [START_REF] Shi | Magnetic Resonance Elastography for the Evaluation of Liver Fibrosis in Chronic Hepatitis B and C by Using Both Gradient-Recalled Echo and Spin-Echo Echo Planar Imaging: A Prospective Study[END_REF], can provide a more precise diagnosis in this subgroup. Our study has included large populations with diagnostic and prognostic analyses, but it was performed in patients referred to hepatology units from tertiary centers. Further works are now required to validate the relevance of the eLIFT-FM VCTE for the screening of advanced liver fibrosis and cirrhosis in at-risk populations (diabetics, alcoholics, psychiatric patients, etc.) and in primary care settings. These works should include cost-effectiveness analyses to evaluate if the costs induced by the eLIFT-FM VCTE algorithm are counterbalanced by a better patient management and less expenses linked to decompensated liver diseases.

In conclusion, eLIFT is a new, user-friendly, at-a-glance fibrosis test available to all physicians, whether specialized in hepatology or not, who manage CLD patients in their daily clinical practice. The eLIFT-FM VCTE algorithm, which sequentially combines eLIFT and FibroMeter VCTE , can discriminate patients having no or mild fibrosis with excellent prognosis from those who have advanced CLD with impaired prognosis and need specialized care. Finally, the sequential eLIFT-FM VCTE algorithm defines a pathway that will help to regulate the large flow of CLD patients between primary care and specialized centers: patients with no or mild liver lesions will remain under the care of their usual physician whereas those with advanced CLD will be redirected to hepatologists for specialized management.

Figure 1 :

 1 Figure 1: Rate of patients with FIB4 results ≥1.20 as a function of age and the presence of advanced fibrosis in the whole cross-sectional population. Patients were ranked by age and categorized into 20 equally-populated, consecutive subgroups (5.0% percentiles). Then, the prevalence of FIB4 ≥1.20 was determined in each of these 20 subgroups, both in patients with no/mild fibrosis and in patients with advanced fibrosis.

Figure 2 :Figure 3 :

 23 Figure 2: New eLIFT-FM VCTE algorithm that combines the eLIFT (first-line "simple" test) and the FibroMeter VCTE (second-line "diagnostic" test) in a stepwise manner

Table 1 :

 1 Patient characteristics at inclusion in the cross-sectional population

		All	Derivation	Validation	p
		(n=3,754)	(n=2,503)	(n=1,251)
	Age (years)	49.5 ± 13.1	49.6 ± 13.1	49.4 ± 13.1	0.798
	Male gender (%)	56.6	56.5	56.8	0.889
	Cause of CLD (%):				0.355
	-Alcohol	7.7	7.6	8.0
	-HCV	45.5	45.0	46.4
	-HBV	3.2	2.8	3.9
	-Co-infection HBV/HCV-HIV	5.0	5.2	4.7
	-NAFLD	34.2	34.8	32.9
	-Others	4.4	4.6	4.1
	Fibrosis (%)				0.513
	-No/mild fibrosis	54.0	54.7	52.8
	-Septal fibrosis	32.8	32.4	33.4
	-Cirrhosis	13.2	12.9	13.8
	Biopsy length (mm)	25 ± 10	25 ± 10	25 ± 10	0.557
	AST (IU/L)	57 ± 58	57 ± 61	56 ± 51	0.654
	ALT (IU/L)	79 ± 86	79 ± 88	78 ± 81	0.963
	GGT (IU/L)	122 ± 201	124 ± 198	119 ± 207	0.730
	Bilirubin (µmol/L)	13 ± 16	12 ± 13	13 ± 20	0.063
	Platelets (G/L)	221 ± 74	221 ± 75	220 ± 73	0.499
	Prothrombin time (%)	93 ± 12	93 ± 12	93 ± 12	0.378
	APRI	0.75 ± 0.96	0.75 ± 1.01	0.73 ± 0.85	0.620
	FIB4	1.84 ± 1.89	1.83 ± 1.87	1.84 ± 1.94	0.767
	CLD: chronic liver disease; HCV: hepatitis C virus; HBV: hepatitis B virus; Co-infection
	HBV/HCV-HIV: co-infection with hepatitis B or C virus and human immunodeficiency virus

Table 2 :

 2 The easy LIver Fibrosis Test (eLIFT)

	ITEM	POINTS
	Age (years)
	-<40	0
	-≥40	3
	Gender
	-Female	0
	-Male	1
	AST (IU/L)
	-<35	0
	-35 -69	2
	-≥70	4
	Gamma-GT (IU/L)
	-<35	0
	-35 -89	1
	-≥90	2
	Platelets (G/L)
	-250 ≤	0
	-170 -249	1
	-<170	4
	Prothrombin time (%)
	-97 ≤	0
	-84 -96	2
	-<84	4

Table 3 :

 3 Accuracy of the eLIFT-FM VCTE algorithm for the diagnosis of advanced fibrosis in the core group of the cross-sectional population.

		All	Derivation	Validation	p
		(n=1,946)	(n=1,282)	(n=664)
	DA	84.4	84.9	83.3	0.357
	Se	78.2	79.4	76.1	0.238
	Spe	91.4	90.9	92.2	0.615
	NPV	78.8	80.4	75.6	-
	PPV	91.1	90.4	92.4	-
	-LR	0.24	0.23	0.26	-
	+LR	9.0	8.8	9.8	-
	OR	37.9	38.6	37.6	-
	LB	19.3	19.3	19.4	0.952

DA: diagnostic accuracy (rate of well-classified patients, %); Se: sensitivity (%); Spe: specificity (Spe); NPV: negative predictive value (%); PPV: positive predictive value (%); -LR: negative likelihood ratio; +LR: positive likelihood ratio; OR: Odds Ratio; LB: rate of liver biopsy required (%)

ACKNOWLEDGMENTS

This work received grand support from the Angers University Hospital