Nader Mamdouh Zaydan
email: mamdouh.zaydan@lau.edu.lb

El Khatib

Amin Ghorbel
email: amineghorbel@yahoo.fr

Dalel Amri
email: dalel.amri9@gmail.com

Nader El Khatib
email: nader.elkhatib@lau.edu.lb

Traveling wave solutions for the

Keywords: AMS Classification: 35B27, 35F20, 45K05, 47G20, 49L25, 35B10 Accelerated Frenkel-Kontorova model, fully non-linear framework, traveling waves, viscosity solutions, monostable non-linearity

come

Introduction

Our motivation is to construct a traveling wave solution for the accelerated Frenkel-Kontorova model (AFK): for i ∈ Z and t > 0,

m 0 U ′′ i (t) + U ′ i (t) = U i+1 (t) + U i-1 (t) -2U i (t) + f (U i (t)) (1.1)
where U i (t) denotes the position of a particle i at time t, U ′ i (t) its velocity and U ′′ i (t) its acceleration. The constant m 0 denotes the mass of the particle and f is the force created by periodic potential (whose period 1). In addition, we assume that f is "positive monostable": f (0) = f (1) = 0 and f > 0 on (0, 1). An example of such force f is given by f (x) = 1 -cos(2πx).

System (1.1) can be used as a model of the motion of a dislocation defect in crystal (see [START_REF] Braun | The Frenkel-Kontorova model: concepts, methods, and applications[END_REF]). A traveling wave solution of (1.1) is a particular solution of the form

U i (t) = ϕ(i + ct) (1.2)
with ϕ is non-decreasing, ϕ(-∞) = 0 and ϕ(+∞) = 1.

(1.3)

Here c is the velocity of propagation of the traveling wave ϕ and (1.3) reflects the existence of a defect of one lattice space called dislocation. Expression (1.2) means that the defect moves with velocity c. In addition, ϕ is a phase transition from the unstable equilibrium ϕ(-∞) to the stable equilibrium ϕ(+∞). To be more precise, we will show that there exists c + ∈ R such that a traveling wave solution for (1.1) exists if and only if c ≥ c + . Our work strongly depends on the results obtained previously in the bistable case by authors in [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF]: our approach is to study the positive monostable case as the limit of a family of bistable cases. The approach will be detailed through the course of the paper. The dependence of our work on [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF] means also that we can present our results in a more general case. We will consider the following system of general and fully non-linear discrete equations of the form

m 0 U ′′ i (t) + U ′ i (t) = F (U i+r0 (t), U i+r1 (t), ..., U i+r N (t)) + σ (1.4)
where F is a real function whose properties are specified later and (r j ) j=0,...,N ∈ R. The parameter σ allows us to distinguish the bistable and monostable regimes and it lies inside an interval [σ -, σ + ].

The precise definition of constants σ + and σ -will be given later (see Definition 2.1). For instance, we just mention that for some fixed θ ∈ (0, 1) and for σ ∈ [σ -, σ + ], there exists m σ ∈ [θ -1, 0] and b σ ∈ [0, θ] such that: i) for σ ∈ (σ -, σ + ), f σ (v) = F (v, v, ..., v) + σ is bistable on [m σ , m σ + 1] : ii) For σ = σ + , f σ + (v) = F (v, v, ..., v) + σ + is positive monostable on [0, 1]:

f σ (m σ ) = f σ (b σ ) = f σ (m σ +
f σ + (m σ + ) = f σ + (m σ + + 1) = 0 and f σ + > 0 on (m σ + , m σ + + 1) with m σ + = 0 . iii) For σ = σ -, f σ -(v) = F (v, v, ..., v) + σ -is negative monostable on [θ -1, θ]: f σ -(m σ -) = f σ -(m σ -+ 1) = 0 and f σ -< 0 on (m σ -, m σ -+ 1) with m σ -= θ -1.
For model (1.4), we look for particular solutions U i of the form U i (t) = ϕ (i + ct) with ϕ is non-decreasing and ϕ(-∞) = m σ and ϕ(+∞) = m σ + 1.

(1.5)

In [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF], authors obtained an existence result for a particular bistability: for a unique velocity c, they constructed a traveling wave solution connecting the stable points 0 and 1. Thanks to this result, with a simple change of variable, we can prove that if σ ∈ (σ -, σ + ), a traveling wave for (1.4) exists for a unique velocity c σ . As a consequence, we can generate the function σ → c σ . We will prove that c σ is continuous, non-decreasing and bounded on (σ -, σ + ). We will exploit this function to study the positive monostable case in two ways: first, the velocity c + is defined by

c + = lim σ→σ + c σ .
Second, thanks to the monotony of c σ , we will show that traveling wave solutions exist if and only if c ≥ c + . Similarly, by taking σ → σ -, we can study the negative monostable case.

Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky, and Piskunov [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] were among the first to study traveling waves in reaction-diffusion equations. Existence of traveling wave solutions has been established in many works, see for example [START_REF] Berestycki | Traveling wave solutions to combustion models and their singular limits[END_REF][START_REF] Fife | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF][START_REF] Smaily | Two-dimensional curved fronts in a periodic shear flow[END_REF][START_REF] Volpert | Reaction-diffusion waves in biology[END_REF]. The existence, uniqueness, and stability of traveling waves with different nonlinearities with applications, particularly in biology and combustion, are the subject of a vast body of research; see, for example, the references given in [START_REF] Shen | Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations[END_REF][START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]. Moreover, a number of works have been done on the discrete version of the reaction-diffusion equations in the bistable case (for examples, see [START_REF] Chen | Traveling waves in discrete periodic media for bistable dynamics[END_REF][START_REF] Haj | Existence and uniqueness of traveling waves for fully overdamped frenkel-kontorova models[END_REF][START_REF] Carpio | Wave solutions for a discrete reaction-diffusion equation[END_REF][START_REF] De Masi | Travelling fronts in non-local evolution equations[END_REF]). There are also several works on different discrete or non-local version of reaction diffusion equations in the positive monostable case (see for instance [START_REF] Coville | Nonlocal anisotropic dispersal with monostable nonlinearity[END_REF][START_REF] Weinberger | Long-time behavior of a class of biological models[END_REF][START_REF] Haj | Velocity diagram of traveling waves for discrete reactiondiffusion equations[END_REF][START_REF] Haj | Traveling waves for discrete reaction-diffusion equations in the general monostable case[END_REF][START_REF] Guo | Front propagation for discrete periodic monostable equations[END_REF][START_REF] Chen | Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices[END_REF]).

Up to our knowledge, our work is the first one to establish the existence of traveling waves for accelerated and discrete monotsable dynamics with general Lipschitz non-linearities. Moreover, we get the existence of solutions under weaker assumptions compared to other works. Finally, we obtain our results in the framework of viscosity solutions and we refer the reader to reference [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Barles | An introduction to the theory of viscosity solutions for first-order hamilton-jacobi equations and applications[END_REF][START_REF] Crandall | Viscosity solutions of hamilton-jacobi equations[END_REF][START_REF] Droniou | Solutions de viscosité et solutions variationnelle pour edp non-linéaires[END_REF] for a full presentation of this theory.

Main results

In the rest of the paper, we will work with an equivalent formulation of (1.4). We borrow the idea from [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF] and consider for all i ∈ Z,

Ξ i (t) = U i (t) + 1 α 0 Ui (t) with α 0 = 1 2m 0 .
Using this new function, we obtain the following system of ODEs equivalent to (1.4) for all i ∈ Z, for all t ∈ (0, +∞),

Ui (t) = α 0 (Ξ i (t) -U i (t)) , Ξi (t) = α 0 (U i (t) -Ξ i (t)) + 2F (U i+r0 (t), U i+r1 (t), ..., U i+r N (t)) + 2σ. (2.1)
We look for particular solutions of (2.1) of the form

U i (t) = ϕ 1 (i + ct) , Ξ i (t) = ϕ 2 (i + ct) , where (ϕ 1 , ϕ 2 ) solves          cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) for z ∈ R, cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F (ϕ 1 (z + r 0 ), ϕ 1 (z + r 1 ), ..., ϕ 1 (z + r N )) + 2σ for z ∈ R, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (-∞) = ϕ 2 (-∞) = m σ and ϕ 1 (+∞) = ϕ 2 (+∞) = m σ + 1.
Next, we first give the assumptions on the non-linearity F and we then define the parameters σ -and σ + . Assumption ( Ã). Let F : R N +1 → R and θ ∈ (0, 1).

• (Regularity) F is globally Lipschitz continuous over R N +1 and C 1 over a neighborhood in R N +1 of the two intervals ]0, Θ[ and ]Θ, E[ where Θ = (θ, ..., θ) and E = (1, ..., 1).

• (Monotonicity) For X = (X 0 , X 1 , ..., X N ) ∈ R N +1 , the function F is non-decreasing w.r.t. each X i for i ̸ = 0 and

2 ∂F ∂X 0 + α 0 > 0. • (Periodicity) For X = (X 0 , X 1 , ..., X N ) ∈ R N +1 , we have F (X 0 + 1, ..., X N + 1) = F (X 0 , ..., X N ). Assumption ( B). Let f (v) = F (v, v, ..., v).
We assume that f is "bistable": f (0) = f (1) and

f ′ > 0 on (0, θ), f ′ < 0 on (θ, 1).
Definition 2.1 (Definition of σ -, σ + and consequences). Under assumptions ( Ã) and ( B), we define

σ + = -min f = -f (0), σ -= -max f = -f (θ). For each σ ∈ [σ -, σ + ], there exists a unique m σ ∈ [θ -1, 0] and a unique b σ ∈ [0, θ] such that m σ and b σ are solutions of f (s) + σ = 0. Moreover, σ → m σ (resp. σ → b σ ) is increasing (resp. decreasing) on [σ -, σ + ].
The main results of this paper is the following theorems. In the first one, we study the velocity function associated to the bistable case. Theorem 2.2 (Velocity function in the bistable case). Consider a function F satisfying assumptions ( Ã) and ( B) and let σ ∈ (σ -, σ + ). Then, there exists a unique c = c σ such that there exists a viscosity solution

(ϕ 1 , ϕ 2 ) of          cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) for z ∈ R, cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F (ϕ 1 (z + r 0 ), ϕ 1 (z + r 1 ), ..., ϕ 1 (z + r N )) + 2σ for z ∈ R, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (-∞) = ϕ 2 (-∞) = m σ and ϕ 1 (+∞) = ϕ 2 (+∞) = m σ + 1.
The map σ → c σ is continuous and non-decreasing on (σ -, σ + ). In addition, c σ is bounded on (σ -, σ + ) and therefore we can define the finite limits

c -= lim σ→σ -c σ and c + = lim σ→σ + c σ . (2.2)
Moreover, we have c -= c + = 0 or c -< c + .

The second main result of this paper is the existence of traveling wave solution in the monostable cases and it's announced in the next theorem. Theorem 2.3 (Existence of traveling wave solutions in the monostable cases). Consider a function F satisfying assumptions ( Ã) and ( B). Let c -and c + be the constants defined in (2.2). i) For any c ≤ c -, there exists a viscosity solution (ϕ 1 , ϕ 2 ) solving for z ∈ R the following system

         cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)), cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F (ϕ 1 (z + r 0 ), ϕ 1 (z + r 1 ), ..., ϕ 1 (z + r N )) + 2σ -, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (-∞) = ϕ 2 (-∞) = m σ -= θ -1 and ϕ 1 (+∞) = ϕ 2 (+∞) = m σ -+ 1 = θ (2.3)
Moreover, if c > c -, there is no viscosity solutions of (2.3).

ii) For any c ≥ c + , there exists a viscosity solution (ϕ 1 , ϕ 2 ) solving for z ∈ R the following system

         cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)), cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F (ϕ 1 (z + r 0 ), ϕ 1 (z + r 1 ), ..., ϕ 1 (z + r N )) + 2σ + , ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (-∞) = ϕ 2 (-∞) = m σ + = 0 and ϕ 1 (+∞) = ϕ 2 (+∞) = m σ + + 1 = 1.
(2.4) Moreover, if c < c + , there is no viscosity solutions of (2.4).

