S Betelu
email: s.betelu@brgm.fr

G Berger

E Giffaut

I Ignatiadis
email: i.ignatiadis@brgm.fr

Mechanistic and kinetic study of pyrite-hydrogen interaction at low temperature using electrochemical techniques

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mechanistic and kinetic study of pyrite-hydrogen interaction at low temperature using electrochemical techniques

Stéphanie Betelu, Gilles Berger, E. Giffaut, Ioannis Ignatiadis

To cite this version: Within the context of the investigation of the impact of the geochemicalevolutions of the nuclear waste storages on the surroundingmaterial, the objective of the work presented here was the characterization of pyrite interaction with hydrogen at low temperature (25-90°C). The emphasis was put on theunderstandingin depth of bothphenomenologyandkinetics.

For this purpose, the use of electrochemical techniques in combination with pyrite mineral electrodes has constituted a novel experimental and complementary approach in comparison with the geochemical research realized by [START_REF] Truche | Kinetics of pyritetopyrrhotitereductionbyhydrogen in calcitebuffered solutions between 90 and 180°C: implicationfor the nuclear waste disposal[END_REF] under hyper-thermaltemperatures, from 90 to 180°C andunder H 2 partialpressurespH 2 rangingfrom 8 to 18 bars.

Experiments wereconductedin a wide range of conditions (pH, temperature, ionic composition, gaseous environment) close to the expected evolution of the repository groundwater chemistry. More specifically, after pyriteelectrodes had been designedandassembled (figure 1A), variouselectrochemicaldisturbanceswereappliedto FeS 2 electrode (andtoplatinumelectrode (Pt) forcomparison) whileit was submerged in a partiallyreconstitutedCOxpore water solution, enclosed in a High PressureThermo-Reactor (HPTR) at 90°C (figure 1A) or in a Low PressureThermo-Reactor (LPTR) (figure 1B) at 25°C, in the absence and in the presence of (i) pyritegrains (particlesizerangingfrom 40 to 63 µm) and (ii) H 2 (pH 2 rangingfrom 0 or 1 bar).

Experiments that were performed in carbonaceous buffers (HCO 3 -/CO 3 2-buffer, pH 8.6 and 10.0) inthe HPTR(at 90°C, which is the maximal temperature expected in the repository) with or without hydrogen led us to determine mechanism and estimate pyrite corrosion rates.

Experiments thatwereperformed in anoxic environment (N 2 and/or H 2 ) in carbonaceous buffer in the LPTR(at 25°C) in whichhydrogen was producedin situbyelectrolysis led to comfort phenomenology at higher pH, from 9.3 to 11.2. Indeed polarizationprovided H 2 and OH -formation in the solution, and thus its progressive alkalization.

Mechanism was established by using potentiometric measurements, linearsweeppolarization(LSP)and Electrochemical Impedance Spectrometry (EIS). Both electrochemical (Tafel plots and Polarisation Resistance (R p ))and gravimetric (mass loss) approaches were used as complementary methodologies to estimate pyrite corrosion rates. The elucidation of the mechanismgoverning the reduction of the sulphur S° of the FeS 2 was the first challenge we answered. Indeed, the S° is reducedtosulphide (HS -/S 2-), which is freed in solution, whereas the pyriteturnssuperficiallyintopyrrhotite (FeS 1+x with 0<x<0.125), forming a layerthat slows down the reductionprocess. Experiments conductedintoHPTR and LPTR lead to approach phenomenology (whichwillbediscussed)versus pH. Investigations conducted by electrochemistry (at 90°C in high pressure thermo-reactor) allowed us to provide Pyrite corrosion rates in the absence and in the presence of hydrogen. It is worth noting that they are of the same order of magnitude 10 -3 mm.year -1 whatever the pH (in the range 0.5 mm.year -1 -3.6 10 -3 mm.year -1 ). In parallel, experiments conducted by gravimetric measurements were of prime importance. Both electrochemical and gravimetric measurements allow us to identify and distinguish the following phenomena on pyrite: (i) corrosion and corrosion-scaling measurable by electrochemistry, (ii) corrosion and chemical dissolution measurable by gravimetric measurements. Results provided by electrochemical and gravimetric measurements will be discussed and compared to the calculated corrosion rates using the rate law determined by Trucheet al.(2010).

Stéphanie Betelu ,

 Betelu Gilles Berger, E. Giffaut, Ioannis Ignatiadis. Mechanistic and kinetic study of pyrite-hydrogen interaction at low temperature using electrochemical techniques. Clays in Natural and Engineered Barriers for Radioactive Waste Confinement : Clay Conference 2015, Mar 2015, Brussels, Belgium. hal-01118763

  Electron Microscopy (SEM)and Energy-dispersive X-ray spectroscopy (EDS).

Figure 1 :

 1 Figure 1: A) FeS 2 electrode made with "pure" Peruvianpyriteforexperiments realized in a Hastelloy HighPressure Thermo-Reactor (HPTR) B) Electrolysis scheme experiment of the Low Pressure Thermo-Reactor (LPTR).

Acknowledgement

The research has receivedfundingfrom i) the European Union's European Atomic Energy Community's (Euratom) FP7/2007-2011 undergrant agreement n° 212287 (RECOSY project) and ii) the ANDRA under the BRGM-ANDRA partnership (TRANSFERT and CAPTANDRA projects).

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 oral presentation Topic of scientific programme:


Christophe Tournassat

Ian C Bourg

Carl I Steefel

Breakdown

teaching and research institutions in France or abroad, or from public or private research centers.   Breakdown of the modified Gouy-Chapman model for clay surfaces in equilibrium witha CaCl 2 electrolyte: a molecular dynamic study

Breakdown of the modified Gouy-Chapman model for clay surfaces in equilibrium witha

Clay minerals have a high specific surface area that enhances their ability to influence water in their vicinity. Their surface charge, which varies as a function of physical-chemical conditions of the surrounding water, results in a complex organization of solute ions as a function of distance from the surface. This organization is known as the electrical double layer where counter-ions are electrostatically attracted to the surface whereas co-ions are repelled from it. Abundant and convincing spectroscopic evidence has emerged of a complex layering of water molecules and ionic species adsorbed at specific distances from the charged surfaces, a feature that can be captured by surface complexation models derived from Stern theory. The modified Gouy-Chapman model (MGC), or its modified version based on the resolution of the full Poisson-Boltzmann equation, is often used to calculate the electrostatic potential value in the diffuse region between the Stern layer and the bulk solution. Based on a comparison of the MGC model with Monte-Carlo simulations of discrete hard-sphere ions immersed in an implicit solvent, it was demonstrated long ago that the resolution of the full Poisson-Boltzmann equations can give accurate results for cations and anions distribution in the vicinity of a charged surface if both anions and cations are monovalent, if their total concentration does not exceed 0.1 mol/dm 3 and if the surface charge is not too high. When divalent (or multivalent) ionic species are present, the MGC model fails at reproducing Monte-Carlo simulations results in most conditions, because the MGC model neglects ion-ion interactions. For example, for 2:1 electrolytes the MGC model is thought to be inaccurate at salt concentrations greater than 0.005 mol/dm 3 [START_REF] Torrie | Electrical double layers. 4. Limitations of the Gouy-Chapman theory[END_REF][START_REF] Carnie | Advance in Chemical Physics[END_REF][START_REF] Boda | Monte Carlo and density functional theory study of the electrical double layer: the dependence of the charge/voltage relation on the diameter of the ions[END_REF].

Deviations from Poisson-Boltzmann model predictions probed by spectroscopic tools are now also available. [START_REF] Bu | How accurate is Poisson-Boltzmann theory for monovalent ions near highly charged interfaces?[END_REF] demonstrated,using anomalous X-ray reflectivity data, that the MGC model is strikingly accurate at a highly charged interface in contact with a homovalent monovalent salt background up to a concentration of 10 -2 mol dm -3 . Another study performed in similar conditions found that while accurate at 0.01 mol dm -3 , the Poisson-Boltzmann model failed to reproduce X-ray reflectivity data at higher concentrations [START_REF] Luo | Ion distributions near a liquid-liquid interface[END_REF]. Molecular dynamics (MD) calculation studies have provided broadly similar results to those obtained by X-ray reflectivity [START_REF] Wang | Electroosmosis in homogeneously charged micro-and nanoscale random porous media[END_REF][START_REF] Lima | Specific ion adsorption and surface forces in colloid science[END_REF][START_REF] Lee | Comparison of Molecular Dynamics with Classical Density Functional and Poisson-Boltzmann Theories of the Electric Double Layer in Nanochannels[END_REF]. However, these studies were performed using homovalent monovalent electrolytes that are not generally representative of environments that are rich in clay minerals, whose surface sites tend to be occupied by divalent cations in natural clay formations.

In the present study, we evaluated the accuracy of MGC model predictions in a clay/water system in the presence of divalent cations and monovalent anions using MD calculations. Clay surfaces present several advantages with regards to molecular simulations owing to the 2D pseudo-periodicity of the clay particles. This system is particularly well suited to enabling accurate comparison between the MGC model prediction and the actual distributionof ions at the solid-water interface. Amongst clay minerals, montmorillonite has been the matter of multiple of MD simulation studies, butthe vast majority of them involved only monovalent cations in interlayer spaces without the presence of anions. Few studies have been focused on divalent cations and/or charged surfaces in contact with large pores where the diffuse layer can fully extend without overlapping [START_REF] Tournassat | Comparison of molecular dynamics simulations with Triple Layer and modified Gouy-Chapman models in a 0.1 M NaClmontmorillonite system[END_REF][START_REF] Bourg | Molecular dynamics simulations of the electrical double layer on smectite surfaces contacting concentrated mixed electrolyte (NaCl-CaCl2) solutions[END_REF]. We will show that the MGC model prediction of the electrostatic potential value at the surface as well as the charge compensation over the distance from the surface are considerably more accurate than previously inferred from its comparison with Monte-Carlo simulations of discrete hard-sphere ions immersed in an implicit solvent.

  CaCl 2 electrolyte: a molecular dynamic study. Christophe Tournassat a* , Ian C. Bourg b , Carl I. Steefel b a Water and Environment division, French Geological Survey (BRGM), Orléans, France b Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720 USA *corresponding author: c.tournassat@brgm.fr


Éric Laporte

Choosing features for classifying multiword expressions

Working Group 1: Lexicon/Grammar Interface A major requirement for a classification of MWEs is that it be explicitly based on features of the MWEs. Designing a satisfactory classification involves choosing such features. Accordingly, the resulting classification may be more or less fruitful for computational use.

Equivalence between features

By feature we mean a property which can differ between MWEs, and therefore be used to assign them to different classes. For example, the French verbal idiom guérir le mal par le mal 'fight fire with fire' admits a cleft construction, as in C'est par le mal qu'on guérit le mal 'It is with fire that you fight fire', whereas rater un éléphant dans un couloir 'be unable to hit the broad side of a barn' does not: *C'est dans un couloir que tu raterais un éléphant.

