Jérémy Firozaly

Homogenization of a 1D pursuit law with delay and a counter-example

Keywords: Hamilton-Jacobi equations, infinite system, delay time, strict comparison principle, homogenization

In this paper, we consider a one dimensional pursuit law with delay which is derived from traffic flow modelling. It takes the form of an infinite system of first order coupled delayed equations. Each equation describes the motion of a driver who interacts with the preceding one, taking into account his reaction time. We derive a macroscopic model, namely a Hamilton-Jacobi equation, by a homogenization process for reaction times that are below an explicit threshold. The key idea is to show, that below this threshold, a strict comparison principle holds for the infinite system. In a second time, for well-chosen dynamics and higher reaction times, we show that there exist some microscopic pursuit laws that do not lead to the previous macroscopic model.

Introduction

In the present paper we consider an infinite system of delay differential equations (DDEs). This form can especially be derived from a one dimensional pursuit law on a straight road with a very simple model. Such models, where we follow the time position of every driver are called microscopic models. The effect of drivers' reaction times on the stability of the corresponding systems of DDEs is studied in [START_REF] Atay | Complex time-delay systems: theory and applications[END_REF] and [START_REF] Nagatani | The physics of traffic jams[END_REF]. Presentations of some classical microscopic traffic flow models of first order or second order are made in [START_REF] Chandler | Traffic dynamics: studies in car following[END_REF], [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] or [START_REF] Costeseque | Contribution to road traffic flow modeling on networks thanks to Hamilton-Jacobi equations[END_REF]. Conversely, macroscopic models describe the time evolution of densities in global traffic flow. We want to study if we can recover a macroscopic model from our microscopic one. Formally, it corresponds to making the interdistance between drivers go to zero or to observe the pursuit law from a height and time that go to infinity. When possible, the passage can be made rigorous by an homogenization process that will be done here in the framework of viscosity solutions of Hamilton-Jacobi equations. The interested reader is referred to the pionnering works [START_REF] Lions | Homogenization of hamilton-jacobi equations[END_REF] or [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear pde[END_REF] about homogenization of Hamilton-Jacobi equations. He can see the works of [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF] or [START_REF] Richards | Shock waves on the highway[END_REF] for the historical approach used to define and analyse macroscopic traffic flow models and links with fluid mechanics. Several examples (and counter-examples) will be exhibited throughout this paper to show that both the initial dynamics and reaction time values will have a huge influence on the homogenization process. This paper derives from Régis Monneau's work [START_REF] Monneau | [END_REF] (see also [START_REF] Costeseque | Contribution to road traffic flow modeling on networks thanks to Hamilton-Jacobi equations[END_REF]).

Description of the model and main results

The common velocity of each driver is supposed to be a Lipschitz continuous, bounded, nondecreasing function, F say, of the distance that separates each driver from the preceding one (with F (0) = 0). Each driver has its own reaction time and we assume that the reaction times are uniformly bounded from above and from below; we denote τ the upper bound and ξ > 0 the lower bound. Let us study the evolution of the system during a time T ∈ (0, +∞] with T > τ . Given a sequence (X i ) i∈Z of drivers' positions on the road, the microscopic model then takes the form of an infinite system of DDEs:

dX i dt (t) = F (X i+1 (t -τ i ) -X i (t -τ i )) (i, t) ∈ Z × (0, T ), (1)
where τ i denotes the individual reaction time.

If we suppose that we know the dynamics of the cars during the initial time interval, we can define time functions (x 0 i ) i∈Z such that:

X i (t) = x 0 i (t) ∀t ∈ [-2τ, 0]. (2)
The first step is to embed the microscopic system dynamics into a single PDE. In order to go from the microscopic scale to the macroscopic one, we have to introduce a small parameter, ε > 0 say, so as to rescale properly the function and to take into account the microscopic space and time oscillations with high frequency.

Hence, by hyperbolic change of variables, we define the function u ε as follows (the space variable will be denoted as x):

u ε (x, t) = εX ⌊ x ε ⌋ ( t ε ). (3)
We suppose that there exists a Lipschitz continuous function u 0 : R × [-2τ, 0] → R such that:

x 0 i (t) = u 0 (iε, tε) ε (i, t) ∈ Z × [-2τ, 0],
and a function τ 0 : R → [ξ, τ ] such that:

τ i = τ 0 (iε) ∀i ∈ Z.
The rescaled model can thus be embedded into the following equation:

∂ t u ε (x, t) = F u ε (x+ε,t-ετ 0 (x))-u ε (x,t-ετ 0 (x)) ε (x, t) ∈ R × (0, T ), u ε (x, s) = u 0 (x, s) (x, s) ∈ R × [-2ετ, 0], (4)
with the following assumptions on F , τ 0 and u 0 :

     F is a non-decreasing, bounded and C F -Lipschitz continuous function on R, τ 0 (R) = [ξ, τ ], u 0 is a globally L -Lipschitz continuous function on R × [-2τ, 0]. (5)
As the argument of F in (4) converges towards a space derivative for vanishing ε, it is natural to introduce the following dynamics in the form of a first-order Hamilton-Jacobi equation:

∂ t u 0 (x, t) = F (∂ x u 0 (x, t)) (x, t) ∈ R × (0, T ), u 0 (x, 0) = u 0 (x, 0) x ∈ R. (6)
The first goal of the article is to prove that u ε tends locally uniformly towards u 0 :

Theorem 1.1 (Homogenization of the delay equation).

For τ ∈ 0; 1 eC F and under assumptions (5), the solution u ε to (4) converges locally uniformly towards the (unique viscosity) solution of (6).

The key idea of the convergence proof relies on a strict comparison principle. This one is stated for the microscopic model for ε = 1:

∂ t u(x, t) = F (u(x + 1, t -τ 0 (x)) -u(x, t -τ 0 (x))) (x, t) ∈ R × (0, T ), u(x, s) = u 0 (x, s) (x, s) ∈ R × [-2τ ; 0]. (7)
In contrast with the classical cases where it is sufficient to keep the initial order of the solutions, here, because of the reaction time, those solutions (namely the vehicles in traffic flow modelling) must be sufficiently spaced out at initial times in the sense made precise in the following theorem:

Theorem 1.2 (Strict comparison principle).
Under the assumptions (5), let v and u be respectively a supersolution and a subsolution of [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF]. Let us assume that there exist δ > 0, t 0 ∈ [0, T ), R ∈ [0, +∞), x 0 ∈ R, and ρ a positive, non decreasing function on R such that:

v(x, t) ≥ u(x, t) for R ≤ |x -x 0 | ≤ R + 1, t ∈ [t 0 -τ, T ) (8) δ ≤ (v -u)(x, t -τ ′ ) ≤ ρ(τ ′ )(v -u)(x, t) for τ ′ ∈ [0, τ ], (x, t) ∈ [x 0 -R, x 0 + R] × [t 0 -τ, t 0 ], (9)
where ρ is such that:

1 + C F ρ(τ ) τ ′ 0 ρ(s)ds < ρ(τ ′ ) τ ′ ∈ [0, τ ]. ( 10
)
Then we have:

0 < δe -C F ρ(τ )(t-t 0 ) ≤ v(x, t) -u(x, t) for (x, t) ∈ [x 0 -R, x 0 + R] × [t 0 , T ). ( 11
)
Remark 1.3. The theorem is still valid when R = +∞ (substituting

[x 0 -R, x 0 + R] by R), (8) is defined on ∅.
Remark 1.4. All the proofs involved in this paper remain the same when dealing with stochastic reaction times provided those are uniformly bounded from above and from below.

The homogenization proof is different from the one performed in [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF] on generalized Frenkel-Kontorova models, concerning an infinite system of particles that interact with a finite number of neighbours and that are subject to a periodic potential (this system takes the form of non-linear ODEs). The convergence proof in [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF] relies on the construction of hull functions in a periodic setting. The case where particles interact with an infinite number of other particles is covered in [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF]. Other homogenization results in traffic flow modelling without considering the reaction time can be found in [START_REF] Forcadel | A junction condition by specified homogenization of a discrete model with a local perturbation and application to traffic flow[END_REF] when adding a junction condition in the microscopic model and in [START_REF] Forcadel | Homogenization of second order discrete model and application to traffic flow[END_REF] for a second order microscopic model.

The second goal of the article is to provide a "counter-example to homogenization" in the following sense: we will display an explicit model for which u ε does not converge locally uniformly towards u 0 in the special case where all the drivers have a common reaction time larger than the threshold:

Theorem 1.5 (A counter-example to homogenization).

In the special case τ 0 ≡ τ , there exist a Lipschitz continuous F , an initial data, and τ > 1 eC F such that the solution u ε to (4) does not converge locally uniformly towards the solution of [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF].

We derive such a counter-example by a small perturbation of a trivial but unstable case where the homogenization holds for all reaction times. This trivial case is a stationnary one where all the vehicles are initially equally spaced. This also highlights the fact that the initial dynamics is at least as important as the value of the reaction time from the homogenization perspective.

Organisation of the article

To get the homogenization result, we will first show the existence and uniqueness of solutions to (4) and (6) in Section 2. We will then prove the strict comparison principle for (4) in Section 3 and we will show that the restriction on τ in Theorem 1.1 is equivalent to the existence of functions ρ that verify [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF]. We will also state and prove a direct corollary of Theorem 1.2 on drivers' positions and give a counter-example to Theorem 1.2 when the vehicles are not suitably spaced out at initial times, regardless of the threshold on τ . We will give the explicit convergence proof in Section 4 and give the unstable example where homogenization holds for all reaction times, provided the vehicles are perfectly spaced at initial times. In section 5, we will give the explicit model and will prove Theorem 1.5.

Existence and uniqueness

This section is devoted to the study of the existence and uniqueness of solutions to (4) and [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF]. Studying existence and uniqueness of solutions to (4) is stricly equivalent to studying [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF].

Proposition 2.1. Under the assumptions (5), there exists a unique classical solution to [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF].

Proof of Proposition 2.1. . The proof is based on an explicit and incremental construction. Indeed, on the time interval (0; ξ] the equation can be written as follows:

∂ t u(x, t) = F (u 0 (x + 1, t -τ 0 (x)) -u 0 (x, t -τ 0 (x))) (x, t) ∈ R × (0, ξ].
Hence, it can be explicitly integrated and we get:

u(x, t) = u 0 (x, 0) + t 0 F (u 0 (x + 1, s -τ 0 (x)) -u 0 (x, s -τ 0 (x)))ds (x, t) ∈ R × (0, ξ].
This enables to define the function u 1 which coincides with u 0 at initial times and that is defined by the previous expression in [0, ξ].