Simplification and notation.

Without loss of generality, we will assume that

r 0 = 0 and r i ̸ = r j if i ̸ = j.
For the whole paper, we define

r * = max i=0,...,N |r i | (2.5)
and as a notation, we set for a general function h:

F ((h(z + r i )) i=0,...,N ) = F (h(z + r 0 ), h(z + r 1 ), ..., h(z + r N )).
Organization of the paper. In Section 3, we give the definition of viscosity solutions and then we recall some useful results which are used all over the paper. In Section 4, we construct traveling wave solutions for large (resp. small) velocities in the positive (resp. negative) monostable case. This result will be used in the proof of Theorem 2.2 to prove that the velocity c + (resp. c -) is finite. In section 5, we prove Theorem 2.2 and in Section 6, we prove Theorem 2.3.

Viscosity solutions and some useful results

In this section, we first give the definition of viscosity solutions and then we present some useful results that we will use in this work. We start with the definition of viscosity solutions. Let -∞ ≤ a < b ≤ +∞ and consider the intervals

I = (a -r * , b + r * ) and I ′ = (a, b). Let ϕ 1 , ϕ 2 : I → R be a locally bounded functions. 1) We say that (ϕ 1 , ϕ 2 ) is a viscosity sub-solution (resp. super-solution) of cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on I ′ , cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) + 2σ on I ′ (3.1)
if for all test function φ ∈ C 1 (R) such that (ϕ 1 ) * -φ attains a local maximum (resp. (ϕ 1 ) * -φ attains a local minimum) at some point z 0 ∈ I ′ , we have

cφ ′ (z 0 ) ≤ α 0 ((ϕ 2 ) * (z 0 ) -(ϕ 1 ) * (z 0 )) (resp. cφ ′ (z 0 ) ≥ α 0 ((ϕ 2 ) * (z 0 ) -(ϕ 1 ) * (z 0 )))
and if for all test function φ ∈ C 1 (R) such that (ϕ 2 ) * -φ attains a local maximum (resp. (ϕ 2 ) * -φ attains a local minimum) at some point z 0 ∈ I ′ , we have

cφ ′ (z 0 ) ≤ α 0 ((ϕ 1 ) * (z 0 ) -(ϕ 2 ) * (z 0 )) + 2F (((ϕ 1 ) * (z 0 + r i )) i=0,...,N ) + 2σ, (resp. cφ ′ (z 0 ) ≥ α 0 ((ϕ 1 ) * (z 0 ) -(ϕ 2 ) * (z 0 )) + 2F (((ϕ 1 ) * (z 0 + r i )) i=0,...,N ) + 2σ).
2) We say that (ϕ 1 , ϕ 2 ) is a viscosity solution of (3.1) if ϕ 1 , ϕ 2 are locally bounded functions and (ϕ 1 , ϕ 2 ) is a viscosity sub-solution and viscosity super-solution of (3.1).

Next, we state Perron's method which is a great tool to construct viscosity solutions.

Proposition 3.2 (Perron's method [START_REF] Ishii | Perron's method for monotone systems of second-order elliptic partial differential equations[END_REF]). Consider a function F satisfying ( Ã). Let -∞ ≤ a < b ≤ +∞ and consider the intervals I = (a -r * , b + r * ) and

I ′ = (a, b). Let (φ 1 , φ 2 ) and (ψ 1 , ψ 2 )
be respectively viscosity sub-solution and viscosity super-solution of (3.1) on I ′ satisfying

φ 1 ≤ ψ 1 on I, φ 2 ≤ ψ 2 on I.
Let S be the set defined by

S = {( ψ1 , ψ2 ) is a super-solution of (3.1) on I ′ such that φ 1 ≤ ψ1 , φ 2 ≤ ψ2 on I}.
We define

ϕ 1 (z) = inf{ ψ1 (z) such that ( ψ1 , ψ2 ) ∈ S }, ϕ 2 (z) = inf{ ψ2 (z) such that ( ψ1 , ψ2 ) ∈ S }.
Then, (ϕ 1 , ϕ 2 ) is a viscosity solution of (3.1) on I ′ and

φ 1 ≤ ϕ 1 ≤ ψ 1 on I, φ 2 ≤ ϕ 2 ≤ ψ 2 on I.
The following proposition allows us to have a stability result under weak assumptions.

Proposition 3.3 (Equivalence between viscosity solutions and almost everywhere solution, see Lemma 2.9 in [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF]). Consider a function F satisfying assumption ( Ã). Let ϕ 1 , ϕ 2 : R → R be non-decreasing functions. Then, (ϕ 1 , ϕ 2 ) is a viscosity solution of

0 = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R, 0 = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) + 2σ on R (3.2)
if and only if (ϕ 1 , ϕ 2 ) is an almost everywhere solution of (3.2).

As a consequence of Proposition 3.3, we have the following useful criterion to pass to the limit.

Proposition 3.4 (Stability by passage to limit). Consider a function F satisfying ( Ã) and ( B).

Let (c n ) n∈N and (σ n ) n∈N be sequences of real numbers and ϕ n 1 , ϕ n 2 : R → R be a sequence of non-decreasing functions. Assume that

(ϕ n 1 , ϕ n 2 ) is a viscosity solution of c n (ϕ n 1 ) ′ (z) = α 0 (ϕ n 2 (z) -ϕ n 1 (z)) on R, c n (ϕ n 2 ) ′ (z) = α 0 (ϕ n 1 (z) -ϕ n 2 (z)) + 2F ((ϕ n 1 (z + r i )) i=0,...,N ) + 2σ n on R. (3.3)
In addition, we assume that there exists C > 0 (not depending on n) such that

|ϕ n 1 | L ∞ (R) ≤ C, |ϕ n 2 | L ∞ (R) ≤ C, |c n | ≤ C and |σ n | ≤ C.
Then, up to a subsequence , we have c n → c, σ n → σ and

ϕ n 1 (resp. ϕ n
2 ) converges at least almost everywhere to some function ϕ 1 (resp. ϕ 2 )

and

(ϕ 1 , ϕ 2 ) is a viscosity solution of cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R, cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) + 2σ on R. (3.4) Proof. First, let us remark that |F ((ϕ n 1 (z + r i )) i=0,...,N )| = |F ((ϕ n 1 (z + r i )) i=0,...,N ) -F ((ϕ n 1 (z)) i=0,...,N ) + F ((ϕ n 1 (z)) i=0,...,N )| ≤ L F max i=0,...,N |ϕ n 1 (z + r i ) -ϕ n 1 (z)| + f (ϕ n 1 (z)) ≤ 2L F C + f (θ) = C 1
where L F is the Lipschitz constant of F . We distinguish two cases.

Case 1: c ̸ = 0: for n large enough, we have |c n | ≥ |c| 2 and using (3.3), we deduce that

     |(ϕ n 1 ) ′ (z)| ≤ 4α 0 C |c| , |(ϕ n 2 ) ′ (z)| ≤ 4α 0 C |c| + 2C 1 + 2C |c| .
Thus ϕ n 1 and ϕ n 2 are uniformly Lipschitz. Using Ascoli's Theorem and the diagonal extraction argument, up to a subsequence, we get that ϕ n 1 (resp. ϕ n 2 ) converges locally uniformly to a function ϕ 1 (resp. ϕ 2 ) on R. By stability of viscosity solution (see [START_REF] Barles | An introduction to the theory of viscosity solutions for first-order hamilton-jacobi equations and applications[END_REF]), (ϕ 1 , ϕ 2 ) is a viscosity solution of (3.4).

Case 2: c = 0: from the classical Helly's Theorem for monotone functions, up to a subsequence, we get that ϕ n 1 (resp. ϕ n 2 ) converges almost everywhere to a function ϕ 1 (resp.

ϕ 2 ) on R. Let b 1 , b 2 ∈ R such that b 1 < b 2 . Integrating the first equation in (3.3) between b 1 and b 2 , we get c n (ϕ n 1 (b 2 ) -ϕ n 1 (b 1 )) = α 0 b2 b1 (ϕ n 2 (z) -ϕ n 1 (z))dz.
Moreover, we have

ϕ n 2 -ϕ n 1 → ϕ 2 -ϕ 1 almost everywhere on R, |ϕ n 2 -ϕ n 1 | ≤ 2C on R.
Thus, using Lebesque's dominated convergence Theorem, we pass to the limit as n goes to +∞ and we get

0 = α 0 b2 b1 (ϕ 2 (z) -ϕ 1 (z))dz
which implies (since b 1 and b 2 are arbitrary) that

ϕ 1 = ϕ 2 almost everywhere on R.
Integrating the second equation in (3.3) between b 1 and b 2 , we get

c n (ϕ n 2 (b 2 ) -ϕ n 2 (b 1 )) = 2 b2 b1 F ((ϕ n 1 (z + r i )) i=0,...,N ) + σ n dz.
Using that

F ((ϕ n 1 (z + r i )) i=0,...,N ) + σ n → F ((ϕ 1 (z + r i )) i=0,...,N ) + σ almost everywhere on R, |F ((ϕ n 1 (z + r i )) i=0,...,N ) + σ n | ≤ C 1 + C on R,
we can proceed as above to deduce that F ((ϕ 1 (z + r i )) i=0,...,N ) + σ = 0 almost everywhere on R. Finally, from Proposition 3.3, we deduce that (ϕ 1 , ϕ 2 ) is a viscosity solution of (3.4).

Traveling wave solutions for large (resp. small) velocities in the positive (resp. negative) monostable case

In this section, we construct traveling wave solutions for large (resp. small) velocities c in the positive (resp. negative) monostable case. Such result is needed to prove that the velocities c - and c + are finite (see Proposition 5.6). It's sufficient to prove this result in the positive case only.

In fact, once the existence is established in the positive monostable case, we obtain the existence in the negative monostable case by a simple variable change. For F satisfying assumptions ( Ã) and ( B), without loss of generality, we assume in this section that σ + = 0. This implies that f (0) = f (1) = 0 and f > 0 on (0, 1).

For σ + = 0, we consider the following system

         cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R, cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) on R, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (-∞) = ϕ 2 (-∞) = 0 and ϕ 1 (+∞) = ϕ 2 (+∞) = 1.
cψ ′ 1 (z) = α 0 (ψ 2 (z) -ψ 1 (z)) on R, cψ ′ 2 (z) = α (ψ 1 (z) -ψ 2 (z)) + 2F ((ψ 1 (z + r i )) i=0,...,N ) on R.
In the following lemma, we show that if a positive viscosity super-solution of (4.1) exists, then we can construct a viscosity solution of (4.1). Lemma 4.2. Consider a function F satisfying ( Ã) and ( B) with σ + = 0. Let c ∈ R fix and for this constant consider system (4.1). Assume that there exists a viscosity super-solution (ψ 1 , ψ 2 ) of (4.1) such that ψ i > 0 for i = 1, 2. Then, there exists a viscosity solution (ϕ 1 , ϕ 2 ) of (4.1).

Proof of Lemma 4.2. Using that ψ 1 > 0, ψ 1 (-∞) = 0, ψ 1 (+∞) = 1 and that ψ 1 is non-decreasing, there exists γ ∈ (0, 1) and x 0 ∈ R such that ψ 1 (x 0 ) = γ. Up to translation of ψ 1 and ψ 2 (to always get a super-solution of (4.1)), we can assume that ψ 1 (0) = γ.

