A S Tchakoutio Nguetcho
email: nguetchoserge@yahoo.fr

Jibin Li
email: jibinli@gmail.com

J M Bilbault
email: bilbault@u-bourgogne.fr

Bifurcations of Phase Portraits of a Singular Nonlinear Equation of the Second Class

Keywords: Hamiltonian system, Nonlinear wave equation, non-convex interparticle interactions, deformability of the substrate potential, Solitary wave solution, Kink wave solution, Periodic wave solution, Breaking wave solution

The soliton dynamics is studied using the Frenkel Kontorova (FK) model with nonconvex interparticle interactions immersed in a parameterized on-site substrate potential. The case of a deformable substrate potential allows theoretical adaptation of the model to various physical situations. Non-convex interactions in lattice systems lead to a number of interesting phenomena that cannot be produced with linear coupling alone. In the continuum limit for such a model, the particles are governed by a Singular Nonlinear Equation of the Second Class. The dynamical behavior of traveling wave solutions is studied by using the theory of bifurcations of dynamical systems. Under different parametric situations, we give various sufficient conditions leading to the existence of propagating wave solutions or dislocation threshold, highlighting namely that the deformability of the substrate potential plays only a minor role.

Introduction

Simplicity and universality are two key requirements of a good physical model.

Universal models, which can be used to describe a variety of different phenomena, are very rare and yet are of major importance. They have been proved useful at several scales and moreover possess an educational value. A simple example of such models is the Frenkel Kontorova (FK) model [START_REF] Braun | The Frenkel-Kontorova Model, Concepts, Methods, and Applications[END_REF]. The FK model has become very popular in many niches of solid state and nonlinear physics. They invented their model in order to describe the motion of a dislocation in crystals. Meanwhile, the FK model has also become a model for an adsorbate layer on the surface of crystal, for ionic conductors, for glassy materials, for charge-density-wave (CDW) transport, and for chains of coupled Josephson junctions. Especially, a number of current researches, such as sliding friction [START_REF] Yuan | Nonlinear Dynamics of a Sliding Chain in a Periodic Potential[END_REF][START_REF] Vanossi | Driven dynamics of simplified tribological models[END_REF] , heat conduction [START_REF] Li | Negative differential thermal resistance and thermal transistor[END_REF][START_REF] Li | Interface Thermal Resistance between Dissimilar Anharmonic Lattices[END_REF][START_REF] Hu | Asymmetric Heat Conduction in Nonlinear Lattices[END_REF] , chaos control and nonlinear coupled pendulum [START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF] are also carried out using the FK model.

The standard FK model describes a chain of coupled atoms subjected to an external periodic potential and its Hamiltonian is defined as

H = T + U. (1)
T and U are the kinetic and potential energies respectively. U consists of interparticle interactions V c that take into account the linear and nonlinear couplings between the nearest neighbors of the chain and the substrate external potential V e along the chain with spatial period a. b stands for the natural spacing of the unperturbed chain. The contribution of the particle i to potential energy U is then defined as follows:

U (x i ) = V c (r i ) + V e (x i ) , 1 ≤ i ≤ n (2)
where r i = x i+1x ib + a represents the distance between the nearest neighbors.

One of the basic restrictions of this traditional model adopted originally by the authors, which makes it applicable for small lattice misfits only, is the purely elastic interaction between neighboring atoms as a substitute for the real inter-atomic forces. In addition, in real physical systems, the shape of the substrate potential can deviate from the standard (sinusoidal or rigid) one and this may affect strongly the transport properties of the system.

As a better approach for modeling real systems, many researchers have considered the FK model while either modifying the external periodic potential [START_REF] Braun | The Frenkel-Kontorova Model, Concepts, Methods, and Applications[END_REF] or modifying interactions between atoms [START_REF] Chacón | Controlling Chaotic Solitons in Frenkel-Kontorova Chains by Disordered Driving Forces[END_REF][START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF][START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF][START_REF] Nguetcho | Kink compactons in models with parametrized periodic double-well and asymmetric substrate potentials[END_REF][START_REF] Nguetcho | Soliton patterns and breakup thresholds in hydrogen-bonded chains[END_REF]. For realistic anharmonic interactions, such as those of Toda, Morse, Markov and Trayanov, besides the position of the minimum there exists a second characteristic length at which the tensile strength of the bond reaches its maximum, that is the inflection point r inf . This makes the behavior of such systems more complex. The mathematical challenges of solving the problem analytically increase since the equations are no more single valued. It has been reported in a numerical study of the discrete FK model with anharmonic interactions (Toda, Morse) [START_REF] Milchev | The effect of anharmonicity in epitaxial interfaces: I. Substrate-induced dissociation of finite epitaxial islands[END_REF] that beyond some critical values of the independent parameters (the natural lattice mismatch and/or amplitude of the external periodic potential), these equations have no solution which was interpreted as a disintegration of the system . This work aims to consider a non-convex pair potential V c [START_REF] Milchev | The effect of anharmonicity in epitaxial interfaces: I. Substrate-induced dissociation of finite epitaxial islands[END_REF][START_REF] Braun | Nonlinear dynamics of the Frenkel-Kontorova model[END_REF][START_REF] Milchev | Solitary waves in a frenkel-kontorova model with non-convex interactions[END_REF] defined as :

V c (r) = V 0 B r r 0 2 1 + B r r 0 2 (3)
with r 0 = a 2π

• B changes the width of the potential well with depth V 0 .

It is a single non-convex even potential with a single minimum and infinitely versus the nearest neighbors distance r r 0 in Fig. 1, is infinitely differentiable and supposed to bear all qualitatively important features of the problem under the influence of a nonlinear periodic deformable substrate V e .

For this purpose, we used an important type of on-site potential that describes many realistic situations (e.g., the dynamics of atoms adsorbed on crystal surfaces, Josephson jonctions, charge density wave condensate, • • • ) [START_REF] Braun | The Frenkel-Kontorova Model, Concepts, Methods, and Applications[END_REF], that is:

V e (x i ) = ε s sin 2 πx i a σ 2 + (1 -σ 2 ) sin 2 πx i a , ( 4
)
where σ is related to the coefficient s introduced by Peyrard and Remoissenet [START_REF] Braun | The Frenkel-Kontorova Model, Concepts, Methods, and Applications[END_REF] by σ = 1 + s 1s . The parameter ε s is the parameter that controls or adjusts the strength of the energy barriers of the substrate potential. In adsystem, ε s is the activation energy for diffusion of an isolated adatom. At σ = 1 the potential V e (x i , σ) has a sinusoidal shape, while at σ < 1, broad wells separate narrow barriers, and at σ > 1, deep narrow wells separate broad gently sloping barriers (see Fig. 2). The next section will be devoted to the presentation of the equations and integrals of motion. The phase portraits will be presented in section 3, while the main behaviors of the wave solutions will be studied in ( §4), followed by a summary and some conclusion remarks.