Hence, in the time interval (ξ; 2ξ], the equation can be written as follows:

∂ t u(x, t) = F (u 1 (x + 1, t -τ 0 (x)) -u 1 (x, t -τ 0 (x))) (x, t) ∈ R × (ξ; 2ξ].
Here again, it can be explicitly integrated:

u(x, t) = u 1 (x, ξ) + t ξ F (u 1 (x + 1, s -τ 0 (x)) -u 1 (x, s -τ 0 (x)))ds (x, t) ∈ R × [ξ, 2ξ].
The process can be iterated to obtain a function u defined piecewise which is, by construction, continuous on R × [0, ∞) and which solves equation [START_REF] Forcadel | Homogenization of fully overdamped Frenkel-Kontorova models[END_REF] in each R × (kξ, (k + 1)ξ), k = 0, .., ⌊ T ξ ⌋ -1. It remains to prove that it solves the equation globally and so, that u is C 1 in time on (0, T ). Indeed, at each kξ, the left and right limits of the derivative coincide and verify the equation.

The uniqueness also comes from the process as we see that on each time interval, the solution is completely determined by its expression on the previous interval which implies global uniqueness for a given initial condition u 0 . Equation ( 6) is an Hamilton-Jacobi equation and is therefore studied in the viscosity solutions' framework. The existence and uniqueness are classical and are respectively given by Perron's method and the usual comparison principle, see [START_REF] Barles | Solutions de viscosité et équations elliptiques du deuxieme ordre[END_REF] for instance.

Strict comparison principle

This section is first devoted to the proof of Theorem 1.2. We will then explain the restriction on τ . We will also prove the conservation of initial order of the vehicles as a corollary when those are well spaced out at initial times. Finally, we will finally present a counter-example to comparison principle for any reaction time τ > 0. Indeed, the existence of ρ functions that verify [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] is equivalent to consider τ under the threshold but both ( 10) and ( 9) are necessary for the strict comparison principle to hold and hence, the restriction on τ is not a sufficient condition to ensure the result. The existence of ρ functions is not enough, it has to be linked with the initial dynamics of the vehicles. Namely, we will show that if the vehicles are not suitably spaced out at initial times (in the sense made precise in ( 9)), their initial order can be disturbed even for reaction times that are below the threshold.

Proof of Theorem 1.2

Proof of Theorem 1.2. . Let us consider d := vu and define T * as:

T * = sup{S ∈ [0, +∞)/∀τ ′ ∈ [0, τ ], ∀(x, t) ∈ [x 0 -R, x 0 + R] × [t 0 -τ, S], d(x, t -τ ′ ) ≤ ρ(τ ′ )d(x, t)}.
The set is not empty as it contains t 0 so T * is well-defined.

• We first claim that to establish [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF], it is sufficient to prove that T * ≥ T .

Thanks to [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] we have ρ(0) > 1. Taking τ ′ = 0 in the definition of T * , with T * ≥ T , implies that:

d(x, t) ≥ 0 (x, t) ∈ [x 0 -R, x 0 + R] × [t 0 -τ, T ). ( 12
)
By combining with (8), we get:

d(x, t) ≥ 0 (x, t) ∈ [x 0 -R -1, x 0 + R + 1] × [t 0 -τ, T ). ( 13
)
Let us consider (x, t) ∈ [x 0 -R, x 0 + R] × (t 0 , T ). By definition of d, we have:

∂ t d(x, t) ≥ F (v(x + 1, t -τ 0 (x)) -v(x, t -τ 0 (x))) -F (u(x + 1, t -τ 0 (x)) -u(x, t -τ 0 (x))). (14)
By combining [START_REF] Monneau | [END_REF] with the fact that F is non-decreasing gives:

∂ t d(x, t) ≥ F (u(x + 1, t -τ 0 (x)) -v(x, t -τ 0 (x))) -F (u(x + 1, t -τ 0 (x)) -u(x, t -τ 0 (x))).
Using the fact that F is C F -Lipschitz, we get:

∂ t d(x, t) ≥ -C F d(x, t -τ 0 (x)).
If T * ≥ T , then we have:

∂ t d(x, t) ≥ -C F ρ(τ 0 (x))d(x, t).
By using the fact that ρ is non-decreasing, we get:

∂ t d(x, t) ≥ -C F ρ(τ )d(x, t). (15)
Hence, d is a supersolution of the problem:

∂ t w(x, t) = -C F ρ(τ )w(x, t), w(x, t 0 ) = δ. ( 16
)
We conclude ( 11) by a standard comparison principle for this ODE.

• Let us now show that T * ≥ T .

By contradiction, if we suppose that T * < T , then for all

β ∈ 0, inf T -T * 2 , 1 , there exists τ β ∈ [0, τ ], (x β , t β ) ∈ [x 0 -R, x 0 + R] × (T * , T * + β] such that: d(x β , t β -τ β ) > ρ(τ β )d(x β , t β ). ( 17
)
Let us notice that [START_REF] Richards | Shock waves on the highway[END_REF] holds true for all t ∈ [t 0 , T * ] (and so does ( 11)).

As a preliminary result, let us first show that:

d(x, t -τ ′ ) ≤ ρ(τ ′ )d(x, t) τ ′ ∈ [0, τ ], (x, t) ∈ [x 0 -R, x 0 + R] × [t 0 , T * ], (18)
with ρ given by:

ρ(τ ′ ) = 1 + C F ρ(τ ) τ ′ 0 ρ(s)ds.
By remarking that ρ(0) = 1, we notice the equality for τ ′ = 0.

For τ ′ ∈ [0, τ ], (x, t) ∈ [x 0 -R, x 0 + R] × [t 0 , T * ],
we have:

∂ τ ′ (ρ(τ ′ )d(x, t)) = C F ρ(τ )ρ(τ ′ )d(x, t).
Let us remark that τ ′ → d(x, tτ ′ ) is a subsolution of this equation. Indeed:

∂ τ ′ d(x, t -τ ′ ) = -∂ t d(x, t -τ ′ ). (19)
We can use (15) as tτ ′ ∈ [t 0τ, T * ]:

∂ τ ′ d(x, t -τ ′ ) ≤ C F ρ(τ )d(x, t -τ ′ ). (20)
Finally, by definition of T * , we get:

∂ τ ′ d(x, t -τ ′ ) ≤ C F ρ(τ )ρ(τ ′ )d(x, t). (21)
The preliminary result is then obtained by a comparison principle on the equation

∂ τ ′ W = C F ρ(τ )ρ(τ ′ )W .
Let us notice that ( 5) and ( 14) imply that:

∂ t d(x, t) ≥ -2||F || ∞ . (22)
By integrating it, we get:

d(x, t) ≥ d(x, s) -2||F || ∞ (t -s) (x, t) ∈ [x 0 -R, x 0 + R] × [t 0 , T ], s ∈ [t 0 , t]. (23) Case 1: t β -τ β ≤ T * .
In this case, by defining τ ′ β := T *t β + τ β ∈ [0, τ ] we get:

d(x β , t β -τ β ) = d(x β , T * -τ ′ β ).
Hence, by (18) we get:

d(x β , t β -τ β ) ≤ ρ(τ ′ β )d(x β , T * ).
Thanks to (23), we have:

d(x β , t β -τ β ) ≤ ρ(τ ′ β )(d(x β , t β ) + 2||F || ∞ β). (24)
In order to get a contradiction with (17), it is sufficient to consider β < min(β 1 , β 2 ) with:

β 1 := δ 2||F ||∞+1 e -C F ρ(τ )(T * -t 0 ) > 0, β 2 := β 1 2||F ||∞ inf [0,τ ] ( ρ ρ -1) > 0. (25)
Indeed, for β < β 1 , thanks to [START_REF] Lighthill | On kinematic waves. ii. a theory of traffic flow on long crowded roads[END_REF]

(which is still valid in [x 0 -R, x 0 + R] × [t 0 , T * ]
) and to (23), we have:

d(x β , t β ) ≥ d(x β , T * ) -2||F || ∞ β ≥ δe -C F ρ(τ )(T * -t 0 ) -2||F || ∞ β 1 = β 1 ,
and for β < min(β 1 , β 2 ) we have:

ρ(τ ′ β )(d(x β , t β ) + 2||F || ∞ β) ≤ ρ(τ ′ β )d(x β , t β ).
Together with (24), this implies:

d(x β , t β -τ β ) ≤ ρ(τ ′ β )d(x β , t β ).
Finally, as ρ is non decreasing and τ ′ β ≤ τ β , we get a contradiction with (17).

Case 2: t βτ β > T * .

As we have

t β < T * + β, this implies that τ β ∈ [0, β].
Then, thanks to (23) and (17) we have:

d(x β , t β ) + 2||F || ∞ β ≥ d(x β , t β ) + 2||F || ∞ τ β ≥ d(x β , t β -τ β ) > ρ(τ β )d(x β , t β ).
In particular, we have:

(ρ(τ β ) -1)d(x β , t β ) < 2||F || ∞ β.
Thanks to (23), this implies:

(ρ(τ β ) -1)d(x β , T * ) < 2||F || ∞ ρ(τ β )β.
Thus,

0 < δe -C F ρ(τ )(T * -t 0 ) < ρ(τ β ) ρ(τ β ) -1 2||F || ∞ β. (26)
For vanishing β this implies δe -C F ρ(τ )(T * -t 0 ) = 0 which is false.

3.2 Restriction on τ and existence of solutions to [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] Condition ( 10) explains why we chose τ < 1 eC F . In fact, we have the following proposition:

Proposition 3.1. (10) admits solutions if and only if τ < 1 eC F .

We now give a proof of Proposition 3.1.

Proof of Proposition 3.1. We want to show that condition (10) implies τ < 1 eC F . Let us define:

R(τ ′ ) = τ ′ 0 ρ(s)ds.
We have R(0) = 0. By defining α = R ′ (τ ) = ρ(τ ), condition [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] is equivalent for R to be a strict supersolution of:

y ′ = C F αy + 1, in [0, τ ], y(0) = 0. ( 27
)
As S :

τ ′ → e C F ατ ′ -1 C F α
is a solution of this Cauchy problem on [0, τ ], by Gronwall's lemma or a standard comparison principle, we deduce that R ≥ S or equivalently that:

1 + C F R(τ ′ )R ′ (τ ) ≥ e C F R ′ (τ )τ ′ τ ′ ∈ [0, τ ].
Moreover, as R is a strict supersolution of (27), we deduce that:

R ′ (τ ′ ) > e C F R ′ (τ )τ ′ for all τ ′ ∈ [0, τ ].
By considering this inequality for τ ′ = τ and taking the logarithm, we get:

C F τ < ln R ′ (τ ) R ′ (τ ) ≤ max x>0 ln x x = 1 e .
Let us now show that (10) admits solutions as soon as τ < 1 eC F . It is equivalent to find positive and non-decreasing solutions R of:

R ′ (τ ′ ) > C F R ′ (τ )R(τ ′ ) + 1 τ ′ ∈ [0, τ ], R(0) = 0. (28)
Let us consider λ ∈ 1,

1 eτ C F .
The function h : γ → ln γλγτ C F is smooth on (0, +∞) and admits a maximum at γ max = 1 λτ C F . As λ < 1 eτ C F , it is straightforward to check that ln λ < h(γ max ) and so, there exists γ 0 > 0 such that h(γ 0 ) = ln λ or equivalently that γ 0 = λe λγ 0 C F τ .

