R Émi

Ément Gallo

WKB ANALYSIS OF NON-ELLIPTIC NONLINEAR SCHR ÖDINGER EQUATIONS

We justify the WKB analysis for generalized nonlinear Schrödinger equations (NLS), including the hyperbolic NLS and the Davey-Stewartson II system. Since the leading order system in this analysis is not hyperbolic, we work with analytic regularity, with a radius of analyticity decaying with time, in order to obtain better energy estimates. This provides qualitative information regarding equations for which global well-posedness in Sobolev spaces is widely open.

i∂ t ψ + 1 2 ∂ 2 1 ψ - 1 2 ∂ 2 2 ψ ± |ψ| 2 ψ = 0, (x 1 , x 2 ) ∈ R 2 ,
appears in nonlinear optics (see e.g. [START_REF] Dumas | Variants of the focusing NLS equation: derivation, justification, and open problems related to filamentation[END_REF][START_REF] Sulem | The nonlinear Schrödinger equation, Self-focusing and wave collapse[END_REF]), but remains quite mysterious as far as analysis is concerned: it is locally well-posed in H s (R 2 ) for any s > 0, it is L 2 -critical, hence locally well-posed in L 2 (R 2 ) (with a suitable definition of local well-posedness in this critical case), but apart from the small data case, the global existence issue remains a delicate issue in such spaces, even though refined Strichartz estimates are available [START_REF] Rogers | A refinement of the Strichartz inequality on the saddle and applications[END_REF], because the conserved energy is not a positive functional,

E = ∂ 1 ψ 2 L 2 (R 2 ) -∂ 2 ψ 2 L 2 (R 2 ) ∓ ψ 4 L 4 (R 2 )
. Note that the sign of the nonlinearity is rather irrelevant, since we may exchange the roles of x 1 and x 2 . However, global existence in H s (R 2 ) for s > 0 is obtained through modulation approximation in [START_REF] Totz | Global well-posedness of 2D non-focusing Schrödinger equations via rigorous modulation approximation[END_REF]. On the other hand, global solutions under the form of spatial standing waves have been studied in [START_REF] Correia | Some L ∞ solutions of the hyperbolic nonlinear Schrödinger equation and their stability[END_REF][START_REF] Kevrekidis | Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D[END_REF], along with their stability.

Similarly, the Davey-Stewartson system (1.2)

   i∂ t ψ + 1 2 ∂ 2 1 ψ - 1 2 ∂ 2 2 ψ = χ|ψ| 2 + ω∂ 1 φ ψ, (x 1 , x 2 ) ∈ R 2 , χ, ω ∈ R, ∂ 2 1 φ + ∂ 2 2 φ = ∂ 1 |ψ| 2 ,
is locally well-posed in the same spaces, L 2 -critical, and enjoys a Hamiltonian structure with an energy whose sign is indefinite. Indeed, (1.2) can be rewritten

i∂ t ψ + 1 2 ∂ 2 1 ψ - 1 2 ∂ 2 2 ψ = χ|ψ| 2 + ωK * |ψ| 2 ψ, 1
where the symmetric kernel is such that

K(ξ) = ξ 2 1 ξ 2 1 + ξ 2 2 .
On the other hand, for a suitable combination of the coefficients χ and ω, that is 2χ + ω = 0, (1.2) is completely integrable (see e.g. [START_REF] Ghidaglia | On the initial value problem for the Davey-Stewartson systems[END_REF][START_REF] Klein | Davey-Stewartson systems[END_REF]). Global well-posedness and scattering in L 2 for the defocusing case were recently established in this specific case thanks to inverse scattering and harmonic analysis techniques, see [START_REF] Nachman | A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey-Stewartson equation and to the inverse boundary value problem of Calderón[END_REF]. In this note, we justify the approximation of such equations in a high frequency regime, known as semi-classical limit, this giving some extra information concerning the dynamics associated to these equations.

1.2. Setting. We consider the equation, including both (1.1) and (1.2), (1.3)

iε∂ t u ε + ε 2 2 D 2 u ε + iε β, ∇ g |u ε | 2 u ε = V u ε + J j=1 K j * |u ε | 2σj u ε ,
in the semi-classical limit ε → 0, where ε > 0, T > 0, d 1 is the spatial dimension, J 1, and (t, x) ∈ [0, T ] × E d . More specifically,

• u ε (t, x) ∈ C is the wave function • E d can be either R d or to the torus

T d = (R/2πZ) d , • D 2 = d j,k=1 η j,k ∂ j ∂ k = ∇, H∇ ,
where H = (η j,k ) 1 j,k d is a symmetric (not necessarily positive or invertible) real matrix, and •, • denotes the inner product on R d , • β = (β j ) 1 j d ∈ R d and g(s) = αs γ , where α ∈ R and γ ∈ N\{0}. We consider such a function g in order to simplify the notations, but our method also works if g is not a monomial. • For j ∈ {1, . . . , J}, σ j ∈ N\{0} is an integer, and K j denotes a tempered distribution with a bounded Fourier transform K j ∈ L ∞ (E d ). This covers the case where K = δ, typically as in (1.1).

• V = V (t, x) is a potential. V is supposed to be analytic in the x variable.
More precisely, we assume that V belongs to the space

L 2 T0 H +1/2 w0
for some T 0 > 0, w 0 > 0, > (d + 1)/2, a space that will be defined below. Our motivation for considering the case E d = T d lies in the fact that numerical simulations are often performed in a periodic box: unless suitable absorbing boundary conditions are imposed, the observed dynamics is that of (1.3) on T d , which is fairly different from the one on R d . Remark 1.1. In view of the assumption that is usually made on V in order to get H s solutions (namely, ∂ α V ∈ L ∞ for 2 |α| s, see e.g. [START_REF] Carles | WKB analysis for nonlinear Schrödinger equations with potential[END_REF]), it is reasonable to ask for analyticity of V in order to get analytic solutions.

Remark 1.2. In a similar fashion as we consider an external potential, our analysis exports to the magnetic case, where

D 2 = d j,k=1 η j,k (∂ j -iA j ) (∂ k -iA k ) ,
provided that the magnetic potentials A j are analytic (in the same sense as for V ).

The initial data that we consider are WKB states:

(1.4) u ε (0, x) = a ε 0 (x)e iφ ε 0 (x)/ε =: u ε 0 (x)
, where φ ε 0 : E d → R is a real-valued phase, and a ε 0 : E d → C is a possibly complexvalued amplitude. We emphasize that our approach is distinct from the polar decomposition known as Madelung transform, hence the possibility for the amplitude to be (or become) complex. Our goal is to understand the semi-classical limit of equation (1.3), that is to describe the behavior in the limit ε → 0 of the solutions to (1.3) with initial data (1.4). Generalizing the idea of [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF], we remark that if (φ ε , a ε ) solves the system (1.4). Therefore, we focus on (1.5). Note that φ ε , there, remains realvalued, while a ε will be complex-valued (even if a ε 0 is real), due to the term iεD 2 a ε , which is a remain of dispersive effects in the initial Schrödinger equation.

(1.5)                    ∂ t φ ε + 1 2 ∇φ ε , H∇φ ε + g(|a ε | 2 ) β, ∇φ ε + J j=1 K j * |a ε | 2σj + V = 0, φ ε |t=0 = φ ε 0 , ∂ t a ε + ∇φ ε , H∇a ε + 1 2 a ε D 2 φ ε + β, ∇ g(|a ε | 2 )a ε = iε 2 D 2 a ε , a ε |t=0 = a ε 0 , then u ε (t, x) = a ε (t, x)e iφ ε (t,x)/ε solves (1.3)-
Example 1.3. In the case of the hyperbolic NLS (1.1), d = 2 and the equations in (1.5) read

     ∂ t φ ε + 1 2 |∂ 1 φ ε | 2 -|∂ 2 φ ε | 2 ∓ |a ε | 2 = 0, ∂ t a ε + ∂ 1 φ ε ∂ 1 a ε -∂ 2 φ ε ∂ 2 a ε + 1 2 a ε ∂ 2 1 φ ε -∂ 2 2 φ ε = iε 2 ∂ 2 1 a ε -∂ 2 2 a ε .
Passing formally to the limit ε → 0, and setting ρ = |a| 2 , v = ∇φ, and ṽ = (∂ 1 φ, -∂ 2 φ) T , we find, respectively,

∂ t v j + v 1 ∂ 1 v j -v 2 ∂ 2 v j ∓ ∂ j ρ = 0, ∂ t ρ + ∂ 1 (ρv 1 ) -∂ 2 (ρv 2 ) = 0,    ∂ t ṽ + ṽ, ∇ ṽ ∓ ∂ 1 ρ -∂ 2 ρ = 0, ∂ t ρ + ∇, ρṽ = 0.
No special structure such as symmetry (ensuring the hyperbolicity of the system) seems to be available here. Because of this, we work with analytic regularity, since Sobolev regularity is hopeless in such a case (see [START_REF] Lerner | The onset of instability in first-order systems[END_REF][START_REF] Métivier | Remarks on the well-posedness of the nonlinear Cauchy problem[END_REF]).