Distinct features are equivalent if they are observed in the same lexical entries. For example, 'decomposable [verbal idioms] tend to be syntactically flexible, in a manner predicted by the nature of the semantic decomposition; non-decomposable [verbal idioms], on the other hand, tend not to be syntactically flexible' (Baldwin & Kim, 2010:277). Such an equivalence invites you to use the features for classification: assigning an entry to a class implicitly specifies both features at the same time. Thus, Baldwin & Kim (2010:279, Fig. 12.1) subdivide verbal idioms into two subclasses: non-decomposable idioms 'with hard restrictions on word order and composition'; decomposable, syntactically flexible idioms.

However, the temptation should be resisted until a systematic investigation confirms the intuition of equivalence between the features. In the case of decomposability and syntactic flexibility, [START_REF] Baldwin | Multiword Expressions[END_REF] do not make use of comprehensive lexical databases of MWEs. An examination of facts in the lists of French verbal idioms (available at http://infolingu.univ-mlv.fr/) by [START_REF] Gross | Une classification des phrases "figées" du français[END_REF] readily reveals counter-examples. Trouver chaussure à son pied 'find the perfect match' seems decomposable as find(x, partner), but does not admit syntactic variations. Conversely, mettre toutes les chances de son côté 'not take any chances' is hardly semantically decomposable, but admits the passive form: Au moins, toutes les chances sont mises de mon côté 'At least, I am not taking any chances'.

When a classification assumes features are equivalent and uses them as a single criterion, it takes the risk of misclassifying the entries for which they diverge. This compromises computational usage: a classification should ensure the members of each class have the corresponding defining features. Thus, it is a good practice to specify each criterion precisely and specifying which entries have which features, like in Lexicon-Grammar tables [START_REF] Freckleton | Sentence idioms in English[END_REF].

Reproducibility of features

Among the features that discriminate MWEs, which should be used for classification, and therefore investigated in priority? Of course, linguistic intuition plays a prominent role in this selection, but reproducibility is relevant too. Reproducible observations are those inherently susceptible to high inter-judge agreement. Features are not equal in this respect. For example, the existence of cleft constructions of a MWE is judged by checking the grammaticality of some sentences, which is relatively factual; semantic decomposability relies on pure semantic intuition: for se mettre le doigt dans l'oeil 'bark up the wrong tree', how else to arbitrate between speakers who associate le doigt with the semantics of 'opinion(x)' and dans l'oeil with 'false', and those that do not? Reproducibility of observation is classically improved by adjusting feature definition, and in particular by resorting to formal or syntactic criteria. Among semantic features, differential semantic evaluation is more reproducible than absolute semantic evaluation (Gross, 1975:391).

Features with high reproducibility of observation are a good basis for classification. 'The exact form of syntactic variation [of verbal idioms] is predicted by the nature of their semantic decomposability' [START_REF] Baldwin | Multiword Expressions[END_REF], but it would not be effectual to infer their syntactic variation from a description of their semantic decomposability, because the former is more reproducibly observable than the latter. Existing classifications and tables of idioms (e.g. [START_REF] Gross | Une classification des phrases "figées" du français[END_REF][START_REF] Freckleton | Sentence idioms in English[END_REF] produced by users of the Lexicon-Grammar method [START_REF] Gross | The Lexicon-Grammar of a Language: Application to French[END_REF] prioritize two types of features particularly easy to observe: syntactic variations and selectional restrictions. Other interesting features could be encoded too, e.g. the possibility of anaphoric reference to a component of a verbal idiom. Applying the criterion of distributionally frozenness, citer un témoin 'call as a witness' is idiomatic, because the verb citer has this meaning only with this noun. Still, anaphoric reference to the noun is possible: La défense a cité un témoin. Il vient de s'exprimer 'The defence called a witness. He has just testified'. This property can be observed in a reasonably reproducible way, due to the formal criterion involved in its definition.

Reproducibly observable features are often useful for language processing when then determine the possibility of occurrence of actual forms, such as a cleft variant, or the anaphoric form above: this is essential to automatically recognising such MWEs.

Cost and benefit of intensive description of the lexicon

The examples above illustrate the benefits of intensive description of the lexicon. It is costly, this might explain why it is still alien to most of NLP research. However, such work is not unfeasible: lexicon-grammars of MWEs with representation of individual features have been published as early as 1974 (Labelle) for support-verb constructions in French, and 1985 (Freckleton) for verbal idioms in English (both available at http://infolingu.univ-mlv.fr/). Some are used in parsing now [START_REF] Constant | Combining Compound Recognition and PCFG-LA Parsing with Word Lattices and Conditional Random Fields[END_REF].


Matthew Paddick
email: paddick@ljll.math.upmc.fr

Transverse nonlinear instability of Euler-Korteweg solitons

Keywords: Euler-Korteweg system, solitary waves, nonlinear instability AMS classification: 35C08, 35Q35, 37K45

We show that solitary waves for the 2D Euler-Korteweg model for capillary fluids display nonlinear instability when subjected to transverse perturbations.

Introduction

We consider the motion of a compressible, inviscid and isentropic planar fluid, in which internal capillarity is taken into account. This phenomenon occurs for example at diffuse interfaces in liquid-vapour mixes [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF]. In this model, the free energy of the fluid depends on both the density of the fluid, the scalar function ρ, and its gradient ∇ρ in the following way:

F (ρ, ∇ρ) = F 0 (ρ) + 1 2 K(ρ)|∇ρ| 2 ,
with K and F 0 two given smooth, positive functions. We then derive the pressure from the free energy as follows:

P (ρ, ∇ρ) = ρ ∂F ∂ρ -F = P 0 (ρ) + 1 2 (ρK ′ (ρ) -K(ρ))|∇ρ| 2 ,
in which P 0 is the standard part of the pressure. Let g 0 (ρ) be the bulk chemical potential of the fluid, so that ρg ′ 0 (ρ) = P ′ 0 (ρ). Then, the principles of classical mechanics yield the Euler-Korteweg equation that we will study:

   ∂ t ρ + div (ρu) = 0 ∂ t u + (u • ∇)u = ∇ K(ρ)∆ρ + 1 2 K ′ (ρ)|∇ρ| 2 -g 0 (ρ) (ρ, u)| t=0 = (ρ 0 , u 0 ). (1)
The variables are t ∈ R + and (x, y) ∈ R 2 ; as is standard, the operators ∇, div and ∆ contain only derivatives with respect to the space variables x and y. The unknowns of equation [START_REF] Benjamin | Impulse, flow force and variational principles[END_REF] are the density ρ and the velocity vector field u : R + × R 2 → R 2 . The scalar functions g 0 and K are given.

In this paper, we will be interested in the transverse stability of solitary wave solutions of [START_REF] Benjamin | Impulse, flow force and variational principles[END_REF]. These are 1D travelling waves written as

Q c (t, x) = ρ c (t, x) u c (t, x) = Q c (x -ct),
with u c scalar (not a 2D vector field). Based on a remark by T. Benjamin [START_REF] Benjamin | Impulse, flow force and variational principles[END_REF], S. Benzoni, R. Danchin, S. Descombes and D. Jamet showed in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] that the hamiltonian structure of the system led to the existence of solitary wave solutions for every c ∈ R in the case of a van der Waals-type pressure law. There are two kinds of travelling-wave solutions, depending on the homoclinic or heteroclinic nature of the connecting orbits. In the homoclinic case, the wave has identical endstates, and we can write

lim |z|→+∞ Q c (z) = Q c,∞ = (ρ c,∞ , u c,∞ ),
and such a travelling wave solution is called a soliton. We will be studying this type of solitary wave. In the heteroclinic case, the endstates are different, but must nonetheless satisfy a Rankine-Hugoniot-type condition. See [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] for more on these solutions, which are called kinks.

From now on, we set c and Q c is a soliton, whose endstate, which we also set (thus we drop the index c), satisfies

ρ ∞ g ′ 0 (ρ ∞ ) > (u ∞ -c) 2 , (2)
which means that Q ∞ is a saddle point for the hamiltonian ODE satisfied by Q c . Under this condition, we have that ρ ′ c vanishes only once.

The standard Lyapunov stability notion is that if solutions of equation [START_REF] Benjamin | Impulse, flow force and variational principles[END_REF] have initial conditions close to Q c , they remain close to Q c (x -ct) at all times. But this notion is not satisfactory in describing the stability of travelling waves. Indeed, let c ′ = c be close to c; we then have that the profile Q c ′ is close to Q c , but, as the speeds are different, Q c ′ (x -c ′ t) and Q c (x -ct) drift apart, despite their profiles remaining very similar. To see this, for a given t, compare Q c ′ (x -c ′ t) with the translated profile

Q c (x -c ′ t) = Q c (x -ct + (c -c ′ )t).
The correct notion of stability therefore stems from taking the difference of solutions with all the translated versions of Q c (x -ct). A travelling wave solution will be considered stable if it is orbitally stable: for every ε > 0, there exists

δ > 0 such that if (ρ 0 , u 0 ) -Q c ≤ δ, then sup t∈R + inf a∈R (ρ(t), u(t)) -Q c (• -a -ct) ≤ ε.
The problem of orbital stability can be divided into two parts, depending on the type of perturbation we consider. 1D perturbations are perturbations of Q c that depend only on x and satisfy u 2 (t, x) = 0. The stability problem associated with these perturbations has been in part dealt with by S. Benzoni et al. in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF], and improved upon by Benzoni in [START_REF] Benzoni-Gavage | Spectral transverse instability of solitary waves in Korteweg fluids[END_REF]. A sufficient condition for orbital stability was obtained in the first paper using an argument by M. Grillakis, J. Shatah and W. Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF], while the second article adds a sufficient condition for linear instability. See also J. Höwing [START_REF] Höwing | Stability of large-and small-amplitude solitary waves in the generalized Korteweg-de Vries and Euler-Korteweg/Boussinesq equations[END_REF][START_REF] Höwing | Standing solitary Euler-Korteweg waves are unstable[END_REF] for other stability results for the 1D Euler-Korteweg system.

The question of transverse stability deals with perturbations that also depend on the transverse variable y and have a 2D velocity field. So far, Benzoni in [START_REF] Benzoni-Gavage | Spectral transverse instability of solitary waves in Korteweg fluids[END_REF] and F. Rousset and N. Tzvetkov in [START_REF] Rousset | A simple criterion of transverse linear instability for solitary waves[END_REF] have proved linear instability. This occurs when the linearised equation around Q c has eigenvalues with positive real part. On one hand, Benzoni used Evans functions computations to get that orbitally stable solitons are transversally linearly unstable. On the other hand, Rousset and Tzvetkov applied a simple, abstract criterion for instability in linearised PDEs with a hamiltonian structure in the case where the endstate of the soliton satisfies [START_REF] Benzoni-Gavage | Spectral transverse instability of solitary waves in Korteweg fluids[END_REF]. We recall this criterion in Theorem 2.2. This criterion was applied to other equations with solitary waves in the same article: KP-I and Gross-Pitaevskii.

The result of this paper is that the spectral instability mentioned above implies nonlinear instability of Euler-Korteweg solitons.