Let us define U

λ : τ ′ → 1 γ 0 C F (e λγ 0 C F τ ′ -1). We have: U ′ λ (τ ) = λe λγ 0 C F τ = γ 0 . As λ > 1, U λ verifies: U ′ λ (τ ′ ) = λγ 0 C F U λ (τ ′ ) + λ > γ 0 C F U λ (τ ′ ) + 1 = C F U ′ λ (τ )U λ (τ ′ ) + 1 τ ′ ∈ [0, τ ], U λ (0) = 0.
For all λ ∈ 1, 1 eτ C F , we can construct a solution U λ to (28). Therefore, as soon as τ < 1 eC F , there exist infinitely many solutions. Remark 3.2. Condition (10) is reduced to:

ρC F τ ′ < 1 - 1 ρ ,
if we consider ρ as a constant. If it is considered for τ ′ = τ , we get τ < 1 4C F because the condition tells that the function x → C F τ x 2x + 1 has a negative part. Conversely, for all fixed τ < 1 4C F , we can always find constant solutions.

Consequence on drivers' positions and a counter-example

A very practical consequence of this strict comparison principle for the microscopic model derived from traffic flow is the conservation of initial order for vehicles, provided they are suitably spaced out at initial times.

Corollary 3.3 (Conservation of initial order). Let us consider (X i ) i∈Z a sequence of drivers' positions that evolve under the dynamics (1) with initial conditions given by (2) and for τ 0 ≡ τ . We consider the solution u to (7) such that:

u 0 (i, t) = x 0 i (t) (i, t) ∈ Z × [-τ, 0
]. We suppose that there exist σ > 0, a time function ρ p which verifies [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] such that:

σ ≤ u 0 (x + 1, t -τ ′ ) -u 0 (x, t -τ ′ ) ≤ ρ p (τ ′ )(u 0 (x + 1, t) -u 0 (x, t)), τ ′ ∈ [0, τ ], (x, t) ∈ R × [-τ, 0]. (29) Then, X i+1 (t) > X i (t) (i, t) ∈ Z × [0, T ).
Proof of Corollary 3.3. . By uniqueness, we have:

X i (t) = u(i, t) (i, t) ∈ Z × [0, T ). (30)
We consider v : (x, t) → u(x + 1, t). In particular, we have:

X i+1 (t) = v(i, t) (i, t) ∈ Z × [0, T ). (31)
For d := vu, thanks to (29), the conditions of Theorem 1.2 (and its remark) are fulfilled with δ = σ, t 0 = 0, R = +∞ and ρ = ρ p . Therefore we have v > u on R × [0, T ) and in particular on Z × [0, T ). Using (30) and (31) this is equivalent to have:

X i+1 (t) > X i (t) (i, t) ∈ Z × [0, T ).
Remark 3.4. Our microscopic model is valid when the drivers always stay in the same order, otherwise it is not adapted anymore to the physical situation. More explicitly, the behaviour of the driver at position X i does not depend anymore on the position X i+1 if at some time the car i + 1 is not in front of the car i.

In contrast with most Hamilton-Jacobi equations, it is not possible to state a classical comparison principle because of the delay time. More accurately, it is compulsory to ensure that the vehicles are suitably spaced out (this space is represented by the function ρ).

Example 3.5. We consider the case: τ 0 ≡ τ . For any delay time τ > 0, it is straightforward to show that the initial order of solutions is sometimes not conserved when the vehicles are not suitably spaced in the sense made precise in Theorem 1.2. Let us consider n 0 ∈ N * such that τ > 2 n 0 . Let us consider the particular case for F being identically equal to the identity on [0; 1] (and that verifies (5) in R). Let us consider two sets of drivers (X i ) i∈Z and (Y i ) i∈Z that evolve under the same dynamics (1) and such that:

y 0 i (t) < x 0 i (t) (i, t) ∈ Z × [-τ, 0],
and there exists j ∈ Z such that for t ∈ [-τ, 0]:

     y 0 j (t) = j -1 + n 0 n 0 +1 e n 0 t < x 0 j (t) = j, y 0 j+1 (t) = j + n 0 n 0 +1 e n 0 t , x 0 j+1 (t) = y 0 j+1 (t) + 1 n 0 +1 (32)
Those two sets of drivers do not see each other, that means that they do not evolve on the same physical road but on two identical copies of the real line. We can integrate (1) on [0, τ ] for the two sets and we find that:

(Y j -X j )(τ ) = n 0 n 0 + 1 τ - 1 n 0 + 1 - 0 -τ e n 0 u du = n 0 n 0 + 1 τ - 1 n 0 + 1 - 1 -e -n 0 τ n 0 + 1 > 0.
Therefore, the initial order is disrupted even if:

1 n 0 + 1 ≤ x 0 i (t) -y 0 i (t) (i, t) ∈ Z × [-τ, 0].

Convergence

This section is mainly devoted to the proof of Theorem 1.1. The unstable stationnary case will be given as the final remark of this section.

Proof of Theorem 1.1

Proof of Theorem 1.1. . Let us first show that u ε is globally Lipschitz continuous in space and time uniformly in ε.

In time: By looking at (4), we remark that u ε ||F || ∞ -Lipschitz continuous in time.

In space: The equation is translation invariant in space and invariant by addition of constants to the solutions. Let h > 0.

The solution corresponding to the initial condition u

1 : (x, t) → u 0 (x + h, t) is the function w : (x, t) → u ε (x + h, t).
The one associated to u 2 : (x, t) → u 0 (x, t) + 2Lh is v : (x, t) → u ε (x, t) + 2Lh.

We define ψ 1 : (x, t) → 1 ε w(εx, εt) and ψ 2 : (x, t) → 1 ε v(εx, εt). ψ 1 solves:

∂ t u(x, t) = F (u(x + 1, t -τ 0 (εx)) -u(x, t -τ 0 (εx))) (x, t) ∈ R × (0, T ), u(x, t) = 1 ε u 0 (εx + h, εt) (x, t) ∈ R × [-2τ ; 0]. (33)
ψ 2 solves:

∂ t u(x, t) = F (u(x + 1, t -τ 0 (εx)) -u(x, t -τ 0 (εx))) (x, t) ∈ R × (0, T ), u(x, t) = 1 ε u 0 (εx, εt) + 2Lh ε (x, t) ∈ R × [-2τ ; 0]. (34)
By ( 5), ψ 1 and ψ 2 are L-Lipschitz continuous functions on R × [-2τ, 0].

Hence, d := ψ 2 -ψ 1 is a 2L-Lipschitz continuous function on R × [-2τ, 0]
. Otherwise, we have:

d(x, t) = 1 ε (u 0 (εx, εt) -u 0 (εx + h, εt)) + 2Lh ε ≥ - Lh ε + 2Lh ε = Lh ε (x, t) ∈ R × [-τ ; 0]. (35)
Let us now consider ρ m any solution to [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] for τ < 1 eC F (see Proposition 3.1). We recall that [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] implies

ρ m ≥ ρ m (0) > 1.
We have:

d(x, t -τ ′ ) ≤ d(x, t) + 2Lτ τ ′ ∈ [0, τ ], (x, t) ∈ R × [-τ, 0].
Therefore, to get:

d(x, t -τ ′ ) ≤ ρ m (τ ′ )d(x, t) τ ′ ∈ [0, τ ], (x, t) ∈ R × [-τ, 0],
it is sufficient to verify:

2Lτ ≤ (ρ m (τ ′ ) -1)d(x, t) τ ′ ∈ [0, τ ], (x, t) ∈ R × [-τ, 0].
As ρ m is a non-decreasing function, and as d ≥ 0, it is finally sufficient to verify:

2Lτ ρ m (0) -1 ≤ d(x, t) τ ′ ∈ [0, τ ], (x, t) ∈ R × [-τ, 0].
Thanks to (35), this will be the case for ε small enough. The conditions of the strict comparison principle are fulfilled with δ = Lh ε , t 0 = 0, R = +∞ and ρ = ρ m . Then, by Theorem 1.2 we have ψ 2ψ 1 > 0 in R × [0, T ) and this implies for all (y, s) ∈ R × [0, T ):

u ε (y + h, s) ≤ u ε (y, s) + 2Lh.
Analogously, we can show that:

u ε (y + h, s) ≥ u ε (y, s) -2Lh.
Hence, u ε is 2L-Lipschitz continuous in space. Therefore, we can define the relaxed upper and lower semi-limits ū(x, t) = lim ε→0 sup * u ε (x, t) and u(x, t) = lim ε→0 inf * u ε (x, t). By definition, we have u ≤ ū.

Given ν > 0, for ε small enough, we can apply Theorem 1.

2 on v b : (x, t) → 1 ε (u 0 (εx, εt) + ν + (||F || ∞ + L)εt) and on u b : (x, t) → 1 ε u ε (εx, εt
) and we get:

u ε (x, t) ≤ u 0 (x, t) + ν + (||F || ∞ + L)t (x, t) ∈ R × [-2ετ, T ),
which gives for vanishing ν:

u ε (x, t) ≤ u 0 (x, t) + (||F || ∞ + L)t (x, t) ∈ R × [-2ετ, T ).
Similarly, we get:

u ε (x, t) ≥ u 0 (x, t) -(||F || ∞ + L)t (x, t) ∈ R × [-2ετ, T ),
and so:

|u ε (x, t) -u 0 (x, t)| ≤ (||F || ∞ + L)t (x, t) ∈ R × [-2ετ, T ).
This implies:

ū(0, x) = u(0, x) = u 0 (x, 0) x ∈ R.
To get the convergence, it is sufficient to show that ū and u are respectively a subsolution and a supersolution of [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF]. Indeed, by using a classical comparison principle we will then get:

u ≤ ū ≤ u 0 ≤ u
Let us only show by contradiction that ū is a subsolution of ( 6) (the other one being very similar). Assume that there exist (x, t), ϕ ∈ C 1 x,t , (r, θ, η) ∈ (0, +∞) 3 such that:

               ū(x, t) = ϕ(x, t), ū < ϕ in B 2r (x, t) \ {(x, t)}, ū ≤ ϕ -2η in B 2r (x, t) \ B r (x, t), ∂ t ϕ(x, t) = 2θ + F (∂ x ϕ(x, t)), ∂ t ϕ(x, t) ≥ θ + F (∂ x ϕ(x, t)) in B r (x, t), (36)
where we set B s (p, q) := (ps, p + s) × (qs, q + s).

