Jean-Paul Chehab

Georges Sadaka

On Damping Rates of dissipative KdV equations

We consider here different models of dissipative Korteweg-de Vries (KdV) equations on the torus. Using a proper wave function Γ, we compare numerically the long time behavior effects of the damping models and we propose a hierarchy between these models. We also introduce a method based on the solution of an inverse problem to rebuild a posteriori the damping operator using only samples of the solution.

1.

Introduction

The modeling and the underlying mathematical analysis of gravity waves is still a challenging topic despite intensive works and progress that have been done, especially during the last decade, see, e.g. [START_REF] Miranville | Mathematical Modeling in Continuum Mechanics[END_REF] for the derivation of the models. The long time behavior of dissipative asymptotic models remains an important issue: capturing damping rates in several norms, measuring regularization effect or pointing out complex asymptotic dynamics (existence of attractors, Hopf bifurcations etc) to name but a few are important points for the understanding of natural phenomena. For many models, several of these questions are still open and the numerical simulation is a way to capture some properties and to select pertinent models.

Damped Korteweg-de Vries equations (KdV) appear in different physical situations and they can be expressed in a large generality on the torus T = T(0, L) as

u t + L(u) + u xxx + uu x = 0, x ∈ T, t > 0, ( 1
)
where L is a linear operator, defined on a Hilbert space V , subspace of L 2 and satisfying

L 0 L(v)vdx ≥ 0, (2)
for all function v ∈ V , regular enough, in such a way the L 2 -norm of the solution is decreasing in time as 1 2

d|u| 2 L 2 dt + L 0 L(v)vdx = 0. ( 3
)
The definition of L depends on the physical situations: in [START_REF] Ott | Nonlinear theory of ion acoustic wave with Landau damping[END_REF], Ott and Sudan have proposed a damped KdV equation as a model of Landau damping for ion acoustic wave, the (linear) damping being nonlocal, and in [START_REF] Ott | Damping of solitary waves[END_REF] they have presented different models of damping taking L(u) = |D| α u, where |D| = √ -∆; in [START_REF] Dias | Viscous potentiel free-surface flows in a fluid layer of finite depth[END_REF] Dias and Dutykh have considered the operators

L(u) = -νu xx + ν 2 t 0 u t (s) √ t -s ds, (4)
L(u) = -νu xx - ν 2 t 0 u x (s) √ t -s ds, (5)
respectively, to model natural damping of water waves, mathematical analysis and simulations can be found also in [START_REF] Cabral | Chaos for a damped and forced KdV equation[END_REF][START_REF] Chen | Decay of solutions to a water wave model with nonlocal viscous dispersive term, discrete and continuous dynamical systems[END_REF][START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity[END_REF]; in [START_REF] Chehab | Numerical study of a family of damped KdV equations[END_REF][START_REF] Sadaka | Etude mathématique et numérique d'équations d'ondes aquatiques amorties[END_REF] it has been proposed a general model of weak damping for spatial nonlocal frequency by frequency damping,

L(u) = k∈Z γ k ûk e 2iπkx L . (6)
Generally, we do not have damping rate estimates so it is difficult to make a hierarchy between these models without using numerical simulations. We are here interested in comparing the long time effect of these different type of damping and for that purpose, we follow [START_REF] Chehab | Numerical study of a family of damped KdV equations[END_REF] using the norm ratio function

Γ(t; u) = < L(u), u > |u| ,
that allows to compare the damping in L 2 norm; here |.| and < ., . > denote respectively the L 2 -norm and the associated scalar product. A consequence of relation is that the L 2 decay depends on L, in fact an application of a Gronwall lemma gives

|u(t)| 2 = e -2 t 0 Γ 2 (s)ds |u(0)| 2 . ( 7
)
so that Γ characterizes the damping in L 2 .

A interesting inverse problem is the following: starting from samples of u at discrete times t m that can be measured (u represents the vertical elevation of the wave with respect to the horizontal position), is it possible to rebuild a given type of damping operator whose the effect on large time interval fits with Γ 2 (t) ?

The present work addresses to these two questions and we propose some numerical techniques that allow in the one hand, to compare the effect of different damping models on large time interval and, on the other hand, starting from samples of the solution, to recover numerically an approximation of the operator L.

The article is organized as follows: in Section 2 we recall some damping properties of the different models and we introduce the tools that we will use for making the hierarchy. After that, in Section 3, we present the numerical schemes in space and in time. Then, in Section 4, we first compare numerically the long time behavior of the damping and we display a time signal analysis of Γ(t). After that, we propose to rebuild in some cases and a posteriori the damping operator by fitting proper parameters with the solution of a constraint least square problem.

2 Damping properties

The different models

We here recall some characteristics and properties of the damping we will consider for the application.

Weakly damped KdV equations

Consider the situation in which the damping operator can be described by Fourier series as

L(u) = k∈Z Z γ k ûk e 2iπkx L ,
where γ k ≥ 0 are the damping weights and ûk denotes the k-th Fourier coefficient of u ; the assumption of positivity of γ k implies that

< L(u), u >= k∈Z Z γ k û2 k ≥ 0,
insuring the decreasing in time of the L 2 -norm of the solution. The choice

γ k = 2πk L α , α > 0 corresponds to L γ (u) = |D| α u, where |D| = √ -∆. More generally, when lim k→+∞ γ k = +∞, (e.g. γ k = k 2
for a parabolic damping [START_REF] Dutykh | Modélisation mathématique des Tsunamis[END_REF]), the equation is regularizing at finite time. When γ k is constant, say L(u) = γu, the damping is said to be "weak" and is not regularizing at finite time but, as proved by Ghidaglia [START_REF] Ghidaglia | Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time[END_REF][START_REF] Ghidaglia | A note on the strong convergence towards attractors for damped forced KdV equations[END_REF] and Goubet [START_REF] Goubet | Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations[END_REF][START_REF] Goubet | Rosa Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line[END_REF], it allows the equation to have a finite dimensional attractor which is in a more regular space than the initial data: this is the asymptotic regularization property. In all these situations γ k ≥ γ > 0 is bounded from below and the damping in L 2 -norm is done at least at an exponential rate, we have indeed, after integration by parts and use of Gronwall's lemma,

u L 2 ≤ e -γt u 0 L 2 .
In [START_REF] Chehab | Numerical study of a family of damped KdV equations[END_REF], the limit case, ( lim k→+∞ γ k = 0) has been considered, and it was pointed out numerically also asymptotic regularization properties but the damping rate is not uniform: it depends on both the sequence γ and of the Fourier coefficients of the initial data, see [START_REF] Chehab | Numerical study of a family of damped KdV equations[END_REF].