In [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF], we have already addressed the issue of the semi-classical limit of (1.3) in the case where d = 1,

E d = R, D 2 = ∂ 2
x , where the nonlinearity is local (that is, K = δ) and where V = 0. We show that the method that was used in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF] Note that at least for the first three aspects evoked above, our approach provides results which can be established by following the same strategy as in [START_REF] Gérard | Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire[END_REF] (case x ∈ T d ) and [START_REF] Thomann | Instabilities for supercritical Schrödinger equations in analytic manifolds[END_REF] (case x ∈ R d ), based on the notion of analytic symbols, as developed by J. Sjöstrand [START_REF] Sjöstrand | Singularités analytiques microlocales[END_REF]. On the other hand, incorporating nonlocal nonlinearities seems to be easier when relying on a notion of (time dependent) analyticity based on Fourier analysis, as in [START_REF] Ginibre | Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions[END_REF] (and [START_REF] Mouhot | On Landau damping[END_REF] for the same idea in a different context); see the next subsection for more details.

1.3. The functional framework. For w 0 and 0, we consider the space

H w = {ψ ∈ L 2 (E d ), ψ H w < ∞},
where

ψ 2 H w :=        R d ξ 2 e 2w ξ | ψ(ξ)| 2 dξ if E d = R d , m∈Z d m 2 e 2w m | ψ(m)| 2 if E d = T d , with ξ = 1 + |ξ| 2
, and where the Fourier transform and series are defined by

ψ(ξ) = Fψ(ξ) = 1 (2π) d/2 R d ψ(y)e -i ξ,y dy if E d = R d , ψ(m) = Fψ(m) = 1 (2π) d/2 T d ψ(y)e -i m,y dy if E d = T d .
We obviously have the monotonicity property, (1.6) 0

w 1 w 2 =⇒ ψ H w 1 ψ H w 2 .
The interest of considering a time-dependent, decreasing, weight w is that energy estimates become similar to parabolic estimates, since

(1.7) d dt ψ 2 H w = 2 Re (ψ, ∂ t ψ) H w + 2 ẇ ψ 2 H +1/2 w
, where (•, •) H w denotes the natural inner product stemming from the above definition. We choose a weight w = w(t) = w 0 -M t, where w 0 > 0 and M > 0 are fixed. For T > 0, we work in spaces such as

C([0, T ], H w ) = ψ | F -1 e w(t) ξ ψ ∈ C([0, T ], H 0 ) = C([0, T ], H ) , where H = H (E d ) is the standard Sobolev space, or L 2 ([0, T ], H w ) = L 2 T H w = ψ | T 0 ψ(t) 2 H w(t) dt < ∞ .
Phases and amplitudes belong to spaces

Y w,T = C([0, T ], H w ) ∩ L 2 T H +1/2 w ,
and the fact that phase and amplitude do not have exactly the same regularity shows up in the introduction of the space

X w,T = Y +1 w,T × Y w,T
, which is reminiscent of the fact that in the case where the operator on the left hand side of (1.5) is hyperbolic (typically, starting from a defocusing cubic Schrödinger equation with the standard Laplacian), the good unknown is (∇φ ε , a ε ) rather than (φ ε , a ε ) (see [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF]). The space X w,T is endowed with the norm (φ, a) X w,T =|||φ||| +1,T +|||a||| ,T , where

(1.8) |||ψ||| 2 ,t = max sup 0 s t ψ(s) 2 H w(s) , 2M t 0 ψ(s) 2 H +1/2 w(s)
ds .

1.4. Main results. Our first result states local well-posedness for (1.5) in this functional framework. [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF], where w(t) = w 0 -M t and T = T ( ) < min (w 0 /M, T 0 ). Moreover, up to the choice of a possibly larger value for M (and consequently a smaller one for T ), we have the estimates

Theorem 1.4. Let w 0 > 0, > (d+1)/2, T 0 > 0, V ∈ L 2 T0 H +1/2 w0 and (φ ε 0 , a ε 0 ) ε∈[0,1] be a bounded family in H +1 w0 × H w0 . Then, provided M = M ( ) > 0 is chosen sufficiently large, for all ε ∈ [0, 1], there is a unique solution (φ ε , a ε ) ∈ X w,T to (1.
|||φ ε ||| 2 +1,T 4 φ ε 0 2 H +1 w 0 + max 1 j J a ε 0 4σj H w 0 + V 2 L 2 T 0 H +1/2 w 0 , |||a ε ||| 2 ,T 2 a ε 0 2 H w 0 .
An important aspect in the above statement is the fact that the local existence time T is uniform in ε ∈ [0, 1]. In view of the discussion in Subsection 1.2, this yields a uniform time of existence for the solution of (1.3). We emphasize that this property is not a consequence of the standard local well-posedness argument (based on a fixed point), which would yield a local existence time T ε = O(ε α ) for some α 1, while we recall that the a priori estimates do not make it possible to extend the local solution to much larger time. In other words, the formulation (1.5) is already helpful at the level of the life-span of the solution to (1.3).

Our second result states the convergence of the phase and of the complex amplitude as ε → 0.

Theorem 1.5. Let w 0 > 0, > (d + 1)/2, T 0 > 0, V ∈ L 2 T0 H +3/2 w0 , (φ 0 , a 0 ) ∈ H +2 w0 × H +1 w0 and (φ ε 0 , a ε 0 ) ε∈(0,1] bounded in H +1 w0 × H w0 such that r ε 0 := φ ε 0 -φ 0 H +1 w 0 + a ε 0 -a 0 H w 0 -→ ε→0 0.
Let M = M ( + 1) and T = T ( + 1), as defined as in Theorem 1.4. Then there is an ε-independent C > 0 such that for all ε ∈ (0, 1],

|||φ ε -φ||| +1,T +|||a ε -a||| ,T C (r ε 0 + ε)
, where (φ ε , a ε ) denotes the solution to (1.5) and (φ, a) is the solution to the formal limit of (1.5) as ε → 0 (1.9)

                   ∂ t φ + 1 2 ∇φ, H∇φ + g |a| 2 β, ∇φ + J j=1 K j * |a| 2σj + V = 0, φ |t=0 = φ 0 , ∂ t a + ∇φ, H∇a + 1 2 aD 2 φ + β, ∇ g |a| 2 a = 0, a |t=0 = a 0 ,
whose existence and uniqueness stem from Theorem 1.4.

However, regarding convergence of the wave function u ε , the previous result is not sufficient. Indeed, as fast as the initial data φ ε 0 and a ε 0 may converge as ε → 0, Theorem 1.5 at most guarantees that φ ε -φ = O(ε), which only ensures that a ε e iφ ε /ε -ae iφ/ε = O(1), due to the rapid oscillations. However, the above convergence result suffices to infer the convergence of quadratic observables: Corollary 1.6. Under the assumptions of Theorem 1.5, the position and momentum densities converge:

|u ε | 2 -→ ε→0 |a| 2 , and Im (εū ε ∂u ε ) -→ ε→0 |a| 2 ∂φ, in L ∞ ([0, T ]; L 1 ∩ L ∞ (E d )), where ∂ = ∂ j , for any j ∈ {1, • • • , d}.
In order to get a good approximation of the wave function a ε e iφ ε /ε , we have to approximate φ ε up to an error which is small compared to ε. It will be done by adding a corrective term to (φ, a). For this purpose, we consider the system obtained by linearizing (1.5) about (φ, a), solution to (1.9), (1.10)

                                 ∂ t φ 1 + ∇φ, H∇φ 1 + g |a| 2 β, ∇φ 1 + 2g |a| 2 β, ∇φ Re (aa 1 ) + 2 J j=1 σ j K j * |a| 2(σj -1) Re (aa 1 ) = 0, φ 1|t=0 = φ 10 , ∂ t a 1 + ∇φ, H∇a 1 + 1 2 a 1 D 2 φ + ∇a, H∇φ 1 + 1 2 aD 2 φ 1 + β, ∇ g |a| 2 a 1 + 2 β, ∇ ag |a| 2 Re (aa 1 ) = i 2 D 2 a, a 1|t=0 = a 10 .
Provided (φ 0 , a 0 ) ∈ H +3 w0 × H +2 w0 (which implies (φ, a) ∈ X +2 w,T according to Theorem 1.4) and (φ 10 , a 10 ) ∈ H +2 w0 × H +1 w0 , we will see that the solution to (1.10) belongs to X +1 w,T , and our final result is the following.