The equations of motion

The Hamiltonian for such a system is defined as:

H = A n i    1 2
dφ i dt 2 + C 2 0 2b 2 (φ i+1 -φ i -P ) 2 1 + B(φ i+1 -φ i -P ) 2 + + ω 2 0 sin 2 φ i 2 σ 2 + (1 -σ 2 ) sin 2 φ i 2            , (5)
where The parameters C 0 and ω 0 are the characteristic velocity and frequency respectively, and are related to the non-convex dimensionless constant B (the parameter that controls the strength of the nonlinear coupling) and the amplitude of the substrate potential

φ i = 2πx i a is
ε s through C 2 0 = 2BV 0 b 2 A and ω 2 0 = 2π a 2 ε s m •
Finally, m represents the mass of atoms.

In the continuum limit, with dimensionless variables X = x b and T = ω 0 t, when φ i (t) is replaced by φ (X, T ), the particles are governed by the following Euler-Lagrange equation:

∂ 2 φ ∂T 2 - C 2 0 b 2 ω 2 0          1 -3B ∂φ ∂X -P 2   1 + B ∂φ ∂X -P 2   3          ∂ 2 φ ∂X 2 + 1 2 σ 2 sin (φ) (σ 2 -1) sin 2 φ 2 -σ 2 2 = 0.
(6) In extreme cases B → 0 and σ → 1, equation ( 6) leads to the wellknown Sine Gordon equation. Using the independent propagating variable u = X -vT , where v is a constant velocity, (6) yields:

         1 -3B dφ du -P 2   1 + B dφ du -P 2   3 -V 2          d 2 φ du 2 = 1 2 λσ 2 sin (φ) (σ 2 -1) sin 2 φ 2 -σ 2
where V = vbω 0 C 0 is a more suitable dimensionless expression to determine the velocity of the travelling waves.

d 0 = C 0 ω 0 is the characteristic length of the system, while λ = b d 0 2
denotes the scaled amplitude of the periodic potential and measures the effective depth of the substrate potential of the system. The study of Eq. ( 7) can be best driven in the phase plane φ, y = dφ du . Eq [START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF] can then be rewritten as:

dφ du = y, dy du = 1 2f (y) λσ 2 sin (φ) (σ 2 -1) sin 2 φ 2 -σ 2 2 , (8)
with

f (y) = 1 -3B(y -P ) 2 [1 + B(y -P ) 2 ] 3 -V 2 . ( 9
)
If the function f (y) has at least a real zero at y = y 0 , ( 8) is a singular nonlinear system of the second class (see [START_REF] Jibin | Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions[END_REF][START_REF] Jibin | Travelling wave solutions in a model of the helix polypeptide chains[END_REF][START_REF] Jibin | On the Breaking Property for the Second Class of Singular Nonlinear Traveling Wave Equations[END_REF]). By using the transformation du = f (y)dξ when y = y 0 , system (8) becomes the associated regular system as follows:

dφ dξ = yf (y), dy dξ = 1 2 λσ 2 sin (φ) (σ 2 -1) sin 2 φ 2 -σ 2 2 ≡ Q(φ). (10)
Systems ( 8) and ( 10) have the same first integral defined as:

H (φ, y) = (1 -3BP 2 ) y 2 + 4BP y 3 -By 4 (1 + BP 2 ) 1 + B(y -P ) 2 2 -V 2 y 2 - 2λsin 2 φ 2 σ 2 + (1 -σ 2 ) sin 2 φ 2 = h, (11)
where h is a constant. System ( 10) is 2π-periodic in φ. Hence, the state (φ, y) can be viewed, for the main properties, on a phase cylinder S 1 × R, where S 1 = ]π, π] and -π is identified with π (see Fig. 3).

Without loosing generality, we will assume in the following that σ is fixed, which leads to consider the systems ( 8) and [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF] as four-parameter systems depending on (B, V, P, λ). The role of σ will be discussed later. Obviously, y = G(φ) can not be derived explicitly from [START_REF] Nguetcho | Kink compactons in models with parametrized periodic double-well and asymmetric substrate potentials[END_REF]. Therefore, it is not possible to obtain explicit solutions by solving equation [START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF]. However with the qualitative theory of differential equations, Eq. ( 7) can be solved and all related informations to its solutions can be derived, as well as phase portraits [START_REF] Saha | Bifurcation of travelling wave solutions for the generalized KP-MEW equations[END_REF][START_REF] Yana | The bifurcation and exact travelling wave solutions for the modified BenjaminBonaMahoney (mBBM) equation[END_REF].

Let Y = (y -P ) 2 . For V = 0, to find the zeros of function f (y) is equivalent to find the roots of the cubic polynomial

F (Y ) = Y 3 + 3 B Y 2 + 3 B 2 1 + 1 V 2 Y + 1 B 3 1 - 1 V 2 ≡ Y 3 + a 2 Y 2 + a 1 Y + a 0 . (12) Let q = 1 3 a 1 - 1 9 a 2 2 , r = 1 6 (a 1 a 2 -3a 0 ) - 1 27 a 3 2 .
Then, the discriminant S =

q 3 + r 2 of cubic equation F (Y ) = 0 is S = 1 + 4V 2 V 6 B 6 > 0. It follows that F (Y ) has only one real root Y = Y 0 .
If and only if a 0 < 0, i.e., 0 < V < 1, the only real root Y 0 of cubic polynomial F (Y ) is positive. Then f (y) has two real roots y 1 and y 2 .

Therefore, for any P = 0, looking for the equilibrium points of system (10),

we have the following conclusions:

(1) For the static (V = 0) or subsonic velocity (0 < V < 1) cases:

For V = 0, f (y) becomes zero for y 1 = P + r inf , and

y 2 = P -r inf , while if 0 < V < 1, y 1 = P - √ Y 0 and y 2 = P + √ Y 0 .
Then, for (0 ≤ V < 1), there exist six equilibrium points of system [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF] in the phase cylinder, at O(0, 0), A(π, 0), B 1 (0, y 1 ), B 2 (π, y 1 ), C 1 (0, y 2 ) and C 2 (π, y 2 ). On the straight lines y = y 1 and y = y 2 , dy du , defined by system (8), becomes infinite. These two straight lines are then two singular straight lines for system (8).