Nonlocal damping in time

The nonlocal damping in time

L(u)(x, t) = -νu xx - √ ν √ π t 0 u t (s) √ t -s ds.
was considered by Dias and Dutykh in [START_REF] Dias | Viscous potentiel free-surface flows in a fluid layer of finite depth[END_REF]. The associated Damped KdV equation

u t + u xxx + uu x -νu xx - √ ν √ π t 0 u t (s) √ t -s ds = 0 (8)
models one way water waves in a fluid layer of finite depth under the influence of viscous effects. A model of the same kind is studied in [START_REF] Bona | Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory[END_REF][START_REF] Dias | Viscous potentiel free-surface flows in a fluid layer of finite depth[END_REF], say

u t + u xxx + uu x -νu xx - √ ν √ π t 0 u x (s) √ t -s ds = 0 (9)
but it will not considered in the present work. A mathematical analysis is presented in [START_REF] Chen | Decay of solutions to a water wave model with nonlocal viscous dispersive term, discrete and continuous dynamical systems[END_REF] a numerical study is realized in [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity[END_REF].

Localized damping in space

It corresponds to

L(u)(x, t) = χ [a,b] u,
where 0 < a < b < L. This situation have been considered in the context of the stabilization of KdV equations, when the domain is the torus T [START_REF] Laurent | Control stabilization of the Korterweg-De-Vries equation in a periodic domain[END_REF] or the half line IR + and L = χ [a 0 ,+∞[ with a 0 > 0, [START_REF] Pazoto | Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-space[END_REF]. A exponential decay in time was established in proper Sobolev norms. We will recover numerically on T this exponential rate of convergence (after a transient time) and compare it to the other damping models, see Section 4.

Comparison of the damping

As stated in the introduction, we would like to compare numerically these damping models for long time intervals and to display a time signal analysis. This is a first step to be considered for making a hierarchy. To this end, we follow [START_REF] Chehab | Numerical study of a family of damped KdV equations[END_REF] and introduce the function Γ(t)

Γ(u)(t) = < L(u), u > < u, u > , (10)
that we will note Γ(t) for simplicity. In the general case, we can prove that |u| L 2 → 0 as t → +∞ but without deriving explicit bounds for the damping rate:

Proposition 1. Assume that -L : V ⊂ L 2 → L 2 is linear continuous -< L(u), u >≥ 0 -< L(u), u >= 0 =⇒ u = 0 then lim t→+∞ |u| L 2 = 0.
Proof. By taking the scalar product in L 2 of the equation with u, we obtain after an integration by parts 1 2

d|u| 2 dt + < L(u), u >= 0, (11)
making the function t → |u| decreasing in time. Assuming that u is smooth enough and arguing that |u| ≥ 0 we deduce that |u| converges towards a limit ℓ as t goes to ∞. This implies that < L(u), u >→ 0 as t goes to +∞, hence that u → 0 a.e.

The function Γ(t) is related to the L 2 norm of the solution: we recall the simple Gronwall lemma from [START_REF] Chehab | Numerical study of a family of damped KdV equations[END_REF] Proposition 2. Let u(x, t) be a regular solution of equation [START_REF] Bona | Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory[END_REF]. We assume that G(u, t) is C 1 in t.

Then we have the relation

|u(t)| 2 L 2 = e - R t 0 Γ 2 (s)ds |u 0 | 2 L 2 . In particular, lim t→+∞ |u| L 2 = 0 iff t → G(t) ∈ L 2 t (0, +∞).
Of course, the damping rate in L 2 depends on the operator L, so it appears that the function Γ(t) is an appropriate tool for comparing two damping in long time intervals. For these reasons, we propose in section 4 to provide a time signal analysis using a proper orthogonal polynomial basis associated to a weighted L 2 scalar product.

Let us consider u and v respectively (regular) solutions of the damped KdV equations

u t + L 1 (u) + u xxx + uu x = 0, x ∈ T, t > 0,
and

v t + L 2 (v) + v xxx + vvx = 0, x ∈ T, t > 0.
We let w = u 1 -u 2 . We can write

w t + w xxx + L 1 (w) + (L 1 -L 2 )(v) + ww x + wv x + vw x = 0.
Then, taking the L 2 -scalar product of these terms with w, we obtain after the usual simplifications, 1 2

d|w| 2 dt + < L 1 (w), w > + 1 2 L 0 w 2 v x dx+ < (L 1 -L 2 )(v), w >= 0.
The comparison of two solutions generated by two different damped KdV models needs to make additional assumption. It is however possible to make a comparison in the case of weak damping, i.e. when L is defined as

L(u) = L γ (u) = k∈Z Z γ k ûk e 2iπkx L .
We have the following result:

Proposition 3. Consider L 1 and and L 2 , two weak damping operators defined by

L 1 = γId and L 2 (u) = k∈Z Z γ k ûk e 2iπkx L .
We denote by u (resp. v) the solution of the damped KdV equation with L 1 (resp. L 2 ) as damping operator. Assume that -

γ k > 0, γ k = γ, |k| ≤ N, γ k ≤ γ, |k| ≥ N -∃β > 0/γ -1 2 |v x | ∞ ≥ β > 0, ∀t > 0 Then |w(t)| 2 ≤ e -βt |w(0)| 2 + γ N +1 1 -e -βt β 2 |v(0)| 2
where w = u -v.

Proof. We have directly 1 2

d|w| 2 dt + β|w| 2 + < (L 1 -L 2 )(v), w >= 0. Now | < (L 1 -L 2 )(v), w > | ≤ 1 2β L 1 -L 2 2 + β 2 |w| 2 , here L 1 -L 2 denotes the norm of the linear operator L 1 -L 2 from L 2 to L 2 . We have for any z ∈ L 2 |(L 1 -L 2 )(z)| 2 = +∞ k=N +1 γ k |ẑ k | 2 ≤ γ N +1 |z| 2 .
Finally,

d|w| 2 dt + β|w| 2 ≤ γ N +1 β |v| 2 ≤ γ N +1 β |v(0)| 2 ,
hence the result by using an integration in time.