As u ε and ϕ are continuous, we can define:

M ε := max B 2r (x, t) (u ε -ϕ) := (u ε -ϕ)(x ε , t ε ). (37)
M ε ≥ -η for ε small enough and hence (x ε , t ε ) ∈ B r (x, t).

Let us define: ϕ ε := ϕ + M ε . This function satisfies:

ϕ ε (x ε , t ε ) = u ε (x ε , t ε ) ∂ t ϕ ε (x, t) ≥ θ + F (∂ x ϕ ε (x, t)) in B r (x, t). (38)
By regularity of ϕ ε and F , for ε small enough, we have in B r (x, t):

∂ t ϕ ε (x, t) ≥ F ϕ ε (x + ε, t -ετ 0 (x)) -ϕ ε (x, t -ετ 0 (x)) ε + θ 2 . (39)
Let us define:

d ε (x, t) = ϕ ε (x, t) -u ε (x, t).
In B 2r (x, t) \ B r (x, t) we have:

d ε (x, t) = ϕ(x, t) -u ε (x, t) + M ε ≥ 3η 2 -η = η 2 . ( 40
)
Using the fact that u ε is ||F || ∞ -Lipschitz continuous in time and that ϕ is smooth, we remark that d ε is Lipschitz continuous in B 2r (x, t). Let K denote its Lipschitz constant. Let us define:

d 1 : (x, t) → 1 ε d ε (εx, εt). d 1 is K-Lipschitz continuous in B 2r ε x ε , t ε . Thanks to (40), we have in B 2r ε x ε , t ε \ B r ε x ε , t ε : d 1 (x, t) ≥ η 2ε . (41)
Let us now consider ρ 1 any solution to [START_REF] Gipps | A behavioural car-following model for computer simulation[END_REF] for τ < 1 eC F (see Proposition 3.1). We recall that (10) implies

ρ 1 ≥ ρ 1 (0) > 1.
We have:

d 1 (x, t -τ ′ ) ≤ d 1 (x, t) + Kτ τ ′ ∈ [0, τ ], (x, t) ∈ B 2r ε x ε , t ε \ B r ε x ε , t ε .
Therefore, to get:

d 1 (x, t -τ ′ ) ≤ ρ 1 (τ ′ )d 1 (x, t) τ ′ ∈ [0, τ ], (x, t) ∈ B 2r ε x ε , t ε \ B r ε x ε , t ε ,
it is sufficient to verify:

Kτ ≤ (ρ 1 (τ ′ ) -1)d 1 (x, t) τ ′ ∈ [0, τ ], (x, t) ∈ B 2r ε x ε , t ε \ B r ε x ε , t ε .
As ρ 1 is a non-decreasing function, and as d 1 ≥ 0, it is finally sufficient to verify:

Kτ ρ 1 (0) -1 ≤ d 1 (x, t) τ ′ ∈ [0, τ ], (x, t) ∈ B 2r ε x ε , t ε \ B r ε x ε , t ε .
Thanks to (41), this will be the case for ε small enough. We choose ε such that max(1, τ ) < r 8ε . The previous inequalities enable us to apply the strict comparison principle with δ = η 2ε , R = 3r 2ε , x 0 = x ε , T = t ε + R and t 0 = t ε -R. Indeed, we have:

   [x 0 -R -1, x 0 + R + 1]\]x 0 -R, x 0 + R[×[t 0 -τ, T ) ⊂ B 2r ε x ε , t ε \ B r ε x ε , t ε [x 0 -R, x 0 + R] × [t 0 -τ, t 0 ] ⊂ B 2r ε x ε , t ε \ B r ε x ε , t ε . (42)
Hence, by Theorem 1.2, we have

d 1 > 0 in [x 0 -R, x 0 + R] × [t 0 , T )
which is in contradiction with the definition of (x ε , t ε ) as we have:

xε ε , tε ε ∈ B r ε x ε , t ε ⊂ [x 0 -R, x 0 + R] × [t 0 , T ), d 1 xε ε , tε ε = 0.

A special case: homogenization for any reaction time

A natural question arises. What happens for higher reaction times? Example 3.5 highlights the fact that the initial dynamics is very important and that reaction times cannot be considered separately. The answer to this question is not invariable and the following example shows that the expected macrosopic model can be derived for any reaction time for a special initial condition.

Example 4.1. For T = +∞, let us consider the case where all drivers have the same reaction time τ ∈ (0, +∞). Let L > 0 be the common interdistance between all the vehicles. We consider that the vehicles do not move at initial times:

x 0 i (t) = Li, t ∈ [-τ, 0].
This corresponds to:

u 0 (x, t) = Lx, (x, t) ∈ R × [-τ, 0].
By incremental construction (or directly by uniqueness), we see that the solution u ε to (4) for this initial condition does not depend on ε and is given by:

u ε (x, t) = Lx + F (L)t, (x, t) ∈ R × [0, +∞).
The unique solution u 0 of (6) corresponding to this initial linear data is also given by the same expression. Therefore we have:

u ε (x, t) = u 0 (x, t) = Lx + F (L)t, (x, t) ∈ R × [0, +∞).

A counter-example to homogenization

The goal of this section is to exhibit a counter-example derived from Example 4.1. We will first give the explicit expressions of F , of the vehicles' initial positions and the value of τ and give a list of lemmas that will be useful to finally prove Theorem 1.5.

For simplicity, we fix T = +∞. Let us consider the same L > 0 and (k, β, α) ∈ (0, +∞) 3 with the condition α > 4βL. We consider the following F :

F (x) = k + β(x -L) 2 + α(x -L) x ∈ [0, 2L],
and we continuously extend F to R by constants; hence F satisfies [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear pde[END_REF].

We here choose the common reaction time: τ = π 4α . Let us consider A ∈ (0, L 2 ). We now introduce the initial positions of the vehicles:

x 0 i (t) = iL + (-1) i A 2 sin(2αt) (i, t) ∈ Z × [-τ, 0]. (43)
Remark 5.1. We have x 0 i+2x 0 i = 2L but x 0 i+1x 0 i = L. We recover the initial data of Example 4.1 for A = 0. This means that in this case, the vehicles alternatively oscillate around the previous stationnary positions at initial times. Those oscillations will be essential in the construction of the counter-example: they will last for all time and will raise the time velocity of the vehicles that will become striclty superior to the value of the velocity function of the corresponding macroscopic space gradient.

The key relationship between τ and α will allow the periodic oscillations to remain for all times as expressed in the following lemma: Lemma 5.2. Under the previous initial conditions, we have:

X i+1 (t) -X i (t) = L + A(-1) i+1 sin(2αt) t ≥ 0.
Remark 5.3. We have X i+2 -X i = 2L for all time.

Proof. The proof is based on the incremental construction of the solutions and is thus an induction proof. We show the result for each [nτ, (n + 1)τ ], n ∈ N. We only perform the first step as the next ones are identical.

Let us define:

d i := X i+1 -X i .
For t ∈ [0, τ ], we have:

d ′ i (t) = F (X i+2 (t -τ ) -X i+1 (t -τ )) -F (d i (t -τ )).
As tτ ∈ [-τ, 0]:

d ′ i (t) = F (x 0 i+2 (t -τ ) -x 0 i+1 (t -τ )) -F (d i (t -τ )).
Thanks to Remark 5.1, we get:

d ′ i (t) = F (2L -d i (t -τ )) -F (d i (t -τ )).
With the expression of F , this equivalently gives for di := d i -L:

di ′ (t) = -2α di (t -τ ).
Thanks to (43), we have:

di ′ (t) = -2αA(-1) i+1 sin(2αt -2ατ ).
We remind that we have chosen τ = π 4α to get: di ′ (t) = 2αA(-1) i+1 cos(2αt).

We then integrate:

di (t) = A(-1) i+1 sin(2αt).
This gives the result for n = 0.

For ε > 0, we now consider u ε associated to this initial data and we recall the relationship:

X i (t) = u ε (iε, tε) ε (i, t) ∈ Z × [-τ, +∞). (44)
Thanks to the previous remark, we have:

u ε ((i + 2)ε, tε) -u ε (iε, tε) 2ε = L (i, t) ∈ Z × [-τ, +∞),
or equivalently:

u ε ((i + 2)ε, s) -u ε (iε, s) 2ε = L (i, s) ∈ Z × [-ετ, +∞).
For s = 0, this gives for the initial condition:

u 0 ((i + 2)ε, 0) -u 0 (iε, 0) 2ε = L (i, ε) ∈ Z × (0, +∞). (45)
Moreover, thanks to (43), we have:

x 0 0 (0) = 0. Thus, we also have:

u 0 (0, 0) = 0. (46)
This leads to the following lemma:

Lemma 5.4. The initial condition is given on the real line by: u 0 (x, 0) = Lx.

Proof. Let us consider ε > 0, h > 0 and x ∈ R. We define:

n := ⌊ h 2ε ⌋, m := ⌊ x 2ε ⌋. (47)
Let us now compute the rate of change:

u 0 (x + h, 0) -u 0 (x, 0) h = u 0 (x + h, 0) -u 0 (2(m + n)ε, 0) h + u 0 (2(m + n)ε, 0) -u 0 (2mε, 0) h + u 0 (2mε, 0) -u 0 (x, 0) h . (48)
We introduce a telescopic sum for the second term: Thanks to (45), we get:

u 0 (2(m + n)ε, 0) -u 0 (2mε, 0) h = 2nε h L.
Thus, (48) becomes:

u 0 (x + h, 0) -u 0 (x, 0) h = u 0 (x + h, 0) -u 0 (2(m + n)ε, 0) h + 2nε h L + u 0 (2mε, 0) -u 0 (x, 0) h . ( 49
)
Let us remind that:

|h -2nε| ≤ 2ε, |x -2mε| ≤ 2ε. (50)
The left-hand side of (49) does not depend on ε. By taking the limit when ε → 0, using (50) and the fact that u 0 is a L-Lipschitz continuous function, we get: u 0 (x + h, 0)u 0 (x, 0) h = L.