(2) For the case of sonic velocities (V = 1) .

There exist four equilibrium points of system [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF] at O(0, 0), A(π, 0), B 1 (0, P ) and B 2 (π, P ). On the straight lines y = P , dy du , defined by system (8), becomes infinite. Thus, the straight line y = P is a singular straight line for system [START_REF] Chacón | Controlling Chaotic Solitons in Frenkel-Kontorova Chains by Disordered Driving Forces[END_REF].

(3) For the case of supersonic velocities

(1 < V < ∞) .
There exist two equilibrium points of system [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF] at O(0, 0) and A(π, 0).

In this case, because f (y) has no real zero, system ( 8) is a regular dynamical system.

We notice that

f (0) = - V 2 F (P 2 )B 3 (1 + BP 2 ) 3 • (a) When 1 < V < ∞, F (P 2 ) > 0 and f (0) < 0. (b) When 0 < V < 1, if P 2 < Y 0 , then f (0) > 0, while if P 2 > Y 0 , then f (0) < 0.
Let M(φ i , y j ) be the matrix of the linearized approximation of system (10) at the equilibrium point (φ i , y j ). Then, we have

detM(φ i , y j ) = -Q ′ (φ i )[y j f ′ (y j ) + f (y j )],
where

f ′ (y) = 12B (y -P ) B(y -P ) 2 -1 B(y -P ) 2 + 1 4 ,
and T race M (φ i , y j ) = 0. Thus,

J(0, 0) = detM(0, 0) = - λ 2σ 2 f (0), J(π, 0) = detM(π, 0) = λσ 2 2 f (0), J(0, y j ) = detM(0, y j ) = - λ 2σ 2 y j f ′ (y j ), J(π, y j ) = detM(π, y j ) = λσ 2 2 y j f ′ (y j ), j = 1, 2.
The theory of planar dynamical systems gives the following results for an equilibrium point of a planar integrable system:

• if J < 0, then the equilibrium point is a saddle point;

• If J > 0 then it is a center point;

• if J = 0 and the Poincaré index of the equilibrium point is 0, then this equilibrium point is a cusp [START_REF] Jibin | On the Breaking Property for the Second Class of Singular Nonlinear Traveling Wave Equations[END_REF][START_REF] Saha | Bifurcation of travelling wave solutions for the generalized KP-MEW equations[END_REF].

For the Hamiltonian values defined by [START_REF] Nguetcho | Kink compactons in models with parametrized periodic double-well and asymmetric substrate potentials[END_REF], we have

h 0 = H(0, 0) = 0, h 1 = H(π, 0) = -2λ, h 2,3 = H(0, y 1,2 ) = (1 -3BP 2 )y 2 1,2 + 4BP y 3 1,2 -By 4 1,2 (1 + BP 2 ) (1 + B(y 1,2 -P ) 2 ) 2 -V 2 y 2 1,2 , h 4,5 = H(π, y 1,2 ) = (1 -3BP 2 )y 2 1,2 + 4BP y 3 1,2 -By 4 1,2 (1 + BP 2 ) (1 + B(y 1,2 -P ) 2 ) 2 -V 2 y 2 1,2 -2λ.
3 The phase portraits of the model Note first that the case P and -P are symmetric with respect to the φ-axe, which allows us to restrict our study to P ≥ 0.

In order to solve the four-parameter systems ( 8) or [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF], and to draw the portrait phases, we will discuss according the different cases V = 0, 0 < V < 1, V = 1 and V > 1. Using the above informations to perform qualitative analysis, we also represent in phase portraits the corresponding singular straight lines. They intersect transversally some families of orbits.

We should emphasize that when these singular straight lines of system [START_REF] Chacón | Controlling Chaotic Solitons in Frenkel-Kontorova Chains by Disordered Driving Forces[END_REF] exist, the vector fields defined by system (8) and system [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF] are different. If an orbit of system (8) intersect a singular straight line y = y j , that is f (y j ) = 0 in a particular point (φ 0 , y j ), where φ 0 is not a multiple of π, then at this point, the second derivative d 2 φ du 2 of φ(u) becomes infinite as shown by equation [START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF]. The direction of vector field defined by system (8) changes rapidly to the inverse direction of the vector field defined by system [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF]. This lets the wave solution φ(u) of (7) associated with system (8) degenerate into a breaking solution, meaning physically a dislocation of the particle lattice, as we will show now.

Considering first the static case V = 0, the coupling potential V c can be rewritten as

i V c (r i ) = i V 0 B (φ i+1 -φ i -P ) 2 1 + B(φ i+1 -φ i -P ) 2 • ( 13
)
Deriving V c versus φ i gives the resulting force from the lattice on particle number i, namely

F i = -2V 0 B    φ i+1 -φ i -P 1 + B(φ i+1 -φ i -P ) 2 2 - φ i -φ i-1 -P 1 + B[φ i -φ i-1 -P ) 2 2    (14)
Using the continuous media approximation φ i+1φ i → ∂φ ∂X gives

F i = -2V 0 B ∂ 2 φ ∂X 2          1 -3B ∂φ ∂X -P 2   1 + B ∂φ ∂X -P 2   3          (15)
This result shows that when ∂φ ∂X = P ± 1 √ 3B , the resulting force on particle i from the lattice vanishes. The attraction on particle i from the left neighbors is exactly balanced by the one from the right neighbors, resulting on a breaking of the lattice cohesion.

Note that the particle remains submitted to the substrate potential V e (x i ) (see Eq. ( 4)) but the links to their neighbors are broken.

Finally, with y = ∂φ ∂X = dφ du , (we recall that u = X -vT = X for V = 0), the lattice submits a dislocation when f (y) = 0, that is on the singular straight lines.

When V = 0, the coupling potential must be replaced by an effective one, taking into consideration the propagative variable u = X -vT . Then, dislocations occur when the slope y = dφ du reaches one of the roots of f (y) = 0, when they exist. In the phase portraits, this corresponds to any intersection of the orbits with the singular straight lines, except at the minima of the potential V e (φ i ).

(i) Phase portraits in the static case (V = 0) with P = 0

For a given B, we have

y 1 = -y 2 = - 1 √ 3B , h 2 = h 3 = 1 8B , h 4 = h 5 = 1 8B -2λ. Hence, when λ = 1 16B
, it follows that h 4 = h 5 = h 0 = 0.