Step 1: construction of a sub-solution. Let ε > 0 small enough such that ε < γ. Using assumption ( B), we can easily verify that (ε, ε) is a sub-solution of

cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R, cϕ ′ 2 (z) = α (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) on R. (4.2)
Step 2: construction of a solution Taking ε small enough, and using that ψ 1 , ψ 2 are nondecreasing, positive and ψ 1 (-∞) = ψ 2 (-∞) = 0, we deduce that for i ∈ {1, 2}, we can find

x ε i < 0 such that ε ∈ [(ψ i ) * (x ε i ), (ψ i ) * (x ε i )] and ψ i (x) > ε if x > x ε i . Let K ε = max i=1,2
x ε i . For x > K ε , we have

ψ i (x) > ε for i = 1, 2.
Moreover, using that

ψ 1 (-∞) = ψ 2 (-∞) = 0 and ψ 1 , ψ 2 > 0, we deduce that K ε → -∞ as ε → 0. Using Perron Method (see Proposition 3.2), we construct a viscosity solution (ϕ ε 1 , ϕ ε 2 ) of (4.2) such that      ϕ ε 1 and ϕ ε 2 are defined on (K ε , +∞) , ε ≤ ϕ ε 1 ≤ ψ 1 and ε ≤ ϕ ε 2 ≤ ψ 2 on (K ε , +∞) , (ϕ ε 1 , ϕ ε 2 ) is viscosity solution of (4.2) on (K ε + r * , +∞) .
Step 3: monotonicity of ϕ ε 1 , ϕ ε 2 . We recall that Perron's method provide an explicit form of the solution. We know that

ϕ ε 1 (z) = inf{u 1 (z) such that (u 1 , u 2 ) ∈ S }, ϕ ε 2 (z) = inf{u 2 (z) such that (u 1 , u 2 ) ∈ S } (4.3)
where the set S is defined by

S = {(u 1 , u 2 ) is a super-solution of (4.2) on (K ε + r * , +∞) such that ε ≤ u 1 , ε ≤ u 2 on (K ε , +∞)}.
We define the functions

   φ1 (z) = inf p≥0 ϕ ε 1 (z + p), φ2 (z) = inf p≥0 ϕ ε 2 (z + p)
and we claim that φi (z) = ϕ ε i (z) for all z ∈ (K ε , +∞). First, let us remark that

φ1 ≤ ϕ ε 1 , φ2 ≤ ϕ ε 2 .
Using the stability of viscosity solutions (see [START_REF] Crandall | Viscosity solutions of hamilton-jacobi equations[END_REF]), we know that (( φ1 ) * , ( φ2 ) * ) is a viscosity supersolution of (4.2) on (K ε + r * , +∞). Since ε ≤ ϕ ε i (z + p) for all p ≥ 0 and z ∈ (K ε , +∞), we get

for i = 1, 2, ε ≤ ( φi ) * (z) for all z ∈ (K ε , +∞).
We deduce that (( φ1 ) * , ( φ2 ) * ) ∈ S and using the definitions in (4.3), we get that

ϕ ε 1 (z) ≤ ( φ1 ) * (z) ≤ ( φ1 )(z) for z ∈ (K ε , +∞), ϕ ε 2 (z) ≤ ( φ2 ) * (z) ≤ ( φ2 )(z) for z ∈ (K ε , +∞).
Therefore, we obtain that φi (z) = ϕ ε i (z) for all z ∈ (K ε , +∞) which implies that ϕ ε i is nondecreasing on (K ε , +∞) for i = 1, 2.

Step 4: passing to the limit as ε → 0. Using that (ϕ ε 1 , ϕ ε 2 ) solves (4.2) on (K ε + r * , +∞) and that ϕ ε 1 , ϕ ε 2 are non-decreasing, we deduce that lim x→+∞ ϕ ε i exists for i = 1, 2. Since ε ≤ ϕ ε i (z) ≤ 1 on (K ε , +∞), passing to the limit as x → ±∞ in (4.2), we get

0 = α 0 (ϕ ε 2 (+∞) -ϕ ε 1 (+∞)), 0 = α(ϕ ε 1 (+∞) -ϕ ε 2 (+∞)) + 2f (ϕ ε 1 (+∞)) which implies that f (ϕ ε 1 (+∞)) = 0 and in particular ϕ ε 1 (+∞) = ϕ ε 2 (+∞) = 1
. Now, on the one hand, using that ψ 1 (0) = γ, we get that ϕ ε 1 (z) ≤ γ for all z ≤ 0. On the other hand, using that ϕ ε 1 (+∞) = 1, we deduce that there exists

x ε ≥ 0 such that γ ∈ [(ϕ ε 1 ) * (x ε ), (ϕ ε 1 ) * (x ε )]. We define φε 1 (z) = ϕ ε 1 (z + x ε ), φε 2 (z) = ϕ ε 2 (z + x ε ).
We have γ ∈ [( φε 1 ) * (0), (ϕ ε 1 ) * (0)] and ( φε 1 , φε 2 ) is a viscosity solution of (4.2) on (r * + K ε -x ε , +∞). Since φε 1 , φε 2 are non-decreasing and uniformly bounded on (K ε -x ε , +∞), we get that φε i → ϕ i (at least) almost everywhere as ε → 0 for i = 1, 2. Moreover, from proposition 3.4, (ϕ 1 , ϕ 2 ) is a viscosity solution of

               cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R, cϕ ′ 2 (z) = α (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) on R, ϕ 1 and ϕ 2 are non-decreasing on R, 0 ≤ ϕ 1 , ϕ 2 ≤ 1 , γ ∈ [(ϕ 1 ) * (0), (ϕ 1 ) * (0)] (4.4)
where we used that K ε → -∞ as ε → 0. Passing to the limit as z → ±∞, we get

0 = α 0 (ϕ 2 (±∞) -ϕ 1 (±∞)), 0 = α(ϕ 1 (±∞) -ϕ 2 (±∞)) + 2f (ϕ 1 (±∞)).
We deduce that ϕ 1 (±∞) ∈ {0, 1}. Using that γ ∈ [(ϕ 1 ) * (0), (ϕ 1 ) * (0)], and that ϕ 1 is nondecreasing, we have that ϕ 1 (-∞) = 0 and ϕ 1 (+∞) = 1. Finally, we recall that ϕ 2 (±∞) -ϕ 1 (±∞) = 0 and this ends the proof.

In view of Lemma 4.2, to prove Proposition 4.1, we need to construct a positive super-solution of (4.1). This can be done by using the next lemma.

h ∈ C 1 (R) such that h is a solution of                  h ′ (z) = 2f (h(z)), h is non-decreasing, h(0) = 1 2 , 0 < h(z) < 1 for z ∈ R, h(-∞) = 0 and h(+∞) = 1. (4.5)
Proof. The proof is divided into 2 steps.

Step

1: construction on [-δ, δ]. Let δ > 0 small such that 2||f ′ || ∞ δ < 1. (4.6)
We define

A = h such that h ∈ C([-δ, δ]) such that h(0) = 1 2 .
Clearly, A is a closed subset of the Banach space C([-δ, δ]). Now, let Φ : A → A the operator defined by

Φ(h)(z) = 2 z 0 f (h(s))ds + 1 2 .
Next, we prove that Φ is a contraction. Let h, g ∈ A: we have

Φ(h)(z) -Φ ( g)(z) ≤ 2||f ′ || ∞ δ||h -g|| L ∞ [-δ,δ] .
In particular, this implies that for δ > 0 small enough as in (4.6), we have

||Φ(h) -Φ(g)|| C([-δ,δ]) < ||h -g|| C([-δ,δ]) .
Hence, the Banach fixed point theorem ensures that there exists h ∈ A such that Φ( h) = h. In particular, h ∈ C 1 ([-δ, δ]) and h is a solution of

   h′ (z) = 2f ( h(z)) for z ∈ (-δ, δ), h(0) = 1 2 .
Step 2: construction on R. First, let us recall that h is non-decreasing on [-δ, δ]. We define o(δ) = inf{x * such that 0 < x * ≤ δ and h ̸ = {0, 1} on [-x * , x * ]}.

By the same arguments used in step 1, we can construct a solution h of

h′ (z) = 2f ( h(z)) for z ∈ (-δ -o(δ), δ + o(δ)), h = h on [-o(δ), o(δ)].
To be more precise, we define the set B by

B = h such that h ∈ C([-δ -o(δ), δ + o(δ)]) such that h = h on [-o(δ), o(δ)] .
Then, we consider the operator Ψ : B → B defined by

Ψ(h)(z) =            h(z) if z ∈ [-o(δ), o(δ)], 2 z -o(δ) f (h(s))ds + h(-o(δ)) if z ∈ [-δ -o(δ), -o(δ)], 2 z o(δ) f (h(s))ds + h(o(δ)) if z ∈ [o(δ), o(δ) + δ].
As in step 1, taking δ as in (4.6), we prove that Ψ is a contraction on B. By an iteration argument, we can construct a solution h of

h ′ (z) = 2f (h(z)) for z ∈ R. (4.7) Moreover, h ∈ C 1 (R), h ′ ≥ 0 and h = h on [-δ, δ], h(z) ∈ [0, 1] and h ̸ = {0, 1} on R.
This implies that h(-∞) = 0 and h(+∞) = 1 and this ends the proof.

Proof of Proposition 4.1. Let M > 0 be large such that

M > 3L F ||f ′ || ∞
where L F is the Lipschitz constant of F . We set c = 1 ε where ε > 0 is small enough such that

e 2ε(1+M ε)r * ||f ′ ||∞ -1 ε < 3.
Let a ε = 1 + M ε and define

h ε (z) = h(a ε z)
where h is the solution of (4.5) defined in Lemma 4.3.

Step 1: equivalence of systems For z ∈ R, we define

ψ 1 (z) = ψ 2 (z) = h ε z c .
Then, it's easy to verify that (ψ 1 , ψ 2 ) is a super-solution of (4.1) if and only if h ε satisfies

     (h ε ) ′ (z) ≥ 2F ((h ε (z + εr i )) i=0,...,N ) for z ∈ R, h ε is non-decreasing, h ε (-∞) = 0 and h ε (+∞) = 1 . (4.8)
Step 2: oscillations of

h ′ . Let z 0 ∈ R such that h(z 0 ) = θ. Since f is a C 1 on (0, θ) ∪ (θ, 1), we have for z ̸ = z 0 , h ′′ (z) = 2h ′ (z)f ′ (h(z)). (4.9)
Using the equivalence between almost everywhere solutions and viscosity ones for ODE, we deduce that h is a viscosity solution of (4.9) on R. Now for z ∈ R and b ̸ = 0 fixed, we consider the function defined for t ≥ 0 by

x(t) = h ′ (z + bt).
Deriving x, we remark that it's a sub-solution of

x ′ (t) = 2|b|||f ′ || ∞ x(t).
Using the comparison principle, we deduce that for t ≥ 0,

x(t) ≤ x(0)e 2|b|||f ′ ||∞t
which implies

h ′ (z + bt) ≤ h ′ (z)e 2|b||f ′ ||∞t . (4.10)
Step 3: h ε solves (4.8). First, let us remark that the second and the third lines in (4.8) are direct consequences of the definition of h. It remains to prove that

(1 + M ε)h ′ (a ε z) ≥ 2F ((h(a ε z + εa ε r i )) i=0,...,N ). ( 4

.11)

Using that h ′ (z) = 2f (h(z)) = 2F (h(z), ..., h(z)), to prove (4.11), it's sufficient to show that

M εh ′ (a ε z) ≥ 2 (F ((h(a ε z + εa ε r i )) i=0,...,N ) -F ((h(a ε z)) i=0,...,N )) .
We have

h(a ε z + εa ε r i ) = h(a ε z) + εa ε r i 1 0 h ′ (a ε z + εa ε r i t)dt which implies that F ((h(a ε z + εa ε r i )) i=0,...,N ) = F h(a ε z) + εa ε r i 1 0 h ′ (a ε z + εa ε r i t)dt i=0,...,N
.