Theorem 1.1. Let Q c (x -ct) be a soliton solution to [START_REF] Benjamin | Impulse, flow force and variational principles[END_REF] such that the endstate Q ∞ satisfies [START_REF] Benzoni-Gavage | Spectral transverse instability of solitary waves in Korteweg fluids[END_REF]. Then there exist δ 0 and ε 0 > 0 such that for every 0 < ε < ε 0 , there exists an initial condition U 0 = (ρ 0 , u 0 ) with

U 0 -Q c H s (R 2 ) ≤ ε
for some s > 0, such that, for every a ∈ R, the solution U = (ρ, u) of ( 1) with this initial condition satisfies, at a time T ε ∼ ln(ε -1 ),

U(T ε ) -Q c (• -cT ε -a) L 2 (R 2 ) ≥ δ 0 .
Moreover, the velocity u can be chosen to be potential: u = ∇ϕ.

The proof relies on an argument originally by E. Grenier [START_REF] Grenier | On the nonlinear instability of Euler and Prandtl equations[END_REF], in which one constructs an approximate solution U ap to the equation based on a WKB expansion starting with the reference solution and unstable eigenmodes. An energy estimate on U -U ap shows that, if there are enough terms in the expansion, this difference is small, thus, for times under T ε , the linear instability is dominant. Primarily used to obtain nonlinear instability of boundary layers in numerous settings (unstable Euler shear flows and Prandtl layers [START_REF] Grenier | On the nonlinear instability of Euler and Prandtl equations[END_REF], Ekman layers for rotating fluids [START_REF] Desjardins | Linear instability implies nonlinear instability for various types of viscous boundary layers[END_REF][START_REF] Rousset | Stability of large Ekman boundary layers in rotating fluids[END_REF][START_REF] Masmoudi | Stability of oscillating boundary layers in rotating fluids[END_REF], Ekman-Hartman layers in MHD [START_REF] Desjardins | Linear instability implies nonlinear instability for various types of viscous boundary layers[END_REF], Navier-Stokes with a boundary-layer-scale slip condition [START_REF] Paddick | Stability and instability of Navier boundary layers[END_REF]), the method relies on building approximate solutions from a compact set of unstable wave numbers, thus the idea has been transposed to showing transverse nonlinear instability of solitary waves, when these can be shown to be linearly unstable. F. Rousset and N. Tzvetkov have thus obtained nonlinear instability of solitary waves in many models: KP-I and NLS [START_REF] Rousset | Transverse nonlinear instability for two-dimensional dispersive models[END_REF], multiple hamiltonian models including generalised KP-I and the Boussinesq equations [START_REF] Rousset | Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's[END_REF], and the free-surface water-waves equation [START_REF] Rousset | Transverse instability of the line solitary water-waves[END_REF].

Similarly to many situations, such as KP-I, our result extends to the y-periodic framework R × T L , where T L is the torus of length L > 0, in the following way: there exists a critical period L 0 (c) > 0, depending on the soliton's speed, such that if L > L 0 (c), we have Theorem 1.1. The proof is identical, and limited to above a critical period due to the loss of linear instability below this period (the set of unstable wavenumbers is bounded).

Outline of the proof. The proof of Theorem 1.1 is in two parts. First, in section 2, we build on Rousset and Tzvetkov's linear instability theorem, deriving an important resolvent estimate for the Euler-Korteweg equation, linearised around Q c . This will allow us, in section 3, to build an approximate solution U ap with the appropriate behaviour of being predominantly unstable for t ∼ T ε . An energy estimate on U -U ap will then be used. This is particularly important ensure that the time of existence of U is large enough to get the desired amplification, as we only have local existence for solutions to (1) (see S. Benzoni, R. Danchin and S. Descombes in [START_REF] Benzoni-Gavage | On the well-posedness for the Euler-Korteweg model in several space dimensions[END_REF]). Article [START_REF] Benzoni-Gavage | On the well-posedness for the Euler-Korteweg model in several space dimensions[END_REF] also provides a blowup criterion, but the instability phenomenon is not related to the blow-up if it occurs; indeed, the mechanism is also observed on systems that have global solutions (most of the previous examples, whether they concern boundary layer or solitary wave instability, fall in this category). Finally, combining the two will lead to the instability result.

Linear analysis

In this part, we consider the linearised problem about (ρ c , u c ) to obtain an essential ingredient in order to apply Grenier's method.

Considering that u is potential, we write the system on (ρ, ϕ), where u = ∇ϕ:

∂ t ρ + ∇ϕ • ∇ρ + ρ∆ϕ = 0 ∂ t ϕ + 1 2 |∇ϕ| 2 = K(ρ)∆ρ + 1 2 K ′ (ρ)|∇ρ| 2 -g 0 (ρ
), which we linearise around (ρ c , u c ). Having changed the space variable from x to x -ct (which turns the solitary wave into a stationary solution), we are interested in

∂ t ρ = (c∂ x -u c ∂ x -u ′ c )ρ -(ρ ′ c ∂ x + ρ c ∆)ϕ ∂ t ϕ = (c∂ x -u c ∂ x )ϕ + (K(ρ c )∆ + K ′ (ρ c )ρ ′ c ∂ x -m)ρ, (3)
with m = K ′ (ρ c )ρ ′′ c + K ′ (ρ c )(ρ ′ c ) 2 -g ′ 0 (ρ c
). We abbreviate the system by defining two operators

J = 0 1 -1 0 and L = -∂ x (K(ρ c )∂ x ) -K(ρ c )∂ 2 yy -m (u c -c)∂ x -∂ x ((u c -c)•) -∂ x (ρ c ∂ x ) -ρ c ∂ 2 yy ,
thus the system (3) can be written synthetically as ∂ t V = JLV , where V = (ρ, ϕ).

The first part of the linear analysis involves finding unstable eigenmodes for (3). These are non-trivial solutions to the equation that can be written as V (t, x, y) = e σt e iky v(x) for k = 0 and Re(σ) > 0. We rewrite (3) using the Fourier transform on the transverse variable y: the equation

∂ t V = JLV becomes σv = JL(k)v with L(k) = -∂ x (K(ρ c )∂ x ) + K(ρ c )k 2 -m (u c -c)∂ x -∂ x ((u c -c)•) -∂ x (ρ c ∂ x ) + ρ c k 2 .
We begin by examining the existence of eigenmodes and the behaviour of σ depending on k, and we follow up with an important resolvent estimate for JL(k).

Proposition 2.1. Properties of the linearised equation.

(a) The linearised equation is unstable, that is there exist eigenmodes written as V (t, x, y) = e σt e iky v(x), with v ∈ H 2 (R) and Re(σ) > 0, that solve [START_REF] Benzoni-Gavage | On the well-posedness for the Euler-Korteweg model in several space dimensions[END_REF]. For each k, the dimension of the subspace of unstable solutions of σv = JL(k)v is at most 1.

The instability is localised in the transverse Fourier space: there exists k max > 0 such that, for |k| ≥ k max , eigenvalues necessarily satisfy Re(σ) ≤ 0. Let σ(k) be the eigenvalue of JL(k) with highest real part. Then the function k → Re(σ(k)) has a global maximum σ 0 > 0 at a certain k 0 > 0.

(b) If V (t, x, y) = e iky U(t, x), we define the following semi-norm for U:

U(t) 2 X j k = U 1 (t) 2 H j+1 (R) + ∂ x U 2 (t) 2 H j (R) + |k| 2 U(t) 2 H j (R) .
It is essentially the

H j norm of |k|U(t) plus the H j+1 norm of U(t), omitting the L 2 norm of U 2 (t). Set γ > σ 0 , n > 0, s ∈ N and let U solve ∂ t U(t, x) = JL(k)U(t, x) + F (t, x, k), (4)
with U(0, x) = 0, and, F satisfying, for every j ≤ s,

∂ s-j t F (k, t) H j+1 (R) ≤ M s e γt (1 + t) n , (5)
uniformly for |k| ≤ k max (M s does not depend on k). Then U satisfies a similar estimate: for every j ≤ s, and uniformly for |k| ≤ k max , we have

∂ s-j t U(t) X j k ≤ C s (1 + t) -n e γt . ( 6
)
Consequences of part (b): a quick energy estimate on the equation of U 2 yields that U 2 (t) ∈ L 2 (R 2 ) (as ϕ(0) = 0), and this L 2 norm also satisfies [START_REF] Dunford | Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space[END_REF]. We will therefore subsequently consider that the result is valid in H s , for any s ≥ 0.

By the Parseval equality, this result also implies identical H s bounds for (ρ, u) when written as (ρ, u) = (ρ, ∇ϕ) and (ρ, ϕ) = R f (y)e iky U(t, x) dk, with f ∈ C ∞ 0 (R). Indeed, norms of |k| 2 U can be replaced, using equation ( 4), by derivatives on x and t that satisfy (6).

Proof of Proposition (a)

Properties of eigenmodes 2.1.1 Existence of unstable eigenmodes.

The existence of unstable eigenmodes was shown by F. Rousset and N. Tzvetkov [START_REF] Rousset | A simple criterion of transverse linear instability for solitary waves[END_REF] using a general criterion for detecting transverse linear instability of solitary waves. We have seen that equation (3) for functions written as V (t, x, y) = e σt e iky v(x) becomes an eigenvalue problem, that is

σv = JL(k)v (7)
with J a skew-symmetric matrix and L a self-adjoint differential operator on (L 2 (R 2 )) 2 whose domain is seen to be (H 2 (R 2 )) 2 . We have the following result for such systems.

Theorem 2.2. (Rousset and Tzvetkov, [START_REF] Rousset | A simple criterion of transverse linear instability for solitary waves[END_REF]) If L has the following properties,

• (H1) there exists k max > 0 and α > 0 such that L(k) ≥ αId for |k| ≥ k max ;

• (H2) for every k = 0, the essential spectrum of

L(k) is included in [α k , +∞[ with α k > 0; • (H3) L ′ (k) is a positive operator;
• (H4) the spectrum of L(0) consists of one isolated negative eigenvalue -λ and a subset of R + ; then there exist σ > 0 and k = 0 such that ( 7) has a non-trivial solution, and, for every unstable wavenumber k, such an eigenvalue σ is unique. This is shown by finding k ′ > 0 such that L(k ′ ) has a one-dimensional kernel, and by using the Lyapunov-Schmidt method in the vicinity of this point; we do not detail the proof of this theorem. Proof that the linearised Euler-Korteweg system satisfies the hypotheses of this theorem was also done in [START_REF] Rousset | A simple criterion of transverse linear instability for solitary waves[END_REF], but we shall briefly recall this, as it contains some useful arguments for the subsequent points of Proposition 2.1 (a).

(H1): using Young's inequality, ab ≤ δ 2 a 2 + 1 2δ b 2 , with δ = K(ρc) 2 , we quickly get that

(L(k)v, v) ≥ R K(ρ c ) 2 |∂ x v 1 | 2 + k 2 - 1 2 |v 1 | 2 + ρ c |∂ x v 2 | 2 + k 2 - 1 K(ρ c ) |v 2 | 2 , ( 8
)
which is greater than α U 2 L 2 for |k| large enough (remember that ρ c and K(ρ c ) are positive).