When λ increases, the cases

h 1 < 0 < h 4 < h 2 , h 1 < h 4 = 0 < h 2 and
h 1 < h 4 < 0 < h 2 are successively obtained, as shown in Fig. 4. . The bifurcation of phase portraits of system (10) for V = 0, P = 0, B = 0.9,

(a) λ < 1 16B , (b) λ = 1 16B , (c) λ > 1 16B and (d) λ > 1 16B
but σ is now 3.

(ii) Phase portraits of the subsonic velocity case (0 < V < 1)

In this case, there exist a fixed positive root Y 0 of F (Y ).

If P = 0, y 1,2 = ± √ Y 0 and h 2 = h 3 = Y 0 (1 -BY 0 ) 4(1 + BY 0 ) 2 -V 2 Y 0 , h 4 = h 5 = Y 0 (1 -BY 0 ) 4(1 + BY 0 ) 2 -V 2 Y 0 -2λ. Let λ b = 1 2 Y 0 (1 -BY 0 ) 4(1 + BY 0 ) 2 -V 2 Y 0 . It is easy to see that respectively the case λ < λ b (λ = λ b , then λ > λ b ) leads to h 1 < 0 < h 4 < h 2 (respectively h 1 < h 4 = 0 < h 2 , then h 1 < h 4 < 0 < h 2 )
, while these different phase portraits are shown in Fig. 5. If P = √ Y 0 , then y 1 = 0, y 2 = 2Y 0 , F (P 2 ) = 0. Consequently f (0) = 0 and the two equilibrium points O(0, 0) and A(π, 0) are cusps.

When P varies from 0 to ∞, there exists a value P = P b such that h 2 = h 5 . On Fig. 6, we present the phase portraits highlighting the bifurcations obtained when the parameter P increases.

(iii) Phase portraits in the sonic velocity case (V = 1)

We have now Y 0 = 0, and y 1 = y 2 = P . The two equilibrium points B 1 (0, P ) and B 2 (π, P ) are cusps, and h 2,3 = -BP 4 1 + BP 2 . Thus, for a given parameter pair (B, λ), h 1 = h 2,3 when P = P a ≡ λ + λ 2 + 2λ B ;

we see in Fig. 7 the phase portraits for P < P a , P = P a and P > P a .

(iv) Phase portraits in the supersonic velocity case (V > 1)

12 x phi (f) P b < P Fig. 6. The bifurcation of phase portraits of system (10) for 0 < V < 1 and for B = 0.9, σ = 1.5, λ = 0.03.

The phase portraits are simple without singular lines. As shown on Fig. 8, no bifurcation occurs when P or λ are varied.

In this section, we have studied the main characteristics of the phase portraits for the systems ( 8) and [START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF]. We notice at this stage that the deformability parameter σ acting on substrate potential (4) does not play an important role.

In fact, it only slightly and qualitatively changes the shapes of the trajectories

-3 -2 -1 0 1 2 3 y -3 -2 -1 1 2 3 x phi A' A O (a) P = 0 -3 -2 -1 0 1 2 3 y -3 -2 -1 1 2 3 x phi B 1 A' A O B' 2 B 2 M φ 0 (b) 0 < P < P a -4 -2 0 2 4 y -3 -2 -1 1 2 3 x phi B 1 A' A B' 2 B 2 (c) P = P a , h 1 = h 2 -4 -2 2 4 y -3 -2 -1 1 2 3 x phi A A' B 1 B' 2 B 2
(d) P a < P Fig. 7. The bifurcation of phase portraits of system (10) for V = 1 (sketch for any given parameter set (B, λ)). In this section, we consider the solutions of equation ( 7) giving the wave profile φ(u) and corresponding to some trajectories of the phase portraits of section 3. For a sake of simplicity, we will consider the velocity as a decreasing parameter.

(I) Case of supersonic velocities: V > 1. (see Fig. 8) .

In this case, the function f (y) has no real zero. Thus, systems ( 8) and ( 10) lead both to the phase portraits of Fig. 8 for any kind of parameter set (B, P, λ).

(i) Corresponding to the family of closed orbits defined by H(φ, y) = h, h ∈ ] -2λ, 0[, there exists a family of oscillating periodic solutions of equation ( 7), as depicted in Fig. 9(a).

- (II) Case of sonic velocities (V = 1, see Fig. 7).

(II-1) If P = 0 (see Fig. 7(a)), there exist only the equilibrium points O, A and A ′ , and the families of solutions for Eq. ( 7) are the same as the ones reported in § 4(I).

(II-2) If 0 < P < P a (see Fig. 7(b)) or P = P a (Fig. 7(c)) we have

h 4 < h 1 < h 2 < 0 or h 4 < h 1 = h 2 < 0 respectively.
• Corresponding to the family of closed orbits defined by H(φ, y) = h, h ∈ ]h 2 , 0[, there exists a family of oscillating periodic solutions of Eq. ( 7) similar to those shown on Fig. 9(a).

• Corresponding to the family of closed orbits defined by

H(φ, y) = h, h ∈ ]h 1 , h 2 ]
, there exists a family of oscillating periodic solutions of Eq. [START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF]. But these orbits intersect the singular straight line y = P first for -π < φ < 0. For example, the orbit starting from the point A ′ (φ = -π, y = 0) in Fig. 7 • Considering the two heteroclinic orbits linking the equi-librium points A and A ′ [H(φ, y) = h 1 ], the upper one intersects also the singular straight line y = P first for -π < φ < 0 (behavior similar to the one depicted in Fig. 10) while the lower one is a standard anti-kink wave (similar to Fig. 9(c)).

• Corresponding to H(φ, y) < h 1 , there exist different families of trajectories. For the lower ones, the profiles φ(u) resemble to the curves on Fig. 9(d) while the upper ones intersect the singular straight line y = P and correspond to breaking wave solution as on Fig. 10.

(II-3) P > P a . In this case h 4 < h 2 < h 1 < 0. In addition to the previous case, we notice the presence of a heteroclinic orbit joining the points A ′ and A, corresponding to a kink wave solution φ(u) similar to the curve on Fig. 9(b).

(III) Case of subsonic velocities (0 < V < 1).

(III-1) Let us consider first the case P = 0 (see Fig. 5 (a,b,c)).

Some families of orbits do not intersect the singular straight lines y = y 1 = √ Y 0 and y = y 2 = -√ Y 0 . There are limit cycles around the equilibrium points A ′ or A. The corresponding wave solution φ(u) of Eq. ( 7) is similar to those depicted on Fig. 9(a), with only a difference concerning the ordinate values.