Theorem 1.7. Let w 0 > 0, > (d + 1)/2, T 0 > 0, V ∈ L 2 T0 H +5/2 w0 , (φ 0 , a 0 ) ∈ H +3 w0 × H +2 w0 , (φ 10 , a 10 ) ∈ H +2 w0 × H +1 w0 and (φ ε 0 , a ε 0 ) ε∈(0,1] bounded in H +1 w0 × H w0 such that r ε 1 := φ ε 0 -φ 0 -εφ 10 H +1 w 0 + a ε 0 -a 0 -εa 10 H w 0 = o(ε) as ε → 0.
Then, for M = M ( + 2) and T = T ( + 2) as in Theorem 1.4, there is an εindependent C > 0 such that for all ε ∈ (0, 1],

(1.11)

|||φ ε -φ -εφ 1 ||| +1,T +|||a ε -a -εa 1 ||| ,T C r ε 1 + ε 2
, where (φ ε , a ε ) denotes the solution to (1.5), (φ, a) is the solution to (1.9), and (φ 1 , a 1 ) is the solution to (1.10). In particular,

u ε -ae iφ1 e iφ/ε L ∞ ([0,T ];L 2 ∩L ∞ (E d )) = O r ε 1 ε + ε -→ ε→0 0.
Outline. In Section 2, we prove Theorem 1.4, by starting with a generalization of key estimates established in [START_REF] Ginibre | Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions[END_REF] to the periodic setting E d = T d . Theorem 1.5 is proved in Section 3, and the proof of Theorem 1.7 is sketched in the final Section 4.

Well-posedness

2.1.

A key bilinear estimate. The following proposition is proved in [START_REF] Ginibre | Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions[END_REF] in the case E d = R d in the context of long range scattering. We have used it in [START_REF] Carles | On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit II. Analytic regularity[END_REF][START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF] in the context of semi-classical analysis. We extend it here to the case

E d = T d .
Proposition 2.1. Let 0 and s > d/2. Then, for every

ψ 1 , ψ 2 ∈ H max( ,s) w , (2.1
)

ψ 1 ψ 2 H w C ,s ψ 1 H w ψ 2 H s w + ψ 1 H s w ψ 2 H w , where
C ,s =      2 (2π) d/2 1 • s L 2 (R d ) if E d = R d , 2 (2π) d/2 1 • s 2 (Z d ) if E d = T d .
We detail the proof only in the case E d = T d . The proof is analogous in the case E d = R d , and can be found in [START_REF] Ginibre | Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions[END_REF] (with different notations, though). Proposition 2.1 in the case E d = T d stems from the following sequence of lemmas. We skip the proofs of the most classical ones.

Lemma 2.2. For all m, n ∈ R d , m + n m + n . Lemma 2.3. If ψ 1 , ψ 2 ∈ L 2 (T d ), for all m ∈ Z d , ψ 1 ψ 2 (m) = 1 (2π) d/2 k∈Z d ψ 1 (k) ψ 2 (m -k) =: 1 (2π) d/2 ψ 1 * ψ 2 (m). Lemma 2.4. For , w 0, if ψ 1 , ψ 2 ∈ H w , then ψ 1 ψ 2 H w 2 (2π) d/2 • e w • | ψ 1 | * e w • | ψ 2 | 2 (Z d ) + e w • | ψ 1 | * • e w • | ψ 2 | 2 (Z d )
Proof. From Lemma 2.3,

ψ 1 ψ 2 2 H w 1 (2π) d m∈Z d   n∈Z d m e w m | ψ 1 (n)|| ψ 2 (m -n)|   2 .
From Lemma 2.2 and because for any m, n ∈ Z d , we have either n m /2 or m -n m /2, we deduce

ψ 1 ψ 2 2 H w 1 (2π) d m∈Z d   n∈Z d , n m /2 m e w n | ψ 1 (n)|e w m-n | ψ 2 (m -n)| + n∈Z d , m-n m /2 m e w n | ψ 1 (n)|e w m-n | ψ 2 (m -n)|   2 2 2 (2π) d m∈Z d   n∈Z d n e w n | ψ 1 (n)|e w m-n | ψ 2 (m -n)| + n∈Z d e w n | ψ 1 (n)| m -n e w m-n | ψ 2 (m -n)|   2 .
The result follows thanks to the triangle inequality in 2 (Z d ).

Lemma 2.5. If u ∈ 2 (Z d ) and v ∈ 1 (Z d ), then u * v ∈ 2 (Z d ), and u * v 2 (Z d ) u 2 (Z d ) v 1 (Z d )
Proof of Proposition 2.1. Let us estimate the first term in the bracket of the right hand side of the inequality in Lemma 2.4. The other term is treated similarly. According to Lemma 2.5, we have

• e w • | ψ 1 | * e w • | ψ 2 | 2 (Z d ) • e w • | ψ 1 | 2 (Z d ) e w • | ψ 2 | 1 (Z d ) • e w • | ψ 1 | 2 (Z d ) 1 • s 2 (Z d ) • s e w • | ψ 2 | 2 (Z d ) 1 • s 2 (Z d ) ψ 1 H w ψ 2 H s w ,
where we have also used the Cauchy-Schwarz inequality.

2.2. The iterative scheme. In this section, ε ∈ [0, 1] is fixed. To lighten the notations, we consider the case J = 1 (only one Fourier multiplier), and leave out the corresponding index: the proof shows that considering finitely many such terms is straightforward. Solutions to (1.5) are constructed as limits of the solutions of the iterative scheme (2.2)

                       ∂ t φ ε j+1 + 1 2 ∇φ ε j , H∇φ ε j+1 + g(|a ε j | 2 ) β, ∇φ ε j+1 = -K * |a ε j | 2σ -V, φ ε j+1|t=0 = φ ε 0 , ∂ t a ε j+1 + ∇φ ε j , H∇a ε j+1 + 1 2 (D 2 φ ε j )a ε j+1 + β, ∇ g(|a ε j | 2 ) a ε j+1 + h(|a ε j | 2 )a ε j β, ∇a ε j a ε j+1 = iε 2 D 2 a ε j+1 , a ε j+1|t=0 = a ε 0 ,
where h(s) = g(s)/s. The scheme is initialized with the time-independent pair (φ ε 0 , a ε 0 ) ∈ H +1 w0 × H w0 ⊂ X w,T for any T > 0. The scheme is well-defined: if > (d + 1)/2, for a given (φ ε j , a ε j ) ∈ X w,T , (2.2) defines (φ ε j+1 , a ε j+1 ). Indeed, in the first equation, φ ε j+1 solves a linear transport equation with smooth coefficients, which guarantees the existence of a solution

φ ε j+1 ∈ L ∞ T L 2 (see e.g [2, Section 3], or [1, Section II.C, Proposition 1.2]
). The same argument provides a solution a ε j+1 ∈ L ∞ T L 2 to the second equation in the case ε = 0. On the other hand, if ε > 0, the second equation is equivalent through the relation v ε j+1 = a ε j+1 e iφ ε j /ε to the equation

iε∂ t v ε j+1 + ε 2 2 D 2 v ε j+1 + W (t, x)v ε j+1 = 0, (2.3) with initial condition v ε j+1|t=0 = v ε 0 = a ε 0 e iφ ε 0 /ε ,
where

W = ∂ t φ ε j + 1 2 ∇φ ε j , H∇φ ε j + iε β, ∇ g |a ε j | 2 + iεh(|a ε j | 2 )a ε j β, ∇a ε j .
This is a linear Schrödinger like equation, with a second order operator D 2 which is not necessarily elliptic, and with a smooth and bounded external time-dependent potential W (t, x). Note that this external potential is complex-valued, so the existence of a solution for (2.3) is not quite standard. On the other hand, a fixed point argument applied to the map

Ψ : C([0, T ε ], L 2 ) -→ C([0, T ε ], L 2 ) u → e iεtD 2 /2 v ε 0 + i ε t 0 e iε(t-s)D 2 /2 [W (s)u(s)] ds, where 0 < T ε T , provides the existence of a C([0, T ε ], L 2 ) solution to (2.3), for some T ε > 0. We actually have v ε j+1 ∈ C([0, T ], L 2 ), since W ∈ L ∞ ([0, T ] × E d ).
The following lemma gives the estimates that will ensure that (φ ε j+1 , a

ε j+1 ) ∈ X w,T provided (φ ε j , a ε j ) ∈ X w,T . It is almost identical to Lemma 2.2 in [5]. Lemma 2.6. Let > (d + 1)/2 and T > 0. Let (φ, a) ∈ X w,T , ã ∈ Y +1 w,T and (F, G) ∈ L 2 ([0, T ], H +1/2 w × H -1/2 w ) such that ∂ t φ = F, φ(0) ∈ H +1 w0 , (2.4) ∂ t a = G + iθ 1 D 2 a + iθ 2 D 2 ã, a(0) ∈ H w0 , (2.5) where θ 1 , θ 2 ∈ R. Then |||φ||| 2 +1,T φ(0) 2 H +1 w 0 + 1 M |||φ||| +1,T √ 2M F L 2 T H +1/2 w , (2.6) |||a||| 2 ,T a(0) 2 H w 0 + 1 M |||a||| ,T √ 2M G L 2 T H -1/2 w + |θ 2 | 2M |||a||| ,T |||ã||| +1,T . (2.7)
Moreover, there exists C > 0 (that depends only on , not on w) such that