(H2): as lim

x→±∞ Q c (x) = Q ∞ = (ρ ∞ , u ∞ )
, with standard arguments [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF], and using the fact that L(k) is self-adjoint, the essential spectrum of L(k) is given by that of

L ∞ (k) = K(ρ ∞ )(-∂ 2 xx + k 2 ) + g ′ 0 (ρ ∞ ) (u ∞ -c)∂ x (c -u ∞ )∂ x ρ ∞ (-∂ 2 xx + k 2 )
, whose essential spectrum can be determined by using the Fourier transform in the xvariable and explicitly writing the eigenvalues µ(ξ, k). We get that these are positive when k = 0. The essential spectrum of L(k) is equal to that of L ∞ (k), so (H2) is verified.

(H3): we easily have L ′ (k) = diag(2kK(ρ c ), 2kρ c ).

(H4): we apply the following lemma to L(0).

Lemma 2.3. Let L be a symmetric operator on a Hilbert space such that

L = L 1 A A * L 2
with L 2 invertible. Then, we can write

(Lv, v) = (L -AL -1 2 A * )v 1 , v 1 + L 2 (v 2 + L -1 2 A * v 1 ), v 2 + L -1 2 A * v 1 .
As a result, we write

(L(0)v, v) = (Mv 1 , v 1 ) + R ρ c ∂ x v 2 + 1 ρ c (u c -c)v 1 2 dx, with M = -∂ x (K(ρ c )∂ x •)-m-(uc-c) 2 ρc
. M is a second-order differential operator on which we can perform Sturm-Liouville analysis [START_REF] Dunford | Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space[END_REF]. First, the essential

spectrum of M is included in [α, +∞[ with α > 0; indeed M is a perturbation of M ∞ = -K(ρ ∞ )∂ 2 xx + g ′ 0 (ρ ∞ ) - (u∞-c) 2 ρ∞
, whose essential spectrum is positive under the assumption that

ρ ∞ g ′ 0 (ρ ∞ ) > (u ∞ -c) 2 . Next, the function ρ ′
c is in the kernel of M, and it has one zero, so by Sturm-Liouville theory, M has a unique negative eigenvalue associated with an eigenfunction

U - 1 . Setting U - 2 such that ∂ x v - 2 = -1 ρc (u c -c)v - 1 ,
we have a generalised eigenfunction for L(0) (the second component is not in L 2 ). By using H 2 approximations of U - 2 , we see that (L(0)v, v) can be negative with U ∈ H 2 , confirming that L(0) has one negative eigenvalue.

Localisation of instability and boundedness of unstable eigenvalues.

We start be taking the real part of the L 2 scalar product of the eigenvalue equation [START_REF] Grenier | On the nonlinear instability of Euler and Prandtl equations[END_REF] by L(k)U: we get that Re(σ)(L(k)U, U) = 0, as J is skew-symmetric. The operator L(k) satisfies (H1) of Theorem 2.2, so, if Re(σ) > 0, we must have 0

= (L(k)U, U) ≥ α U 2 L 2
for |k| ≥ k max . Thus, the only function satisfying σU = JL(k)U with Re(σ) > 0 and |k| large is U = 0; there are no unstable eigenfunctions for |k| large.

In order to get the boundedness of the unstable eigenvalues, we decompose L(k) as follows:

L(k) = L 0 (k) + L 1 with L 0 (k) := -∂ x (K(ρ c )∂ x ) + K(ρ c )k 2 -m 0 0 0 -∂ x (ρ c ∂ x ) + ρ c k 2 ,
where -m 0 (x) = max -m(x), 1 2 g ′ 0 (ρ ∞ ) . We compute the scalar product of (7) and L 0 (k)v, and take the real part, which gives us

Re(σ)(L 0 (k)v, v) = Re(JL 1 v, L 0 (k)v). ( 9
)
It is quickly noticed that there exists α > 0 such that

Re(σ)(L 0 (k)v, v) ≥ αRe(σ) ∂ x v 2 L 2 + k 2 v 2 L 2 + v 1 2 L 2 . ( 10
)
We shall now bound |Re(JL 1 v, L 0 (k)v)| by the same norms as on the right. Note that

JL 1 = -∂ x ((u c -c)•) 0 m -m 0 -(u c -c)∂ x .
Computing the scalar product directly, and bounding the terms involving ρ c and u c in L ∞ , we get

|Re(JL 1 v, L 0 (k)v)| ≤ C (1 + k 2 )(|Re(v 1 , ∂ x v 1 )| + |Re(∂ x v 2 , v 2 )|) + |Re(v 1 , ∂ x v 2 )| + ∂ x v 1 2 L 2 + k 2 v 1 2 L 2 + ∂ x v 2 2 L 2 +|(v 1 , ∂ xx v 1 )| + |(v 1 , ∂ 2 xx v 2 )| + |(|k|v 1 , |k|v 2 )|
The top line vanishes by integration by parts, and, also by integrating by parts, the first two terms in the final line are repeats of terms in the second line. Finally, by using Young's inequality, the last term is bounded by k 2 v 2 L 2 , thus proving that, combining with ( 9) and ( 10), there exists C > 0 such that

Re(σ) ∂ x v 2 L 2 + k 2 v 2 L 2 + v 1 2 L 2 ≤ C ∂ x v 2 L 2 + k 2 v 2 L 2 + v 1 2
L 2 , which implies that Re(σ) cannot be unbounded when positive. This ends the proof of part (a) of Proposition 2.1.

Proof of Proposition 2.1 (b) Resolvent estimate

The proof of part (b) is split in two. First, we get the result for s = 0; we bound the X 0 k norm of U by similar norms of F by using the Laplace transform and spectral arguments. The case s > 0 is then obtained by induction on s, the number of total derivatives (time and space).

2.2.1

The case s = 0

The proof of (6) for s = 0 relies on the Laplace transform, and is similar to the resolvent estimate proofs in [START_REF] Rousset | Transverse nonlinear instability for two-dimensional dispersive models[END_REF] and [START_REF] Paddick | Stability and instability of Navier boundary layers[END_REF]. Let σ 0 < γ 0 < γ. For f (t), we denote by f (τ ) the following Laplace transform:

f (τ ) := +∞ 0 exp(-(γ 0 + iτ )t)f (t) dt.
Using the Laplace transform turns equation ( 4),

∂ t U = JL(k)U + F , into an eigenvalue problem: (γ 0 + iτ ) Ũ (τ ) = JL(k) Ũ (τ ) + F (τ ). ( 11
)
As γ 0 > σ 0 , γ 0 + iτ is not in the spectrum of JL(k). Indeed, we can use the strategy employed to prove that hypothesis (H2) is satisfied to show that the essential spectrum of JL(k) is embedded in iR. Once again using the argument from [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF], we can examine the spectrum of the Fourier transform in x of JL ∞ (k),

F x (JL ∞ )(ξ, k) = -i(u ∞ -c)ξ ρ ∞ (ξ 2 + k 2 ) -K(ρ ∞ )(ξ 2 + k 2 ) -g ′ 0 (ρ ∞ ) -i(u ∞ -c)ξ
, which contains the solutions of the equation

X 2 -2iξ(u ∞ -c)X + ρ ∞ K(ρ ∞ )(ξ 2 + k 2 ) 2 + ρ ∞ g ′ 0 (ρ ∞ )k 2 + (ρ ∞ g ′ 0 (ρ ∞ ) -(u ∞ -c) 2 )ξ 2 = 0,
which depend on (ξ, k). Using the positiveness of ρ ∞ , K(ρ ∞ ) and condition (2), we get that the discriminant of this equation is negative for (ξ, k) = 0, and clearly the only eigenvalue at (ξ, k) = (0, 0) is zero, so the essential spectrum of JL(k) is imaginary.

As γ 0 + iτ is not in the spectrum of JL(k) for any τ ∈ R, the norm of the resolvent ((γ 0 + iτ )Id -JL(k)) -1 is uniformly bounded for (τ, k) in any compact subset of R 2 . It remains to show that, for |k| ≤ k max , there exists the following bound for |τ | large.

Lemma 2.4. If Ũ solves [START_REF] Höwing | Standing solitary Euler-Korteweg waves are unstable[END_REF], then there exists C, M > 0 such that, for |τ | ≥ M,

|| Ũ(τ )|| X 0 k ≤ C|| F (τ )|| H 1 . ( 12
)
Proof: we consider the scalar product of the Laplace-transformed equation ( 11) with L(k) Ũ , and write

(γ 0 + iτ )(L(k) Ũ , Ũ ) = ( F , L(k) Ũ ). ( 13
) Note that (L(k) Ũ , Ũ) = (L(0) Ũ , Ũ) + K(ρ c )k 2 || Ũ1 || 2 L 2 + ρ c k 2 || Ũ2 || 2 L 2 .
Let us concentrate on the term (L(0) Ũ, Ũ). Using Lemma 2.3, we know that it is equal to

(L(0) Ũ , Ũ) = (M Ũ1 , Ũ1 ) + R ρ c ∂ x Ũ2 + 1 ρ c (u c -c) Ũ1 2 dx,
The operator M, which we remind the reader is equal to

-∂ x (K(ρ c )∂ x •) -m -(uc-c) 2
ρc , and whose quadratic form is defined on H 1 , has one simple negative eigenvalue, as well as a one-dimensional kernel containing ρ ′ c . Recall U -the generalised eigenfunction corresponding to the negative eigenvalue of L(0). We do not have U - 2 ∈ L 2 , so we consider U -= (U - 1 , 0). We denote U 0 = (ρ ′ c , 0), which is in the kernel of M ⊗ Id. We renormalise U -and U 0 so that their L 2 norms are equal to 1. Let U + be orthogonal to U -and U 0 ; we show that (L(0)U + , U + ) ≥ η U + X 0 0 [START_REF] Piccinini | Ordinary differential equations in R n[END_REF] for some η > 0. First, as U + 1 is not in the kernel or the negative eigenspace of M, we have (MU

+ 1 , U + 1 ) ≥ α U + 1 2 H 1 , since the essential spectrum of M is included in [α, +∞[. Thus, we already have (L(0)U + , U + ) ≥ α U + 1 2 H 1 . (15)
But this does not suffice to get the X 0 0 norm. Using the fact that ρ c is positive, there is a positive

β such that R ρ c ∂ x U + 2 + 1 ρ c (u c -c)U + 1 2 dx ≥ β ∂ x U + 2 + 1 ρ c (u c -c)U + 1 2 L 2 ≥ β ∂ x U + 2 2 L 2 + Cβ U + 1 2 L 2 -2C|(∂ x U + 2 , U + 1 )|
for some C > 0. We use Young's inequality on the final term with δ = β/2C, thus there exists

C ′ ∈ R such that (L(0)U + , U + ) ≥ (α + C ′ ) U + 1 2 H 1 + β 2 ∂ x U + 2 2 L 2 .
If perchance C ′ is negative, we add |C ′ | α × (15) to the above, and obtain that there does indeed exist η > 0 such that we have [START_REF] Piccinini | Ordinary differential equations in R n[END_REF], and (L(k)U + , U + ) ≥ η U + 2 X 0 k . We now write the orthogonal decomposition in L 2 of the first component, Ũ1 = aU - 1 + bU 0 1 + U + 1 , and replace in [START_REF] Paddick | Stability and instability of Navier boundary layers[END_REF]. The eigenfunctions U -and U 0 are fixed, so their H 1 norms are given constants. On the right-hand side, using integration by parts and basic estimates including Young's inequality, we have

|( F , L(k) Ũ )| ≤ η 4 U + 2 X 0 k + C(|| F || 2 H 1 + a 2 + b 2 ),
while on the left-hand side, we have

(L(k) Ũ , Ũ) ≥ η(|k| 2 ||U + || 2 L 2 + U + 2 X 0 0 ) -C(a 2 + (|a| + |b|) U + 2 X 0 0 ).
Taking the real part of ( 13) and moving the negative part of the above to the right-hand side and once again applying Young's inequality to absorb U + X 0 0 , we get

U + 2 X 0 k ≤ C(|| F || 2 H 1 + a 2 + b 2 ). ( 16
)
To finish off, we take the dot product of ( 11) with U -and U 0 . We quickly get

(γ 0 + iτ )a = -( Ũ , L(k)JU -) + ( F , U -) and (γ 0 + iτ )b = -( Ũ , L(k)JU 0 ) + ( F , U 0 ).
Integrating by parts as usual, we get

a 2 + b 2 ≤ C γ 2 0 + |τ | 2 (a 2 + b 2 + U + 2 X 0 k + || F || 2 H 1 ).
We see that if |τ | is large enough, a 2 + b 2 can be absorbed by the left-hand side. Combining with ( 16), we get the result.