3 Numerical Schemes

Discretization in space

The Damped KdV equation is studied here on the torus, it is then natural to use pseudo-spectral Fourier collocation techniques for the discretization in space; this produces a high accurate discretization and moreover it allows to take advantage of FFT and to make the computations more faster. We shall follow this approach for all the damping we consider except for the local space damping L(u) = χ [a,b] u. Indeed, since this term is spatially localized, it is very badly localized in frequencies, so the Fourier approach can not be efficient. A way to overcome this drawback while having a high level of accuracy, that can be compared with the spectral one, is to implement finite difference compact schemes [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF]. In two words, these schemes consist in approaching a linear operator (differentiation, interpolation) by a rational (instead of polynomial-like) finite differences scheme: let U = (U 1 , • • • , U n ) T be a vector whose the components are the approximations of a regular function u at (regularly spaced) grid points

x i = ih, i = 1, • • • , n.
The classical finite difference schemes consists in computing T u = f at x i with the matrix-vector product

F = T U,
where T is the discretization matrix of T n, a derivation or an interpolation operator. The relation f (x i ) = (T U ) i is then satisfied up to a certain order of accuracy. The compact schemes consist in approaching T u as

P F = QU,
in such a way the relation f (x i ) = (P -1 QU ) i is satisfied at the most possible higher order. The matrix P represents the implicit part of the scheme while Q is the explicit part. We reconsider here the compact schemes introduced by Lele [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF]. These schemes are defined with a compact stencil: the implicit part consists of a banded matrix, easy to invert, and thanks to the implicit part, the discretization scheme uses only few points near the boundaries, when the non-periodic case is considered, see [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF]. To approach ∂u ∂x (x i ) we consider the relations

u ′ (x i ) + α(u ′ (x i+1 ) + u ′ (x i-1 )) + β (u ′ (x i+2 ) + u ′ (x i-2 )) = a u(x i+1 ) -u(x i-1 ) 2h + b u(x i+2 ) -u(x i-2 ) 4h + c u(x i+3 ) -u(x i-3 ) 8h . ( 12
)
Here h is the spatial step size. Coefficients α, β a, b et c are computed in such a way [START_REF] Ghidaglia | Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time[END_REF] to be satisfied at a given order.

-Order 10 (error in h

= c = β = 0, α = 1/4, a = 3/2. -Order 2 (error in h 2 ) :a = 1, b = c = α = β = 0.
We use the periodic boundary conditions to close the system. In the same way we define the compact scheme for the third derivative in space as

u (3) (x i ) + α(u (3) (x i+1 ) + u (3) (x i-1 )) + β(u (3) (x i+2 ) + u (3) (x i-2 )) = a -2 • u(x i+1 ) + 2 • u(x i-1 ) + u(x i+2 ) -u(x i-2 ) 2h 3 +b -2 • u(x i+2 ) + 2 • u(x i-2 ) + u(x i+4 ) -u(x i-4 ) 16h 3 + c -2 • u(x i+3 ) + 2 • u(x i-3 ) + u(x i+6 ) -u(x i-6 ) 54h 3 . ( 13
)
-Order 6 (error in h 6 ) : b = 1/9, α = 1/3, a = 14/9, c = 0, β = 0.

-Order 4 (error in h 4 ) :

b = c = β = 0, α = 1/4, a = 3/2. -Order 2 (error in h 2 ) :a = 1, b = c = α = β = 0.

Time discretization schemes

Crank Nicolson schemes

Let D be the differential operator associated to the first derivative in space. The classical Crank-Nicolson scheme writes as

u (n+1) -u (n) ∆t + L γ u (n+1) + u (n) 2 + D 3 u (n+1) + u (n) 2 + 1 4 D (u (n+1) ) 2 + (u (n) ) 2 = f, (14)
We will use it for the simulations with non time depending damping. In practice it will be implemented using at each time step a Picard fixed point method, other numerical schemes are presented an studied in [START_REF] Chehab | Numerical study of a family of damped KdV equations[END_REF].

Generalized Gear schemes

The Gear scheme is a popular second order semi implicit scheme to solve evolution problems such as parabolic ones, see e.g. [START_REF] Guermond | An Overview of Projection Methods for Incompressible Flows[END_REF] and the reference therein for the solution of incompressible Navier-Stokes equations ; it addresses to the approximation of the first derivative in time. Dubois et al proposed to adapt it to the numerical computation of fractional derivative, particularly the "half-time" derivative.

Here, we aim at approaching the nonlocal operator G

1/2 t u(t) = 1 √ π t 0 u t (s) √ t -s ds.
In [START_REF] Dubois | Introduction à la dérivation fractionnaire ; théorie et applications[END_REF], the authors propose the following scheme

(G 1/2 t u) n = 3 2∆t • +∞ j=0 g j+1 u n-j = 3 2∆t • n j=0 g n+1-j u j ,
where g j are the Gear coefficients computed in [7]. Taking the Fourier series of each term we have

a 0 d dt u k + √ ν • G 1/2 t u k + a 1 • i • 2πk L -a 2 • i • 2πk L 3 + a 4 • 2πk L 2 LT • u k + a 3 • i • 2πk L NLT • u 2 k 2 = 0. We now set C = 2πk L , LT = C • a 1 • i -a 2 • i • C 2 + a 4 • C and NLT = a 3 • i • C.
Here a 0 , a 1 , a 2 , a 3 , a 4 and ν are given real numbers. The system to be solved is then:

   a 0 d dt u k + √ ν • G 1/2 t u k + LT • u k + NLT • u 2 k 2 = 0, u k (x, 0) = u k (0).
We will set for the sequel U (x, t) = u(x, t) -u 0 (x), so

     a 0 d dt U k + √ ν • G 1/2 t U k + LT • U k + u 0 + NLT • (U + u 0 ) 2 k 2 = 0, U k (x, 0) = 0.
For simplicity, we consider the following explicit scheme

     a 0 U n+1 k -U n k ∆t + √ ν • G 1/2 t U n+1 k + LT • U n+1 k + u 0 + NLT • (U n+1 ) 2 k 2 = 0, U k (x, 0) = 0.
We have