Using that F is globally Lipschitz continuous, we have

F ((h(a ε z + εa ε r i )) i=0,...,N ) -F ((h(a ε z)) i=0,...,N ) ≤ L F εa ε r * 1 0 h ′ (a ε z + εa ε r i0 t)dt
where

1 0 h ′ (a ε z + εa ε r i0 t)dt = max i=0,...,N 1 0 h ′ (a ε z + εa ε r i t)dt .
Using (4.10), we have

2 (F ((h(a ε z + εa ε r i )) i=0,...,N ) -F ((h(a ε z)) i=0,...,N )) ≤ 2L F εa ε r * h ′ (a ε z) 1 0 e 2εaεr * ||f ′ ||∞t dt ≤ L F ||f ′ || ∞ e 2εaεr * ||f ′ ||∞ -1 h ′ (a ε z) ≤ M εh ′ (a ε z)
where we use that

M > 3L F ||f ′ || ∞ and e 2ε(1+M ε)r * ||f ′ ||∞ -1 ε < 3.
Finally, using that h > 0, we deduce that (ψ 1 , ψ 2 ) is a positive viscosity super-solution of (4.1). Thanks to Lemma 4.2, this ends the proof of this proposition.

To achieve this section's goals, it remains to construct traveling wave solutions in the negative monostable case. This is announced in the following corollary.

Corollary 4.4 (Traveling wave solutions for negative small velocities in the negative monostable case). Consider a function F satisfying ( Ã) and ( B). Then, for c < 0 small enough, there exists a viscosity solution of

         cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R, cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) + 2σ -on R, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (-∞) = ϕ 2 (-∞) = θ -1 and ϕ 1 (+∞) = ϕ 2 (+∞) = θ. (4.12)
Proof. We define F (X) = -F (θ -X) -σ -and ri = -r i for i = 0, ..., N . It's straightforward to remark that F satisfy ( Ã) and ( B) where

f (v) = F (v, ..., v) satisfies f (0) = f (1) = 0.
Hence, σ+ = -min f = 0. Using Proposition 4.1, for c > 0 large enough, there exists a viscosity solution ( φ1 , φ2 ) of

         c φ′ 1 (z) = α 0 ( φ2 (z) -φ1 (z)) on R, c φ′ 2 (z) = α 0 ( φ1 (z) -φ2 (z)) + 2 F ( φ1 (z + ri )) i=0,...,N
on R, φ1 and φ2 are non-decreasing, φ1 (-∞) = φ2 (-∞) = 0 and φ1 (+∞) = φ2 (+∞) = 1.

Finally, we define

ϕ 1 (z) = θ -φ1 (-z), ϕ 2 (z) = θ -φ2 (-z)
and we can easily verify that (ϕ 1 , ϕ 2 ) is a viscosity solution of (4.12) for c = -c,

Proof of Theorem 2.2

Goal of this section is to prove Theorem 2.2. For a function F satisfying assumptions ( Ã) and ( B), let σ ∈ (σ -, σ + ). We will consider the following system

         cϕ ′ 1 (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R, cϕ ′ 2 (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) + 2σ on R, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (-∞) = ϕ 2 (-∞) = m σ and ϕ 1 (+∞) = ϕ 2 (+∞) = m σ + 1.
(5.1)

Our major interest is to study the velocity function c = c σ . The existence of traveling wave solutions for system (5.1) could be easily obtained by using a variable change as we will next show.

Proposition 5.1 (Existence and uniqueness of traveling wave: the fix bistable case (see Proposition 2.3 and Theorem 1.3 in [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF])). Consider a function F satisfying assumption ( Ã). We set

f (v) = F (v, ..., v).
Assume that F satisfies the following assumptions (B) and (C):

Assumption (B)
1) f (0) = f (1) = 0 and ∃b ∈ (0, 1) such that

f (b) = 0, f | (0,b) < 0, f | (b,1) > 0 and f ′ (b) > 0. 2) F ∈ C 1 in a neighborhood of {b} N +1 .
Assumption (C) There exists β 0 > 0 such that for a > 0, we have

F (X + (a, ..., a)) < F (X) if X, X + (a, ..., a) ∈ [0, β 0 ] N +1 , F (X + (a, ..., a)) < F (X) if X, X + (a, ..., a) ∈ [1 -β 0 , 1] N +1 . (5.2)
Then, there exists a unique velocity c such that there exists a viscosity solution ( φ1 , φ2 ) of

         c φ′ 1 (z) = α 0 ( φ2 (z) -φ1 (z)) on R, c φ′ 2 (z) = α 0 ( φ1 (z) -φ2 (z)) + 2 F ( φ1 (z + r i )) i=0,...,N
on R, φ1 and φ2 are non-decreasing, φ1 (-∞) = φ2 (-∞) = 0 and φ1 (+∞) = φ2 (+∞) = 1.

(5.3) Now, we prove the existence of solutions for (5.1). Proposition 5.2 (Traveling wave solutions for system (5.1)). Consider a function F satisfying assumptions ( Ã) and ( B) and let σ ∈ (σ -, σ + ). Then, there exists a unique c = c σ such that there exists a viscosity solution (ϕ 1 , ϕ 2 ) of (5.1). on R + × R. Also, (v 1 , v 2 ) is a viscosity super-solution of (5.7) replacing σ 1 by σ 2 . In particular, using that σ 1 ≤ σ 2 , we have that (v 1 , v 2 ) is a viscosity super-solution of (5.7). Using (5.6), we have for i ∈ {1, 2},

u i (0, ±∞) < v i (0, ±∞).
Up to translation, we can assume that for i ∈ {1, 2},

u i (0, x) < v i (0, x) for all x ∈ R.
Using the comparison principle for system (5.7) (see Proposition 2.6 in [START_REF] Forcadel | Homogenization of accelerated frenkelkontorova models with n types of particles[END_REF]), we deduce that for i ∈ {1, 2},

u i (t, x) ≤ v i (t, x) for (t, x) ∈ R + × R.
(

Taking i = 1 and x = -c σ1 + c σ2 2 t in (5.8), we get

ϕ 1 c σ1 -c σ2 2 t ≤ ψ 1 c σ2 -c σ1 2 t .
Taking t → +∞, we get

ϕ 1 (+∞) ≤ ψ 1 (-∞)
which gives a contradiction.

Corollary 5.4 (Monotony of c σ ). Consider a function F satisfying assumptions ( Ã) and ( B).

For σ ∈ (σ -, σ + ), let c σ be the unique velocity such that there exists a viscosity solution of (5.1) (see Proposition 5.2). If σ 1 , σ 2 ∈ (σ -, σ + ) such that

σ 1 < σ 2 ,
then, we have

c σ1 ≤ c σ2 .
Proof. The proof of this corollary could be easily derived from Proposition 5.3. In fact, for σ 1 , σ 2 ∈ (σ -, σ + ) such that σ 1 < σ 2 , we have

m σ1 < m σ2 , m σ1 + 1 ≥ θ > 0 ≥ m σ2 .
In the next lemma we prove a strict monotony property for the function σ → c σ . In particular, this result allows us to prove that if c -= c + , then c σ = 0 for any σ ∈ [σ -, σ + ]. Lemma 5.5. Consider a function F satisfying assumptions ( Ã) and ( B). For σ ∈ (σ -, σ + ), let c σ be the unique velocity such that there exists a viscosity solution of (5.1) (see Proposition 5.2). Let σ 1 , σ 2 ∈ (σ -, σ + ) such that

σ 1 < σ 2 .
Then, there exists K = K(σ 1 , σ 2 ) > 0 such that

c σ1 + K|c σ1 |(σ 2 -σ 1 ) ≤ c σ2 .
(5.9)

Proof. Let C = α 0 + 2 max X∈[-1,1] N +1
F (X) + σ + and consider a constant γ such that

0 < γ < 2(σ 2 -σ 1 ) α 0 + 2L F (5.10)
where L F is the Lipschitz constant of F . As a notation, we set c 1 = c σ1 , c 2 = c σ2 and

c = c 1 + K|c 1 |(σ 2 -σ 1 ).
Our goal is to prove (5.9) for any constant K such that

0 < K < min α 0 γ (σ 2 -σ 1 )C , 2(σ 2 -σ 1 ) -α 0 γ -2L F γ (σ 2 -σ 1 )C . ( 5.11)
If c 1 = 0, then (5.9) is true by Corollary 5.4. Assume that c 1 ̸ = 0. Using the two first equations in (5.1) for σ = σ 1 , we have

|c 1 |ψ ′ 1 (z) ≤ C, |c 1 |ψ ′ 2 (z) ≤ C.
(5.12)

For t ≥ 0 and x ∈ R, we define the functions

u 1 (t, x) = ψ 1 (x + ct) -γ, u 2 (t, x) = ψ 2 (x + ct)
where c is defined in (5.9). We will show that for (u 1 , u 2 ) is a viscosity sub-solution of (u 1 ) t (t, x) = α 0 (u 2 (t, x) -u 1 (t, x)), (u 2 ) t (t, x) = α 0 (u 1 (t, x) -u 2 (t, x)) + 2F ((u 1 (t, x + r i )) i=0,...,N ) + 2σ 2 (5.13) on R + × R. Assume that ψ 1 and ψ 2 are C 1 functions (if not we proceed similarly in the viscosity sense). On the one hand, we have for z = x + ct,

(u 1 ) t (t, x) = cψ ′ 1 (z) = (c -c 1 )ψ ′ 1 (z) + c 1 ψ ′ 1 (z) = (c -c 1 )ψ ′ 1 (z) + α 0 (ψ 2 (z) -ψ 1 (z)) ≤ K(σ 2 -σ 1 )C + α 0 (u 2 (t, x) -u 1 (t, x)) -α 0 γ ≤ α 0 (u 2 -u 1 )
where we use (5.12) in the third line and (5.10), (5.11) in the last line. On the other hand, we have

(u 2 ) t (t, x) = cψ ′ 2 (z) = (c -c 1 )ψ ′ 2 (z) + c 1 ψ ′ 2 (z) = (c -c 1 )ψ ′ 2 (z) + α 0 (ψ 1 (z) -ψ 2 (z)) + 2F ((ψ 1 (z + r i )) i=0,...,N ) + 2σ 1 = (c -c 1 )ψ ′ 2 (z) + α 0 (u 1 (t, x) -u 2 (t, x)) + α 0 γ + 2F ((u 1 (t, z + r i )) i=0,...,N + γ) + 2σ 1 ≤ K(σ 2 -σ 1 )C + α 0 (u 1 (t, x) -u 2 (t, x)) + α 0 γ + 2F ((u 1 (t, z + r i )) i=0,...,N ) + 2L F γ + 2σ 1 ≤ α 0 (u 1 (t, x) -u 2 (t, x)) + 2F ((u 1 (t, z + r i )) i=0,...,N ) + 2σ 2
where we use that F is Lipschitz in the forth line and (5.10), (5.11) in the last line. Finally, we define the functions

v 1 (t, x) = ϕ 1 (x + c 2 t), v 2 (t, x) = ϕ 2 (x + c 2 t).
Clearly, (v 1 , v 2 ) is a viscosity super-solution of (5.13) and proceeding as in the proof of Proposition 5.3, we get (5.9). Proposition 5.6. Consider a function F satisfying assumptions ( Ã) and ( B). For σ ∈ (σ -, σ + ), let c σ be the unique velocity such that there exists a viscosity solution of (5.1) (see Proposition 5.2). Then, we have Taking σ → σ + , we get that c + is finite. In the same way, we can prove that c -is finite. Now, assume that c -= c + . Using that σ → c σ is non-decreasing on (σ -, σ + ), we deduce that

c -= lim
c σ = c + for all σ ∈ (σ -, σ + )
and thanks to Lemma 5.5, we obtain c -= c + = 0.