For the end of the proof of Proposition 2.1 (b), we start by using the Parseval equality in the following,

T 0 e -2γ 0 t U(t) 2 X 0 k dt = T 0 || Ũ(t)|| 2 X 0 k dt ≤ C T 0 || F (t)|| H 1 dt = C T 0 e -2γ 0 t F (t) H 1 dt.

Now recall the assumption on F , (5): we have

T 0 e -2γ 0 t U(t) 2 X 0 k dt ≤ CM 0 T 0 e 2(γ-γ 0 )t (1 + t) α dt ≤ C 0 e 2(γ-γ 0 )T (1 + T ) n .
We inject this in the energy estimate on (4), that is

d dt ( U(t) 2 X 0 k ) ≤ C( U(t) 2 X 0 k + F (t) 2 H 1 ),
and multiply the result by e -2γ 0 t , integrate in time and we get the result.

The induction for s > 0

The extension of Proposition 2.1 (b) to every s ≥ 0 will follow the lines of the similar result on the linearised water-waves equation in [START_REF] Rousset | Transverse instability of the line solitary water-waves[END_REF]; it is done with a double induction, double in the sense that one is embedded in the other. The first induction is on s, the total number of derivatives. Set s > 0, and we assume that, for every s ′ < s and j ≤ s ′ , we have [START_REF] Dunford | Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space[END_REF], that is

∂ s ′ -j t U(t) X j k ≤ C s ′ e γt (1 + t) n .
To get the wanted result, we must prove that, for every 0 ≤ j ≤ s,

U 2 Ẋk := ∂ s-j t ∂ j x U 2 Ḣ1 + |k| 2 ∂ s-j t ∂ j x U 2 L 2 ≤ C s e 2γt (1 + t) 2n , ( 17
)
where Ḣ1 is the usual homogeneous Sobolev norm on R. This is done by induction on the number of space derivatives, j. Note that the Ẋk norm (semi-norm if k = 0) defined here is a sort of homogeneous Sobolev norm expressed in the Fourier space. Morally, at rank s of the induction, we must get bounds for the L 2 norms of terms involving s+1 derivatives.

Starting with j = 0, we are interested in the X 0 k norm of ∂ s t U. Simply differentiate the equation, ( 4), s times with respect to t, and notice that

W (t) = ∂ s t U(t) -∂ s t U(0) satisfies ∂ t W = JL(k)W + G, with W | t=0 = 0 and G(t) H s+1 ≤ 2M s (1 + t)
-n e γt : we can re-use the case s = 0 shown above. Now, let j > 0. To lighten the notations, we will write V s,j = ∂ s-j t ∂ j

x V . Note that we want to control the Ẋk norm of U s,j , which means s + 1 derivatives in total, of which j + 1 space or Fourier derivatives. We apply ∂ s-j t ∂ j

x to the equation. This time, the derivatives do not commute with JL(k), hence we consider

∂ t U s,j = JL(k)U s,j + J[∂ j
x , L(k)]U s-j,0 + F s,j := JM s,j (k)U + F s,j . We take the real part of the scalar product of this equation with M s,j (k)U, which yields

1 2 d dt (U s,j , L(k)U s,j ) = -Re(U s+1,j , [∂ j x , L(k)]U s-j,0 ) + Re(F s,j , M s,j (k)U).
To bound the second part of the right-hand side, we look more closely at the commutator term in M(k)U. We notice that there exist two sets of

L ∞ matrices (m 1 i , m 2 i ) 0≤i≤j+1 such that [∂ j x , L(k)]∂ s-j t U = j+1 i=0 m 1 i (x)U s-j+i,i + j-1 i=0 m 2 i (x)k 2 U s-j+i,i . (18)
We notice that all the terms, except the one with i = j + 1, have a total of s derivatives or less, and thus, using k 2 ≤ k max |k|, they are controlled by our induction hypothesis.

Integrating the terms of (F s,j , L(k)U s,j ) involving j + 2 space derivatives and using assumption (5) and Young's inequality with a parameter η to be chosen later, we obtain that the right-hand side is bounded by

|(F s,j , M s,j (k)U)| ≤ η 2 U s,j 2 Ẋk + C e 2γt (1 + t) 2n .
To deal with the first term of the right-hand side, we notice that U s+1,j has s + 1 derivatives, of which j space derivatives, hence the L 2 norm of U s+1,j falls under our induction hypothesis. We can thus use Young's inequality and use decomposition [START_REF] Rousset | A simple criterion of transverse linear instability for solitary waves[END_REF] once again, and get 1 2

d dt (U s,j , L(k)U s,j ) ≤ η U s,j 2 Ẋk + C e 2γt (1 + t) 2n .
Finally, we integrate this in time, and recall (8) from the verification of the (H1) hypothesis of Theorem 2.2, which says that

(L(k)U s,j , U s,j ) ≥ η ′ U s,j 2 Ẋk -C U s-1,j-1 2 L 2 ,
in which the final term can be moved to the right-hand side and controlled by the induction hypothesis. Note that we can take the L 2 norm of U s-1,j-1 since the number of time derivatives is preserved (we do not encounter the second component of U which is not assumed to be in L 2 ). In total, we therefore have

U s,j (T ) 2 Ẋk ≤ η η ′ T 0 U s,j (t) 2 Ẋk dt + C e 2γt (1 + t) 2n .
We choose η in the Young inequalities above so that η/η ′ ≤ 2γ, and the Grönwall lemma gives us [START_REF] Rousset | Transverse nonlinear instability for two-dimensional dispersive models[END_REF] for the couple (s, j). Both inductions are now complete.

Nonlinear instability

In this part, U = (ρ, u). Obtaining Theorem 1.1 relies on the construction of an approximate solution U ap built around unstable eigenmodes of the linearised equation. In our case, this construction is classical and we will not write all the details of the calculations (see also, for instance, [START_REF] Grenier | On the nonlinear instability of Euler and Prandtl equations[END_REF][START_REF] Desjardins | Linear instability implies nonlinear instability for various types of viscous boundary layers[END_REF][START_REF] Rousset | Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's[END_REF][START_REF] Rousset | Transverse instability of the line solitary water-waves[END_REF]). Energy estimates must then be obtained on U -U ap to ensure that the approximate solution is close enough to the exact solution for long enough to see the difference between U ap and Q c reach an amplitude O(1). This is the more delicate part, as we remind the reader that only local existence is guaranteed for solutions of the Euler-Korteweg equation, thus we must also ensure that the solution U still exists when the instability appears. The closing argument is standard for this method.

Construction and properties of the approximate solution

For a whole number N independent of ε to be chosen later, we will set

U ap = Q c (t, x) + N j=1 ε j U j (t, x, y),
with u j potential: we set V ap = (ρ ap , ϕ ap ), V j = (ρ j , ϕ j ). This is expected to solve the Euler-Korteweg system leaving an error of order ε N +1 , as follows,

∂ t ρ ap + div (ρ ap ∇ϕ ap ) = ε N +1 R ap 1 ∂ t ϕ ap + 1 2 |∇ϕ ap | 2 + g 0 (ρ ap ) = K(ρ ap )∆ρ ap + 1 2 K ′ (ρ ap )|∇ρ ap | 2 + ε N +1 R ap 2 ,
thus, expanding V ap in these equations and isolating the terms of order ε, we see that V j solves the linearised Euler-Korteweg equation around Q c with a source term:

∂ t V j = JLV j + R j , (19)
with R j containing nonlinear interaction terms between the V n with n < j, but with the sum on indices in each interaction term equal to j. For instance, we have R 1 = 0, and then, in the equation on ρ 2 , we see that R 2,1 = div (ρ 1 ∇ϕ 1 ). In the equation on ϕ j , we notice that there will be quadratic terms similar to the one we have mentioned, but also cubic interaction terms, in the sense that they involve V n 1 , V n 2 and V n 3 with n 1 + n 2 + n 3 = j. Likewise, R ap contains all the interaction terms whose sum of indices is greater than N. We do not detail the remainders any further.

We now construct V 1 as a wavepacket of unstable eigenmodes of the linearised equation. Recall that k 0 > 0 is the global maximum for the function

σ : k → max{Re(λ) | λ ∈ σ(JL(k))},
where σ(JL(k)) is the spectrum of the operator JL(k). We assume that this maximum is nondegenerate (although the method is easily adapted to the degenerate case, see [START_REF] Rousset | Transverse instability of the line solitary water-waves[END_REF]). The function σ is continuous, hence we can define

V 1 (t, x, y) = R f 1 (k)e iky e σ(k)t v 1 (k, x) dk,
with f 1 (k) smooth, even, equal to 1 in the vicinity of k 0 and supported in the set {k | σ(k) > 3σ 0 /4}, and w(t, x) = e σ(k)t v 1 (k, x) solving ∂ t w = JL(k)w. For any s ≥ 0, we consider the quantity

s s ′ =0 R f 2 1 (k)|k| 2s ′ C s,s ′ e 2σ(k)t dk = s s ′ =0 R f 2 1 (k)|k| 2s ′ w(k, t) 2 H s-s ′ (R) dk = V 1 (t) 2 H s (R 2 ) ,
the second equality being the Parseval equality. We use the Laplace method around the critical point

k 0 to get that V 1 (t) 2 H s ∼ t→+∞ t -1/2 e 2σ 0 t , so 1 C 1,s e σ 0 t (1 + t) 1/4 ≤ V 1 (t) H s (R 2 ) ≤ C 1,s e σ 0 t (1 + t) 1/4 . (20)
We get estimates on V j by induction. Assume that I = supp(f 1 ) is made up of two separate intervals around ±k 0 , and we set

V j (t, x, y) = I • • • I v j (k 1 , • • • , k j ; t, x)e ik 1 y • • • e ik j y dk 1 • • • dk j .
Assuming that, for every n < j,

v n (k 1 , • • • , k i ; t) H s ≤ C n exp[n(σ(k 1 ) + • • • + σ(k n ))t], (21)
we get that v j solves the linearised Fourier-transformed equation