• If F = ∂ j ψ 1 ∂ k ψ 2 with ψ 1 , ψ 2 ∈ Y +1,T , then (2.8)
√ 2M F L 2 T H +1/2 w C|||ψ 1 ||| +1,T |||ψ 2 ||| +1,T . • If F = 2n j=1 b j ∂ k ψ with n 1, ψ ∈ Y +1,T and b j ∈ Y ,T for all j, then (2.9)
√ 2M F L 2 T H +1/2 w C   2n j=1 |||b j ||| ,T   |||ψ||| +1,T . • If F = K * 2n j=1 b j with n 1, b j ∈ Y ,T
for all j and K uniformly bounded, then

(2.10)

√ 2M F L 2 T H +1/2 w C   2n j=1 |||b j ||| ,T   . • If G = ∂ j ψ∂ k b with ψ ∈ Y +1,T and b ∈ Y ,T , then (2.11) √ 2M G L 2 T H -1/2 w C|||ψ||| +1,T |||b||| ,T . • If G = bD 2 ψ with ψ ∈ Y +1,T and b ∈ Y ,T , then (2.12) √ 2M G L 2 T H -1/2 w C|||ψ||| +1,T |||b||| ,T . • If G = 2n j=1 b j ∂ k b with n 1, b, b j ∈ Y ,T for all j, then (2.13)
√ 2M G L 2 T H -1/2 w C   2n j=1 |||b j ||| ,T   |||b||| ,T .
Proof. The proof of (2.6) and (2.7) is identical to the one given in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF]. The new constraint > (d + 1)/2 plays no role here. Inequalities similar to (2.8)-(2.13) were proved in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF]. The only differences with [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF]

K *   2n j=1 b j   H +1/2 w K L ∞ 2n j=1 b j H +1/2 w C 2n j=1 j-1 k=1 b k H w b j H +1/2 w 2n k=j+1 b k H w .
In order to prove (2.11)-(2.13), we use (2.1) with m = s = -1/2 > d/2 which is possible thanks to the assumption > (d+1)/2. Actually, even (2.11)-(2.13) can be proved under the condition > d/2, thanks to a refined version of Lemma 2.1 (see [START_REF] Carles | On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit II. Analytic regularity[END_REF]). However, since it is not useful in the sequel to sharpen this assumption, we choose to make the stronger assumption > (d+1)/2 for the sake of conciseness.

First step: boundedness of the sequence. In view of the equation satisfied by φ ε j+1 in (2.2), Lemma 2.6 yields

|||φ ε j+1 ||| 2 +1,T φ ε 0 2 H +1 w 0 + C M |||φ ε j+1 ||| 2 +1,T |||φ ε j ||| +1,T + C M |||φ ε j+1 ||| +1,T |||a ε j ||| 2σ ,T + C M |||φ ε j+1 ||| 2 +1,T |||a ε j ||| 2γ ,T + C M |||φ ε j+1 ||| +1,T √ 2M V L 2 T H +1/2 w .
As for a ε j+1 , we obtain in a similar way

|||a ε j+1 ||| 2 ,T a ε 0 2 H w 0 + C M |||a ε j+1 ||| 2 ,T |||φ ε j ||| +1,T + C M |||a ε j+1 ||| 2 ,T |||a ε j ||| 2γ ,T
. Up to the term with V in the first one, the last two estimates are exactly the ones we had in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF]. The proof of the boundedness of the sequence (φ ε j , a ε j ) in X w,T is quite similar to what was done in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF]. Indeed, under the assumption

(2.14) C M |||φ ε j ||| +1,T 1 4 , C M |||a ε j ||| 2γ ,T 1 4 ,
we have

1 4 |||φ ε j+1 ||| 2 +1,T φ ε 0 2 H +1 w 0 + 2C 2 M 2 |||a ε j ||| 4σ ,T + 4C 2 M V 2 L 2 T H +1/2 w (2.15) and 1 2 |||a ε j+1 ||| 2 ,T a ε 0 2 H w 0 . (2.16)
Note that the monotonicity property (1.6) implies

V L 2 T H +1/2 w V L 2 T 0 H +1/2 w 0 .
We next show by induction that, provided M is sufficiently large, we can construct a sequence (φ ε j , a ε j ) j∈N such that for every j ∈ N,

|||φ ε j ||| 2 +1,T 4 φ ε 0 2 H +1 w 0 + 8C 2 M 2 2 a ε 0 2 H w 0 2σ + 16C 2 M V 2 L 2 T 0 H +1/2 w 0 , (2.17) |||a ε j ||| 2 ,T 2 a ε 0 2 H w 0 . (2.18)
For that purpose, we choose M sufficiently large such that (2.14) holds for j = 0 and such that

4 φ ε 0 2 H +1 w 0 + 8C 2 M 2 2 a ε 0 2 H w 0 2σ + 16C 2 M V 2 L 2 T 0 H +1/2 w 0 M 2 16C 2 , (2.19) and (2 a ε 0 2 H w 0 ) γ M 4C . (2.20)
Then, (2.17)-(2.18) hold for j = 0, since with

(φ ε 0 , a ε 0 )(t, x) = (φ ε 0 , a ε 0 )(x) indepen- dent of time, it is easy to check that |||φ ε 0 ||| +1,T = φ ε 0 H +1 w 0 and |||a ε 0 ||| ,T = a ε 0 H w 0
. Let j 0 and assume that (2.17)-(2.18) hold. Then (2.17)-(2.18) and (2.19)-(2.20) ensure that the condition (2.14) is satisfied, and therefore (2.15)-(2.16) hold, from which we infer easily that (2.17)-(2.18) are true for j replaced by j + 1 (for the estimate on the norm of V , we use here the fact that T T 0 and w w 0 ). Second step: convergence. For j 1, we set δφ ε j = φ ε j -φ ε j-1 , and δa ε j = a ε j -a ε j-1 . Then, for every j 1, we have

∂ t δφ ε j+1 + 1 2 ∇φ ε j , H∇δφ ε j+1 + ∇δφ ε j , H∇φ ε j + g |a ε j | 2 β, ∇δφ ε j+1 + g |a ε j | 2 -g |a ε j-1 | 2 β, ∇φ ε j + K * |a ε j | 2σ -|a ε j-1 | 2σ = 0. and ∂ t δa ε j+1 + ∇φ ε j , H∇δa ε j+1 + ∇δφ ε j , H∇a ε j + 1 2 δa ε j+1 D 2 φ ε j + 1 2 a ε j D 2 δφ ε j + β, ∇ g |a ε j | 2 δa ε j+1 + β, ∇ g |a ε j | 2 -g |a ε j-1 | 2 a ε j + h |a ε j | 2 β, ∇a ε j a ε j δa ε j+1 + h |a ε j | 2 β, ∇a ε j δa ε j a ε j + h |a ε j | 2 β, ∇δa ε j a ε j-1 a ε j + h |a ε j | 2 -h |a ε j-1 | 2 β, ∇a ε j-1 a ε j-1 a ε j = i ε 2 D 2 δa ε j+1 ,
Lemma 2.6 and the boundedness of (φ ε j , a ε j ) in X w,T imply like in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF] that for M large enough,

max |||δφ ε j+1 ||| 2 +1,T , |||δa ε j+1 ||| 2 ,T K M |||δφ ε j ||| 2 +1,T +|||δa ε j ||| 2 ,T
for some K > 0 which does not depend on ε provided (φ ε 0 , a ε 0 ) ε∈[0,1] is uniformly bounded in H +1 w0 × H w0 . We conclude as in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF] that provided > (d + 1)/2, possibly increasing M , (φ ε j , a ε j ) converges geometrically in X w,T as j → ∞. Uniqueness of the solution (φ ε , a ε ) to (1.5) follows from the same kind of estimates as the ones which prove the convergence.

First order approximation

As in the previous section, we assume J = 1 for the sake of conciseness.

Proof of Theorem 1.5. Next, assume that (φ 0 , a 0 ) ∈ H +2 w0 × H +1 w0 . Then, in view of Theorem 1.4, the solution (φ, a) to (1.9) belongs to X +1 w,T . Given ε > 0, if (φ ε 0 , a ε 0 ) ∈ H +1 w0 × H w0 , we denote by (φ ε , a ε ) the solution to (1.5). We also denote (δφ ε , δa ε ) = (φ ε -φ, a ε -a). Then, in the same fashion as above, we have

∂ t δφ ε + 1 2 ( ∇δφ ε , H∇φ ε + ∇φ, H∇δφ ε ) + g |a ε | 2 β, ∇δφ ε + g |a ε | 2 -g |a| 2 β, ∇φ + K * |a ε | 2σ -|a| 2σ = 0 and ∂ t δa ε + ∇δφ ε , H∇a ε + ∇φ, H∇δa ε + 1 2 δa ε D 2 φ ε + 1 2 aD 2 δφ ε + β, ∇ g |a ε | 2 δa ε + β, ∇ g |a ε | 2 -g |a| 2 a + h |a ε | 2 β, ∇a ε a ε δa ε + h |a ε | 2 β, ∇a ε δa ε a + h |a ε | 2 β, ∇δa ε |a| 2 + h |a ε | 2 -h |a| 2 β, ∇a |a| 2 = i ε 2 D 2 δa ε + i ε 2 D 2 a.
Like in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF], for some new constant k, Lemma 2.6 and Theorem 1.4 imply, for M large enough,

|||δφ ε ||| 2 +1,T k φ ε 0 -φ 0 2 H +1 w 0 + k M |||δa ε ||| 2 ,T , and
|||δa ε ||| 2 ,T k a ε 0 -a 0 2 H w 0 + k M |||δφ ε ||| 2 +1,T + k M ε|||δa ε ||| ,T |||a||| +1,T .
Possibly increasing the value of M and adding the last two inequalities, we deduce

|||δφ ε ||| 2 +1,T +|||δa ε ||| 2 ,T C φ ε 0 -φ 0 2 H +1 w 0 + C a ε 0 -a 0 2 H w 0 + Cε 2 ,
hence Theorem 1.5. As for the choice of M , a careful examination of the previous inequalities shows that aside from the assumption M M ( + 1), which enables to estimate the source term, M can be chosen as in Theorem 1.4, namely such that M M ( ).