Next, we show that the map σ → c σ is continuous on (σ -, σ + ). To this end, we first need the following comparison principles on the half-line.

Theorem 5.7 (Comparison principle on (-∞, r * ) (see Theorem 4.1 in [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF]).). Consider a function F satisfying assumption ( Ã) and ( B) and let σ ∈ (σ -, σ + ). For c ∈ R, let (u 1 , u 2 ) and (v 1 , v 2 ) be respectively a viscosity sub-solution and a viscosity super-solution of

cu ′ 1 (z) = α 0 (u 2 (z) -u 1 (z)) on (-∞, 0), cu ′ 2 (z) = α 0 (u 1 (z) -u 2 (z)) + 2F ((u 1 (z + r i )) i=0,.
..,N ) + 2σ on (-∞, 0).

(5.14)

Assume that u i ∈ [m σ , m σ + 1] and v i ∈ [m σ , m σ + 1] for i = 1, 2.
We suppose that ∃β 0 > 0 such that

u 1 ≤ m σ + β 0 on (-∞, 2r * ], u 2 ≤ m σ + β 0 on (-∞, 2r * ]
and

u 1 ≤ v 1 on [0, r * ], u 2 ≤ v 2 on [0, r * ].
Then,

u 1 ≤ v 1 on (-∞, r * ], u 2 ≤ v 2 on (-∞, r * ]. Corollary
i ∈ [m σ , m σ + 1] and v i ∈ [m σ , m σ + 1] for i = 1, 2. We suppose that ∃β 0 > 0 such that v 1 ≥ m σ + 1 -β 0 on [-2r * , +∞), v 2 ≥ m σ + 1 -β 0 on [-2r * , +∞)
and

u 1 ≤ v 1 on [-r * , 0], u 2 ≤ v 2 on [-r * , 0].
Then,

u 1 ≤ v 1 on [-r * , +∞), u 2 ≤ v 2 on [-r * , +∞).
Remark 5.9. The reader can remark that the previous theorem and corollary are not stated exactly as in [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF]. Nevertheless, we can deduce them by a simple change of variable: since F satisfies assumptions ( Ã) and ( B), as in the proof of Proposition 5.2, we define the new function F (X) = F (X + m σ ) + σ and for i = 1, 2, we consider the new functions ũi and ṽi defined by

ũi (z) = u i (z) -m σ and ṽi (z) = v i (z) -m σ .
Then, we apply Theorem 4.1 in [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF] considering these new functions to obtain our Theorem 5.7.

In the same way, we can derive Corollary 5.8 from Corollary 4.2 in [START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF].

Proposition 5.10 (Continuity of the velocity function). Consider a function F satisfying assumptions ( Ã) and ( B). For σ ∈ (σ -, σ + ), let c σ be the unique velocity such that there exists a viscosity solution of (5.1) (see Proposition 5.2). Then, the map σ → c σ is continuous.

Proof. Let σ 0 ∈ (σ -, σ + ) and consider a sequence σ n ∈ (σ -, σ + ) such that σ n → σ 0 as n → +∞.

We want to show that c n = c σn → c 0 = c σ0 . Up to extract a sub-sequence, we have

c n → c ∞ .
We want to show that c ∞ = c 0 . Let (ϕ n 1 , ϕ n 2 ) (resp. (ϕ 1 , ϕ 2 )) be the viscosity solution of (5.1) replacing σ by σ n (resp. σ 0 ) and c by c n (resp. c 0 ).

Case 1: c

0 < c ∞ . Since ϕ n 1 is non-decreasing and ϕ n 1 (-∞) = m σn < m σ0 + b σ0 2 < ϕ n 1 (+∞) = m σn + 1, there exists x n ∈ R such that m σ0 + b σ0 2 ∈ [(ϕ n 1 ) * (x n ), (ϕ n 1 ) * (x n )] .
We define

φn 1 (x) = ϕ n 1 (x + x n ), φn 2 (x) = ϕ n 2 (x + x n ).
Because our system (5.1) is invariant by translation, ( φn 1 , φn 2 ) is a viscosity solution of (5.1) and

m σ0 + b σ0 2 ∈ ( φn 1 ) * (0), ( φn 1 ) * (0) .
Up to extract a subsequence, φn 1 (resp. φn 2 ) converges almost everywhere to ϕ ∞ 1 (resp. ϕ ∞ 2 ). Thanks to Proposition 3.4, (ϕ ∞ 1 , ϕ ∞ 2 ) satisfies Then, Theorem 5.7 implies that ϕ ∞ 1 ≤ ϕ 1 on (-∞, r * ), ϕ ∞ 2 ≤ ϕ 2 on (-∞, r * ).

                 c ∞ (ϕ ∞ 1 ) ′ (z) = α 0 (ϕ ∞ 2 (z) -ϕ ∞ 1 (z)) on R, c ∞ (ϕ ∞ 2 ) ′ (z) = α 0 (ϕ ∞ 1 (z) -ϕ ∞ 2 (
Since ϕ i (+∞) > ϕ ∞ i (+∞), we can assume (up to translation) that

ϕ ∞ 1 ≤ ϕ 1 on R, ϕ ∞ 2 ≤ ϕ 2 on R.
Finally, arguing as in the proof of Proposition 5.3, we get

b σ0 = ϕ ∞ 1 (+∞) ≤ ϕ 1 (-∞) = m σ0
which gives a contradiction. Then, using the forth line in (6.2) and that ϕ p j is non-decreasing for j = 1, 2, we deduce that 0 ≤ ϕ p j (1) -ϕ p j (0) ≤ ϕ p j 1 p -ϕ p j (0) ≤ 1 (6.5)

and for z ∈ R and i ∈ {0, ..., N }, we have Passing to the limit as p → 0, we get the convergence of ϕ p j to a function ϕ j almost everywhere for j = 1, 2. Moreover, (ϕ 1 , ϕ 2 ) satisfies

-1 = ϕ p j z -
         c(ϕ 1 ) ′ (z) = α 0 (ϕ 2 (z) -ϕ 1 (z)) on R,

  1) = 0 and f σ < 0 on (m σ , b σ ) and f σ > 0 on (b σ , m σ + 1) with b σ ∈ (0, θ) and m σ ∈ (θ -1, 0).

Definition 3 . 1 (

 31 Definition of viscosity solutions). Consider a function F defined on R N +1 and let c, σ ∈ R. Let ϕ 1 , ϕ 2 be two locally bounded functions. We define the upper and lower semicontinuous envelopes of ϕ 1 and ϕ 2 by (ϕ 1 ) * (z) = lim sup y→z ϕ 1 (y) and (ϕ 1 ) * (z) = lim inf y→z ϕ 1 (y), (ϕ 2 ) * (z) = lim sup y→z ϕ 2 (y) and (ϕ 2 ) * (z) = lim inf y→z ϕ 2 (y).

(4. 1 )

 1 Proposition 4.1 (Existence of traveling wave for large velocities in the positive monostable case). Consider a function F satisfying ( Ã) and ( B) with σ + = 0. Then, for c > 0 large enough, there exists a viscosity solution (ϕ 1 , ϕ 2 ) of (4.1). The proof of Proposition 4.1 requires to construct positive viscosity super-solution. We say that the couple (ψ 1 , ψ 2 ) is a viscosity super-solution of (4.1) if ψ 1 and ψ 2 are non-decreasing on R, ψ 1 (-∞) = ψ 2 (-∞) = 0 and ψ 1 (+∞) = ψ 2 (+∞) = 1 and (ψ 1 , ψ 2 ) is a viscosity super-solution in the sense of Definition 3.1 of

Lemma 4 . 3 .

 43 Consider a function F satisfying ( Ã) and ( B) with σ + = 0. Then, there exists

  σ→σ -c σ and c + = lim σ→σ + c σ exist and are finites and c -= c + = 0 or c -< c + . Proof. Thanks to Corollary 5.4, σ → c σ is non-decreasing, the limits c -= lim σ→σ - c σ and c + = lim σ→σ + c σ exist. From Proposition 4.1, we know that for c >> 1 big enough, there exists a viscosity solution (ϕ 1 , ϕ 2 ) of (5.1) replacing σ by σ + . Then, from Corollary 5.4, we get for any σ ∈ (σ -, σ + ), c σ ≤ c.

  z)) + 2F ((ϕ ∞ 1 (z + r i )) i=0,...,N ) + 2σ 0 on R, ϕ ∞ 1 and ϕ ∞ 2 are non-decreasing, m σ0 ≤ ϕ ∞ 1 ≤ m σ0 + 1 and m σ0 ≤ ϕ ∞ 2 ≤ m σ0 + 1, m σ0 + b σ0 2 ∈ [(ϕ ∞ 1 ) * (0), (ϕ ∞ 1 ) * (0)] .Passing to the limits as x → ±∞, we deduce thatϕ ∞ 1 (±∞) = ϕ ∞ 2 (±∞) and f (ϕ ∞ 1 (±∞)) + σ 0 = 0. Using that m σ0 + b σ0 2 < b σ0 , we obtain ϕ ∞ 1 (-∞) = m σ0 and ϕ ∞ 1 (+∞) ∈ {b σ0 , m σ0 + 1}. If ϕ ∞ 1 (+∞) = m σ0 +1, we obtain c ∞ = c 0 using the uniqueness of c 0 and this gives a contradiction. Now, assume that ϕ ∞ 1 (+∞) = b σ0 . Using that ϕ ∞ i (-∞) = m σ0 and ϕ i (+∞) = m σ0 + 1 for i ∈ {1, 2}, then up to translation, we can assume thatϕ ∞ 1 ≤ m σ0 + β 0 on (-∞, 2r * ), ϕ ∞ 2 ≤ m σ0 + β 0 on (-∞, 2r * )andϕ ∞ 1 ≤ ϕ 1 on [0, r * ), ϕ ∞2 ≤ ϕ 2 on [0, r * ).

Case 2 :< ϕ n 1 ( 1 0F ((ϕ p 1 (

 2111 c 0 > c ∞ . Since ϕ n 1 is non-decreasing andϕ n 1 (-∞) = m σn < m σ0 + b σ0 + 1 2 +∞) = m σn + 1, there exists x n ∈ R such that m σ0 + b σ0 + 1 2 ∈ [(ϕ n 1 ) * (x n ), (ϕ n 1 ) * (x n )] .Integrating (6.3) between 0 and 1, we get c ((ϕ p 1 (1) -ϕ p 1 (0)) + (ϕ p 2 (1) -ϕ p 2 (0))) = 2 z + r i )) i=0,...,N )dz + 2σ(c, p).