∂ t v j (k 1 , • • • , k j ) = JL(k 1 + • • • + k j )v(k 1 , • • • , k j ) + r j (k 1 , • • • , k j ), in which r j (k 1 , • • • , k j ) H s ≤ C exp[n(σ(k 1 ) + • • • + σ(k j ))t
] by the structure of the remainder (combination of products between v n 's with n < j) and (21). Then, since, for k ∈ I, σ(k) > 3σ 0 /4, the sum in the exponential is greater than σ 0 , and setting v j | t=0 = 0, we can apply Proposition 2.1 (b) to get that v j (k 1 , • • • , k j ; t) satisfies (21). We then use Parseval's equality and the Taylor expansion of σ around the critical point k 0 to write that, for some β > 0,

V j (t, x, y) 2 H s = jk∈jI k 1 +•••+k j =jk v j (k 1 , • • • , k j ; t, x)e ijky dk 1 • • • dk j-1 2
H s dk ≤ jk∈jI Ce 2(jσ 0 -jβ(k-k 0 ) 2 )t j m=1 km=jk e -2β j m=1 (km-k) 2 t dk 1 • • • dk j-1 dk
Integrate these gaussian functions (remembering that k j = jk -n=1 k n ), and we get the desired inequality: for every j ≤ 1,

V j (t) H s (R 2 ) ≤ C j e jσ 0 t (1 + t) j/4 . (22)
We now take a look at the remainder of the equation on V ap , R ap , which contains the interaction terms of the equation whose sum of indices is greater than N. Similarly to our proof of (22), we have

ε N +1 R ap (t) H s ≤ 3N j=N +1 C j ε j e jσ 0 t (1 + t) j/4 , (23)
and, in what follows, we will be interested in times for which the smaller powers of ε(1 + t) -1/2 e σ 0 t are dominant. We set T * ε such that εe σ 0 T * ε (1 + T * ε ) 1/4 = κ, for 0 < κ < 1 to be chosen later, and t = T * ε -τ . Replace in (23), and we have

ε N +1 R ap (T * ε -τ ) H s ≤ max j∈{N +1,•••,3N } C j 3N j=N +1 κ j e -jσ 0 τ ≤ C R κ N +1 e -(N +1)σ 0 τ ,
which, returning to the original time variable t, gives us, for t ≤ T * ε ,

ε N +1 R ap (t) H s ≤ C R ε N +1 e (N +1)σ 0 t (1 + t) (N +1)/4 . (24)

Getting the instability

If U is the solution of the Euler-Korteweg system (1) with the initial condition U(0) = U ap (0), we will observe the instability by studying

U(t) -Q c (t) L 2 (R 2 ) ≥ U ap (t) -Q c (t) L 2 (R 2 ) -U(t) -U ap (t) L 2 (R 2 ) .
On one hand, we have U ap -Q c = N j=1 ε j U j , and

N j=1 ε j U j L 2 (R 2 ) ≥ εU 1 (t) L 2 (R 2 ) - N j=2 ε j U j L 2 (R 2 ) ≥ C ′ 1 εe σ 0 t (1 + t) 1/4 - N j=2 C j ε j e jσ 0 t (1 + t) j/4
by ( 20) and ( 22). Taking times smaller than T * ε , we can consider that the sum on the right behaves like ε 2 (1 + t) -1 e 2σ 0 t , and, replacing t by T * ε -τ , we write

(U ap -Q c )(T * ε -τ ) L 2 ≥ κ C ′ 1 e -σ 0 τ -κC ′ 2 e -2σ 0 τ ≥ κC ′ 1 e -σ 0 τ (1 - κC ′ 2 C ′ 1 e -σ 0 τ ).
We notice that, for a given C > 0, there exists τ C > 0 such that, for τ ≥ τ C , 1 -Ce -σ 0 τ ≥ 1/2, so we set τ 1 > 0, independent of ε, such that, for τ ≥ τ 1 ,

(U ap -Q c )(T * ε -τ ) L 2 (R 2 ) ≥ κC ′ 1 2 e -σ 0 τ . (25)
On the other hand, we will use energy estimates to ensure that U(t) -U ap (t) L 2 (R 2 ) is small. We will readily use those shown by S. Benzoni, R. Danchin and S. Descombes in [START_REF] Benzoni-Gavage | On the well-posedness for the Euler-Korteweg model in several space dimensions[END_REF], which are obtained by considering the equation on (G, z) = (G, u + iw), with G a primitive of the function ρ → K(ρ)/ρ and w = ∇(G(ρ)). The equation on z is a Schrödinger-type equation, written as

∂ t z + u • ∇z + i∇z • w + i∇(A(ρ)div z) = Q(ρ),
with functions A and Q that we do not detail, while G satisfies

∂ t G + (u • ∇)G + A(ρ)div u = 0.
The approximate solution satisfies a similar system with a remainder term which we will denote R = (R G , R z ). From now on, we use tildes to designate the difference between the exact and approximate terms in this system, e.g. ũ = u -u ap , Ã = A(ρ) -A(ρ ap ). The difference ( G, z) satisfies the equation

   ∂ t G + u • ∇ G + ũ • ∇G ap + A(ρ)div ũ + Ãdiv u ap = -R G ∂ t z + u • ∇z + ũ • ∇z ap + i∇z • w + i∇z ap • w +i∇(A(ρ)div z) + i∇( Ãdiv z ap ) = Q -R z . (26)
The energy estimates in H s on this equation involve multiplying by an adequately chosen gauge ψ s . In the potential case, this gauge is A(ρ) s/2 (whereas in the non-potential case, a additional term is needed), and the weighted norm ψ s Λ s • L 2 , in which Λ s is the standard Fourier multiplier for s derivatives, is equivalent to the standard H s norm. We send the reader to article [START_REF] Benzoni-Gavage | On the well-posedness for the Euler-Korteweg model in several space dimensions[END_REF] for details, in particular part 3 where the reasons for the simpler norm is explained, and part 6 where energy estimates on the difference between a solution and a reference solution ((26) We rewrite this as follows: let W = U -U ap ; there exists a polynomial function Q such that, for s > 0 large enough (to handle the L ∞ norms by Sobolev embedding),

W (t) 2 H s ≤ t 0 Q( U ap (z) -Q ∞ H s + U(z) H s ) W (z) 2 H s + R ap (z) 2 H s dz.
We will now choose N to get the right growth in time for W , as well as κ to get the existence up to T * ε of the exact solution U. First of all, in the same way that we get (25), we note that

U ap (t) -Q ∞ H s ≤ Q -Q ∞ H s + N j=1 C j ε j e jσ 0 t (1 + t) j/4 ≤ Q -Q ∞ H s + κ when t ≤ T *
ε -τ 2 , with τ 2 ≥ τ 1 independent of ε. We consider times t ≤ T W so that W (t) H s ≤ 1 and ρ(t, x, y) > 0 (no vacuum on the exact solution), and choose N so that, for t ≤ T W , 2Nσ

0 > Q( Q -Q ∞ H s + 1 + κ).
A variant of the Grönwall inequality from [START_REF] Piccinini | Ordinary differential equations in R n[END_REF] then provides us with

W (t) 2 H s ≤ C
ε N +1 e 2(N +1)σ 0 t (1 + t) (N +1)/2 (27)

for t ≤ T W . Now, take t = T * ε -τ : we notice that the right-hand side is smaller than C(N)κ 2(N +1) , which is therefore smaller than κ if κ < 1 is small enough. We now choose κ so that 2κ < min ρ c , to ensure that there is no vacuum. We therefore have T W ≥ T * ε by a bootstrap argument. So, (27) is valid for t = T * ε -τ with τ ≥ τ 2 , and we have

W (T * ε -τ ) H s ≤ C ′ 0 κ N +1 e -(N +1)σ 0 τ . (28)
Recall that f 1 , which is the localising function for the unstable term U 1 , is supported in I which is made up of two closed intervals that do not contain 0. Set f a smooth function such that f (k) = 1 for k ∈ I and f = 0 in the vicinity of 0, and let us define Π, a Fourier projector on frequencies in I, by

F y (Πu)(x, k) = f (k)(F y u)(x, k).
As U(0, x, y) = Q c (x)+εU 1 (0, x, y), ΠU| t=0 = U 0 . Moreover, for any a ∈ R, the difference Q c (x -ct -a) -Q c (x -ct) does not depend on y, hence Π(Q c (• -ct -a) -Q c (• -ct)) = 0. We can now combine (25) and (28) to show the instability: we have, for any a,

U(t) -Q c (• -a -ct) L 2 (R 2 ) | t=T * ε -τ ≥ Π(U(t) -Q c (• -ct) L 2 (R 2 ) | t=T * ε -τ ≥ κC ′ 1 2 e -σ 0 τ 1 - 2κ N C ′ 0 C ′ 1 e -N σ 0 τ ,
For τ ≥ τ 3 ≥ τ 2 , we have the last exponential on the right smaller than 1/2, and, as a result, letting τ ′ ≥ τ 3 be fixed, independent of ε,

(U -Q c )(T * ε -τ ′ ) L 2 ≥ κC ′ 1 4
e -σ 0 τ ′ := δ 0 .

The number δ 0 we have found is positive and does not depend on ε: Theorem 1.1 is proved.

  without the source term R) are obtained.Using the H s norms of Q c -Q ∞ as constants, the energy estimate readsd dt z 2 H s ≤ C z H s (|| G|| L 2 + z H s )(1+ ∇z ap H s+1 )(1+ ∇z ap H s-1 + z H s )+C R 2 H s .

Acknowledgements: this result was obtained during my Ph.D at Rennes 1 University, under the supervision of Frédéric Rousset, who I warmly thank for the opportunity to work on the topic, and for his guidance throughout the preparation of this article.

The author is partially supported by the ANR project DYFICOLTI, Agence Nationale de la Recherche grant ANR-13-BS01-0003-01.


Anthony Bigot
email: anthony.bigot@irit.fr

Sébastien Déjean
email: sebastien.dejean@math.univ-tlse.fr

Josiane Mothe
email: josiane.mothe@irit.fr

Learning to Choose Automatic Selection of the Information Retrieval Parameters

Keywords: Information retrieval, Meta search, Evaluation, Learning in IR, System combination, Repeated queries

In this paper we promote a selective information retrieval process to be applied in the context of repeated queries. The method is based on a training phase in which the meta search system learns the best parameters to use on a per query basis. The training phase uses a sample of annotated documents for which document relevance is known. When an equal-query is submitted to the system, it automatically knows which parameters it should use to treat the query. This Learning to choose method is evaluated using simulated data from TREC campaigns. We show that system performance highly increases in terms of precision (MAP), specifically for the queries that are difficult to answer, when compared to any unique system configuration applied to all the queries.

Introduction

In information retrieval (IR), a ranking function is used to decide the document to retrieve according to a query as well as their order. Document and query similarity is a key component of the ranking function. Many parameters are involved in this process: the way documents are indexed, the possible query expansion, the similarity function itself, ... Systems vary because they use different parameters throughout retrieval.