Thanks to (46), we obtain the desired result on [0, +∞):

u 0 (h, 0) = Lh h ≥ 0.
For h < 0, we define h ′ := -h > 0, and we have for x = h: u 0 (h + h ′ , 0)u 0 (h, 0) h ′ = L, and thus: u 0 (h, 0) = -Lh ′ = Lh.

Corollary 5.5. The solution of the macroscopic equation (6) corresponding to this initial data is:

u 0 (x, t) = Lx + F (L)t (x, t) ∈ R × [0, +∞).
Proof. This is a classical solution and the solution is unique.

Remark 5.6. We see that we have:

∂ t u 0 (0, t) = k = F (L) t > 0.
That was also the case for u ε in Example 4.1 but it is not true anymore for A > 0.

Indeed, we have for s > 0:

∂ t u ε (0, s) = F u ε (ε, sετ )u ε (0, sετ ) ε

Thanks to (44), we get:

∂ t u ε (0, s) = F X 1 s ε -τ -X 0 s ε -τ .
From Lemma 5.2, we get for the velocity function F considered:

∂ t u ε (0, s) = F (L) + βA 2 sin 2 2α s ε -τ -αA sin 2α s ε -τ ,
or equivalently, for τ = π 4α :

∂ t u ε (0, s) = F (L) + βA 2 cos 2 2α s ε + αA cos 2α s ε . (51)
We are now ready to prove Theorem 1.5.

Proof of Theorem 1.5. By contradiction, we suppose that u ε converges locally uniformly towards the solution u 0 of (6) whose expression is given in corollary 5.5.

Let us consider s > 0 and h > 0. We then have: lim ε→0 u ε (0, s + h)u ε (0, s) h = u 0 (0, s + h)u 0 (0, s) h = F (L).

(52)

Otherwise, we have for ε > 0:

u ε (0, s + h) -u ε (0, s) h = 1 h s+h s ∂ t u ε (0, s ′ )ds ′ .
From (51), we get:

u ε (0, s + h) -u ε (0, s) h = F (L) + β A 2 2 +
lim ε→0 u ε (0, s + h) -u ε (0, s) h = F (L) + β A 2 2 ,
which is in contradiction with (52).

Acknowledgements. I would like to thank Cyril Imbert for all his support, help and remarks on this article. I would also like to thank Régis Monneau for all his contribution. This work was partly supported by Labex Bézout, (ANR-10-LABX-58-01) and ANR HJnet, (ANR-12-BS01-0008-01).

u 0 ( 2 ( 1 i=0u 0

 210 m + n)ε, 0)u 0 (2mε, 0) h = n-(2mε + 2(i + 1)ε, 0)u 0 (2mε + 2iε, 0) h ,or equivalently:u 0 (2(m + n)ε, 0)u 0 (2mε, 0) 2mε + 2iε + 2ε, 0)u 0 (2mε + 2iε, 0) 2ε .


Songmuang

C Himwas

M Den Hertog

Le Si Dang

E Monroy

R Songmuang

Alloy inhomogeneity and carrier localization in AlGaN sections and AlGaN/AlN nanodisks in nanowires with 240-350 nm emission

Alloy inhomogeneity and carrier localization in AlGaN sections and AlGaN/AlN nanodisks in nanowires with 240-350 nm emission

The Al-Ga intermixing at Al(Ga)N/GaN interfaces in nanowires and the chemical inhomogeneity in Al x Ga 1-x N/AlN nanodisks (NDs) are attributed to the strain relaxation process. This interpretation is supported by the three-dimensional strain distribution calculated by minimizing the elastic energy in the structure. The alloy inhomogeneity increases with Al content, leading to enhanced carrier localization signatures in their optical characteristics i.e. red shift of the emission, s-shaped temperature dependence and linewidth broadening. Despite these alloy fluctuations, the emission energy of AlGaN/AlN NDs can be tuned in the 240-350 nm range with internal quantum efficiencies around 30%.

AlGaN/AlN quantum wells (QWs) 1 and quantum dots (QDs) 2,3,4 have become promising active media for solid-state ultraviolet lighting. Nanodisks (NDs) in nanowires (NWs) have also emerged as alternative nanostructures which offer not only strong carrier localization 5,6 but also a possibility to improve photon extraction efficiency 7 . However, synthesis of ternary alloys in NWs generally suffers from strong chemical inhomogeneities 8 , which modify their electronic properties with respect to random alloys.

Here, we present structural and optical studies of AlGaN sections and AlGaN/AlN NDs in NWs. In the case of AlGaN sections, alloy gradient and spontaneous chemical ordering were found along the wire axis, in addition to the formation of an Alrich shell. We found that alloy fluctuations increase in AlGaN sections with higher Al content, as evidenced by stronger carrier localization in their optical characteristics. At GaN/Al(Ga)N interfaces, we identify a transition zone with strong Al-Ga intermixing whose formation is attributed to the strain relaxation process. This interpretation is supported by a three-dimensional (3D) strain simulation. Despite the radial and axial alloy inhomogeneity, the photon emission energy of AlGaN/AlN NDs can be tuned by adjusting the nominal thickness and Al content.

AlGaN sections and stacks of AlGaN/AlN NDs on GaN NWs were grown on Si (111) by plasma-assisted molecular beam epitaxy (PAMBE). Initially, a thin (~2 nm) AlN buffer layer 9 was grown at the substrate temperature T s = 840°C. Then, T s was decreased to 795 °C for the growth of GaN NWs using a Ga flux Φ Ga ~ 0.17 ML/s while the flux of active nitrogen was set at Φ N = 0.34 ML/s to maintain N-rich conditions. The deposition time of GaN NWs was t = 2.5 hours. Then, the Al x Ga 1-x N sections or the Al x Ga 1-x N/AlN NDs were synthesized on the top of these GaN NWs. The deposition time of AlGaN sections was 40 min, while that of AlGaN NDs was between 10 and 40 s with ~1 min AlN barriers. To adjust the nominal Al content (x = Φ Al /Φ N ) in the Al x Ga 1-x N, the Al flux was varied in the Φ Al = 0.07-0.17 ML/s range, while keeping Φ Ga = 0.09 ML/s and Φ N = 0.34 ML/s for the entire growth process. The Al, Ga and N deposition rates were calibrated by reflection high-energy electron diffraction intensity oscillations during the growth of two-dimensional layers of GaN or AlN on GaN/sapphire substrates at 720°C. The substrate temperature for the calibration is lower than the growth temperature of the NWs. Thus, the real deposition rate should be lower than the calibration due to the thermal activation of Ga desorption and GaN decomposition.

The NW morphology was characterized by field emission scanning electron microscopy using a Zeiss Ultra55. The typical NW diameter is around 60 nm, with a length of 600 nm. The polarity of GaN NWs is found to be N-polar 10 . Detailed structural properties were measured by high angle annular dark field scanning transmission electron microscopy (HAADF-STEM) using a probe-corrected FEI Titan operated at 300 kV. Photoluminescence (PL) was performed by exciting the samples with a continuous-wave frequency-doubled Ar + laser (λ=244 nm). The excitation power was around 100 µW with a spot diameter of 100 µm. The emission from the sample was collected by a Jobin Yvon HR460 monochromator equipped with a UV-enhanced charge-coupled device (CCD) camera. Cathodoluminescence (CL) measurements were performed in a FEI quanta 200 CL system, using an acceleration voltage of 10 kV and a current of ~100 pA. The sample emission was collected by a parabolic mirror, which focused it onto a CCD camera.

Deduced from HAADF-STEM images of several wires, the average length of the Al x Ga 1-x N sections on the GaN NWs was ~180 nm. This value corresponds to an AlGaN vertical growth rate R AlGaN ~ 0.30 ML/s 11 , independent of Φ Al and nearly equal to Φ N , indicating that the AlGaN growth is limited by the N-flux. 12 In addition, R AlGaN is larger than the summation of the Al and Ga fluxes (Φ Ga +Φ Al ≤ 0.26 ML/s). This fact confirms that the adatom diffusion at the NW sidewall contributes to the NW growth, in addition to the direct material deposition on the NW top. The majority of the diffusing adatoms are supposed to be Ga since their diffusion length is longer than that of Al and N adatoms. 12, 13 Figure 1(a) presents an HAADF-STEM image of an Al 0.5 Ga 0.5 N section on a GaN NW. The bright contrast is GaN whereas the darker contrast implies higher Al content. Along the radius of the NWs, we observe an Al-rich shell encapsulating both the AlGaN section and GaN NW base, which spontaneously forms during the deposition of AlGaN section. This Al-rich shell 14,15 , was observed in all our growth conditions (i.e. various metal/N ratio and growth temperature). Its formation is explained by the higher incorporation rate of Al at the sidewalls caused by the shorter diffusion length of the Al adatoms than that of Ga atoms.

Figure 1(b) shows the bright contrast of Fig. 1(a) taken at the NW center along the wire axis. Most of the investigated NWs show a transition zone [labeled (2)] between the GaN stem [labeled (1)] and the AlGaN section. The typical thickness of this area with a lower Al content is around ~10 nm. By reducing the growth temperature from 790°C to 735°C, the HAADF-STEM images show negligible changes in this zone in terms of chemical contrast and thickness. This fact indicates that the formation of this zone is not related to a surface reservoir of Ga atoms at the NW sidewalls, as Ga desorption/diffusion is highly temperature dependence in the range of our growth temperatures. After the transition zone, an Al-rich area [region (3)] was found, followed by an area displaying chemical ordering [region (4)]. The lengths of these two regions fluctuate from wire to wire. In the region (4), we usually find the 1:1 ratio ordering with the thickness of Al-rich and Ga-rich layers of about ~ 3 ML. However, it is difficult to determine precisely the Al content in both regions. The spontaneous chemical ordering is similar to what was observed in AlGaN 16,17,18 and InGaN films 19 . Theoretically, the ordering in wurtzite Al x Ga 1-x N was suggested to be driven by the different binding energy of Al-N bond and Ga-N bond 20,21 . The adatom surface diffusion generally allows the incorporation of Al and Ga atoms at their different preferential sites, leading to the ordering sequence of Al-rich and Ga rich areas along the growth axis. Such a description is highly probable for the AlGaN growth in NWs where the contribution of adatom diffusion is significant. After region (4), several wires show the formation of a random alloy.

To understand the transition zone formation, we investigate an AlN/GaN NW superlattice by HAADF-STEM as shown in Fig. 2. Similarly, the image contrast reveals a transition zone with a length of ~ 8 nm at the first AlN/GaN interface [Label (1) in Fig. 2(a)]. The transition zones at the second, third and forth interfaces [Labels (2), ( 3) and ( 4)] are significantly shorter. From the image contrast, these regions should be Al-Ga intermixing areas. Besides, the intermixing process is seemingly more pronounced on the top of GaN rather than at the interface below them.