Proof of Corollary 1.6. Notice that, provided w 0,

(3.1) ψ H (E d ) ψ H w .
In particular, Sobolev embedding yields, for > (d + 1)/2 1,

ψ L ∞ (E d ) C ψ H w ,
where C is independent of w 0. With these remarks in mind, the L 1 estimates of Corollary 1.6 follow from Theorem 1.5 and Cauchy-Schwarz inequality, since

|u ε | 2 -|a| 2 L ∞ T L 1 = |a ε | 2 -|a| 2 L ∞ T L 1 a ε + a L ∞ T L 2 δa ε L ∞ T L 2 ,
and

Im (εū ε ∂u ε ) -|a| 2 ∂φ L ∞ T L 1 ε Im āε ∂a ε L ∞ T L 1 + |a ε | 2 ∂φ ε -|a| 2 ∂φ L ∞ T L 1 ε a ε 2 L ∞ T H 1 + a ε + a L ∞ T L 2 δa ε L ∞ T L 2 ∂φ L ∞ T L ∞ + a ε L ∞ T L ∞ a ε L ∞ T L 2 δφ ε L ∞ T H 1 .
The L ∞ estimates in space follow by replacing L 1 and L 2 by L ∞ in the above inequalities, and using Sobolev embedding again.

Convergence of the wave function

Again, we assume J = 1 for the sake of conciseness.

Proof of Theorem 1.7. Let > (d + 1)/2, and (φ 0 , a 0 ) ∈ H +2 w0 × H +1 w0 . Theorem 1.4 yields a unique solution (φ, a) ∈ X +1 w,T to (1.9). Let (φ 10 , a 10 ) ∈ H +1 w0 × H w0 . Note that (1.10) is a system of linear transport equations in the unknown (∇φ 1 , a 1 ), whose coefficients are smooth functions. The general theory of transport equations (see e.g. [2, Section 3]) then shows that (1.10) has a unique solution (φ 1 , a 1 ) ∈ C([0, T ], L 2 × L 2 ). We already know by this argument that the solution is actually more regular (in terms of Sobolev regularity), but we shall directly use a priori estimates in H w spaces. Indeed, Lemma 2.6 implies that (φ 1 , a 1 ) ∈ X w,T with, exactly as in [START_REF]WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity[END_REF],

|||φ 1 ||| 2 +1,T φ 10 2 H +1 w 0 + C M |||φ 1 ||| 2 +1,T |||φ||| +1,T + C M |||φ 1 ||| 2 +1,T |||a||| 2γ ,T + C M |||φ 1 ||| +1,T |||φ||| +1,T |||a||| 2γ-1 ,T |||a 1 ||| ,T + C M |||φ 1 ||| +1,T |||a||| 2σ-1 ,T |||a 1 ||| ,T , along with |||a 1 ||| 2 ,T a 10 2 H w 0 + C M |||a 1 ||| ,T |||a||| ,T |||φ 1 ||| +1,T + C M |||a 1 ||| 2 ,T |||φ||| +1,T + C M |||a 1 ||| 2 ,T |||a||| 2γ ,T + C M |||a 1 ||| ,T |||a||| +1,T ,
for some C > 0.

Let > (d + 1)/2. For (φ 0 , a 0 ) ∈ H +3 w0 × H +2 w0 , (φ 10 , a 10 ) ∈ H +2 w0 × H +1 w0 and (φ ε 0 , a ε 0 ) ∈ H +1 w0 × H w0 , we consider:

• (φ, a) ∈ X +2
w,T the solution to (1.9). • (φ 1 , a 1 ) ∈ X +1 w,T the solution to (1.10). • (φ ε app , a ε app ) = (φ, a) + ε(φ 1 , a 1 ). • (φ ε , a ε ) ∈ X w,T the solution to (1.5).

We assume that φ ε 0 -φ 0 -εφ 10 H +1 w 0

= o(ε) and a ε 0 -a 0 -εa 10 H w 0 = o(ε). Set

δφ ε 1 = φ ε -φ ε app = φ ε -φ -εφ 1 = δφ ε -εφ 1 , δa ε 1 = a ε -a ε app = a ε -a -εa 1 = δa ε -εa 1 .
The equation satisfied by δφ ε 1 writes

∂ t δφ ε 1 + ∇φ, H∇δφ
|||δφ ε 1 ||| 2 +1,T φ ε 0 -φ 0 -εφ 1,0 2 H +1 w 0 + C M |||δφ ε 1 ||| +1,T |||δφ ε 1 ||| +1,T + ε 2 +|||δa ε 1 ||| ,T .
We deduce, for M large enough, (1.11) follows. Like in the proof of Theorem 1.5, a careful examination of the inequalities that we have used shows that all the above estimates are valid provided that we assume M M ( ), the constant provided by Theorem 1.4, and also M max(M ( + 1), M ( + 2)) in order to estimate the source terms.

|||δφ ε 1 ||| 2 +1,T C φ ε 0 -φ 0 -
To complete the proof of Theorem 1.7, consider the point-wise estimate a ε e iφ ε /ε -ae iφ1 e iφ/ε |a ε -a| + |a ε | e iφ ε /ε -e i(φ+εφ1)/ε |a ε -a| + |a ε | 2 sin φ ε -φ -εφ 1 2ε

|δa ε | + 1 ε |a ε | |δφ ε 1 | .
We then conclude like in the proof of Corollary 1.6, by using Cauchy-Schwarz inequality, (3.1), and Sobolev embedding.

1 . Introduction 1 . 1 .

 111 Motivation. The two-dimensional "hyperbolic" nonlinear Schrödinger equation,(1.1)

ε 1 + 1 2 1 + 2 + |a ε | 2 -|a| 2 2 1 0( 1 -

 112211 ∇δφ ε , H∇δφ ε + g(|a| 2 ) β, ∇δφ ε g(|a ε | 2 ) -g(|a| 2 ) β, ∇δφ ε + g(|a ε | 2 ) -g(|a| 2 ) -2g (|a| 2 ) Re(aεa 1 ) β, ∇φ + K * |a ε | 2σ -|a| 2σ -2σ|a| 2σ-2 Re(aεa 1 ) = 0.Moreover, the Taylor formula yieldsg(|a ε | 2 ) -g(|a| 2 ) -2g (|a| 2 ) Re(aεa 1 ) = 2 Re(aδa ε 1 ) + |δa ε | 2 g |a| s)g |a| 2 + s |a ε | 2 -|a| 2 ds, (4.1) and the same identity holds for g replaced by f (r) = r σ . Thus, taking into account Theorem 1.4, which implies |||φ ε ||| +1,T , |||a ε ||| ,T = O(1), and Theorem 1.5, which implies |||δφ ε ||| +1,T , |||δa ε ||| ,T = O(ε), it follows from Lemma 2.6 that

  H∇δa ε 1 + ∇δφ ε 1 , H∇a + ∇δφ ε , H∇δa ε

	(4.2)					εφ 10	2 H +1 w 0	+	C M	ε 4 +	C M	|||δa ε 1 ||| 2 ,T .
	Similarly, δa ε 1 solves
	∂ t δa ε 1 + ∇φ, + 1 2 aD 2 δφ ε 1 +	1 2	δa ε 1 D 2 φ +	1 2	δa ε D 2 δφ 1	=	iε 2 2	D 2 a 1 +	iε 2	D 2 δa ε 1 .
	From (4.1), Theorems 1.4 and 1.5, and Lemma 2.6, we deduce
	(4.3)	|||δa ε 1 ||| 2 ,T ≤ C a ε 0 -a 0 -εa 10	2 H w 0	+	C M	ε 4 +	C M	|||δφ ε 1 ||| 2 +1,T .
	Adding (4.2) and (4.3),

ε + β, ∇ g(|a ε | 2 ) -g(|a| 2 ) -2εg (|a| 2 ) Re(aa 1 ) a + β, ∇ g(|a ε | 2 ) -g(|a| 2 ) εa 1 + β, ∇ g(|a ε | 2 )δa ε