Over queries, systems do not behave homogeneously and a system S 1 that answers well a query Q 1 can answer query Q 2 poorly, while another system S 2 does oppositely. As a result, it is impossible to define the set of parameters that will be used for any queries. Rather, most of the approaches optimize the parameters so the results are best in average over queries. Alternatively we propose to chose among a set of systems or system configurations the one that fits the best, on a per query basis. This principle implies to learn a link between queries and systems. The aim of the learning can be to associate query characteristics to systems. For example it is possible to represent queries by features, learn which system is best for each training query or query cluster. A new query can then be compared to learn queries considering the same features to select the system to be applied. [START_REF] Ounis | Terrier: A high performance and scalable information retrieval platform[END_REF] use this principal to associate weigthing schema to queries. We consider an alternative process in which learning is applied to individual query: in that case the training consists in associating the best system to each query. In practice, this latest principle is possible if the queries are known in advance only. This is indeed the case for repeated queries. In this paper we focus on repeated queries.

Several studies have shown repeated queries occur frequently in real applications. Teevan et al. have shown that 33% of queries are exact repeats of a query issued by the same user (equal-query queries) and that one query out of four leads to the same single user click [START_REF] Teevan | Information re-retrieval: repeat queries in yahoo's logs[END_REF]. Sanderson and Dumais studied 3.3 million queries and showed more than a half of them were repeated queries [START_REF] Sanderson | Examining repetition in user search behavior[END_REF].

In this paper, we consider a method that could be used in real systems to treat repeated queries using a selective approach. We call the method "learning to choose". The learning to choose method implements a meta system that makes use of systems variability. It aims at deciding which system configuration should be used to treat a given query. To do so, the learning to choose method learns the best query-system association: it learns the system configuration to be used for a given query in order to optimize the results. This training phase uses a subset of documents for which the document relevance is known. After training, whenever equal-query queries occur, the learnt system configuration will treat the query over the entire document collection.

The paper is organized as follows. In section 2, we report some related works. Section 3 presents the learning to choose method.Section 4 describes the evaluation framework. Section 5 details a preliminary analysis that aims at selecting a subset of system configurations. In section 6, we consider the only systems selected as depicted in section 5 and analyse the results of Learning to chose according to topic difficulty. We show that learning to choose improves the results on the hardest topics. The last section concludes the paper.

Related works

There are various ways to take advantage of system varaibility. Without being exhaustive, this section presents the different types of methods. Data fusion. Data fusion takes advantage of system variability by combining the ranked lists retrieved by different systems for a given information need [START_REF] Fox | Combination of multiple searches[END_REF]. The issues of data fusion are to decide both the ranked lists to fuse and the fusing function. Some works have shown that the systems to fuse should be independent [START_REF] Croft | Combining approaches to information retrieval[END_REF], [START_REF] Wu | Assigning appropriate weights for the linear combination data fusion method in information retrieval[END_REF]. The same fusing function is generally applied whatever the queries are. However, the fusing model can also be learnt on a per query basis; in that case, the fusing function is query dependent [START_REF] Wilkins | Using score distributions for querytime fusion in multimediaretrieval[END_REF]. Unlike data fusion which fuse various system results, learning to choose selects the best system configuration that should treat a query among several.

Learning to rank. Variability and learning are also core elements of learning to rank methods [START_REF] Liu | Introduction to special issue on learning to rank for information retrieval[END_REF]. In the training phase of learning to rank, the examples to learn consist of ranked lists of relevant documents associated with the corresponding information needs. The ranking function is trained on these examples. The testing phase uses the single learned ranking function on new queries [START_REF] Cao | Learning to rank: from pairwise approach to listwise approach[END_REF]. Unlike learning to rank, learning to choose does not learn a general function that would work whatever the query is. Rather, the principle of the method is to select the best system configuration on a per information need basis by analyzing the performance on a subset of documents; the learnt system configuration is then used on the entire set of documents. Repeated queries. Some approaches process repeated queries by storing documents that should be retrieved. This is done for frequently asked queries for which the results are cached [START_REF] Baeza-Yates | The impact of caching on search engines[END_REF], mainly to answer time response issue. In that case, one assumes that the relevant documents remain the same. In addition, because new documents are continuously added or deleted, index updates lead to cache invalidation: cached results correspond to entries whose results have changed [START_REF] Blanco | Caching search engine results over incremental indices[END_REF]. This cache method can also be used to answer queries that are close to previous queries [START_REF] Baeza-Yates | The impact of caching on search engines[END_REF]. Alternatively, some specific treatments can be done on a per query basis [START_REF] Mothe | Linguistic analysis of users' queries: towards an adaptive information retrieval system[END_REF]. Since learning to choose does not store the results but rather the best way to obtain them, it can be applied in the context of dynamic collections. Selective approaches. Finally, the closest related work is selective approaches. For example, He and Ounis [START_REF] He | Inferring query performance using pre-retrieval predictors[END_REF] suggest a method to select among several termweighting models depending on the query. Queries are described by features that are used to cluster them. Training associates the best term-weighting to each query cluster. When a new query is submitted to the system, it decides which cluster the query belongs to and process it by the corresponding system. In the same way Bigot et al. [START_REF] Bigot | Fusing different information retrieval systems according to query-topics: a study based on correlation in information retrieval systems and TREC topics[END_REF] suggest three methods that aim at deciding which system(s) would be the best to process a given query considering TREC participant systems as system candidates. All the three methods are based on a training stage in which for each query the decision is learnt using a sub-set of the document collection and the maximization of a performance measure (e.g. precision). Once trained the meta-system knows which system should process the query when the query will occur again; this system will then process the query over the entire document collection. The first method they called OneT2OneS (for One Topic to One System) selects the best retrieval system for each query. The two other methods are based on a first stage that clusters the systems prior to the training phase, then choose an ambassador for each cluster. They finally select the best cluster by selecting the ambassador that maximizes the performance measure. The two approaches that use system clustering are OneT2ClusterS (for One Topic to System Cluster) and ClusterT2ClusterS (for Cluster Topic to Cluster Systems). They differ in that ClusterT2ClusterS clusters topics (queries) according to their difficulty. Compared to [START_REF] He | Inferring query performance using pre-retrieval predictors[END_REF] that learns some queries over the entire collection, [START_REF] Bigot | Fusing different information retrieval systems according to query-topics: a study based on correlation in information retrieval systems and TREC topics[END_REF] learns all the queries on a sample of documents. Our method is closer to the second approach since it targets re-peated queries. In that context, queries are known and the problem is to chose the best system configuration to apply. Compared to [START_REF] He | Inferring query performance using pre-retrieval predictors[END_REF] and [START_REF] Bigot | Fusing different information retrieval systems according to query-topics: a study based on correlation in information retrieval systems and TREC topics[END_REF] we do not use various systems but rather various retrieval configurations that do not imply various indexing. We think that our method is easier to use in real systems because of this. Moreover, we evaluate the method and show that a limited number of configurations are needed to improve results, specifically on difficult queries.

3 Learning to choose method for repeated queries

Description of the method

In our method, we make the hypothesis that system parameters lead to different system performances depending on the queries. Moreover, the method is based on the fact that for a given query, some system parameters will optimize the results. Thus our method aims at learning which system parameters should be used for a given query. However, IR parameters are not equally resource demanding. For example, making indexing parameters vary implies to index several times the collection which can be costly in terms of time and storage. For this reason, we rather make retrieval parameters vary. A given set of parameter values is called a system configuration. Our method implements a meta-system that choses among pre-defined system configurations on a per-query basis.

The next sections describe the steps of the method. Training phase. The training phase consists in choosing which configuration should process each (repeated) query. Notice that the training query set consists on all the queries that have been identified as repeated and for which the meta-system should learn the best configuration. During the training phase we consider the various system configurations and the results they obtained on the query set using a sample of annotated documents. For these documents, we know if they are relevant to the query or not. Each system configuration treats the training document set over the query set. From the retrieved document lists we compute the evaluation measure for each query and each system configuration. The best configuration is then chosen for each query. The set of system configurations is an important parameter of our method which is analysed in detail in this paper. Testing phase. After training, the meta-system knows which system configuration should process each query. Whenever an equal-query is submitted to the meta-system, it processes it using the learnt system configuration.

Defining the set of system configurations

Principle. The set of system configurations should be composed of configurations that perform well on average over queries. However, as we said previously, system variability can be important. For this reason, we add an additional constraint: to be kept a system configuration should obtain better results than the average on a subset of queries. Moreover, it is not mandatory that a given query is well performed by several system configurations; one being enough to ensure users' satisfaction. As a matter of fact, it is not needed to consider all possible system configurations; what is needed is to ensure that each query is well performed by at least one available system configuration. Considering the example given in table 1, we would like to select either system configuration S 1 or S 2 to ensure query Q 1 is well processed by at least one system configuration. Same goes for system configuration S 3 or S 4 for query Q 2 . System configuration selection. To start with, we dispose of various system configurations and the corresponding performance over the query set when using the document set. As we wanted to perform a quite large analysis, in our experiments, we consider 100 initial system configurations. However, the idea is to decide which system configurations are the most important ones. For defining the configurations to keep, we iteratively select the most important configuration as follows:

1. For each query, we compute the value of the performance measure that corresponds to the highest centile; the system configuration that gets this value becomes a candidate. For example, if we consider the mean average precision (MAP) as the performance measure, we calculate the MAP value M AP 001 for which 1% of the MAP values will be higher than M AP 001 and 99% of the MAP values will be lower than M AP 001 for this query, whatever the system configuration is. Let us consider that the processed query is an easy one and that over 100 system configurations, the best system S b got a MAP of 0.85, M AP 001 for this query is set to 0.85; S b becomes a candidate. 2. System configurations that have been selected in step 1 constitute the set of candidates for this first iteration; 3. For each configuration candidate, we compute the number of times it has been selected; that is to say the number of queries that selected each configuration. We keep the system configuration which is the most frequent. An additional condition to keep the system configuration is that its frequency is higher than a threshold. For example, S b has been selected in step 1 by 10 queries and is the most selected configuration and given a threshold of 6 queries minimum, S b is definitively kept; 4. The queries that selected this configuration are removed from the query set.

Considering our example, the 10 queries that choose configuration S b are removed.

5. We go back to step 1 considering all system configurations and the remaining queries (in our example using N umber of queries -10).

If there is no system configuration selected the process is iterated considering the second centile (the performance value for which 98% of the values are higher than it), third centile, ... If less than T queries remain unmatched, they have to all be selected at once (the centile criteria is relaxed). This rule is used to avoid selecting hardest topics one by one while centile is decreasing.

Evaluation framework

Evaluation collection. Evaluating our method in a real context implies to have access to real data (repeated queries, documents, document relevance); such a collection is not freely available. For this reason, we simulated the environment by evaluating our method on data from TREC (Text Retrieval Conference 4 ). The collection we used (TREC7 and TREC8 adhoc sets) is composed of approximately 528155 documents (2 gigabytes), a set of 100 natural language topic statements and the associated relevance judgments. In our experiment, any TREC topics are considered as repeated information needs. Topics are composed of a title, a description and a narrative part and correspond to information needs. Notice that, since the methods are evaluated on TREC, we distinguish topics and queries in the rest of the paper: topic is used to refer to a TREC topic; a query is the internal representation used by the search system and is built from a topic. QRel consists of document relevance judgements for each topic. By confronting them to the list of retrieved documents, it is possible to compute performance measures to evaluate system configuration performance. We ran different runs with the Terrier platform [START_REF] Ounis | Terrier: A high performance and scalable information retrieval platform[END_REF] by making parameters varying.