We propose that the intermixing at the AlN/GaN interface forms in order to partially relieve the strain induced by the lattice mismatch between AlN and GaN. We have explored the role of the strain via 3D strain simulations using NextNano 3 .The strain distribution was obtained by the minimization of the elastic strain energy through the application of zero-stress boundary conditions at the NW surface, which allows the NW to deform in all three spatial directions. In our case, the structure of interest is a 50 nmdiameter hexagonal GaN NW overgrown by various periods of 10-nm GaN/10 nm-AlN, i.e. the geometry corresponding to the structure in Fig. 2(a). We do not consider the formation of an Al shell as it is not observed in this HAADF-STEM image. The number of GaN/AlN periods in the simulation was increased sequentially, to assess the strain distribution at the topmost AlN/GaN interface after the deposition of each period. Note that the Al-Ga intermixing at the AlN/GaN interface was not taken into account for the simulation.

Figures 2(b) and (c) present the color map of the strain in x direction (e xx ) in a hexagonal GaN NW capped with a single AlN layer. Figure 2(b) illustrates the in-plane distribution of tensile strain at the interface of GaN NW and AlN layer on the AlN side. The cross section in Fig. 2(c); shows the high degree of compressive strain (blue contrast) at the interface in the GaN region, and the tensile strain (red contrast) in the AlN region. From the simulation, the first AlN layer is strongly tensile strained by the GaN NW base. Figure 2(d) describes the elastic energy density at the topmost AlN/GaN interface on the AlN (■) side taken at the center of a GaN NW, plotted as a function of the number of AlN sections in the stack.

The stronger tensile strain in the first AlN layer probably enhances Al-Ga intermixing at the AlN/GaN interface. As the NW growth proceeds, the higher AlN barriers are less strained, thus giving lower driving force for the intermixing process, and consequently a shorter transition zone. The lower intermixing effect at the interface below each GaN layer than the one above them is possibly caused by the higher binding energy of the Al-N bond with respect to the Ga-N bond.

Figure 3 shows the PL spectra measured at 5 K and 300 K of GaN NWs with an Al x Ga 1-x N section with nominal x= 0.21, 0.36 and 0.50 grown at 795°C. At 5 K, the PL peak which remains constant at around 3.47-3.5 eV is assigned to the GaN NW base, whereas those attributed to Al x Ga 1-x N sections monotonically blue shift from 3.76 to 5 eV as a function of x. The full width at half maximum (FWHM) also increases from 78 meV to 445 meV when x is increased to 0.50. These observed linewidths are larger than those reported in AlGaN 2D films 22 . The inset of Fig. 3(a) shows that the PL peak at 5K is lower and progressively deviates (up to 200 meV) from the expected bandgap value 23 . The integrated PL intensity of Al x Ga 1-x N sections as a function of temperature where x = 0.21, 0.36, and 0.50 is shown in Fig. 3(b). The intensity evolution as a function of temperature is described by an equation with two activation energies: I(T) = I 0 /(1+ae -Ea1/kT +be -Ea2/kT ),where I is the PL intensity, E a1 and E a2 are the activation energies, kT is the thermal energy, a and b are fitting parameters. The fits are represented by the solid lines in this figure . The extracted E a1 in the range of 75-135 meV is attributed to the localization energy corresponding to the potential barriers which confine the carriers in Ga-rich area of Al x Ga 1-x N section. The E a2 with the value in the range of 15-20 meV could represent the activation energy accounting for the detrapping of excitons from the interface roughness fluctuation in the ordered alloy. Here, the internal quantum efficiency (IQE) is estimated by comparing the integrated intensity measured at room temperature with respect to that obtained at low temperature, assuming that IQE(5K) is 100%. The extracted IQE of AlGaN section increases from 0.7 to 5 % when x increases.

The PL spectral shifts with temperature can provide information on the carrier localization in potential fluctuations in the alloy. The inset of Fig. 3(b) depicts the evolution of PL peak energy as a function of temperature, showing an s-shape behavior which deviates significantly from Varshni's equation 24 . This trend is explained by the exciton freeze-out in the local potential minima in the alloy at low temperature, followed by the onset of exciton thermalization with increasing temperature. The deviation becomes more pronounced with increasing x, similar to the case of AlGaN 2D films 25 . Potential fluctuations in ternary alloys can reduce the probability of carrier capture by nonradiative recombination centers. Therefore, the higher IQE in AlGaN section with larger Al content (x<0.5) could be associated to the larger alloy inhomogeneity. To further enhance carrier localization in the AlGaN NW UVemitters, Al x Ga 1-x N/AlN NDs were grown on the top of GaN NWs. From the growth conditions and the HAADF-STEM images, the thickness of the AlGaN NDs was found in the range of 1-4 nm and the thickness of the AlN barrier is around 4 nm. Figure 4 shows HAADF-STEM of a 25 nm diameter NW containing Al 0.38 Ga 0.62 N/AlN NDs. The ND thickness is the range of 2 nm, whereas that of the AlN barriers is ~4 nm. The diameter of the first Al x Ga 1-x N ND is similar to the one of GaN NW stem. Due to the lateral growth of AlN barrier, the ND diameter progressively increases along the growth axis. Moreover, the shape of the NDs evolves to the pyramid-like shape with {1-103} facets, similar to GaN SK QDs 26 . From the image contrast, we found a radial compositional gradient, that is, the center of the ND has higher Al content than the edge area [Fig. 4(a)]. The contrast profile taken at the center of the NW in Fig. 4(b) reveals that the Al composition at the center of the NDs gradually increases along the growth axis (the image contrast become darker) and finally saturates at the third ND. In addition, the profile in Fig. 4(c) taken at the edge of the NW shows that the Ga-rich area becomes better defined. This result suggests that the degree of lateral phase separation is more pronounced in the higher ND layer.

From our data, we suggest that that the strain can modify the shape, the axial and the radial distribution of Al composition of the AlGaN NDs. We performed 3D strain simulations of a 25 nmdiameter hexagonal GaN NW with various periods of 2 nm Al 0.38 Ga 0.62 N/ 4 nm AlN, to imitate our result shown in Fig 4(a). In this case, we consider the formation of the Al shell which encapsulates NDs and NW. The number of periods in the simulation was increased sequentially, to assess the strain distribution after the growth of an AlN barrier, following 0, 1, 2, 3 periods of Al 0.38 Ga 0.62 N/AlN ND. For simplicity, we assume random alloy in the AlGaN NDs. The inset of Fig. 4

Figure 1 (

 1 Figure 1 (a) HAADF-STEM image of a GaN NW with a single AlxGa1-xN section. (b) Corresponding image contrast profile taken at the center of the NW shown in (a).

Figure 2 (

 2 Figure 2 (a) HAADF-STEM image of a GaN NW with an AlN/GaN superlattice. The arrows indicate the transition zone at the AlN/GaN interfaces. (b,c) Contour plot exx in a GaN NW with an AlN section on top: (b) in-plane view of the interface at the AlN side, and (c) cross-section at the center of GaN NW. (d) Plot of the elastic energy density measured at the topmost AlN/GaN interface on the AlN sides as a function of the number of AlN layers in the stack.

Figure 3 (

 3 Figure 3 (a) PL spectra measured at 5 K and 300 K of GaN NWs with a 180 nm Al x Ga 1-x N section with x= 0.21, 0.36 and 0.50. Inset: PL peak position as a function of x in comparison with the expected bandgap value (solid line). The error bars represent the PL FWHM (b) Normalized integrated PL intensity of Al x Ga 1-x N sections on GaN NWs as a function of temperature. Inset: Temperature dependent PL peak position. The solid line represents the shift expected from Varshini's equation

Figure 4 (

 4 Figure 4 (a) HAADF-STEM image of Al0.38Ga0.62N/AlN NDs. (b) and (c) Image contrast taken at the center and at the edge of the NW in (a), respectively. (d) The plot of the elastic energy density taken at the center of the NW on the surface of the topmost 4-nm AlN barrier prior to the growth of the Al0.38Ga0.62N ND, as a function of the number of AlN barriers. The inset shows the cross-section contour plot of exx in the structure with 3 layers of 2-nm AlxGa1-xN /4 nm AlN on GaN NW.

  (d) presents the cross-section contour plot of e xx taken at the center of the NW after the growth of the 4 th AlN barrier.

Figure 4 (

 4 d) presents the summary of the elastic energy density taken at the surface of the topmost AlN barrier and at the center of the NW. The values are plotted as a function of the number of

AlN barriers. The first 4-nm AlN barrier is highly tensile strained by the GaN NW base. Then, the strain gradually decreases and saturates after the fourth Al 0.38 Ga 0.62 N/AlN layer. We propose that the higher tensile strain in the first AlN barrier should facilitate Ga incorporation in the first Al 0.38 Ga 0.62 N ND. As the NW growth proceeds, the tensile strain in the AlN barriers monotonically decreases, which enhances the Al incorporation at the center of the Al x Ga 1-x N NDs. Radially, the efficient strain relaxation at the edge of the NDs possibly results in the higher accumulation of Ga atoms at that area.

Despite the chemical inhomogeneity in the Al x Ga 1-x N NDs, we demonstrate that the emission wavelength can be tuned by varying the nominal Al content in the NDs and their thicknesses. Their CL spectra measured at 300K are presented in Fig. 5(a). By increasing x from 0 to 0.38 and/or reducing the ND thickness from 4 to 1 nm, the CL peak emission can be adjusted from 350 nm to 240 nm, with a FWHM in the range of 50-500 meV. The broadening and structuration of the CL line in thinner NDs with higher Al content is attributed to the pronounced chemical inhomogeneity in the NDs due to the larger lattice mismatch with the GaN stem. The integrated CL intensity as a function of temperature of Al x Ga 1-x N NDs with 1-2 nm thicknesses are shown in Fig. 5(b), in comparison to that of GaN QWs and 180 nm AlGaN/GaN NWs with similar Al content. The IQE of AlGaN NDs is in the range of 30-40%, significantly higher than that of 180 nm AlGaN sections and that of GaN QWs, which are around 5% and 0.5%, respectively.

In conclusion, we present the structural and optical studies of AlGaN sections and AlGaN/AlN NDs on GaN NWs. The Al-Ga intermixing at the Al(Ga)N/GaN hetero-interfaces in NWs and the chemical inhomogeneity in the Al x Ga 1-x N NDs are associated to the strain relaxation process. The alloy inhomogeneity on nm length scales in AlGaN sections leads to potential fluctuations that induce carrier localization, which determines their optical behavior. Furthermore, it also causes the multi-peak spectral emission characteristic of AlGaN/AlN NDs. Despite the difficulty to control the alloy uniformity in AlGaN NDs, we demonstrate that the emission energy can be tuned from 240 to 350 nm with IQE around 30% by adjusting their thickness and Al content.