MD Aline Corvol
email: acorvol@hotmail.com

PhD Adina Dreier
email: adina.dreier@uni-greifswald.derene.thyrian@dzne.dewolfgang.hoffmann@dzne.de

Joachim Prudhomm

MD Jochen Rene Thyrian

MD, PhD Wolfgang Hoffmann

MD, PhD Dominique Somme
email: dominique.somme@chu-rennes.fr

Consequences of clinical case management for caregivers: a systematic review

Keywords: Systematic review, older persons, clinical case management, caregiver, persons with dementia

Consequences of clinical case management for caregivers a systematic review

Introduction

Informal caregivers play a critical role in long-term home care and have been described as the "backbone" of the long-term care workforce [START_REF] Wolff | Caregivers of frail elders: updating a national profile[END_REF]. Because the stresses of modern society make the traditional mechanisms of providing informal care more difficult, public policy-makers have sought to find efficient ways to support caregivers (Bodsky et al., 2003). A 2002 meta-analysis [START_REF] Sörensen | How effective are interventions with caregivers? An updated meta-analysis[END_REF], pooling a wide range of interventions with caregivers, found that appropriate interventions can lead to significant improvements in caregiver burden, level of depression, and subjective well-being, with a small to moderate effect size. Psychoeducational and psychotherapeutic interventions showed more consistent results, whereas intervention effects tended to be smaller for caregivers of dementia patients. A 2008 systematic review of literature on dementia caregivers (Parker et al.) found only weak evidence that caregivers benefit from such support. Multidisciplinary case management, as analyzed in the three trials examined by the study, seemed to decrease depression scores in caregivers, but this result was not statistically significant. Paradoxically, interventions such as respite care increased caregiver burden. A recent review [START_REF] Berthelsen | The content, dissemination and effects of case management interventions for informal caregivers of older adults: a systematic review[END_REF] considered effects of case management addressed to informal caregiver of older adults, both on patients and caregivers. Because of a restrictive search algorithm, with a mandatory focus on health information or shared decision making, and methodological limitations (inconsistence in the definition of case management and of caregiver), this review don't provide reliable results concerning caregivers. However, case management, which can be defined as a "targeted, community-based and pro-active approach to care that involves case-finding, assessment, care planning, and care coordination," always involves both the patient and the caregivers [START_REF] Ross | Case management : What it is and how it can best be implemented[END_REF]. This is especially true considering [START_REF] Applebaum | Long Term Care Case Management: Design and Evaluation[END_REF] definition of case management, characterized by three factors: intensity of involvement, breadth of services spanned, and long-term commitment.

Thus, there are two reasons to monitor consequences of case management on caregivers.

First, the caregiver can be seen as a part of the caring system, and caregiver support can enhance quality of the care the patient receives. Second, personal interests of the caregiver should be taken into account. Consequences could be beneficial or detrimental, as case management may either reduce or increase caregiver burden.

In keeping with this perspective, the present review has studied the full range of consequences of case management for caregivers of frail elderly persons or persons with dementia. A systematic retrieval of randomized clinical trials and quasi-experimental studies comparing case management and usual care was conducted, in order to answer the following questions: 1) How does case management affect caregivers? 2) What aspects of the case management program could explain the variations in consequences for caregivers?

3) What patient or caregiver characteristics could explain variations in consequences for caregivers? 4) Is case management efficiency for care recipients associated with a positive or negative effect on caregivers?

Methods

The systematic review presented here was undertaken by one of the authors and was not preregistered as a protocol. This report meets the criteria of the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) statement for systematic reviews [START_REF] Moher | Preferred reporting items for systematic reviews and meta-analyses: the PRISMA statement[END_REF] .

The present study focuses on primary caregivers of elderly patients with loss of autonomy or persons with dementia without age limit. Patients included in the retrieved studies were considered "elderly" if their average age was more than 75 years. The definition of caregiver was the one chosen by the researchers in each trial, which did not necessarily include a family link or a shared home.

In order to make valid comparisons among the explored interventions, only studies in which "a designated care manager combines the planning and coordination with a therapeutic, supportive role" [START_REF] Challis | Care management, dementia care and specialist mental health services: an evaluation[END_REF], under a model referred to as clinical case management [START_REF] Kanter | Clinical case management: definition, principles, components[END_REF][START_REF] Surber | Clinical case management: A guide to comprehensive treatment of serious mental illness[END_REF] or as intensive care management [START_REF] Challis | Care management, dementia care and specialist mental health services: an evaluation[END_REF], were included. Only studies comparing the specified case management program approach to standard care without any form of case management were included. Quasi-experimental studies were retained because those studies can have a high external validity [START_REF] Koch | The potential of case management for people with dementia: a commentary[END_REF]. Case management programs that provided only distance services or were limited to education were excluded, as were short term discharge intervention (i. e. transitional care) lasting less than 3 months. All relevant studies published in peer-reviewed journals were considered, regardless of publication date and language. Concerning time-limited interventions, only the outcome assessment at the end of the program was retained; post hoc studies on part of a sample were discarded.

We began our electronic search on the Scopus Database, which includes all MEDLINE articles and indexes 18,500 peer-reviewed journals, including human sciences and non-English journals (http://www.info.sciverse.com/scopus/scopus-in-detail/facts). We developed our search algorithm from a first non-systematic literature review, using index mapping [START_REF] Strech | Systematic reviews of empirical bioethics[END_REF]. We searched in title, abstract, and keywords for the terms "case management" or "care management" or "disease management" or "guided care" or "family intervention" or "integrated care," in association with "controlled study" or "comparative study" or "prospective experimental design," associated with "caregiver" in the text. This search, conducted in May 2016, was limited to articles, and articles indexed with "child" as a keyword were excluded. Complementary searches were performed in additional databases, that is, EMBASE, ISI Web of Knowledge, the Cochrane Central Register of Controlled Trials , PsycINFO, CSA Social Services Abstract, and FRANCIS.

All articles were screened by title, and all abstracts of possibly relevant articles were read. Full texts of potentially relevant studies were then retrieved and checked against inclusion and exclusion criteria. Reference lists of selected articles were checked in order to identify other relevant studies. No authors were contacted.

The methodological quality of the randomized studies was systematically assessed for each outcome using the Cochrane Collaboration tool for assessing risk of bias in randomized trials [START_REF] Higgins | The Cochrane Collaboration's tool for assessing risk of bias in randomised trials[END_REF]. This tool focuses on internal validity and suggests judging the risk of bias in five domains: selection, performance, detection, attrition, and reporting. We added a systematic examination of implementation [START_REF] Boutron | Extending the CONSORT statement to randomized trials of nonpharmacologic treatment: explanation and elaboration[END_REF] and contamination biases, because of their relevance to the evaluation of socially complex services (Wolff et al., 2000).

Results were then summarized as low, acceptable, high, or unclear.

The potential biases resulting from the absence of blinding differed according to the outcome considered. We maintained that there was a high risk of performance bias on the outcome "satisfaction with care" as caregivers who felt they had "won" the randomization process, or get the "best" service in quasi-experimental studies, might well feel more satisfied. In addition, we saw a high risk of detection bias on the outcome "declared hours of caring," especially when no in-depth research interview were undertaken, since a case manager's assessment could influence the caregiver's awareness of his/her involvement in everyday care. We considered the performance bias on the caregiver's quality of life to be acceptable, as it has been demonstrated that the Hawthorne effect (i.e., the bias linked with attention and multiple assessments), is not large enough to significantly affect quality of life for caregivers of dementia patients (Mc Carney et al., 2007) . By extension, we considered the potential performance bias linked with the absence of blinding on caregiver burden and depression, for both dementia and non-dementia patients, to be acceptable. Non-randomized studies were classified according to the same criteria, taking into account their inherent risk of selection bias. Internal validity assessment was conducted by two reviewers independently, both experienced researchers in the area of case management. Inconsistencies were resolved by discussion. Results of studies with low internal validity were excluded from further analysis. In included studies, outcomes with high risk of bias were not considered.

As quality cannot be reduced to internal validity, external validity was assessed as well [START_REF] Verhagen | The art of quality assessment of RCTs included in systematic reviews[END_REF]. In keeping with Wolff's recommendation on socially complex services evaluation (Wolff et al.,2000), we defined two important domains: the representativeness of the research sample, and the accuracy of the description of the interventions, that allows replication of the intervention in similar contexts. External validity was classified as low, acceptable, high, or unclear.

In order to identify elements that might explain differences in outcomes, we systematically collected program and population characteristics. Program descriptions were analyzed in order to assess program intensity and level of integration. The simplest way to evaluate case management intensity is to consider the case manager's case load (the greater the case load, the lower the intensity) [START_REF] Somme | Analysis of case management programs for patients with dementia: A systematic review[END_REF]. To refine this approach, we also considered the frequency of home visits, the range of services [START_REF] Pacala | Case management of older adults in health maintenance organizations[END_REF] , and the complexity of the needs of the target population [START_REF] Challis | Care management, dementia care and specialist mental health services: an evaluation[END_REF]. We estimated level of integration on the basis of case managers' links with primary care practice, service providers, acute-care facilities, and long-term-care facilities [START_REF] Kodner | The quest for integrated systems of care for frail older persons[END_REF]. Intensity and integration were estimated as low, moderate, high, or unclear. As no consensually validated tool exists to assess external validity, case management intensity, and case management integration, these assessments were made by the two reviewers through continuous discussions to achieve agreement.