We notice even, when λ < λ b (Fig. 5 (III-2) 0 < V < 1 and P = 0 (see Fig. 6(a)→ 6(f)).

In this case, almost all orbits of systems ( 8) and ( 10

Conclusion

In summary, we have studied the dynamic behaviors of traveling wave solutions of the generalized Frenkel-Kontorova model. Our model extends previous lattice soliton-bearing one-dimensional models discussed in the literature, by considering the contribution of anharmonic couplings. Such couplings are achieved through a non-convex interaction potential and a deformable substrate potential. In the continuum limit, a Singular Nonlinear Equation of the Second Class governs the displacement of the particles. Under different parametric conditions, we have shown that various sufficient conditions lead to the existence of a rich diversity of solitary and periodic wave solutions. Another effect is related to an opening of phase trajectories of the system taking place beyond some threshold values of the lattice parameters, and depending on the velocity of the soliton patterns. Analytical expression for the breakdown threshold has been derived although it is quite indifferent of the deformability of the substrate (σ being not a bifurcation parameter). The deformability of substrate in such model gradually has only an influence on the width of the wave solutions. The parameter P , as well as other defects, may locally influence the breakdown threshold and thus plays a major role with respect to nonlinear excitations in such systems.

The bifurcations and dynamic behaviors of traveling wave solutions can be suitable for another nonlinear wave equation. Although it is tremendously challenging to analytically determine the explicit travelling wave solutions, our preliminaries results suggest that they can be combined with a bifurcation method in order to provide an insight of at least implicit solutions. The findings reported in this paper might be of critical impacts in several areas of applied sciences. For instance, they can help to explain the formation of cracks originating from dislocations that are observed in semiconductor heterostructures [START_REF] Braun | The Frenkel-Kontorova Model, Concepts, Methods, and Applications[END_REF][START_REF] Braun | Nonlinear dynamics of the Frenkel-Kontorova model[END_REF][START_REF] Milchev | Solitary waves in a frenkel-kontorova model with non-convex interactions[END_REF][START_REF] Malomed | Interaction of dislocations with a local defect in an atomic chain with a nonconvex interparticle potential[END_REF]. Moreover, they can help to better understand the break up (rupture) of a stretched polymer chain by pulling and its relation to soliton destruction [START_REF] Paturej | Thermal degradation of unstrained single polymer chain: Non-linear effects at work[END_REF].

Fig. 1 .

 1 Fig. 1. Plot of the coupling potential V c (r)

12 Fig. 2 .

 122 Fig. 2. Substrate potential V e (x i ) at different values of parameter σ

  the scalar dimensionless displacement of particles, P = 2π b a -1 stands for the natural lattice incompatibility between substrate and overlayer, while the factor A = m a 2π 2 sets the energy scale.

Fig. 3 .

 3 Fig.3. The phase cylinder S 1 × R of system[START_REF] Xu | Influence of the bond defect in driven Frenkel-Kontorova chains[END_REF].

Fig. 4

 4 Fig.[START_REF] Li | Negative differential thermal resistance and thermal transistor[END_REF]. The bifurcation of phase portraits of system (10) for V = 0, P = 0, B = 0.9,

Fig. 5 .

 5 Fig. 5. The bifurcation of phase portraits of system (10) for P = 0, 0 < V < 1. (Parameters: B = 0.9, V = 0.2, σ = 1.5, (a) λ = 0.025, (b) λ = λ b = 0.03118632263, (c) λ = 0.05.

  P = P b , h 2 = h 5

Fig. 8 .4

 8 Fig.8. Example of phase portraits of system (10) for V > 1 (sketch for any given parameter set (B, P, λ)).

Fig. 9 .

 9 Fig. 9. Solution wave φ(u) of equation (7) corresponding to the case V > 1. The abscisse scale is arbitrary. [The initial values for the pair (φ, y) are respectively: a) (-2.4, 0). b) (-π, 0). c) (π, 0). d) (-3π, 0.1).]

(Fig. 8 ,

 8 Fig.8, such as H(φ, y) = -2λ, there exist two kink solutions as shown in Fig. 9(b) and 9(c).(iii) Corresponding to open orbits defined by H(φ, y) = h, h ∈ ] -∞, -2λ[, there exist rotating solutions of equation (7) with only periodic changes of positive slope y. Although these solutions may not correspond to realistic behaviors in the context of lattice systems, they are represented on figure 9(d).

  (b) intersects the singular straight line at point M. The wave profile, as shown on Fig. 10 leads to the lattice dislocation when φ = φ 0 and y = P . At this point, this gives rise to a breaking solutions φ(u).

Fig. 10 .

 10 Fig. 10. Sketch of a dislocation. Starting from the point A ′ in Fig. 7(b), the orbit intersects the singular straight line at point M , when the slope dφ(u) du = P . (Same parameter set as for Fig. 7).

  (a)), that a kink (antikink) solution links the point O to the point O ′ (2π, 0), with the corresponding wave solutions φ(u) sketched in Fig. 11. Note here the influence of parameter σ on the shape of the wave φ(u) : it only concerns the width of the kink (compare Fiq. 11 (a), (b) and (d)). The other families of orbits in this case all intersect one of the singular straight lines. They correspond thus to breaking waves solution φ(u), with the dislocation happening precisely when their slope y = dφ du reaches either √ Y 0 or -√ Y 0 .

Fig. 11 .

 11 Fig.11. The kink wave and anti-kink wave of Eq[START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF] for 0 < V < 1 (Parameters: P = 0, V = 0.2, B = 0.9, λ = 0.025. For (a) , (b) and (d) φ(-∞) = 0, y(-∞) = 0 but σ takes different values. For (c) φ(-∞) = 2π, y(-∞) = 0).

This author is supported by the Regional Council of Burgundy.

,[START_REF] Martínez | Taming Chaotic Solitons in Frenkel-Kontorova Chains by Weak Periodic Excitations[END_REF]

 [START_REF] Braun | The Frenkel-Kontorova Model, Concepts, Methods, and Applications[END_REF]This author is supported by the National Natural Science Foundation of China (10831003).


Francisco Contreras

Editors Fco Javier Melero

P 12

E Leclercq

C Sapin

M Savonnet

CAA'2010 Fusion of Cultures

Keywords:

This paper details the main concepts and the architecture of WikiBridge, a semantic wiki, developed for the project CARE (Corpus Architecturae Religiosae Europeae -IV-X saec.). The aim of the project CARE is the constitution of integrated corpus of the European Christian buildings dated from the 4th to the beginning of the 11th century. A semantic wiki, called WikiBridge, is developed in order to: 1) allow collaborative work of researchers involved in the project and 2) open the corpus to a large public. WikiBridge combines collaborative and traceability aspects of wiki with semantic consistency and query capabilities. Semantics is guaranteed by an application ontology based on CIDOC-CRM.