√ ν • G 1/2 t U n+1 k = √ ν • 3 2∆t • n+1 j=0 g n+1-j U j = √ ν • 3 2∆t •   g 0 U n+1 k + n j=0 g n+1-j U j   .
Therefore, we obtain

       U n+1 k =   - 3ν∆t 2 • n j=0 g n+1-j U j + a 0 • U n k -∆t •   LT • u 0 + NLT • (U n+1 ) 2 k 2     /M 1, U k (x, 0) = 0.
where

M 1 = a 0 + ∆t • LT + 3ν∆t 2 • g 0 .
Remark 4. The Gear scheme proposed here can be generalized to the approximation of general fractional derivatives of order α expanding formally the αth power of the operator scheme expressed with translation operators [START_REF] Dumont | Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity[END_REF][START_REF] Hirsch | Numerical Computation of Internal and External Flows , The Fundamentals of Computational Fluid Dynamics[END_REF][START_REF] Jordan | Calculus of Finite Differences[END_REF]. We have indeed

G = 1 ∆t 3 2 I -2δ -+ 1 2 (δ -) 2
where δ -is the time backward translation operator :

δ -u n = u n-1 . Hence (G α u) n = 3 2∆t α +∞ j=0 j ℓ=0 4 3 j 41 4 ℓ A α j+1 B j ℓ+1 u n-j-ℓ .
The coefficients A α j+1 and B j ℓ+1 being computed recursively by the formula

A α j+1 = j -α -1 j A α j and B k ℓ+1 = ℓ -j -1 ℓ B j ℓ starting from A α 1 = 1 and B j 1 = 1.
We recover the scheme proposed by Dubois for α = 1 2 reordering the sum as

(G α u) n = 3 2∆t α +∞ j=0 g α j+1 u n-j .

Numerical results

The numerical results we present here are obtained using Matlab. We will have use of the following notations for the initial data:

u 0 (x) = S1 = 3 c sech c 2 (x -pL) 2
, with c = 1, p = 0.4 is the soliton,

u 0 (x) = S2 = χ [a,b] corresponds to the crennel, u 0 (x) = S3 = sin 2πx L is the sine data.
4.1 The different damping

L γ damping

In Figure 1, we display (on the left) the time evolution of the L 2 -norm of the solution when considering various weak damping operators, the corresponding Γ function is also represented (on the right). The initial data are the same in all cases is soliton-like, say u 0 (x) = 3 sech x -0.4 L 2

2

+ c with c chosen such as u 0 is zero mean. We observe that, of course, the weaker is the damping the slowest the L 2 norm of the solution converges towards 0. We report in Figure 2, the time evolution of the L 2 norm, we note that the damping rate is larger with large values of b -a. After a transient time, we see that the L 2 decreases linearly in log. scale, so the damping is exponential. Similar results are obtained with different initial data. In Figure 3 we compare the time evolution of the L 2 norm of the solution when considering L = χ [0,L/2] and various initial data. After a transient time, we note that the rate are the same for each case. Finally, in Figure 4, we have represented the time evolution of Γ 2 (t) for different initial data. We observe that, after a transient time, Γ 2 (t) converge quickly to a constant: this means that the damping rate for large times is the same for all initial data.

Nonlocal integral damping

We display hereafter the solution at different times, varying µ and ν. The numerical results agree with whose in [START_REF] Chen | Decay of solutions to a water wave model with nonlocal viscous dispersive term, discrete and continuous dynamical systems[END_REF] (Figure 5).

Comparison of the damping rates for the different models

In Figures 6 and7, we compare the long time effect of the different damping using the function Γ(t). This approach allows to point out a hierarchy between these different damping models. To this end, we consider the damping operator in its largest generality:

L(u) = µL γ (u) + √ ν t 0 u t √ t -s ds + ωχ [a,b] u (15)
The case ν = ω = 0, allows to recover the weak damping when γ k is bounded and the parabolic damping when γ k = k 2 , a comparison have been made in Figure 1; when ν = µ = 0, the nonlocal spatial damping considered for the stabilization is obtained; finally, when ω = µ = 0, ν = 0, L is the nonlocal damping in time.

To compare the damping effect in L 2 norm of these operators, we distinguish two groups of them -

L(u) = L γ (u) or L(u) = χ [a,b] u
As seen above, the damping in L 2 -norm is exponential in time, after eventually a transient time. For making the comparison coherent, we have scaled the models L γ (u) in such a way the coefficient associated to the frequency k = 1, γ 1 is equal to 1, that is µ = L 2π

2

; we take ω = 1 since asymptotically, the local spatial damping is to be compare asymptotically with

L γ=1 = χ [0,L] .
-

L(u) = -µu xx + √ ν t 0 u t √ t -s ds.
Here we have to take into account of the competition between -µu xx and √ ν t 0 u t √ t -s ds acording to the values of µ and ν.

Overall, we can range the operators in the following way, by decreasing damping rate

1. L(u) = ν t 0 u t √ t -s ds 2. L(u) = -µ∂ 2 x u 3. L(u) = γu 4. L(u) = γuχ [a,b] 5. L(u) = L γ (u) = k∈Z γ k ûk e 2ikπx L , with γ k = 1 1 + |k|
In Figure 6, we observe a hierarchy between the operators of dissipation through the damping they produce in long time interval, we can distinguish 2 groups, from the most to the less dissipative: first the laplacian-like and the weak operators L γ with γ bounded, then the damping of type χ In Figure 7, we observe that the damping provided by the nonlocal operator is stronger than that produced by the negative laplacian.

(t) (log. scale) t γ k =1 γ k =1/k γ k =1/k 2 γ k =k 2 χ [0.2L,0.8L] χ [0.4L,0.6L]
L(u) = -µu xx + √ ν t 0 u t √
t -s ds for different ν and µ with L = 500.