Finally, in order to identify elements that might explain differences in outcomes, we systematically collected program and population characteristics. Program descriptions were analyzed in order to assess program intensity and level of integration. Relevant characteristics of the caregiver population were determined -in particular, the proportion of women, and the proportion of spouses, since those characteristics are known to influence caregiver burden [START_REF] Carretero | The informal caregiver's burden of dependent people: theory and empirical review[END_REF]. We also collected statistically significant results concerning patients, in order to assess the relationship between patient and caregiver outcomes.

A vote-count review was performed on caregiver outcomes, summarizing those reflecting burden (or strain or distress), satisfaction, health-related quality of life (or psychological health), depression, and social support. Sub-group analyses were carried out in terms of patient characteristics (persons with dementia versus elderly persons) and patient results, caregiver characteristics, and intensity and integration levels for case management.

Results

Selection of Studies

Our bibliographic search in Scopus identified 481 articles. The selection process (Figure 1) led us to include 16 studies (Table 1). Frequently assessed outcomes were caregiver burden (13 studies), caregiver health-related quality of life or psychological health (8 studies), caregiver depression (4 studies), satisfaction with care (3 studies), or social support (3 studies). Other outcomes considered by at least one study were caring hours, caregiver sense of competence, personal well-being, family functioning, desire to institutionalize, productivity loss and loneliness .

Eleven randomized controlled trial studies and five quasi-experimental studies were analyzed. No randomized trial was blinded for participants, which makes senses, as such blinding would be difficult, if not impossible, for case management. Assessment was blinded in 11 studies, non-blinded in 5, and unclear in one. Three of the five quasi-experimental studies were considered to have low internal validity, mostly because of significant differences between experimental and control groups at inclusion. One randomized controlled trial with only one caregiver outcome, declared hours of caring, was considered to have low internal validity for this outcome because of a detection bias. These four studies were excluded from further analysis. Reviewers agreed on conclusions concerning internal validity for all randomized trials, but discussions were necessary for the two quasiexperimental studies that have finally been accepted.

There were no studies with high internal validity. The 12 remaining studies were judged to have unclear (2 studies) or acceptable (10 studies) internal validity, in spite of moderate risks of performance, detection, implementation, or contamination bias. Five studies had low external validity, mainly because they lacked a systematic recruitment strategy. Table 2 summarizes the results concerning the internal and external validity of each study, and specifies the kinds of bias that could affect the results. We included two studies published by Chien andLee (2008,2011) despite some questions about these reports: they describe two very similar trials without discussion on the additional value of the second study.

Outcomes for Caregivers

Seven of the 12 studies found at least one valid positive outcome for caregivers [Table 3].

Four of the 11 studies assessing caregiver burden showed a reduction. Four of the 7 studies assessing quality of life for caregivers yielded positive results; one of 3 on caregiver social support showed positive results. The 4 studies that assessed depression in caregivers found no significant differences at the end of the intervention. [START_REF] Callahan | Effectiveness of collaborative care for older adults with Alzheimer disease in primary care: a randomized controlled trial[END_REF] found a significant result on depression score 6 months after the end of the intervention. No studies demonstrate any negative effects of case management on caregivers.

Pre-specified Sub-group Analysis

In further analyses we tried to understand which aspects of a case management program could explain variations in the program's efficacy for caregivers. As it was not possible to control for all confounding factors, especially the variability of the context, the further analyses should be regarded as exploratory.

To analyze intensity and integration influence, we separate here two different case management models compared with control group by McNeil : the "intensive" one, and the "linkage model".

Patient profiles are important determinants of efficacy. As Table 4 shows, none of the programs aimed at elderly patients showed positive outcomes for caregivers. Selection criteria were different (table1) according to the studies, but all these, except in Leung study (2004), can be qualified of "frail", because of a loss of autonomy or a high risk of using health service resource. Six of the ten programs targeted to dementia patients found positive outcomes for caregivers, the other having both a low or moderate integration and a moderate or unclear intensity level. The two studies aimed at mild-dementia patients only demonstrated no positive outcome. Five of the six studies with a high intensity level of case management achieved positive outcomes for caregivers whereas only one of the five studies with a moderate level of intensity achieved positive outcomes for caregivers. The level of integration of a case management program, patient age, or caregiver characteristics did not seem to figure highly in program efficacy.

It is important to emphasize that these results are based only on caregiver outcomes. In terms of patient outcomes, 9 of the 12 studies reported at least one positive outcome for patients, such as delaying institutionalization, reducing hospitalization, enhancing quality of life, or reducing behavioral troubles. Some programs may yield significant effects for patients, especially frail elderly, without benefit for caregivers. Parsons' study (2012) demonstrates the possibility of benefit in the form of delayed institutionalization, which was not mediated by a reduction of caregiver burden.

Discussion

Evidence concerning socially complex services is difficult to gather (Wolff, 2000). Because of the heterogeneity of study designs, interventions, contexts, and outcomes considered, we did not attempt to pool the results in a meta-analysis. We believe that this overview obtained through careful individual assessment of study quality, program, and outcomes can be more informative than any summary measure obtained via global pooling of different interventions.

Although some uncertainty remained, due to the impossibility of conducting blinded trials, this systematic review allows us to conclude that some case management programs aimed at dementia patients can have positive outcomes for caregivers, in terms of reduction of caregiver burden, enhanced quality of life, or greater social support, but not on depression.

Caregivers of patients with mild dementia may not be the persons who benefit most from case management programs [START_REF] Jansen | Effectiveness of case management among older adults with early symptoms of dementia and their primary informal caregivers: a randomized clinical trial[END_REF]. There is no evidence that case management programs may have positive outcomes for caregivers of frail elderly, but such programs can reduce hospitalization [START_REF] Leung | Cost-Benefit Analysis of a Case Management Project for the Community-Dwelling Frail Elderly in Hong Kong[END_REF][START_REF] Béland | A system of integrated care for older persons with disabilities in Canada: results from a randomized controlled trial[END_REF] or delay institutionalization (Parson et al., 2012) without increasing caregiver burden.

Our systematic review has several limitations. In addition to the usual publication bias, identification of all the relevant studies has proven challenging, since case management programs can be described by different terms. Despite the care we took in building a systematic search algorithm, it is possible that some relevant studies were not identified.

Ultimately, however, any real weakness in this systematic review lies in the quality of the studies themselves, all of which were based on subjective outcomes which could have been biased by the absence of blinding. Another potentially important bias concerns implementation, as most of the studies reported no implementation results. Three studies attributed disappointing results to inadequate program implementation [START_REF] Vickrey | The effect of a disease management intervention on quality and outcomes of dementia care: a randomized, controlled trial[END_REF][START_REF] Jansen | Effectiveness of case management among older adults with early symptoms of dementia and their primary informal caregivers: a randomized clinical trial[END_REF][START_REF] Fox | Lessons learned from the Medicare Alzheimer Disease Demonstration[END_REF]. Contamination may also have occurred, especially in trials with individualized randomization: good practices implemented by case managers could have also benefitted the control group, if the same professionals were involved in both groups. Whereas the implementation and contamination bias may have reduced the results, the non-blinded design might have increased them.

It has proved difficult to identify aspects of case management program and patients selection associated with better results. This is mainly due to an insufficient reporting, and to the heterogeneity of studies. Intensity and integration levels were especially difficult to determine. When MacNeil compares with control group two different models of case management, named "intensive" and "linkage", it happens that the case load is similar in the two models (more important in the linkage model, though non significant), the intensive model referring in fact to a more integrated model, more efficient on caregiver psychological health. This stresses the necessity of a complete description of implemented interventions.

The value of quasi-experimental design in evaluation of case management programs is somewhat controversial [START_REF] Koch | The potential of case management for people with dementia: a commentary[END_REF]Wolff, 2000). Hébert (2000) was able to provide interesting data on the strength of an extensive (4 year) follow-up of an "actually implemented" program, and it emphasizes the time required to implement a program before assessing its efficacy. Quasi-experimental studies can avoid contamination bias, when different territories are compared, and their implementation, even when only partial, corresponds to "real-life" implementation. We believe that the performance bias linked with the absence of patient blinding could be decreased by the absence of randomization, as patients would not have felt they had won something others had not. However, the detection bias is increased by the absence of a blinding assessment, usually infeasible in such studies [8]. The main problem, as observed in four such studies [START_REF] Morales-Asencio | Effectiveness of a nurse-led case management home care model in Primary Health Care. A quasi-experimental, controlled, multi-centre study[END_REF][START_REF] Hébert | Impact of PRISMA, a coordination-type integrated service delivery system for frail older people in Quebec [Canada]: A quasi-experimental study[END_REF][START_REF] Janse | The effects of an integrated care intervention for the frail elderly on informal caregivers: a quasi-experimental study[END_REF], McNeil et al., 2015) is the difficulty to obtain two similar groups that could allow comparison. Despite our initial choice of including quasiexperimental study ,three of five were finally not included because of significant differences in the compared groups, added with at least one other identified bias, and for the two included trial there was initially a disagreement between the expert. [START_REF] Challis | Care management, dementia care and specialist mental health services: an evaluation[END_REF] design, with similar group obtained by matching patient pairs, and MacNeil (2015) statistical analysis using propensity score, seems valuable ways to enhance internal validity of quasiexperimental studies. Further studies could consider cluster randomization, eventually with stepped wedge design to model the effect of time [START_REF] Hemming | The stepped wedge cluster randomised trial: rationale, design, analysis, and reporting[END_REF].