1.Introduction

The aim of the international project CARE (Corpus Architecturae Religiosae Europeae -IV-X saec.) is the constitution of integrated corpus of the European Christian buildings dated from the 4th to the beginning of the 11th century. This corpus will greatly facilitate work of comparisons, exchanges and discussions with numerous foreign researchers and specialists. This wide-ranging European program was introduced by the IRCLAMA (International Research Center for Late Antiquity and Middle Ages) at the University of Zagreb (Croatia). Several countries among which Italy, Spain, Czech Republic, Slovakia, Poland and Croatia began 3 or 4 years ago to work on the preliminary documentations of this ambitious project; moreover, Hungary has just joined the Middle-European group; Benelux, United Kingdom, Ireland and Greece are interested, as well as Albania for a common volume with Kosovo and Montenegro; Switzerland and Germany already have published complete catalogs in recent years. The programs of corpus in every country progress on different pace and reveal disparities inherent to sources. For example, Mediterranean regions are often more concerned with the analysis of Early Christian monuments. The most advanced work concerns Northern Italy and Croatia who fall into this category of analysis.

The project has been launched in France on January 1st, 2008 after acceptance of the French National Agency for Research. Managed by Christian Sapin and Pascale Chevalier (UMR 5594-ARTeHIS of the CNRS, Dijon), the project will last 4 years (2008)(2009)(2010)(2011). More than sixty researchers from about twenty universities, diverse research institutions and heritage management institutions are working on. Various categories of staffs are involved: field archaeologists, historians, art historians, draftsmen, topographers, etc. They are collecting and analysing data concerning approximately 2700 monuments. Each of the 22 French regions will form a task force before 2011, 9 of them are already active. With new studies and recent excavations relating to all the period, the French team defines protocols covering all the buildings included in the diverse chronological periods. The accent is placed on the 7th-8th centuries and the decades around the year 1000. The corpus of multimedia documents (including texts, maps, and photographies) concerning every known building will be gradually published in the form of classic books (one for each modern region).

A Web 2.0 application is developed in order to: 1) allow collaborative work of researchers involved in the project and 2) open the corpus to a large public with a little knowledge on the European religious culture. Data driven web application technologies can be used to gen-erate dynamic web content by using databases, but have some major drawbacks in terms of collaboration and traceability. Our approach is based on a combination of collaborative and traceability aspects of wiki with semantic consistency and query capabilities that database can provide. This part of the project is developed by the CNRS LE2I laboratory (Dijon). Some geomaticians of the Social Sciences and Humanities Research Institute of Dijon will conduct specific spatial analysis by providing GIS tools from end of 2010.

The rest of the paper is organized as follow: Section 2 presents our motivation, Section 3 describes the state of art and Section 4 presents our architecture. Finally, Section 5 concludes the paper.

2.Motivation

The CARE corpus needs a spatial and temporal specific models as well as a representation of domain knowledge.

Ontologies and data models have similarities for knowledge representation. They both offer means of description based on concepts and relationships between these concepts. In both cases, knowledge represents a consensus among applications that cooperate. Knowledge representation in an information system may be considered as two dimensions introduced by Spear (SPEAR 2006):

-horizontal dimension (or relevance) aims at determining the extent of information that should be included in the representation of knowledge. For example, if we represent knowledge in the field of archeology, relevance is the choice whether to specify knowledge on liturgical installations, on construction techniques, on religious environment such as diocese (figure 1.a); -vertical dimension (or granularity) aims at determining the degree refinement of knowledge representation. In the archaeological area, the granularity is the choice whether to include a building description from walls structure to decor elements, pavements, etc.

It is problematic to include in a single data model general description for elements and fine details for others, except to take the risk of building a data model difficult to read and to maintain. In contrast, a data model can use multiple sources of knowledge representation, and can therefore adjust the extent of knowledge it covers.

Furthermore, ontologies offer a great freedom in managing the granularity (vertical dimension) of knowledge representation, albeit they cover a limited area. In figure 1.b, each sub-tree from THING may be considered as an ontology of a particular domain. To cover a wider field of knowledge it should compulsorily consider the relationships between several sub-trees. [START_REF] Grenon | SNAP and SPAN: Towards Dynamic Spatial Ontology[END_REF] propose the definition of three kinds of relationships:

-intra-ontology: relationship with two concepts of same part of an ontology; -trans-ontology: relationship with a concept of a sub-tree and a concept of another sub-tree. For instance, a building is consecrated to a saint, in DL we can write: Building ∃isConsecrated.-Saint -meta-ontology: relationship with a concept of an ontology and another ontology (considered as a whole). Moreover, if data models as well as ontologies allow representations based on concepts and relations, their organization in terms of relationships is different. Ontologies focus on specialization relations and strictly control the other relations being used. The data models in turn leave a great freedom in the choice of relations to use.

In short, data models and ontologies for knowledge representation can be combined to mix their specificities: the scope of coverage for the models, the granularity for ontologies. Ontologies are conceptualization of a domain, data models specify an implementation of structure and behaviour according to stated functionality requirements [START_REF] Spyns | Data modelling versus Ontology engineering[END_REF].

Data models are used to implement relational Data Base Management System (DBMS) such as Oracle or PostgreSQL. Semantic data consistency in DBMS is carried out by embedded controls such as triggers or stored procedures. Developing a typical stand alone application based on a database has two major drawbacks: 1) archaeologists have a purely document-based approach, away from the concepts of atomic decomposition and fully structured information imposed by database; 2) research conducted by archaeologists requires an open environment which allows to aggregate knowledge pro- duced by different teams involved in the project. This environment must also provide sharing, exchange and evolution capabilities. The knowledge evolution leads to a dynamic evolution of database schema. It is difficult to subsequently modify a previously defined database schema and its content. A high degree of flexibility is then required. These two drawbacks make the construction of such an application at reasonable cost, difficult [START_REF] Bonomi | Integrating a Wiki in an Ontology Driven Web Site: Approach, Architecture and Application in the Archaelogical Domain[END_REF].