1 p

 1 -ϕ p j (z) ≤ ϕ p j (z + r i ) -ϕ p j (z) ≤ ϕ p j z + 1 p -ϕ p j (z) = 1.Moreover, the Lipschitz continuity of F implies that|F ((ϕ p 1 (z + r i )) i=0,...,N )| = |F ((ϕ p 1 (z + r i )) i=0,...,N ) -F ((ϕ p 1 (z)) i=0,...,N ) + F ((ϕ p 1 (z)) i=0,...,N )| ≤ L F max i=0,...,N ||ϕ p 1 (z + r i ) -ϕ p 1 (z)| + f (ϕ p 1 (z)) ≤ L F + f (θ) (6.6)where L F is the Lipschitz constant of F . Injecting (6.5) and (6.6) in (6.4), we deduce that there exists a constantC 0 = C 0 (c, L F , θ) such that |σ(c, p)| ≤ C 0 .Therefore, up to subsequence, we have σ(c, p) → σ 0 as p → 0.Using that |h 1 (z) -z| ≤ C, we deduce that up to translation of ϕ p 1 and ϕ p 2 , we can assume that   b σ0 + m σ0 2 ∈ [(ϕ p 1 ) * (0), (ϕ p 1 ) * (0)] if σ 0 < σ + , θ ∈ [(ϕ p1 ) * (0), (ϕ p 1 ) * (0)] if σ 0 = σ + .

c(ϕ 2 ) 1 ( 6 b

 216 ′ (z) = α 0 (ϕ 1 (z) -ϕ 2 (z)) + 2F ((ϕ 1 (z + r i )) i=0,...,N ) + 2σ 0 on R, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 (+∞) -ϕ 1 (-∞) ≤ 1 and ϕ 2 (+∞) -ϕ 2 (-∞) ≤ σ0 + m σ0 2 ∈ [(ϕ 1 ) * (0), (ϕ 1 ) * (0)] if σ 0 < σ + , θ ∈ [(ϕ 1 ) * (0), (ϕ 1 ) * (0)] if σ 0 = σ + .(6.8)

  5.8 (Comparison principle on [-r * , +∞) (see Corollary 4.2 in[START_REF] Forcadel | Existence and uniqueness of traveling wave for accelerated frenkel-kontorova model[END_REF]).). Consider a function F satisfying assumption ( Ã) and ( B) and let σ ∈ (σ -, σ + ). For c ∈ R, let (u 1 , u 2 ) and (v 1 , v 2 ) be respectively a viscosity sub-solution and a viscosity super-solution of (5.14) on (0, +∞).

	Assume that u

Proof. Let σ ∈ (σ -, σ + ). We define for X ∈ R N +1 , F (X) = F (X + m σ ) + σ.

By definition of m σ and b σ (see Definition 2.1), we have for σ ∈ (σ -, σ + ), m σ ∈ (θ -1, 0) and b σ ∈ (0, θ). This implies that f ′ (b σ ) > 0 and F is C 1 in a neighborhood of {b σ } N +1 .

(5.4)

Therefore, F satisfies assumption ( Ã) and assumption (B) given in Proposition 5.1 with b = b σ -m σ . Moreover, using that f ′ < 0 in a neighborhood of m σ , we deduce that F satisfies assumption (C). Hence, thanks to Proposition 5.1, there exists a unique velocity c = c σ such that there exists a viscosity solution ( φ1 , φ2 ) of (5.3). Finally, we define

Then, (ϕ 1 , ϕ 2 ) is a viscosity solution of (5.1) for the same velocity c = c σ .

Next, we prove that the map σ → c σ is non-decreasing on (σ -, σ + ). To do this, we need the following proposition.

Proposition 5.3 (Comparison of the velocities). Consider a function F satisfying ( Ã) and consider the following system

..,N ) + 2σ on R, ϕ 1 and ϕ 2 are non-decreasing, ϕ 1 and ϕ 2 are bounded, ϕ 1 (-∞) = ϕ 2 (-∞) and ϕ 1 (+∞) = ϕ 2 (+∞).

(5.5)

Assume that σ 1 ≤ σ 2 and assume that there exists a viscosity sub-solution solution (ϕ 1 , ϕ 2 ) of (5.5) for σ = σ 1 and a viscosity super-solution (ψ 1 , ψ 2 ) of (5.5) for σ = σ 2 . Moreover, we assume that

(5.6) Then, we have

Proof. By contradiction, let us assume that c σ1 > c σ2 . For i ∈ {1, 2}, we define the functions

Then, (u 1 , u 2 ) is a viscosity sub-solution of

We define

Because our system (5.1) is invariant by translation, ( φn 1 , φn 2 ) is a viscosity solution of (5.1) and

Up to extract a subsequence, φn

Passing to the limits as x → ±∞, we deduce that

, we obtain c ∞ = c 0 using the uniqueness of c 0 and this gives a contradiction. Now, assume that ϕ ∞ 1 (-∞) = b σ0 . Using that ϕ ∞ i (+∞) = m σ0 + 1 and ϕ i (+∞) = m σ0 + 1 for i ∈ {1, 2}, then up to translation, we can assume that

Finally, arguing as in the proof of Proposition 5.3, we get

Proof of Theorem 2.2. Theorem 2.2 is now proved thanks to Propositions 5.2, 5.10, 5.6 and Corollary 5.4.

Proof of Theorem 2.3

In this section, we construct traveling wave solution in the positive and negative monostable case. To do this, we will use the so-called "hull functions". These functions were constructed in [START_REF] Forcadel | Homogenization of accelerated frenkelkontorova models with n types of particles[END_REF] in order to pass from the discrete accelerated Frenkel Kontorova model with n types of particles to a macroscopic Hamilton-Jacobi equation. In [START_REF] Forcadel | Homogenization of accelerated frenkelkontorova models with n types of particles[END_REF], the existence of hull functions was obtained for n types of particles and for a non-linear functions F j depending on time . In our case, we need a simplified version since we consider one type of particle and the non-linearity F does not depend on time. The definition and the existence of hull functions is stated in the following lemma: Lemma 6.1 (Existence of hull functions (see Theorem 1.10 and Theorem 1.11 in [START_REF] Forcadel | Homogenization of accelerated frenkelkontorova models with n types of particles[END_REF])). Consider a function F satisfying assumption ( Ã). Let p > 0 and σ ∈ R. There exists a unique λ(σ, p) = λ p (σ) such that there exists a locally bounded functions h 1 : R → R and h 2 : R → R such that (h 1 , h 2 ) satisfies in the viscosity sense

Moreover, the function σ → λ p (σ) is continuous and we have the following properties:

2) λ p is non decreasing in σ.

Next, we prove Theorem 2.3.

Proof of Theorem 2.3. We will only do the proof part ii) since the proof of part i) is similar.

Step 1: traveling wave for c > c + . Let us consider a velocity c such that c > c + .

From the properties of λ p (σ), we deduce that there exists σ(c, p) such that

We define

where (h 1 , h 2 ) satisfies (6.1). Hence, (ϕ p 1 , ϕ p 2 ) satisfies

The sum of the first two equations in (6.2) gives that for almost every z ∈ R, we have

Passing to the limit as z → ±∞, we get that where c σ is defined in Proposition 5.2. Taking σ → σ + , we get c ≤ c + which gives a contradiction because c > c + . We deduce that σ 0 = σ + , and using (6.8), we have ϕ 1 (-∞) = 0 and ϕ 1 (+∞) = 1 and this ends the proof for c > c + .

Step 2: traveling wave for c = c + . Let c > c + such that c → c + . There exists a viscosity solution (

Up to translation, we can assume that θ ∈ [(ϕ c 1 ) * (0), (ϕ c 1 ) * (0)].

Passing to the limit as c → c + , we get the convergence almost everywhere of ϕ c i to ϕ i for i = 1, 2. Then, thanks to Proposition 3.4, we get that (ϕ 1 , ϕ 2 ) solves

Finally, using the last line in (6.9), we get ϕ 1 (-∞) = ϕ 2 (-∞) = 0 and ϕ 1 (+∞) = ϕ 2 (+∞) = 1 .

Step 3: non-existence of traveling wave for c < c + . Assume that there exists a viscosity solution (ϕ 1 , ϕ 2 ) of (2.4) with c < c + . Using Proposition 5.3, for any σ ∈ (σ -, σ + ), we have c σ ≤ c.

Taking σ to σ + , we obtain c + ≤ c which gives a contradiction.

Declarations

Ethical Approval. Not applicable.

Competing Interests. The authors have no relevant financial or non-financial interests to disclose.

Authors' contributions. All authors have contributed equally.

Funding. The authors received no financial support for the research, authorship, and/or publication of this article.

Availability of data and materials. Not applicable.


Issa Bou Zeid

Gülgün Alpan

Bernard Penz
email: bernard.penz@grenoble-inp.fr

Alain Nguyen

Christian Serrano
email: christian.serrano@renault.comgulgun.alpan

Selecting transportation strategies in transportation planning for automotive supply chain

Keywords: inbound logistics network, optimisation, vehicle routing problem

Defining a transportation plan for inbound logistics in the automotive industry represents a major economic challenge for the manufacturers. A three phased algorithm, where in the second, three mixed integer linear programming models (MILP) are developed for the problem. To test the performances of the proposed approach, computational experiments are done on real-world instances, and the results are reported.

Introduction

In the automotive industry, the Supply Chain (SC) is categorically divided into two major logistic processes, the inbound and the outbound SC. While, the inbound SC secures the effective supply of raw materials between suppliers and the plants, the outbound SC includes the delivery of final products to the clients. Recent increasing in global competition, has prompted manufacturers to be innovative in order to reduce costs, by optimizing the logistic processes. This article considers the problem of defining a least-cost inbound transportation plan subject to various operational and network constraints for a leading automobile manufacturer. The inbound logistics of the plants involves a significant number of suppliers as many of the parts needed to produce one vehicle are made by hundreds if not thousands of suppliers, forming the inbound logistics network. To supply the plants, a fleet of vehicles is used that is generally do not belong the automotive manufacturers due to, among others, cost reasons [START_REF] Repoussis | A hybrid evolution strategy for the open vehicle routing problem[END_REF]. Thus the SC activities are outsourced to third party logistics (3PLs). A mixed inbound network with milk-run and indirect strategy transportation strategies is considered. A milk-run is a pickedup tour starts at one supplier, visits a subset of suppliers exactly once before ending at the plant. A special case of it, is the direct strategy, where the number of suppliers is equal to one. Finally, the indirect strategy delivers the plant through an intermediate facility, called crossdock. Hence, in our problem, we aim on finding the transportation strategy for each supplier, and the decision variables related to optimal pick-up sequence, volumes, and frequencies.

This article is organized as follows. Section 2 provides a brief literature review. Section 3 describes in details the problem. Section 4 presents the solution approach. Section 5 presents the results of computational experiments, and finally, in section 6 conclusions are drawn, and future works are discussed.

Literature review

The studied problem is the transportation planning problem (TPP) with transportation strategies selection. The decision variables on transportation strategies selection is related to the class of Designing Transportation networks in the literature. [START_REF] Meyer | Transport concept selection considering supplier milk runs-an integrated model and a case study from the automotive industry[END_REF] proposes an integrated mathematical model that addresses the assignment of frequencies, selection of transportation strategies, and determination of milk-runs scheduling to mixed inbound networks with three strategies. Point-to-Point, Area Forwarding services, and milk runs are solved using a standard solver. [START_REF] Baller | Optimizing automotive inbound logistics : A mixed-integer linear programming approach[END_REF] considers an inbound logistics planning problem with courier, express services and milk-runs transportation modes. A MILP is proposed together with valid inequalities to improve the convergence of the model towards optimal solutions. When suppliers are assigned to direct or milk-run strategies, decisions on pick-up sequence, frequency, and volumes have to be made for each vehicle route. These decisions are related to the vehicle routing problem (VRP), and its variants depending on the possible decisions that might be involved. The Flexible Periodic VRP allows service policies that are flexible with respect to the frequency of visits and the amount delivered at each visit [START_REF] Lucia | The flexible periodic vehicle routing problem : modeling alternatives and solution techniques[END_REF]. The periodic VRP with service choice deals with flexibility in the minimum visit frequency [START_REF] Francis | The period vehicle routing problem with service choice[END_REF]. The Inventory Routing Problem [START_REF] Bertazzi | Inventory routing problems with multiple customers[END_REF][START_REF] Coelho | Thirty years of inventory routing[END_REF], and the split-delivery VRP [START_REF] Dror | Savings by split delivery routing[END_REF] allows flexible service policy on the amount picked-up at each visit. All these VRP classes find the optimal pickup sequence.