Time signal analysis and damping modeling

Time signal analysis

As pointed out in section 2, Γ(t) does not belong in L 2 t (0, +∞). However, it is possible in some cases to find a weight ω such that G ∈ L 2 ω (0, +∞) = {v/ +∞ 0 v 2 ωdx < +∞}. In such a case a time signal analysis can be displayed in the orthogonal polynomial basis associated to the scalar product in L 2 ω ; for example, if G(t) is at most at polynomial growth, then it belongs in L 2 ω with ω = e -t so that the time analysis can be done with Laguerre polynomials. This behavior in time was observed for the various damping models, see above.

To display a time signal analysis, we proceed as follows: formally the time signal Γ 2 (t), is decomposed into the Polynomial Laguerre basis as

Γ 2 (t) = +∞ k=0 g k L k (t),
where L k is the k-th Laguerre polynomial, defined as the orthogonal sequence with respect to the scalar product < f, g >= +∞ 0 f ge -t dt with L 0 = 1. We have then

g k = +∞ 0 Γ 2 (t)L k (t)e -t dt +∞ 0 L 2 k (t)e -t dt .
Of course, such integral terms can not be computed numerically and they are approached by using Laguerre rule associated to Gauss formula, see [START_REF] Trefethen | Spectral Methods in MATLAB[END_REF]. To post-treat sampled values of Γ 2 (t) (t i , Γ 2 (t i )), i = 1, • • • , n produced by a numerical solver of Damped KdV equation, we first fit the discrete time t i to Gauss points of the Laguerre rule formula, ξ j , j = 0, • • • , m, with m << n, the weight are noted w j ; then we use as data (ξ j , Γ 2 (ξ j ) to make signal time analysis and synthesis:

g n k = m j=0 Γ 2 (ξ j )L k (ξ j )w j m j=0 L 2 k (ξ j )w j , so that Π n (Γ 2 )(t) = m k=0 g n k L k (t),
is the interpolation polynomial of Γ 2 at the points ξ j , j = 0, • • • , m. We display hereafter in Figures [START_REF] Dubois | Introduction à la dérivation fractionnaire ; théorie et applications[END_REF] and ( 9) the time signal analysis and synthesis of Γ 2 (t) for the damping operators considered above. Comparison Laguerre power spectra of Γ 2 (t) for different damping models and we get

γ k =1 γ k =1/k γ k =1/k 2 γ k =k 2 χ [0.2L,0.8L] χ [0.4L,0.6L]
Γ 2 (t m+1 ) ≃ |u(t m )| 2 -|u(t m+1 )| 2 ∆t(|u(t m+1 )| 2 + |u(t m )| 2 )
.

We first consider the case of the local damping in space L = χ [a,b] . Let x i be the (regularly spaced) grid points and ℓ i the vector defined by

ℓ i = 1 if x i ∈ [a, b], 0 otherwise.
The computed values of Γ 2 (t m ) at discrete times t m are the numbers

Γ 2 (t m ) = N i=1 ℓ i u 2 i h N i=1 u 2 i h
, hence, we look to positive real numbers γ i solution of the constraint least square problem Inf

γ i ≥0 M m=0        N i=1 γ i (u m i ) 2 N i=1 (u m i ) 2 -Γ 2 (t m )        2 .
This minimization problem will be solved numerically using the Matlab function fmincon.

In the case of the weak damping operator

L(u) = k∈Z Z γ k ûk e 2iπkx L
we proceed as follows: given the

samples Γ(t m ), m = 0, • • • , M we can write Γ 2 (t) = k∈Z Z γ k |û k (t)| 2 k∈Z Z |û k (t)| 2 .
We want to find the coefficients γ k that fit these relations. Of course we'll look to a finite number of them, so that satisfying at the least square sense the relation

M m=0   |k|≤N γ k | ûk (t m )| 2 |k|≤N |û k (t m )| 2 -Γ 2 (t m )   2
with the constraints γ k > 0.

In summary the sequence γ k solves the problem Inf

γ k ≥0 M m=0   |k|≤N γ k |û k (t m )| 2 |k|≤N |û k (t m )| 2 -Γ 2 (t m )   . (16)
We also solve this problem using the Matlab's function fmincon.

The results we give hereafter concern two situations -Linear KdV equation u t + L(u) + u xxx = 0.

Here the frequencies are not melted by a nonlinear term, and the damping is active independently frequency by frequency. -the nonlinear KdV u t + L(u) + u xxx + uu x = 0.

Here the nonlinear term melts the frequencies In both the cases we have 1 2

d|u| 2 dt + < L(u), u >= 0.

Linear KdV

We start from a soliton-like data u(x, 0) = 2c 1 sech( √ c 1 (x -p 1 L)) 2 with c 1 = 0.16, p 1 = 0.5; here L = 20, N = 2 7 and ∆t = 0.01. The original operator is L γ with γ k = γ = 2.7.

In Figure 10, we observe that the operator is rebuild in a relatively satisfactory way. Now, we take the same data as before, but here γ k = 1 (1 + |k|) 2 . We observe in Figure 11 that the reconstruction is correct only for the low frequencies. This is due to the ill-conditioning of the problem.

Nonlinear KdV

First we give numerical results concerning the reconstruction of the local damping in space L = χ [0.4L,0.6L] . In Figure 12, the result obtained is encouraging: the reconstruction of L is rather satisfactory. Figure 13 represents the reconstruction of a weak damping operator. We point out that this numerical recovering of the damping operator is a first step and improvement will be considered in further works. The technique we used for rebuilding the damping operator is the same in the two situations we have considered. However, if the results are encouraging when considering the local damping in space, it is not the case for the weak damping operator. |j|≤N |û j (t m )| 2 is ill-conditioned so, for instance, regularization techniques should be considered together with the minimization solver. In a larger way, the reconstruction of L should be done using recent techniques for the solution of inverse problems, see, e.g., [START_REF] Guerrero | A hybrid optimization method for large-scale non-negative full regularization in image restoration[END_REF].