The request of a web application with a collaborative component led us to choose a solution based on a wiki. Despite the power of wiki (free input, rich user-interface, traceability, bi-directional links between pages, etc.), it is difficult to answer a specific query because of the purely textual information stored. One way to address this problem is to implement social tagging. Social tagging, a key characteristic of Web 2.0, allows users to index contents by their own keywords. Moreover, users do not understand an annotation schema since they decide themselves what keywords to use. The list of commonly used keywords are called a folksonomy. Nevertheless, keywords have no explicit links between them (hierarchy, similarity, synonym) and management of ambiguity and heterogeneity of keywords is not made. In our context, social tagging cannot provide enough semantic quality. Consequently, a semantic annotation approach of content based on ontologies is more relevant. The Semantic Web that can represent a complex knowledge is based on languages (RDF, OWL), tools, reasoners but requires knowledge experts. We generally consider that Semantic Web and social web are competitors. Some authors suggest to combine these two approaches (ANKOLEKAR et al. 2007). In addition, we believe that requirements for interoperability and data exchange (connection with other communities such as historians) must be taken into account since the design phase of the application. The Semantic Web thereby provides such kind of solutions by increasing the expressiveness of data representation, and by allowing reasoning tools and semantic search.

Wiki engines provide tools to manipulate document in a collaborative environment and ensure the cost of developing and maintaining reasonable. Our proposal is to use MediaWiki to develop a numerical corpus by integration of individual contributions. We have extended MediaWiki with some DBMS capabilities: form based acquisition interface, annotations, query engine. The form based acquisition interface allows users to add data with a global structure (specific fields such as location and free text based fields such as the description of liturgical installations). We use annotations to make links between semi-structured data manipulated by Medi-aWiki and structured data necessary to query engine. Annotation semantics is guaranteed by an ontology which allows to describe concepts and their relations. As stated by [START_REF] Abiteboul | Data on the Web: from relations to semistructured data and XML[END_REF]) one of the strengths of semi-structured data is "... the ability to accommodate variations in structure". Our dual approach allows to cope with evolution of knowledge by modify-ing the ontology and annotations dynamically without modifying database schema. Moreover, ontologies can represent concepts at different levels of abstraction (granularity). For example, for some archaeological remains the type of mortar can be described, for others only the presence of marks is recorded.

Figure 2 presents an overview of the interactions between different types of users and our system. Yellow arrow describes data capture, red arrows present semantic queries and blue arrows symbolize links between semantics and semi-structured data.

3.State of art

In traditional wiki, semantics is not explicit, but is implicitly described by links between pages and by the context of the link (surrounding text). In the following subsections we give a short overview of semantic wiki and their theoretical background.

3.1.Semantic wiki engines

A semantic wiki is a wiki which includes semantic web technologies to cope with domain knowledge generally represented by ontologies. Semantic wikis can be built on top of existing wiki or created from scratch. In [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF] authors have identified two approaches of wiki based on their relationship to the ontology: 1) wiki centric approaches use the wiki to organize knowledge i.e ontology emerges from the wiki through categories and links or 2) ontology based approaches allow to import an existing ontology and use it in the annotation process.

However, we consider that this classification is too restrictive and we propose to define a third category of approaches which combines the first two ones.

Platypus, Rise and Rhizome fall in the first category. The first system was probably Platypus [START_REF] Campanini | Platypus Wiki: A Semantic Wiki Wiki Web[END_REF]) which has focused on the creation of RDF meta-data. Rhizome (SOUZIS 2005) allows to edit content, structure and meta-data in RDF format. URL represent elements such as structural components, abstract entities, and relationships. Rise wiki [START_REF] Decker | Self-organized Reuse of Software Engineering Know-ledge Supported by Semantic Wikis[END_REF]) allows user to create and edit the ontology with wiki pages (concepts) and links (relations). Moreover the same approach has been used in numerous wiki engines. For example in MediaWiki, Category is the simplest form of annotation used to classify wiki pages. Semantic MediaWiki [START_REF] Krötzsch | Semantic MediaWiki[END_REF] extends MediaWiki and provides new features such as: 1) Relations are used to describe relationships between two pages by assigning annotation to existing links and 2) Attributes allow users to specify relationships between pages and literals. Table 1 gives a concrete example of a page in MediaWiki using links and categories and the same page using Semantic MediaWiki capabilities. In [START_REF] Vrandečić | Reusing Ontological Background Knowledge in Semantic Wikis[END_REF], the authors propose an equivalent representation between OWL concepts and Semantic MediaWiki constructs (Table 2). This approach mainly produces assertions which correspond to ABox statements. In short this first category of semantic wikis can be used to present knowledge by structuring concepts through pages, categories and links.

The city of

OWL

Makna [START_REF] Dello | Creating and using Semantic Web Information wih MAKNA[END_REF] and BOWiki (BACK-HAUS et al. 2007) are two examples of the second category. In Makna, users can create semantic content using RDF statements referencing pre-existing ontologies. They are provided with an extended Wiki syntax and with assistant tools to simplify ontologies based annotation process. A specific application MannWiki has been developed for sharing knowledge on micro-array in LifeScience domain and allows to make references to GeneOntology. BOWiki is used to collaboratively create knowledge base in biological domains. Moreover, BOWiki allows to access to several ontologies like the GeneOntology and ontologies about cell types or anatomy. This second category of semantic wikis based on pre-existing ontologies can be used as a platform to build applications that require a global consensus over knowledge in order to maintain the quality of data.

The third category of semantic wiki is an hybrid approach that can be used both to build ontologies in a collaborative ontology engineering process or to import pre-existing ontologies to annotate documents. Sweet-Wiki (BUFFA et al. 2006) allows users to tag pages, (called social tagging) but also integrate external ontologies. The set of users tags generates a folksonomy. In addition, SweetWiki adds a WYSIWYG editor for managing content and meta-data, a reasoning engine used for querying the wiki content. Semantic MediaWiki [START_REF] Krötzsch | Semantic MediaWiki[END_REF]) is an hybrid engine which enables to load ontology and to consult it as wiki pages. Semantic MediaWiki does not provide yet a complete set of tools in particular for semantic constraints management.

Our approach of semantic wiki is directed towards scientific application domains which contribute to produce knowledge. This kind of application relies on core ontologies that act as a consensus. Knowledge is enhanced by querying and analyzing data, new concepts can emerge and new constraints can be found out. As a result, ontologies can be modified dynamically and semantic checks are necessary to find inconsistent annotation with regards to an ontology version.