The authors would like to thank F. Donatini for technical assistance on CL measurement. We benefited from the access to the technological platform NanoCarac of CEA-Minatec for HAADF-STEM characterization. This work is partially supported by the French National Research Agency "UVLamp" project (ANR-2011 NANO-027), (#233950), JCJC project COSMOS (ANR-12-JS10-0002) and by the EU ERC-StG "TeraGaN" (#278428) project.


Takeki Kamiyama
email: takeki.kamiyama@univ-paris8.fr

Yayoi Nakamura-Delloye

NATIVE FRENCH SPEAKERS' PERCEPTION OF THE JAPANESE /h/: HA PIECE HOF CAKE?

Keywords: Japanese /h/, French-speaking learners, perception, identification, discrimination

It is commonly observed that native French speakers tend to drop nonnative /h/-like phonemes, or to insert them where unexpected, in the "/h/ languages" they learn (e.g. English). The perception of the Japanese /h/ by French listeners was tested by way of an AXB discrimination task on non-words and a word identification task on Japanese minimal pairs. For naïve listeners not learning Japanese (N=9), the error rate in the discrimination task ranged from 0% to 12.5%; for elementary level learners of Japanese (N=8) the error rate in the identification task ranged from 0% to 8%. 12 errors out of 17 in the latter task occurred when a less familiar word was identified as a more familiar one. These findings suggest that the French perception of the Japanese /h/ is not as difficult as other reported major difficulties in L2 acquisition, and that it interacts with lexical acquisition.

INTRODUCTION

It has been observed and reported that native French speakers learning other languages as a foreign or second language tend to omit the English glottal fricative /h/ [START_REF] Ostiguy | Introduction à la phonétique comparée: les sons: le français et l'anglais nord-américains[END_REF] [START_REF] Capliez | Typologie des erreurs de production d'anglais des francophones : segments vs. suprasegments[END_REF] or similar phonemes in other languages ( [START_REF] Isomura | Onsei o oshieru [Teaching pronunciation[END_REF] for the Japanese /h/), or to insert a glottal fricative where it is unexpected [START_REF] Exare | Étude Phonétique de la résolution du hiatus en anglais langue maternelle et dans l'anglais des francophones[END_REF]. However, the number of experimental studies on the perception and production of this type of phoneme seems to be quite limited, compared to some other widely reported difficulties in the perception and production of second language (L2) pronunciation, such as the English /r/-/l/ contrast by Japanesespeaking learners ( [START_REF] Goto | Auditory perception by normal Japanese adults of the sounds "l" and "r[END_REF], [START_REF] Hattori | Examination of the relationship between L2 perception and production: an investigation of English /r/-/l/ perception and production by adult Japanese speakers[END_REF], inter alii.), or the Catalan /e/-/ɛ/ by Spanish-speaking learners ( [START_REF] Bosch | First and second language vowel perception in early bilinguals[END_REF], [START_REF] Mora | Why are the Catalan contrasts between /e/-/ɛ/ and /o/-/ɔ/ so difficult for even early Spanish-Catalan bilinguals to perceive?[END_REF], inter alii.).

A small number of empirical studies have been reported on the acquisition of the English /h/ by French-speaking learners. It has been suggested that when /h/ is omitted by these learners, the glottis tends to be closed, instead of being open for a fricative sound to be produced [START_REF] Kamiyama | Do French-speaking learners simply omit the English /h/?[END_REF]. [START_REF] Exare | Étude Phonétique de la résolution du hiatus en anglais langue maternelle et dans l'anglais des francophones[END_REF] illustrated French speakers' tendency to insert a [h] in the case of hiatus (e.g. 'to [h]eat'). [START_REF] Watelle-Campbell | Perception et production du phonème /h/ par des apprenants francophones de l'anglais[END_REF] compared the results of an auditory discrimination task (AX) and three different production tasks (repetition; word naming; reading) and found that the least advanced group of learners dropped /h/ more frequently in the reading task, in which the orthography was presented, than in the word naming task (and still less often in the repetition task). This tendency was not observed in more advanced learners and those who were more accustomed to a writing system in which the letter h corresponds to a glottal fricative (German). These findings suggest that the writing system plays a nonnegligible role in the acquisition of a /h/-like phoneme by French-speaking learners. What, then, happens when they acquire a language that has a similar phoneme but a writing system where the phoneme /h/ is not transparent, such as Japanese?

BACKGROUND

In traditional phonological accounts of Japanese, applied to Yamato (native) and sino-Japanese lexical items, the phoneme /h/ is commonly described to have 3 complementary allophones: bilabial [ɸ] before /u/ ([ɯ] in broad phonetic transcription), palatal [ç] before /i/ and /j/, glottal [h] in other contexts ( [START_REF] Amanuma | Nihongo Onseigaku[END_REF], [START_REF] Saito | Nihongo onseigaku nyûmon[END_REF], [START_REF] Vance | The sounds of Japanese[END_REF], [START_REF] Labrune | The phonology of Japanese[END_REF], inter alii.). Besides, it may be realized as voiced [ɦ] intervocalically ( [START_REF] Amanuma | Nihongo Onseigaku[END_REF], [START_REF] Saito | Nihongo onseigaku nyûmon[END_REF], [START_REF] Vance | The sounds of Japanese[END_REF], inter alii.), and [x] ( [START_REF] Amanuma | Nihongo Onseigaku[END_REF], [START_REF] Saito | Nihongo onseigaku nyûmon[END_REF]) or [χ] ( [START_REF] Saito | Nihongo onseigaku nyûmon[END_REF]) may replace [h]. Certain speakers show neutralization with /s/ before /i/ and /j/, leading to the alveolo-palatal realization [ɕ] in eastern (Kantô) varieties (and [ç] in western (Kansai) varieties) [START_REF] Labrune | The phonology of Japanese[END_REF]. However, no other regional variation in the phonetic realization of /h/ has been reported, to our knowledge. Note also that in present-day Japanese, the phone [ɸ] has undergone a "phonemic split" [START_REF] Vance | The sounds of Japanese[END_REF], or "phonemization" [START_REF] Labrune | The phonology of Japanese[END_REF] in recent loan words, contrasting fitto [ɸitːo] (< fit) and hitto [çitːo] (< hit).

By contrast, most present-day varieties of French do not have glottal phonemes any longer ( [START_REF] Léon | Phonétisme et prononciations du français[END_REF], [START_REF] Fougeron | Illustrations of the IPA: French[END_REF], inter alii.), even though a glottal stop [ʔ] may be inserted to mark an initial boundary with an empty onset, as in the case of so-called "h-aspiré" (e.g. la hauteur [la ʔotoeʁ] "the height", where [ʔ] is optional ( [START_REF] Dell | Les règles et les sons[END_REF], inter alii.) and tends to be inserted in the case of emphasis). The French uvular fricative /ʁ/ was shown to be categorized as Japanese /h/ by Japanese listeners in a perceptual assimilation task [START_REF] Yamasaki | How do native speakers of Japanese discriminate and categorize French /r/ and /l/?[END_REF] [21], especially when its phonetic realization is voiceless [χ] [START_REF] Ooigawa | Individual differences in production of voicing of French /R/ sounds and the perception by Japanese adult listeners[END_REF].

The writing system of Japanese is comprised mainly of three scripts, namely, sinograms (kanji) and two syllabaries (hiragana and katakana). The characters of the syllabaries basically represent a combination of a consonant and a vowel (or only a vowel) and they are not decomposable phonemically. Roman transcription (rômaji) is also used in advertisements, street signs or when typing (sequences of roman characters are converted into syllabary characters). Some textbooks of Japanese for self study are provided with roman transcription, but in the great majority of L2 Japanese classrooms, only sinograms and syllabaries are taught. The use of the IPA is rare.

METHOD

Experiment 1: AXB discrimination

The goal of this experiment was to test French speakers' perception of the Japanese /h/ independently of the lexical knowledge of Japanese.

The profiles of the listeners were as follows: 1) 9 "naïve listeners" of French who had never learned Japanese (F-NAI), aged between 18 and 25; they all grew up in the northern half of France, and have never lived abroad.

2) 8 native listeners of French and learners of Japanese (F-LRN), aged between 17 and 23, all firstyear students in Japanese at INALCO (National Institute of Oriental Languages and Civilizations) in Paris. They had all finished their first semester in Japanese studies (162.5 hours of language teaching) at the time of the experiment. Three of them had studied Japanese before the semester (4, 2, and 1 year(s) of learning experience at the time of the study). Their speaking proficiency level corresponds to A1-A2 in the Common European Framework of Reference (CEFR).

3) 8 native listeners of Japanese (J-NAT), aged between 27 and 44, originally from several different regions of Japan (let us remember that no regional variation in the phonetic realization of /h/ has been reported, to our knowledge).

Concerning the linguistic material, a total of 20 non-words were created (Tab. 1):

Word-initial /h/: /hV 1 bjV 2 / vs. /V 1 bjV 2 / Intervocalic /h/: /bjV 2 hV 1 / vs. /bjV 2 V 1 / (V 1 : /i/ /e/ /a/ /o/ /u/; V 2 : /i/ /a/ /u/ (due to the phonotactic constraints), not identical to V 1 ) These words were pronounced in the carrier sentence /sore o __ to iu/ 'I call that __' by a native speaker of Japanese born and raised in Tokyo, and recorded at 44100 Hz, 16 bits, and downsampled to 22050 Hz. Each carrier sentence containing a nonword was pronounced 3 times (but not consecutively), with 2 different pitch accent patterns each (low-high 'LH' and high-low 'HL'). The target non-words were extracted and 4 different combinations of triplets (two different tokens were selected for the same non-word in a triplet) were prepared for each of the 10 minimal pairs and the 2 pitch accent patterns. Thus, a total of 80 triplets were created along with 20 distracters. These triplets were arranged in a semi-random order. In the first half of the list, non-words pronounced with HL pattern were placed, and LH in the latter half. The experimental interface used was "ExperimentMFC" (Praat [START_REF] Boersma | Praat: doing phonetics by computer[END_REF]). The participants were asked to listen to the triplets through a headphone, to decide if the first or the last (non-)word was identical to the second, and then to choose from the two boxes presented on the screen the one that corresponded to their decision. The experiment was preceded by a training session.

Experiment 2: word identification

Experiment 2 consisted of a word identification task administered to the 8 F-LRNs and the 8 J-NATs who participated in Experiment 1. The goal of this experiment was to test the identification of Japanese minimal-pair words containing or not /h/ by learners who are more less familiar with them.