Concluding remarks

The function Γ(t) seems to be an interesting way to compare the large time effect of various damping models for KdV equations and more widely for damped conservative equations of type u t + L(u) + F (u) = 0 [START_REF] Guerrero | A hybrid optimization method for large-scale non-negative full regularization in image restoration[END_REF] with < F (u), u >= 0. The knowledge of samplings of the solution can be exploited in a encouraging way to rebuild a parametrized damping operator. Of course, this last approach as presented here is only a first step and improvements should be obtained using techniques coming from inverse problem area such as those presented recently in [START_REF] Guerrero | A hybrid optimization method for large-scale non-negative full regularization in image restoration[END_REF], particularly for coupling regularization techniques with the solution of ill-conditioned constrained quadratic minimization problems. This is addressed to further developments.

  10 ) : c = 1/75, β = 1/20, b = 101/150, α = 1/2, a = 17/12. -Ordre 8 (error in h 8 ) : β = 1/36, b = 25/54, α = 4/9, a = 40/27, c = 0. -Order 6 (error in h 6 ) : b = 1/9, α = 1/3, a = 14/9, c = 0, β = 0. -Order 4 (error in h 4 ) : b

2 Figure 1 :

 21 Figure 1: Comparison of L 2 -norm of the solution for different damping on [0, L] with L = 20 (left) and of Γ 2 (t) (right).

4. 1 . 2

 12 Local damping in space L = χ [a,b] We now represent the evolution of the solution when considering different damping operators localized in space, say of the form L = χ [a,b] with [a, b] ⊂ [0, L]. We consider the following situations: L = χ [0,L/4] , L = χ [0,L/2] , L = χ [0,3L/2] , L = χ [0,L] . Damping χ [a,b] model with sixth order finite difference scheme with sine initial data b]=[0,L/4] [a,b]=[0,L/2] [a,b]=[0,L/1.5] [a,b]=[0,L]

Figure 2 :

 2 Figure 2: Norm L 2 for the damping χ [a,b] using a sixth order compact scheme.

Figure 3 :

 3 Figure 3: Norm |u| L 2 for the damping χ [0,L/2] using a sixth order compact scheme for initial data: S1, S2 = χ [0.5•L,0.7•L] et S3.

  ) with L(u)=χ [0,L/2] with sixth order finite difference scheme

Figure 4 :

 4 Figure 4: Γ 2 (t) for the damping χ [0,L/2] using a sixth order compact scheme for initial data: S1, S2 = χ [0.5•L,0.7•L] et S3.

1 Figure 5 :

 15 Figure 5: T = 10, N = 2499, 0 < x < 500, a 0 = 1, a 1 = 1, a 2 = 1, a 3 = 6, a 4 = µ.

1 Γ

 1 2 (t) vs time: comparison for different dampings

Γ

2

 2

Figure 6 :

 6 Figure 6: Comparison of the time evolution of L 2 -norm (left) and of Γ 2 (t) (right) for the damping operators L γ and χ [a,b] on [0, L] with L = 20.

1 Γ 2 1 Figure 7 :

 1217 Figure 7: Comparison of the time evolution of L 2 -norm (left) and of Γ 2 (t) (right) for the damping operators

Figure 8 :

 8 Figure 8: Comparison of the power spectrum of Γ 2 (t) for the damping operators L γ and χ [a,b] on [0, L] with L = 20 (left), reconstruction of the signal (right)

Figure 9 :

 9 Figure 9: Comparison of the power spectrum of Γ 2 (t) for the damping operators L(u) = -µu xx + √ ν t

Figure 10 :

 10 Figure 10: Γ 2 (t) (left), comparison of the original and the rebuilt coefficients γ k (right)

Figure 11 :

 11 Figure 11: Γ 2 (t) (left), comparison of the original and the rebuilt coefficients γ k = 1 (|k| + 1) 2 (right)

Figure 12 :

 12 Figure 12: Γ 2 (t) (left), comparison of the original and the rebuilt coefficients L = χ [0.4L,0.6L] (right)

Figure 13 :

 13 Figure 13: Γ 2 (t) (left), comparison of the original and the rebuilt coefficients γ k (right)

  , • • • M ; these values can be computed by a numerical code or measured experimentally. Indeed, with theses data we are able to compute approximate values of Γ 2 (t m ): by integrating the differential equation

	10 -1				Comparison Laguerre power spectra of Γ 2 (t) for different damping models						10 -1					Reconstruction Γ 2 (t) for different damping models
											µ=0.1,ν=0													µ=0.1,ν=0
											µ=0,ν=0.1													µ=0,ν=0.1
											µ=0.1,ν=0.1													µ=0.1,ν=0.1
	10 -2
	10 -3
	Lk(Γ2(t))												Log(Γ2(t))	10 -2
	10 -4
	10 -5
	10 -6	0	2	4	6	8	10	12	14	16		18		10 -3	0	2	4	6	8	10	12	14	16	18	20
						t														t
									1 2	d|u| 2 dt	+ Γ 2 (t)|u| 2 = 0,
	between t m and t m+1 with, e.g., the semi-implicit scheme
					|u(t m+1 )| 2 -|u(t m )| 2 ∆t	+ Γ 2 (t m+1 )	1 2	(|u(t m+1 )| 2 + |u(t m )| 2 ),

The second author is supported by the program "appui à l


Samuel Huron
email: samuel.huron@cybunk.com

Jean-Daniel Fekete

User-Supplied Sentiments in Tweets

Keywords: live event, Twitter, tagging, video annotation, crowdsourcing, Natural Language Processing (NLP). ACM Classification H.5.3 Information Interfaces and Presentation: Group and Organization Interfaces Human Factors, Design, Measurement

Microblogging has become very popular among web users: Twitter broadcasted 200 millions of them every day in 2011. These tweets can be used for social studies, opinion mining, and sentiment analysis. Tweets have been effectively used to analyze opinions and sentiments from specific events such as TV political debates or conference presentations. To analyze tweets, researchers either use Natural Language Processing or human-analyses, e. g. with Amazon Mechanical Turks. In this paper we describe a third method that can replace or complement the first two: convincing the tweet authors to explicitly express their opinion using a simple unambiguous syntax-a tag-while they tweet.

We report on our experience with the PolemicTweet system where authors tag the tweets they send during conferences and TV shows. We explain how we implemented PolemicTweet and discuss briefly the pro and cons.

We believe that author-tagging technologies can effectively complement automated analysis, in particular for analyzing tweets that are by essence ambiguous, and could be ironic, sarcastic or cryptic. We describe the incentive we used to convince the authors and the result we obtained.