Problem description

The TTP is described as a set of destinations called receiving docks d ∈ D, located in the same plant, which are supplied by a set of geographically dispersed sources s ∈ S called suppliers. Demand requests r sd exist from d to s expressed by a vector of three measuring units, l sd in loading meter (LDM), w sd in kilogram (Kg), and cm sd in cubic meter (m 3 ) that can be transported with a fleet of unlimited homogeneous vehicles V having two loading capacity constraints Q for LDM and KG for Kg.

To transport r sd , either indirect strategy or milk-run strategy is used. For the indirect strategy, suppliers are assigned to their respective cross-docks. For the milk-run strategy, the fleet V is used to transport the demands from suppliers to their respective receiving docks. Each transportation strategy has its own cost.

On one hand, indirect cost δ s for supplier s, is composed of the upstream cost between supplier s and its respective cross-dock that depends on d cm sd and the distance s to its cross-dock, the handling cost of d cm sd for supplier s inside the cross-dock calculated for every 1 m 3 , and the downstream cost between the cross-dock and the plant calculated by first adding the fixed cost of using one vehicle to the cost of traveling between cross-dock and plant, then multiplied by the number of vehicles needed. Given r sd , δ s is precalculated. On the other hand, milk-run cost is calculated for every used v ∈ V. It is the sum of the variable cost of the route between a set of suppliers and the plant, the fixed cost of using one vehicle, and the fixed cost of the number of stops (suppliers).

We define the TTP on an undirected graph G = (N , A), where N is the vertex set, and A = {(s i , s j ) : s i , s j ∈ N , s i ̸ = s j } is the arcs set that connects the nodes in the graph. Let N = S ∪ {0}, with S = {s 1 , ..., s |S| }, and {0} is the plant where docks in D are located. Every node s i ∈ S, is associated with a pick-up vector of r sd to every d ∈ D. Three kinds of arcs exists, each associated with a cost. Arcs (0, s i ) between the plant and suppliers have a cost c 0,si = 0 (because the pick-up tour starts at the suppliers), arcs (s i , s j ) between the suppliers have a cost c sisj ≥ 0, which is the travel cost from s i to s j , and arcs (s i , 0) connecting the suppliers and the plant have a cost of c si0 ≥ 0, the travel cost from s i to 0. For milk-run suppliers, the optimal elementary routes that minimize the total costs are obtained for every v ∈ V. We find the elementary routes, by assuring that it starts and ends at the plant {0}, and visits a set of suppliers exactly once with no cycles. Fig. 1a illustrates an example of a small graph G, with N = 4, and D = 2, and Fig. 1b shows an example of a feasible solution for vehicle v in G.

Our objective is therefore to define a transportation plan that satisfies the constraints and minimizes total milk-run and indirect costs. First, it is assumed that if an indirect strategy is chosen for a supplier, an explicit route need not be found, and the precomputed cost δ s is applied. For each supplier s and receiving dock d, with r sd > 0, a minimum visit frequency f sd is imposed in order to have a higher number of small deliveries, and consequently a low inventory at the plant [START_REF] Harris | How many parts to make at once[END_REF]. In addition, for practical industrial reasons, suppliers are divided into clusters of maximum size M axC, and one supplier belongs to exactly one cluster. Finally, for each vehicle v and its associated route, M axS is the maximum number of stops (suppliers in the route), M axD is the maximum number of docks supplied, and the distance between two consecutive stops cannot exceed M axQ.

Solution Approach

Due to the complexity of the problem, we propose a three-phase approach. In phase 1, the clustering phase, we generate all possible clusters from the set of suppliers S (c ∈ C), respecting the distance constraints. For each cluster c ∈ C, the subgraph G ′ c is constructed. In phase 2, the optimal transportation plan for each cluster is found by a MILP. In phase 3, a Set Partitioning Problem is solved to find the best set of clusters that minimizes the total cost.

Phase 2 : Routing and Transportation Strategies

The optimal transportation plan cost for each cluster c ∈ C is obtained by determining the transportation strategy for the suppliers, by solving independently a MILP on each subgraph G ′ c . Before introducing the MILP, parameters and decision variables are presented. For the parameters, c ij ((i, j) ∈ N ) is the cost of the arc (i, j), δ s (s ∈ S) is the cost of assigning supplier s to indirect strategy, S and F are the fixed cost per stop at a supplier and the fixed cost per vehicle respectively. A big M quantity is mandatory in the MILP for each supplier s ∈ S : Ūs = min(Q, d l sd ).

Decision variables are as follows :

x v ij : 1 if the arc (i, j) ∈ N is visited by vehicle v ∈ V, 0 otherwise ; y v s : 1 if supplier s ∈ S is visited by vehicle v ∈ V, 0 otherwise ; z v sd : Quantity supplied from supplier s ∈ S to dock d ∈ D by vehicle v ∈ V (z v sd ∈ R + ) ; t v sd : 1 if supplier s ∈ S and receiving dock d are visited by vehicle v ∈ V, 0 otherwise ; p v d : 1 if dock d ∈ D is visited by vehicle v ∈ V, 0 otherwise ; u v : 1 if vehicle v ∈ V is used, 0 otherwise ; s v : Number of stops for vehicle v ∈ V (s v ∈ N + ) ;
γ s : 1 if supplier s ∈ S is assigned to milk-run strategy, 0 for the indirect strategy.

M inimize

i,j∈N v∈V

x v ij c ij + S v∈V s v + F v∈V u v + s∈S δ s (1 -γ s ) (1)
γ s ≤ v y v s ∀ s ∈ S (2)
s d z v sd ≤ Q ∀ v ∈ V (3) s d w sd d sd z v sd ≤ KG ∀ v ∈ V (4) d z v sd ≤ Ūs y v s ∀ s ∈ S, v ∈ V (5) v z v sd = l sd γ s ∀ s ∈ S, d ∈ D (6) s y v s ≤ M axS u v ∀ v ∈ V (7) u v ≤ s y v s ∀ v ∈ V (8) M p v d ≥ s t v sd ∀ d ∈ D, v ∈ V (9) p v d ≤ s t v sd ∀ d ∈ D, v ∈ V (10) d p v d ≤ M axD u v ∀ v ∈ V (11) z v sd ≤ Q t v sd ∀ s ∈ S, d ∈ D, v ∈ V (12) z v sd ≥ t v sd ∀ s ∈ S, d ∈ D, v ∈ V (13) v t v sd ≥ f sd γ s ∀ s ∈ S, d ∈ D ( 14
)
y v 0 ≥ u v ∀ v ∈ V (15) j∈N x v ij = y v i ∀ i ∈ N , i ̸ = j, v ∈ V (16) i∈N x v ij = y v j ∀ j ∈ N , i ̸ = j, v ∈ V (17) i,j∈I x v ij ≤ |I| -1 ∀ I ⊆ S, i ̸ = j, v ∈ V (18) s v ≥ s∈S y v s -1 ∀ v ∈ V (19)
The objective function (1) minimizes the total milk-run and indirect costs for the given cluster. Constraint (2) guarantees that if supplier s is not visited in any vehicle v, then it is assigned to indirect strategy. Constraints (3,4) guarantees that the vehicle capacities in LDM and Kg respectively are respected. Constraint (5) ensures that if supplier s is visited by vehicle v, at most Ūs can be collected. Constraint (6) satisfies the total demand request for the pair (s, d). Constraint [START_REF] Coelho | Thirty years of inventory routing[END_REF][START_REF] Dror | Savings by split delivery routing[END_REF] respects the maximum number of suppliers in vehicle v, and forces u v to be 1 if at least one supplier is visited. Constraints (9,10) determines when a receiving dock d is visited by vehicle v. Constraint (11) respects the maximum number of receiving dock per vehicle v. Constraints (12,13) ensures that if a positive amount of demand for dock d is collected from supplier s by vehicle v, then vehicle v contains both supplier s and receiving dock d. Constraint (14) respects the minimum frequency for the pair (s, d). Constraint (15) guarantees that the plant is visited only if vehicle is used. Constraints (16,17) guarantee that for the plant, and for each supplier there's one sorting, and one entering arc. Constraint (18) eliminates the sub-tours. Constraint (19) finds the number of stops in vehicle v.

Phase 3 : Set Partitioning Problem (SPP)

The Set Partitioning Problem (SPP), determines the best subset of clusters C ′ ⊆ C, such that each supplier s ∈ S belongs exactly to one cluster c ∈ C ′ , and the total cost is minimum. Let α c be the cost of every c ∈ C, parameter a sc = 1 if supplier s is in cluster c, and variable x c = 1 if cluster c is selected in the optimal solution, 0 otherwise. The integer linear programming model (ILP) for the [SPP] is :

[SPP] M inimize c ∈ C α c x c (20) c ∈ C a sc x c = 1 ∀ s ∈ S (21)

Enhanced formulation

The above model can be further strengthened by using reformulation techniques for some constraints and introducing new ones. The first technique used is disaggregation. The basic idea is to separate one constraint (big-M type constraints) into many without changing the meaning of a MIP model. We use this technique to reformulate constraints 2, 7, and 9. The sub-tour elimination in 18 is also reformulated. It prevents subtours by enumerating all subsets I of size 2 ≤ |I| ≤ |N | -1, and ensures the number of arcs selected is less than the number of nodes -1. However, due to the M axS constraint, it is sufficient to only eliminate subsets I : 2 ≤ |I| ≤ M axS. It be done by finding cliques and cycles of size k : 2 ≤ k ≤ M axS. In addition, the right-hand side of the constraint is replaced by summing on y v s where s ∈ ϕ, with ϕ is the subset of all combination of suppliers from I with |ϕ| = |I| -1 providing a tighter formulation. The new constraint is :

(i,j)∈I x v ij ≤ s∈ϕ y v s , ∀ I ⊆ S, 2 ≤ |I| ≤ M axS, ϕ ⊆ I, |ϕ| = |I| -1, v ∈ V.
We also introduce a valid inequality. Let ψ be the set of all shortest paths for every pair (s, s ′ ), s, s ′ ∈ S with a number of nodes greater than M axS -1. Then :

y v s + y v s ′ ≤ 1, ∀ s, s ′ ∈ ψ, s ̸ = s ′ , v ∈ V
is valid. In other words, if for every pair (s, s ′ ), supplier s cannot reach supplier s ′ in less than M axS -1 stops, then s and s ′ cannot be in the same vehicle v. A symmetry break constraint is also introduced. It imposes that vehicle v + 1 cannot be used if vehicle v is not yet used,

u v ≥ u v+1 , ∀ v ∈ V.

Computational Experiments and Results

To evaluate the performance of the formulations, computational experiments are done on real-world instances conducted on a personal PC equipped with an 11 th generation Intel Core i7 3.00 GHz , 8 logical processor, and 32GB of random-access memory (RAM). The MILP and ILP models are solved using the commercial solver CPLEX, with a maximum time limit for every cluster (TLC) in phase 2, of 180 seconds, and a relative tolerance on the gap between the best integer objective and the objective of the best node remaining of 0.5% for small and medium-sized instances, and 1% to large-sized instances. In addition, a maximum time limit for every instance (TLI) of 7200 seconds (2 hours) for small and medium-sized instances, and 25200 seconds (7 hours) for large-sized instances is also set. Besides, phase 2 of the algorithm is parallelized on two processors using the Fork/Join framework in Java.

The real-world instances are divided into small, medium and large-sized instances, distinctive in the size of S, D, N (graph density), C, r sd (number of V). For example, the size of S varies between 10 and 54, and the minimum, maximum and number of V for clusters are 1 and 85 respectively. The average number of vehicles per cluster varies between, 3.33 and 14.4.