A total of 25 minimal pairs (50 words) were chosen for this experiment (Tab. 3). It should be noted that words with a word-medial, intervocalic /h/ are much smaller in number than those with an initial /h/, partly due to the phonetic changes that occurred in the language (p > h word-initially, but p > w or disappearance word-medially: [START_REF] Labrune | The phonology of Japanese[END_REF], inter alii.). These words were pronounced by the same native speaker in the same manner as for the nonwords used in Experiment 1. The target words, along with 10 distracter words, were arranged in 2 different semi-random orders.

The participants were asked to listen to a word and to choose the corresponding one from the two forms presented in hiragana syllabary on the screen. The list of words was presented in one of the two orders in the first half of the test, and in the other in the second half, with another token of each word. The other procedures were identical to those of Experiment 1.

When the identification task was completed, the F-LRNs were asked to fill in a vocabulary questionnaire, in which they reported their familiarity with the test words (a French translation was provided).

RESULTS

Experiment 1: AXB discrimination

The overall error rate, shown in Fig. 1, was as follows: for F-NAI 4.31% (mean), ranging from 0% (2 listeners our of 9) to 12.5%; for F-LRN 2.50% (mean), ranging from 0% (3 listeners out of 8) to 11.3%; for J-NAT 0.16% (only 1 error out of the 80 non-distracter trials, made by only 1 listener out of 8). A Kruskal-Wallis test shows a significant effect of group on the mean error rates (H (2) = 7.45, p < .05). The difference between F-NAI and J-NAT is also significant (Mann-Whitney's U = 9.5, p < .05). Mean error rates were then calculated for each position (word-initial or word-medial) and for each vowel following /h/. For F-NAI, the error rate was higher for word-medial position (5.56%) than for word-initial position (3.06%), whereas no difference was found for F-LRN (2.5% for both conditions). Concerning the following vowel, for F-NAIs, the error rate was the highest for /u/ (9.03%), followed by /a/ (5.56%), /e/ (2.78%), /o/ and /i/ (2.08% for both) (Friedman test F r corrected for the existence of ties 12.76, p < .05); for F-LRNs, it was the highest for /o/ (4.69%), followed by /u/ and /e/ (3.13% for both), /a/ (1.56), and /i/ (no error).

Experiment 2: word identification

The overall error rate (Fig. 2) was as follows: for F-LRN 2.13% (mean), ranging from 0% (4 listeners out of 8) to 8%; for J-NAT 0% (no error made by any of the 8 listeners). Out of the 17 incorrect answers (out of 800 answers given by the 8 F-LRNs), 11 stimulus words had /h/ but the participant chose a word without /h/, and 6 others did not have /h/ but the one containing /h/ was selected. Two errors were observed for words with or without intervocalic /h/ (Tab. 3).

Table 2: Number of responses in identification and familiarity gap: i) 'positive': the target word was more familiar to the learner than the other word of the minimal pair; ii) 'equal': familiarity was equal for both words of the minimal pair; iii) 'negative': the target word was less familiar than the other word of the minimal pair. The data of the 3 F-LRNs who obtained a 0% error rate both in discrimination and identification were excluded. For each incorrect answer, the lexical familiarity of the participant was checked, except for the 3 F-LRNs who made no errors in the discrimination task (they made no errors in word identification either). For 12 out of the 17 Ωincorrect answers, the participants chose a word more familiar to them when they heard a less familiar word (Tab. 2). When the categories "positive" and "equal" are merged into one single category, the correctness and the familiarity gap show a phi value of 0.18 (Phi coefficient of association: p < 0.001). By contrast, [ɸ] in /hu/, which produced the highest error rate for F-NAIs, may be less familiar, even if it is used as a non-linguistic sound (when blowing out a candle or as a sigh of relief). Also, it is acoustically weak (low intensity). The higher error rate in the word-medial position for F-NAI may be due to the voicing of /h/ in the stimuli, but the same does not hold for F-LRN, suggesting that learners get accustomed to this phone rather quickly.

It is worth noting that in the discrimination task, the error rate was not high even for naïve listeners (F-NAIs' mean: 4.31%). This is fairly low, compared to other major difficulties in crosslanguage perception reported in the literature, and close to the case of the French contrast /ʁ/-/l/ [22] [START_REF] Yamasaki | How do native speakers of Japanese discriminate and categorize French /r/ and /l/?[END_REF]. Let us remember that this French contrast, not very difficult in perception, is known to be a major difficulty in production, and is reported as general observation [START_REF] Kamiyama | To appear. Les japonophones[END_REF] [START_REF] Lauret | Enseigner la prononciation du français : questions et outils[END_REF]. It could thus be argued that the case of French speaker's /h/ presents a similar difficulty. Indeed, F-LRNs in the present study showed a low error rates in both tasks (2.50% in discrimination, 2.13% in identification), but Frenchspeaking learners in general have been reported to show h-omission or insertion even in the written language [START_REF] Kamiyama | L'omission et l'insertion du /h/ dans des expressions verbales écrites par des apprenants de japonais langue étrangère en France[END_REF].

The difficulty in production, however, needs to be investigated further. It may be due to articulatory reasons, as for the fully back French [u] by Japanese-speaking learners who are not used to articulating high back rounded vowels [START_REF] Kamiyama | Perception and production of French close and close-mid rounded vowels by Japanese-speaking learners. Acquisition et interaction en langue étrangère Aile[END_REF]; but since French speakers are known to insert [h], this may not be the case. Is it related to other linguistic levels, such as vocabulary and semantic acquisition? The error patterns in word identification observed in the present study suggest that lexical acquisition may interact with phonological acquisition of second or foreign languages [START_REF] Best | Nonnative and secondlanguage speech perception: Commonalities and complementarities[END_REF] [START_REF] Ota | The KEY to the ROCK: Near-homophony in nonnative visual word recognition[END_REF]. The writing system may also exert a non-negligible influence [7] [27]. In [START_REF] Watelle-Campbell | Perception et production du phonème /h/ par des apprenants francophones de l'anglais[END_REF], French-speaking learners' error rates in an AX discrimination task on English short, basic words containing /h/ were much higher (11% -39% depending on the group of listeners) than those of the present study. It could be argued that once a word is recognized, linguistic processes could exert an influence on the task. It could also be that Japanese words, mainly written in sinograms or syllabaries, are treated differently from English words, written in roman alphabet as in French, and many of which are similar to French words, inciting learners to proceed in a similar decoding mode.

In order to approach these issues, it will be necessary to conduct production studies requiring different processes (repetition, naming, map task, reading, etc.), as well as further perceptual discrimination and identification experiments in more difficult conditions (with noise, or with stimuli produced by multiple talkers), with learners of different proficiency levels.

Figure 1 :

 1 Figure 1: Error rate in AXB discrimination: F-NAI (N=9), F-LRN (N=8), J-NAT (N=8); 80 trials.

Figure 2 :

 2 Figure 2: Error rate in word identification: F-LRN (N=8), J-NAT (N=8); 100 trials.

Table 1 :

 1 Non-words used in the AXB discrimination task and the mean error rates.

	Initial	Initia	Medial	Medial	Error	Error
	/h/	l /Ø/	/h/	/Ø/	F-NAI	F-LRN
	/hibja/ /ibja/ /bjohi/	/bjoi/	2.08% 0
	/hebjo/ /ebjo/ /bjohe/ /bjoe/	2.78% 3.13%
	/habju/ /abju/ /bjuha/ /bjua/	5.56% 1.56%
	/hobja/ /obja/ /bjaho/ /bjao/	2.08% 4.69%
	/hubja/ /ubja/ /bjahu/ /bjau/	9.03% 3.13%
	F-NAI: 3.06%	F-NAI: 5.56%
	F-LRN: 2.5%	F-LRN: 2.5%

Table 3 :

 3 50 words used in the identification task. /H/, /N/, and /Q/ represent respectively the second half of a long vowel, mora nasal, the first half of a geminate. Note that (lexically) unaccented verbs were pronounced with a fall after the final mora of the word in the carrier sentence /sore o __ to iu/ 'I call that __'. * shows an incorrect answer in identification.

	With /h/			Without /h/
	/hiˈ ru/	'daytime'		/iru/	'(to) stay'
	/hiku/	'(to) pull'		/iku/	'(to) go'
	/ˈ heN/	'strange'		/ˈeN/	'circle, yen'
	/ˈ hai/	'yes'		/ˈai/	'love'
	/haˈ na/	'flower'		/aˈ na/	'hole'
	*/haˈ si/	'bridge'		*/aˈsi/	'foot, leg'
	*/ˈ haru/	'spring' (season) */ˈ aru/	'be, exist'
	**/hoˈ soi/ 'thin'		/oˈ soi/	'late'
	/ˈ hiNdo/	'frequency'		/ˈiNdo/	'India'
	**/ˈ heH/	'soldier'		/ˈeH/	'yes'
	*/ˈ hau/	'crawl'		/ˈau/	'(to) meet'
	/hane/	'feather, wing'	/ane/	'elder sister'
	/ˈ hoN/	'book'		**/ˈ oN/ 'obligation'
	/hoQto/	feeling relieved	*/oQto/ 'husband'
	/hoˈ sii/	'desired'		*/oˈ sii/	'regrettable'
	/ˈ humi/	'letter'		/ˈ umi/	'sea'
	/ˈ hutjuH/ '(city of) Fuchû' /ˈ utjuH/ 'universe'
	/ˈ huzi/	'(Mt) Fuji'		/ˈ uzi/	'(city of) Uji'
	/ˈ hema/	'blunder'		/ˈema/	'votive picture'
	*/hizjoH/ 'emergency'		/izjoH/	'unusual'
	/ˈ haha/	'mother'		/ˈ haH/	'yes'
				[haː]
	*/taiheN/ 'serious, awful'	/taieN/	'leaving a park'
	*/koHheN/ 'second and last	/koHeN/ 'park'
		part'
	/ˈ ɡohaN/	'cooked	rice,	/ˈ ɡoaN/	(place name)
		meal'
	*/suiheH/ 'horizontal'		/suieH/	'swimming'
	5. DISCUSSION AND CONCLUSION
	In the discrimination task, stimuli with /i/ produced a
	relatively low error rate, possibly because French
	listeners are familiar with the allophone [ç] (close to
	the German [ç], even though this latter is usually
	replaced by [ʃ] in loans and proper nouns in French).

ACKNOWLEDGEMENTS

The authors would like to express their gratitude to Dr. Pierre Hallé and Prof. Jacqueline Vaissière for their advice and suggestions.