INTRODUCTION

In the recent years, microposts such as tweets have been used to analyze data about particular events, such as political opinions during political TV debates [START_REF] Diakopoulos | Diamonds in the rough: Social media visual analytics for journalistic inquiry[END_REF], sport events [START_REF] Tang | #epicplay: crowd-sourcing sports video highlights[END_REF], and crisis coordination [START_REF] Cameron | Emergency situation awareness from twitter for crisis management[END_REF]. Analyzing tweets is not only useful for the sake of understanding what people think about particular topics in general, it also informs about particular events Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. happening at some place in the world (e. g. a tornado hitting a place) or broadcasted on TV and visible on a large scale (e. g. the Olympics or a political TV debate). As such, tweetsas well as other social networks such as Facebook-provide real-time sources of annotation for spatio-temporal events.

To analyze tweets, researchers have been using Natural Language Processing (NLP) or human-made analyses e. g. with Amazon Mechanical Turks. While useful, NLP analyses suffer from several pitfalls:

• Their cost is proportional to the number of messages to process and can become substantial;

• Most NLP research is targeted at English and not usable with other languages that still use tweets heavily and are of great interest (e. g. Arabic speaking countries);

• Even for English, sentiment analysis or topic classification on tweets is a complex process without "off the shelf' solution yet;

• NLP methods suffer from precision1 problems due to the frequent use of irony, sarcasm, slang, and abbreviations.

Human analysts are better at understanding the language than NLP but are more costly both in time and price. They are not immune of precision problems due to the ambiguity of the natural language and the difficulty of interpreting irony, sarcasm, slang, and abbreviations.

In this position paper we describe a third method that complements the first two: persuading tweet authors to express their intents using an unambiguous coding-a tag-in their microposts. This method avoids having to interpret the intents of the tweets afterwards. In our current system, we ask authors to express 4 intents: agreement, disagreement, a question, or that they provide a reference information (e. g. a quote) related to a topic. To be effective, our method requires a careful mix of incentive and simplicity, as reminded by Shipman et al. [START_REF] Shipman | Formality considered harmful: Experiences, emerging themes, and directions on the use of formal representations in interactive systems[END_REF]: without incentive, people will not spend more time adding explicit statements in their microposts; if expressing their intents is complex or cumbersome, they will not do it either.

Our method enriches the micropost's stream with information that is cheap, easy to parse, work in every language, and is not sensitive to bad interpretation. We report on our experience with author-tagging of conferences and TV shows, we explain how we implemented our method, and discuss its pro and cons. The method has been used for about 30 conferences so far and a national TV shows in the last two years.

Our method-because it is cheap and reliable-can be deployed to provide an initial set of annotations related to space and time for visual analytics.

RELATED WORK NLP for Sentiment Analysis on Tweets

In a few years, NLP techniques on tweets have improved greatly due to the availability of large corpora and the design of appropriate language models with short words and non standard English [START_REF] Pang | Opinion mining and sentiment analysis[END_REF]. O'Connor [START_REF] O'connor | From tweets to polls: Linking text sentiment to public opinion time series[END_REF] shows that lexiconbased classification techniques have a very low recall when using a lexicon of standard English while many tweets are written in an informal dialect. Today the combination of different techniques, such as machine learning based approach [START_REF] Barbosa | Robust sentiment detection on twitter from biased and noisy data[END_REF], semantic rule based approach [START_REF] Jiang | Target-dependent twitter sentiment classification[END_REF], and graph based optimization [START_REF] Speriosu | Twitter polarity classification with label propagation over lexical links and the follower graph[END_REF], provide a significant improvement of sentiment classification, achieving about 86% of accuracy [START_REF] Jiang | Target-dependent twitter sentiment classification[END_REF].

Still, NLP suffers from the pitfalls described in the introduction: a cost proportional to the number of messages to process, limited to English, not readily available from "off the shelf" product, and limited in recall.

Human Annotation via Crowdsourcing

According to MITNews [START_REF] Hardesty | Making crowdsourcing easier[END_REF] "Crowdsourcing is a technique for processing a task over the internet by splitting it into small chunks that dozens, hundreds or even thousands of people complete." This technique is particularly useful for tasks that are trivial for humans but difficult, for computers [START_REF] Quinn | Human computation: a survey and taxonomy of a growing field[END_REF].

Crowdsourcing systems have demonstrated their effectiveness with several applications such as photo selection [START_REF] Bernstein | Crowds in two seconds: Enabling realtime crowd-powered interfaces[END_REF], data analysis [START_REF] Willett | Strategies for crowdsourcing social data analysis[END_REF], and question answering services [START_REF] Bigham | Vizwiz: nearly real-time answers to visual questions[END_REF].

Diakopoulos and Shamma [START_REF] Diakopoulos | Characterizing debate performance via aggregated twitter sentiment[END_REF] have crowdsourced sentiment classification for tweets using Amazon Mechanical Turks. Turkers were paid $0.05 for ten classifications and have to recognize 4 types of sentiments: negative, positive, mixed, and "other". The corpus of tweet was in English, on a binary choice topic (2008 US presidential debate). In a second article [START_REF] Diakopoulos | Diamonds in the rough: Social media visual analytics for journalistic inquiry[END_REF], they applied machine-learning algorithms to perform the same analysis with lower precision and cost but higher speed. Crowdsourcing sentiment classification on tweets is now a standard service provided by the CrowdFlower [START_REF]Senti crowdflower[END_REF] crowdsourcing platform, although it is limited to English.

Crowdsourcing suffers from the following pitfalls: cost, speed, (it is much slower than automated algorithms), and it is not immune from ambiguity. We have designed PolemicTweet [START_REF] Huron | Demo: Polemical video annotation by twitter[END_REF] to address some of these pitfalls.

PolemicTweet

PolemicTweet (PT) was developed in response to the need for tagging videos of conferences to facilitate their browsing and analysis. A large number of organizations (universities, companies, institutions) organize middle-sized events, from workshop to conferences, gathering 20 to 300 attendees, each session lasting about 1 to 2 hours.