Lastly, all studies included only one caregiver per patient, with different definition and often little details provided; so the effect of case management on "second line" caregivers or according to the characteristics of caregivers (spouses versus relatives versus friends) can't be specified.

Despite these limitations, we believe we have been able to gather valuable evidence on the potential benefits for caregivers of case management. Our rigorous selection of studies, based on a clear definition of case management, allow confidence in the conclusions drawn.

Positive results have been obtained in very different cultural and organizational contexts, in Asia, Europe, and America. Despite the limited external applicability of some individual studies, this review provides a good representation of different contexts overall. Of course, the literature cannot provide evidence that every case management program will work everywhere, nor that any given single model will work in every context. In contrast to drug trials, case management is an intervention that cannot be completely standardized (Wolff, 2000), and researchers studying case management have to make compromises between internal and external validity [START_REF] Godwin | Pragmatic controlled clinical trials in primary care: the struggle between external and internal validity[END_REF]. So our review can only stress the possibility of implementing efficient case management programs for caregivers, and highlight the characteristics that seem to be associated with better outcomes. Until now, such programs have been nationally implemented in very different contexts, for example in Japan [START_REF] Matsuda | Long-term care insurance and integrated care for the aged in Japan[END_REF], France (Somme et al, 2009), or Québec, Canada [START_REF] Vedel | Ten years of integrated care: backwards and forwards. The case of the province of Québec, Canada[END_REF].

No study showed any effect of a program on caregiver depression. That said, the relevance of such an outcome for a case management program can be questioned: if depression is envisaged as a systemic disease, involving the mind, brain and body (Gold and Charney, 2002) such intervention could address only social, external components of depression. The choice of assessment instruments for depression can also be questioned [START_REF] Snaith | What do depression rating scales measure?[END_REF][START_REF] Roman | Screening Instruments for Older Adult Depressive Disorders: Updating the Evidence-Based Toolbox[END_REF], the scales chosen to rate caregiver burden. The construction of scales is "necessarily influenced by a variety of value judgements" [START_REF] Janse | The effects of an integrated care intervention for the frail elderly on informal caregivers: a quasi-experimental study[END_REF]. Burden scales that focus on the negative impacts of caregiving have been criticized, because that restriction prevents researchers from assessing sources of satisfaction in caregiving [START_REF] Nolan | Issues and challenges in carer support: A consideration of the literature. In: Services for Supporting Family Carers of Elderly People in Europe: Characteristics, Coverage and Usage[END_REF]. Other scales that include both negative and positive impacts of caregiving could be more relevant to assessment of the global consequences of caregiving (McKee et al., 2003).

The finding that studies aimed at frail elderly can yield positive results for patients without significantly affecting caregivers is in itself valuable. As such programs involve both caregivers and patients, it seems good practice to evaluate the respective effects on both.

Five studies raise questions about possible harmful effects of case management. In the Hébert study (2010), the burden increase was significantly greater in the intervention group.

One hypothesis is that patient empowerment was achieved at the expense of caregiver burden. Two studies [START_REF] Leung | Cost-Benefit Analysis of a Case Management Project for the Community-Dwelling Frail Elderly in Hong Kong[END_REF][START_REF] Vickrey | The effect of a disease management intervention on quality and outcomes of dementia care: a randomized, controlled trial[END_REF] found a non-significant trend in favor of an increase of caregiving hours in the intervention group, which conflicts with Challis (2001) results. It is possible that in some contexts case managers might ask caregivers to perform new caregiving tasks. [START_REF] Lam | A randomized controlled trial to examine the effectiveness of case management model for community dwelling older persons with mild dementia in Hong Kong[END_REF] found a slight negative effect at 12 months on caregivers' psychological health (our review focused on the effect at the end of the intervention at 4 months), and assumed that domestic helpers' lack of experience with dementia patients, and the cost of day care facilities, may have in fact been burdensome to some family caregivers. Finally, Janse (2014) complementary study on caregiver satisfaction demonstrates the possibility of counterproductive effect of case management due to information overload or decrease clarity on "who do what" for caregiver. These results, although a minority, emphasize the necessity of considering caregiver outcomes in all case management intervention, at least as a possible side effect.

Conclusion

Our results show reasonable evidence that clinical case management can lead to positive outcomes for caregivers of dementia patients, and that positive outcomes for frail elderly

Table 3 :

 3 Caregiver outcomes with acceptable or unclear internal validity detailed by study.

	Publication	Burden	Depression	Health Related Quality of life	Social support	Other caregiver outcomes
	Newcomer 1999	Adapted from Zarit Carer Burden Interview	Geriatric (short form) depression Scale	Not assessed	Not assessed	No
		Adapted from the
		Social Behaviour
	Challis2002	Assessment Schedule	Not assessed	Not assessed	Not assessed	No
		Caregiver strain
		(Malaise Scale)
	Béland
	2006	Zarit Carer Burden Interview	Not assessed	Not assessed	Not assessed	No
					Medical	Confidence in
	Vickrey 2006	Not assessed	Not assessed	Euroquol 5D	Outcome Study Social	caregiving; Caregiving
					Support	Mastery
		Caregiver
	Callahan 2006	Neuropsychiatric inventory (caregiver	Patient Health Questionnaire 9	Not assessed	Not assessed	No
		distress)
	Chien 2008	Family Caregiver Burden Inventory	Not assessed	World Health Organization Scale Quality of Life	Six-item Social Questionnaire Support	No
			Center for
	Wolff 2009	Modified Caregiver Strain Index	Epidemiologic Studies	Not assessed	Not assessed	Productivity loss
			Depression Scale
				General Health
	Lam. 2010	Zarit Carer Burden Interview	Not assessed	Questionnaire 30; Personal Well-	Not assessed	No
				Being Index
	Chien 2011	Family Caregiver Burden Inventory	Not assessed	World Health Organization Scale Quality of Life	Six-item Social Questionnaire Support	No
	Jansen. 2011	Self Perceived Pressure by Informal Carer	Center for Epidemiologic Studies Depression Scale	Medical Outcome Study 36 items Short form	Not assessed	Caregiver sense of competence (Sense of Competence Questionnaire)
	Parsons 2012	Caregiver Reaction Assessment	Not assessed	Medical Outcome Short form Study 36 items	Not assessed	No
				Euroquol 5D		Pearlin Mastery Scale
	MacNeil 2015	CarerQOL	Not assesssed	General Health Questionnaire 12	Not assessed	Short sense of competence scale
						Loneliness scale

Table 4 :

 4 Program characteristics and statistically significant outcomes for caregivers and patients.

		Patient Profile	Duration of the Case Management months) Program (in	Intensity	Integration	Positive Caregivers Outcomes Concerning	Positive Outcomes Concerning Patients
							less
							institutionalisation
	Challis 2002	dementia	12	high	high	lower burden	improvement in social contacts, need
							reduction, level of
							risk reduction
	Callahan 2006	dementia	12	high	low	lower burden	lower Neuro-psychiatric inventory
	Chien 2008	dementia	6	high	moderate	lower burden, better QoL	less institutionalisation; decrease of symptom severity
	Chien 2011	dementia	6	high	moderate	lower better QoL burden,	less institutionalisation
	MacNeil2015 Intensive CM	dementia	24	Unclear	High	better QoL	no
	Vickrey 2006	dementia	18	moderate	moderate	better social support and confidence in caregiving	higher care quality; higher quality of life; better social support; fewer unmet needs
	Newcomer 1999	dementia	36	unclear	low	no	fewer unmet needs
	MacNeil2015
	Linkage	dementia
	model

patients can be obtained without increasing caregiver burden. Further research is needed to determine which characteristics of a program, in addition to intensity, could support the program's efficacy, and to determine which caregivers could benefit most from those programs. Even though researchers face numerous methodological challenges in building valid trials, evidence can be strengthened via accurate description of the context, of the intervention itself, and of its actual implementation, as well as through a careful choice of trial design and outcomes. The authors declare that they have no competing interests.

Key points

-Informal caregivers are deeply involved in the case management process -Case management programs can achieve positive results for caregivers, especially intensive programs addressed to persons with dementia.

-Positive results for older patients, such as delaying institutionalization and reducing hospitalization, occur without impacting caregivers' outcome.