Table 1 shows the results of the enhanced model compared to the original one. The minimum, maximum and average improvement are 43.16%, 95.91% and 75.06%. In addition, the enhanced model has improved the bound of several clusters, solving them optimally. For example, for instance number 4, 9 clusters that exceeded the TLC are solved to optimality with the enhanced model. Finally, the number of unsolved clusters has also been improved. the minimum, maximum and average improvement percentage is, 9.3%, 100% and 23.7% respectively. In summary, although the enhanced model has improved the computational time of 57% of the instances, it has limitation on medium and large-sized instances.

Conclusions and perspectives

A three-phase approach is proposed for an inbound SC problem, giving good results small industrial instances. However, solving medium and large-sized instances is challenging because of the : 1) significant number of clusters, 2) clusters with high volumes are more difficult to be solved. The weakness of the approach, comes from solving the MILP that jointly finds for every

  FIG.1-An example of a small graph G, with a feasible solution. into clusters of maximum size M axC, and one supplier belongs to exactly one cluster. Finally, for each vehicle v and its associated route, M axS is the maximum number of stops (suppliers in the route), M axD is the maximum number of docks supplied, and the distance between two consecutive stops cannot exceed M axQ.

4. 1

 1 Phase 1 : Clustering For each subset S ′ ⊆ S such that 1 ≤ |S ′ | ≤ M axC, the corresponding cluster c is validated (c ∈ C) if the associated subgraph G ′ c is connected.

  TAB. 1 -Computational comparison between the two models.

	Type	I 1 S 2 D 3 N 4	C 5	Original model T 6 ̸ O 7 ̸ S 8	Enhanced model T ̸ O ̸ S	∆ T	9	∆ ̸ O 10 ∆ ̸ S 11
		1 10 4 37	293	28.71 0	0	15.05 0	0	47.58	0	0
		2 12 3 51	688 3541.20 2	0	144.66 0	0	95.91 100	0
		3 13 2 55	846	89.78 0	0	28.74 0	0	67.99	0	0
	Small	4 14 5 91 1470 7651.19 9	0	649.31 0	0	91.51 100	0
		5 14 4 46	285 3993.76 3	0	215.37 0	0	94.61 100	0
		6 15 7 86 1776 2045.46 1	0	260.96 0	0	87.24 100	0
		7 15 5 104 1939 519.66 0	0	130.47		0	74.89	0	0
		8 19 4 171 5035	TLI 12 8 3505	TLI	0 831	0	100 76.3
		9 25 8 276 9617	TLI 11 2225	TLI 11 823	0	0	63.0
	Medium	10 33 10 217 35243	TLI	4 15587 TLI	3 7646	0	25 50.9
		11 34 5 520 51887	TLI 12 31808 TLI 12 27192	0	0	14.5
		12 34 6 553 52932	TLI	3 29726 4092.17 0	0	43.16 100 100
	Large	13 50 9 687 118788 TLI 12 76145 TLI 11 69041 14 54 10 721 128147 TLI 6 85793 TLI 1 70222	0 0	8.33 9.3 83.33 18.1

† Institute of Engineering Univ. Grenoble Alpes

Instance number,

Num. suppliers,

Num. receiving docks,

Num. edges,

Num. of clusters,

Computational time in seconds,

Num. of clusters exceeded TLC,

Num. of clusters unsolved within TLI,

Percentage decrease in computational time = (Worst time -Best time ) / Worst time ×

100, 10 Percentage decrease in the number of not optimal clusters, 11 Percentage decrease in the number of unsolved clusters, 12 Instances exceeded TLI.supplier the optimal transportation strategy, and for every vehicle the shortest elementary paths, and its optimal loading plan. Hence, another approach that first directly generates interesting routes, and then choose the optimal ones has to be considered.


Keywords: Electromagnetic field, Modeling, FDTD, Lightning strike, Conductivity of the ground

Contribution to the analysis of the electromagnetic field radiated by lightning taking into account the conductivity of the ground

Abdelghani Chahmi, Mouhoub Birane

I. INTRODUCTION

In recent years, several models of the return arc, with different degrees of complexity, have been developed in order to quantify its electromagnetic radiation. One of the major difficulties in modeling the lightning channel is that the current can only be measured at the base of the channel; however, in order to determine the radiated electric and magnetic fields, it is necessary to know the distribution of the current along the channel, an important property that makes the difference between the models of the proposed return arc on the spatial and temporal distribution of the current. along the lightning channel i (z', t) [START_REF] Bourgois | Modeling electromagnetic disturbances generated on a telecommunications network by a lightning type approval[END_REF][START_REF] Habri | Calculation of the Electromagnetic Field Radiated by Lightning in the Presence of a Laminate Soil[END_REF][START_REF]2D and 3D modeling of electric arcs in argon at atmospheric pressure with the consideration of the thermal and electrical arc-electrode coupling and the influence of metallic vapors[END_REF][START_REF] Hermant | Atmospheric electricity and lightning[END_REF]. Most recent researches are conducted for the computation of the electromagnetic field have been conducted towards the Heidler model and the biexponential model. For this our approach in this work is that we adopt the model MTLL (Modified Transmission Line) also named: modified engineers model proposed by Nucci and Rachidi [START_REF] Ianovici | Electromagnetic Compatibility[END_REF] and approved by convincing results by several authors in different works [START_REF] Rachidi | Electromagnetic Effects of Lightning on Power Lines, EMC Electromagnetic Compatibility, Course Duplication, LRE[END_REF]. The passage of the lightning current from the base of the channel to the cloud is the main source of the electromagnetic field. The calculation of the electromagnetic field requires:

 An analytical expression of the shape of the lightning current measured at the base of the channel.  Spatio-temporal modeling of the current distribution in the Lightning Channel.

Among these EDP algorithms, it is the FDTD method that has the largest number of application domains at the level of the structures that can be modeled as at the level of the frequency domains explored. This method was proposed by Yee in 1966 [START_REF] Bermudez-Arboleda | Lightning currents and electromagnetic fields with strokes to elevated strike Object[END_REF], then more or less forgotten until 1975. That year, Allen Taflove [START_REF] Rachidi | Electromagnetic Effects of Lightning on Electrical Lines, EMC Electromagnetic Compatibility, Course Mimeography, LRE[END_REF] - [START_REF] Rachidi | Response of multiconductor[END_REF], who could not solve a problem of penetration of microwaves on the human eye (many cataracts had been observed on radar technicians during the Second World War), decides to use Yee's algorithm. Despite the success of its modeling and the various improvements it brings to the method, it falls back into oblivion, no doubt, for lack of sufficient computer means (only the American defense continues to use it). By the end of the 1980s, the computer revolution made it possible to generalize the use of the FDTD method [10] [11] [12]. This success is mainly due to the simplicity of the basic principle of the method. It is a question of implementing the Maxwell equations and applying them on the whole modeling space for any moment of simulation.

Algorithm for calculations with matlab

II. THE APPLICATION UNDER STUDY

A. Geometric model chosen for the calculation of the Electromagnetic field

The lightning channel is considered a one-dimensional vertical antenna of height H. The return arc propagates vertically from the ground with a velocity v. It is traversed by a current whose spatiotemporal distribution i (zc; t) determines the electromagnetic field at any point in space. Moreover, the cylindrical symmetry of the problem makes the components of the electromagnetic field independent of the azimuthal angle. The field will be defined by:  the vertical electric field Ez ;  the radial electric field Er;* ;  the radial magnetic field H_φ.

B. Analytical representation of the arc current back to the base of the lightning channel

The shape of the discharge current in time. The theory shows, and the experiment verifies, that the discharge current has a temporal evolution of the form :

            

III. MODIFIED TRANSMISSION LINE MODELS WITH LINEAR ATTENUATION (MTLL)

In this model the current is attenuated according to the height

I (z', t) =I (0, t-z'/c) (1-z'⁄ (H tot ) ∀ z' ≤vt (3)
I (z', t) = 0 ∀ z' ≤vt
Where H_tot is the total height of the lightning channel.

A. Time Domain

The study of disturbances generated by lightning directly involves us in the field of electromagnetic compatibility, the ultimate goal of which is to make compatible the functioning of a sensitive electrical / electronic system in a disturbed electromagnetic environment while respecting the 03 criteria. following:

• No interference with other systems.

• No susceptibility to emissions from other systems.

• No interference of the system with itself.

Dans un problème de CEM, on trouvera deux éléments: une source de perturbation (la foudre), un moyen de couplage (conduction, rayonnement).

In an EMC problem, we will find two elements: a source of disturbance (lightning), a means of coupling (conduction, radiation). For the general case of a finite conductivity sol, the application of the Maxwell equations to the geometry adopted above makes it possible to obtain the equations of the electromagnetic field. These equations involve the Sommerfeld integrals whose numerical evaluation is a delicate task. Assuming a perfectly conducting ground, simpler expressions of the vertical and horizontal components of the electric field and the azimuthal component of the magnetic field can be developed. These expressions are written in the time domain according to M.A Uman [START_REF] Tabara | Electromagnetic Effects of Maneuvers and Lightning in Posts Under Metal Envelope[END_REF] [14]:

                  H H I r dz d c R z i R z z r t z r E 0 ' ' 5 ' 0 , 3 4 1 , ,    + ' ' 3 2 2 ' 4 ' , ( ) , ( ) ( 3 dz t c R t z i R c r dz c R t z i cR z z H H H H                                H H I Z dz d c R z i R r z z r t z r E 0 ' ' 5 2 2 ' 0 , 3 4 1 , ,    + ' ' 3 2 ' ' ' 4 2 2 ' , ( ) ( ) , ( ) ( 2 dz t c R t z i R c z z r dz c R z i cR r z z r H H H H                                 H H H H dz t c R t z i cR r dz c R z R r t z r B ' ' 2 ' ' 3 0 , ( ) , ( 4
1 , ,    R= (6)
Er, Ez: the Horizontal field and the vertical field; Hφ is the azimuthal magnetic field.

The three terms in Equations ( 4) and ( 5) represent the Electrostatic, Induction and Radiation Fields, respectively, while the first term in Equation ( 6) represents the field of induction and the second is the radiation field. Whereas the vertical electric field and the magnetic field are practically independent of the height z of the observation point (for z varying from 0 to about 30 m), the horizontal electric field, which is zero at ground level, increases almost linearly. with z [15] [START_REF] Gary | Lightning: from ancient mythology to modern research[END_REF].

V. CONCLUSION

The purpose of this paper was to develop a computation code of the electromagnetic field radiated by the arc phase in return for a thunderbolt. The determination of the components of the electromagnetic field requires knowledge of the spatiotemporal distribution of the lightning current along the channel. Allowing low temporal 3D problem solving. And for its speed of calculation by avoiding the use of solid numerical methods (FDTD) to improve the calculation efficiency, which often require computation time and important memory space.

Fig. 1 2 1

 12 Fig.1 Simplified organizational chart of the FDTD program

Fig. 2

 2 Fig.2 Study domain geometry

( 1 )

 1 The simulated current is represented by the expression developed byNucci et al. [47] :

Fig. 3

 3 Fig.3 Current at the base of the lightning channel (a) calculated by the measured FDTD (b) methodAt the end of this comparison, we can conclude that the simulation results (Current at the base of the channel) are in agreement with the experimental results (figure above). To calculate the vertical electric field, we adopt the MTLL model to represent the current of the arc back. The speed of the return arc is assumed to be 150 m / μs. The curves of the calculated electric field are shown in Figure5.

Fig. 5 Fig. 7 Fig. 8 Fig. 9

 5789 Fig. 5 vertical electric field 15 m of lightning canal (a) Measured adapted the [78] and (b) calculWe obtain a passage zone between the measured electric field lower than that obtained by the FDTD method FIG. horizontal electric field (a) the FDTD-method, (b) adopted from .