Many of these events are video recorded for archival or for sharing to a broader audience. When we started to develop PolemicTweet, automatic sentiment classification methods on tweets were not as effective as they are now, at least for English; since PT is mainly deployed in France, automatic sentiment classification is still not an effective option.

Therefore, PolemicTweet relies on tweet authors to insert special tags in their tweets to express their intents with regard to the presentation. PolemicTweet currently supports 4 intents, expressed by inserting two repeated characters anywhere in the tweet: agreement ++, disagreement --, question ??, and reference material added ==. This information is collected and used to annotate the produced video (Figure 1), as can be seen on the web site [START_REF] Samuel Huron | Polemic tweet[END_REF] that already contains about 30 annotated videos.

The colored bar chart added to the video navigation panel is a navigation aid that shows every tweet about the event as a colored 5 pixels square. The color code is straightforward: green for agreement, red for disagreement, orange for a question, and blue for reference material added. Grey squares represent tweets with no PolemicTweet tag.

Protocol

To increase the chance that participants use our syntax in their tweets, we have designed a special social protocol and some tools. It is difficult to know precisely how much each of these contribute to convincing the participants but, overall, we obtain an average around 40% and a peak at 77% when the audience is very active. We consider the protocol crucial to the success of PolemicTweet: incentives must be used to convince tweet authors to use the syntax.

The social protocol consists of:

Before the event we have designed a "connection package" to make sure the audience is properly informed. It consists of a flyer, given to attendees during the registration process, containing information about the wireless network, the PT syntax, the URL of the event web page, and the Twitter hashtag for the event.

During the event the web page of the event can be used to read and send tweets with support for the syntax. It includes colored feedback about the tweets that include the syntax. We also record all the tweets that use the defined hashtag.

After the event a tweet is sent to the participants notifying them of the availability of the recorded video on the web site and thanking them for their participation. The video captured during the event is published on the PT web site with the visualization of the tweets activity.

We have implemented two tools to support our protocol and process:

A PT-enhanced tweet client: a web application that allows reading and sending tweets with support for our syntax (Figure 2). By visually highlighting tagged tweets in real time, this web application creates an awareness of the other opinions and incentive to tag her/his own tweets. The Metadata Player: a web application that extends existing video player by adding the color tweets bar chart (Figure 1) [?]. This video player allows navigation on the video using either the standard controls or by interacting with the tweets. By moving the mouse over a square, the text of the tweet is displayed on a popup window. Clicking on a square cues the video at the time it was sent.

DISCUSSION

We designed to PolemicTweet to annotate tweets but is has become a more complete system, because the live feedback provided by the tweets reader in our client adds a lot to the experience of the audience. Participants can share their feelings with others, get a quick overview of the mood and reactions, organizers can gather feedback, questions and reference material about the talks, and presenters can collect feedback on their talk too. So far, the feedback has never been insulting or embarrassing, even if, at times, comments have been negative. PolemicTweet is used on a regular basis and other institutions have been asking to use it for their own events, which is a good sign of success.

The cost of PolemicTweet is different than using NLP or human annotation: it is mostly independent of the number of tweets since tweet authors enter the annotations with no extra cost for the conference organizers. The quality is also different: less ambiguous since authors express their intents, but with only 40% of annotated tweets. As raised in the introduction, nothing prevents NLP or human annotations to be performed to the remaining tweets if needed.

PolemicTweet has been also used to annotate national TV shows, under the name Bubble TV. We have modified the social protocol and tools to present a more appealing interface with animations to be attractive on TV. Here again, tweet authors have used a variant syntax for the tags so the resulting stream of tweets can be used to create an annotated a time line of each show. In that particular case, the goal was more to receive live feedback about the show presenters than to annotate it. Still, both feedbacks are useful and usable with no extra cost.

With our experience deploying PolemicTweet, we see it as a practical tool to generate an annotated timeline to help the exploration and analysis of recorded events. However, it also has limitations.

Limited number of tags: tweet authors will never remember more than a few tags: PolemicTweet cannot provide richly encoded annotations. Yet, other methods based on NLP or crowdsourcing have been very limited too up to now, so PolemicTweet remains useful and could be complemented by entity recognition or other technologies when they become robust enough on tweets.

Tags should be known in advance: exploration of topics and alternative sentiments on tweets cannot be done with our technique, and changing the tag set during the event would certainly confuse the participants and lower the quality and quantity of tagged tweets.

Non technical communities are excluded: we noticed a strong variation on the number of tagged tweets when the audience changed from high-tech communities to more traditional ones.

CONCLUSION AND FUTURE WORK

This position paper explains how we implemented PolemicTweet, a system inciting people to tag their tweets with unambiguous expressions of some of their intents. PolemicTweet specifies a simple syntax for agreement, disagreement, questions, and the adding of reference material in a tweet.

The deployment of PolemicTweet for a series of about 30 events during 2 years shows the practicality of our approach and very positive results in annotating videos with a limited amount of effort. Beyond the relative success of our system, we have witnessed a great interest from our audience and from other event organizers; we assume it is due to the increased engagement offered by PolemicTweet.

There are many events and text posting systems that could benefit from user-supplied sentiments to mine opinions in real time, e. g. social TV, classrooms, or public debates. Our technique would provide some level of unambiguous feedback to the audience and organizers. This position paper also introduced a first step towards the real-time exploration and classification, of sentiments using visualization. We believe our method is applicable to a wide variety of different scenarios and events.

For future work, we would like to test if we can apply this type of method to more complex tasks (e. g. sharing classroom annotations). We also would like to better understand the degree of incentive induced by each component and how to improve it, as well as how we can enhance the degree of tag expressivity without compromising the simplicity of the interface. We will also distribute an open implementation of PolemicTweet to facilitate its wider deployment, adoption, and foster improvements.

Figure 1 .

 1 Figure 1. The Metadata Player: a Video Player with the Polemic Timeline.

Figure 2 .

 2 Figure 2. Enhanced Tweeter client used during the live conference to visualize tweets and remind the PT syntax.

We refer to precision and recall as in Pattern Recognition or Machine Learning. Precision is the fraction of retrieved instances that are relevant (i. e. well classified), while recall is the fraction of relevant instances that are retrieved[START_REF]Precision and recall[END_REF].