3.2.Theoretical background

Semantic web technologies such as RDF and OWL ontologies are based on well founded theoretical background. RDF is based on conceptual graphs and semantic networks [START_REF] Sowa | Conceptual Structures Information Processing in Mind and Machine[END_REF]. OWL is based on description logics [START_REF] Baader | The Description Logic Handbook: Theory, Implementation and Applications[END_REF]. Some features of description logics make it difficult to use for validating data or annotations through integrity constraints: 1) OWL-DL works in open world assumption; 2) OWL does not use the unique name assumption. Finding inconsistent annotations requires to evaluate OWL rules in a closed world assumption to detect violation. Some compelling solutions are described in [START_REF] Sirin | Opening, Closing Worlds -On Integrity Constraints[END_REF].

4.WikiBridge's architecture

In the next section we present WikiBridge's architecture and detail some key features for archaeological application.

4.1.Acquisition

Two types of acquisition form have been created: a form for entering a record corresponding to atomic building and a form corresponding to a group of buildings. These two forms are a simplified version of paper forms filled by archaeologists to publish their research results. Electronic forms are created by using Semantic Forms extension for MediaWiki. It allows users to fill in fields through a model (figure 3). A non-expert in archeology can then easily feed the wiki from paper forms already made.

4.2.Annotation

To improve quality of search, we expanded MediaWiki with a semantic component. The semantic component consists of annotations made by experts, that are guaranteed by an application ontology.

WikiBridge restricts the access to ontological knowledge management to a predefined set of Wiki users: we argue that implementing such functionality without adequate process-level support might have uncontrolled consequences on the operation of the overall wiki system. Knowledge engineers interacting with archaeologists create the ontology with standard tools like Protégé. Ontology contains concepts of the domain, instances and rules. This assertion is considered as a rule in the ontology. Knowledge engineers can test consistency of the ontological representation with reasoners such as Racer or Pellet.

Within the cultural heritage domain, the CIDOC Conceptual Reference Model (CIDOC) has emerged as a domain ontology. CIDOC CRM deals with concepts at a high level of generality. Its scope encompasses the general culture heritage domain and it is envisaged as "semantic glue" useful for exchange among diverse information sources. Application ontologies contain all the definitions that are needed to model the knowledge required for a particular application. Typically, application ontologies are a mix of concepts that are taken from domain ontologies and specific application. We have developed an application ontology as a CIDOC CRM extension covering the Christian European buildings (figure 5). The CARE ontology contains concepts and relations about:

1) Man made objects (type of buildings, architectural elements, liturgical installations)

2) temporal data and 3) others entities that are specific to Christian buildings.

For instance, building is a concept, it has liturgical functions which is another concept. Cathedral and episcopal are instances of building and liturgical functions. A cathedral has one function, that function can be episcopal or archiepiscopal.

Experts directly enter and modify annotations through an extension of the wiki's editing interface (figure 6). The consistency of annotation in relation to the context (field in the form) is checked by specific application module. At this stage of development, this functionality is directly implemented in the application. As a result, annotations are stored in an ad-hoc RDF triple store. Ontology is then stored in a relational database and queried by PHP programs to fill in the annotation wizard.

4.3.Query engine

Although, most of wikis includes by default an engine that can only query full-text. The aim of our engine is to provide semantic search by filling in parameters associated with ontology concepts. Three types of interfaces (figure 7) for building semantic queries are developed: 1) a wizard lets users to specify search parameters to engine; 2) users can create query models that are then stored; and 3) user can navigate through an ontology tree.

Three kinds of results can be displayed: 1) results can appear in a list containing links to articles, at the right annotation place, so where the information is given. User can then manually navigate through articles interlinked; 2) factbox, each article is displayed with its annotations and 3) users can select annotation to be dis-played in the result. From this result, users can obtain the list of the articles in which have the same annotation. This third kind of display is a mix of result list and factbox and allows more sophisticated analysis.

5.Conclusion

A feasible combination of wiki and Semantic Web technologies should preserve the key advantages of both technologies: the simplicity of wiki systems as shared content authoring tool, the power of Semantic Web technologies with.respect.to. structuring and retrieving knowledge. In this article, we have demonstrated that flexibility required by scientific applications can be achieved by using wiki with semantic web technologies. At this stage of development, data quality is maintained by ad-hoc programs.

Ontologies can also include logical rules representing the domain constraints. Reasoners are used to verify the semantics contained in ontology. For example, the following constraint "a building can be consecrated to a saint only if the construction date is later than the death of the holy person" is represented by the following rule: isConsecrated(?b,?p) ← hasConstruc-tionDate(?b,?d1) ∧ hasDateDead(?p,?d2) ∧ d1 ≥ d2

In the next version of WikiBridge, automated verification of integrity constraints will be performed by a reasoning tool. For spatial analysis, we are developing in parallel web services to export data to PostGIS.

Figure 1 :

 1 Figure 1: Horizontal and vertical dimensions of knowledge representation.

Figure 2 :

 2 Figure 2: Outline of WikiBridge users interaction.

  Moulis is located on the [[Medoc region]]. The building of the High [[Middle Ages]] was discovered in 1993 under the present parish church, largely Romane, surrounded by a parish cemetery until 1901, then transformed in the public square... [Category:ArchaelogicalSite]} The city of Moulis is located on the [[region::Medoc]] region. The building of the High [[Middle Ages]] was discovered in [[date::1993]] under the present [[building-type::parish church]], largely Romane, surrounded by a parish cemetery until [[date::1901]], then transformed in the public square... [Category:ArchaelogicalSite]

Figure 3 :

 3 Figure 3: Methodology for entering a form.

Figure 4 :

 4 Figure 4: Multimedia enrichment.

Figure 5 :

 5 Figure 5: CARE ontology (extract).

Figure 6 :

 6 Figure 6: Annotation interface.

Figure 7 :

 7 Figure 7: Query Interface.

Table 1 :

 1 Concepts, relationships and attributes in MediaWiki and Semantic MediaWiki.

Table 2 :

 2 OWL Concepts and Semantic MediaWiki constructs.

		Semantic MediaWiki
	OWL individual	normal article page
	owl:Class	article in namespace Category
	owl:ObjectProperty	article in namespace Relation
	owl:DatatypeProperty	article in namespace Attribute

Proceedings of the 38 th Conference on Computer Applications and Quantitative Methods in Archaeology Granada, Spain, April 2010

Proceedings of the 38th Conference on Computer Applications and Quantitative Methods in Archaeology Granada, Spain, April 2010

Proceedings of the 38st Conference on Computer Applications and Quantitative Methods in Archaeology Granada, Spain, April 2010

Acknowledgements This work is supported by the ANR (ANR-07-CORP-011).