To be more realistic in usability, we considered a single indexing (which is the most resource demanding); and thus indexing parameters are set once. Those runs correspond to system configurations. Evaluation Measures. We use the TREC evaluation software trec eval5 , version 9.0 that computes many measures. Evaluation measures have been shown to be highly correlated [START_REF] Baccini | How many performance measures to evaluate information retrieval systems ?[END_REF]. In this work, we only consider average precision (AP) for each topic and mean average precision (MAP) over topics. AP is the result of averaging precision (number of relevant retrieved documents over retrieved documents) each time a relevant document is retrieved. MAP characterises a system configuration by averaging AP over topics.

Training. The training phase of the learning to choose method is applied to a subset of documents (training set). On this document subset, the various system configurations are used to process the 100 topics which are all considered as repeated queries to be learnt. To decide the best configuration for each topic, we used the QRel from which we removed the documents that do not belong to the sample of training documents. In our experiments, we partition the document collection according to a training part of 2 ⁄3 and a testing part of 1 ⁄3. The repetition of queries is simulated by treating the same topic on test documents. Evaluation uses the QRel (after removing the training documents from it). Cross Validation. To evaluate the learning to choose method, we applied cross validation. Indeed, a single partitioning of the data as presented in the previous subsection is not enough to make solid conclusions. For that reason, we performed 10 different partitions of the collection each composed of a training part of 2 ⁄3 and a testing part of 1 ⁄3 randomly selected [START_REF] Freund | Experiments with a new boosting algorithm[END_REF]. Then we average the results over the 10 collections.

Baselines. The baseline is the best system configuration over the queries. The mean of all ten baselines is the general baseline (it was mandatory since the set of documents varies and thus the MAP varies as well depending on the draw).

System configuration selection

Preliminary study to select system configurations. Terrier allows to choose modules from the three steps of the IR process:

-Indexing (stemmer, size of blocks, empty documents ignored or not); -Matching (retrieving model, topic fields to use, low idf terms ignored or not); -Query expansion (QE) (QE model, number of terms to add to the query, number of top documents to consider for QE, number of top documents a term should appear in to be used during QE).

Indeed, indexing is costly in terms of time processing, disk space and updates when creating an IR stream. For that reason, we consider a single index and the only parameters that we make varying are matching and query expansion parameters. The indexing phase uses the default Terrier stopword list, Porter stemmer algorithm, a block size of 1 and ignores empty documents. Indeed using different stopword lists does not have significant impact on the results but using one (rather than none) does significantly improve the results [?]). Fuller et al. [START_REF] Fuller | Conflation-based comparison of stemming algorithms[END_REF] showed Porter stemmer is the most accurate. Also Compaore et al. [START_REF] Compaoré | Mining information retrieval results: Significant IR parameters[END_REF] showed that block size different from 1 and empty documents does not change the results significantly.

Concretely, we considered many configurations but kept only the 100 ones that get a MAP over 0.2 for at least 50% of the topics. This threshold of 0.2 is set to ensure that the performance of the selected system configurations are good enough. Impact of the number of system configurations. In real applications, using 100 system configurations can be costly. For that reason, we study the impact of the number of system configurations to use. We consider various selection thresholds (10, 20, 30...100 configurations). One solution could be to randomly select the systems. However this solution would not take advantage of system variability. For this reason, we prefer to select system configurations that behave differently on the topics. To do so, an hybrid clustering (hierarchical clustering combined to K-means) is applied so system configurations of each cluster lead to close performance in terms of AP for the topic set on the training document set. For each cluster the best configuration is selected to be used in the Learning to choose process. This method insures that the selected configurations are different enough. We then consider several thresholds: we analyse the results when 100% of the configurations are used (no clustering), when 90% of the intial configurations are used (meaning that we cluster the configurations into 90 clusters), 80% ... until 10% (meaning that only 10 configurations are used, corresponding to 10 different clusters or configuration profiles).

Figure 1 shows the results of this experiment. Grey lines corresponds to the training phase, and black lines to the testing phase. Dashed line is the baseline and plain line with dots correspond to the average over all ten sub-collections as explained in 4. Dotted lines reprensent the result of a random selection. Notice that a random selection leads to the expected value of AP and so to the averaged MAP over systems. Results presented in figure 1 show that using only 10 different system configurations, improves the results by 22%. Using more than 10 sytem configuration candidates does not improve the results very much. From this preliminary study, we conclude that 10 configurations are enough to significantly improve the performance but less can also be enough.

In the following, we select a minimal set of system configurations (less than 10) to make further analysis on the behaviour of the selection technique on a small sample of configurations. Minimal System Configuration Set. To select a minimal set of systems, we apply the method presented in 3.2 to the previous set of 100 systems.

In figure 2, the black line is the number of selected system configurations and the gray line is the average MAP; average MAP is considering the MAP of the systems selected for each topic during the selection. Figure 2 shows the lower the T is, the more specific the selection is, and the better the results are. However, we do not want to be too specific because that leads to selecting too many system configurations; for that reason we consider 10 configurations or less. Figure 3 displays the average MAP in function of the number of system configurations; 10 being the maximum number of system configurations we allow to be selected by the algorithm. As a compromise between effectiveness and the number of selected system configurations, we choose 7 possible configurations. This corresponds to 3 different possibilities of T (see the dots hovered by horizontal dashed line on figure 2) and the best MAP (0.528) is obtained for T=13 (vertical dashed line). The set of these 7 configurations is used for evaluating the learning to choose method in section 6.

6 Evaluation of Learning To Choose in a Close-to-Reality Context

In this section we evaluate learning to choose on a small set of systems. Topics are seen as new during the training phase and they are considered repeated information needs during the testing phase. To avoid randomness effect that could lead to one lucky or unlucky experiment, we proceed a cross validation on ten different splits of the document collection as explained in 4.

Topic Difficulty. As we ran 10 experiments, the number of topic clusters may change from one experiment to another. To be able to merge the results, we need to keep the same number of clusters for all 10 experiments. Each cluster is characterised by a difficulty label according to AP systems obtained. Among the 10 experiments, we observed that 4 to 6 levels of topic difficulty would be optimal for the Learning to chose method. To fix the number of difficulty level to consider, we run the hierarchical clustering on data computed with the full collection of documents and conclude that 5 difficulty levels is an optimal choice. Results of the Selection. As the number of topics in a difficulty level varies along the 10 experiments, the overall average MAP by cluster is weighted with the number of topics included in the considered level. This is done for both the metasystem and the baseline. Figure 4 displays the average mean of MAP by topic difficulty level during the test phase. Topics on which systems performed poorly are qualified "hard", those on which they performed less poorly are qualified "medium hard" and so on for "medium easy" level, "easy" level and finally the "easiest" level on which systems performed greatly. Black dots correspond to the metasystem's map and grey dash show the baseline MAP and are read on the left axis (with black guidelines). Grey squares show the relative difference between the metasystem and the baseline and are read on the right axis (with grey guidelines).

We show that performances are poorly increased on easy topics; that can be explained by the fact it is hard to improve system performances when they already are really high. At least, performances are not degraded. Performances on medium easy topics are improved by 12%; according to the t-test, the increase is significant for 6 experiments out of 10.

The best improvment is observed for the medium hard topics with an increase of 44% (increase is significant for 9 experiments). Finally, the hardest topics are improved by 43%. However the absolute increase is little (0.04), it is still significant for 8 experiments.

Conclusion and Future Works

Using more than ten different systems for a fusion technique is not realistic. First, we showed a method easy to reproduce to select a subset of systems. Systems are selected on the base that whether or not their performances belong to the highest centile for a defined number of topics. Matched topics are successively deleted from the selection until no topics remain. The analysis of this selection Fig. 4: Fusion Results according to Topic difficulty method leads to select seven systems in order to balance performances and real conditions. Even if the last matched topics are poorly performed by the selected systems, due to the selection method itself, the evaluation is not degraded.

Learning to choose is then evaluated on the seven systems and cluster of topics are defined according to system performances: the better system performances are, the easier the topic are qualified. The evaluation shows that, based on a given set of systems, a fusion method is able to improve retrieval performances from 12% on medium difficulty topic to 44% on hard topics. Since results of the information retrieval process are not stored, the method is able to deal with dynamic collections.

In this study, query clusters are defined after queries have been processed at least once. This is possible in the context of repeated queries. However, such a method is not able to deal with queries submitted for the first time. Actually, future works should focus on applying difficulty prediction techniques. Such techniques can be based on linguistic predictors [START_REF] Mothe | Linguistic analysis of users' queries: towards an adaptive information retrieval system[END_REF] or on different predictors such as query ambiguity [START_REF] Chifu | Prédire la difficulté des requêtes : la combinaison de mesures statistiques et sémantiques (short paper)[END_REF].

Fig. 1 :

 1 Fig. 1: Analysis of the number of different system configurations.

Fig. 2 :

 2 Fig. 2: Number of selected system configurations (right axis) & performance (left axis) according to T.

Fig. 3 :

 3 Fig.3: Average performance according to number of selected system configurations.

Table 1 :

 1 Illustration-Performances of 4 system configurations over two queries.

	Q1	S1 ✞ 0.8	S2 ✞ ☎ 0.9 0.2 0.1 S3 S4 ☎
	✝ Q2 0.2 0.1 ✆ ✝	✆ ✞ 0.9	☎ ✞ 0.8	☎
				✝	✆ ✝	✆

http://trec.nist.gov/

http://trec.nist.gov/trec eval/

Acknowledgments. We would like to acknowledge the French ANR agency for their support through the CAAS-Contextual Analysis and Adaptive Search project (ANR-10-CORD-001 01)


Marco Tognon

Antonio Franchi

Extended Simulations for the Link Stress and Elevation Control of a Tethered Aerial Robot

This document is a technical attachment to [1] as an extension of the simulation's section.

VII. EXTENDED SIMULATIONS

We want to show here some additional simulations of the closed loop system for a better validation of the proposed method. In the following we present the results with different type of references for both the outputs, i.e., A) step, B) smooth step, C) and sinusoidal.

The simulation are done using Simulink and modeling the system with the toolbox SimMechanics, thus obtaining a more realistic validation independently from the model used for the control and observer design. The parameters used for these simulations are the same presented in Sec. V of [1], i.e., aerial robot mass: m R = 1 [Kg]; moment of inertia J R = 0.15 [Kg m 2 ]; link length: l = 2 [m]; gains k 1 and k 2 chosen such that the poles of the error dynamics are (-3, -6, -12, -24) and (-6, -12) respectively; observer gain ε = 0.1 and (α 1 , α 2 , α 3 ) chosen in order to have a stable error dynamics; observer discount rate: λ = 20.

A. Step Response

The Figs. 1, 2 show the step response results of the closed loop system. The reference details are reported on the captions of each figure. The plots showing the observer results are focused only on the initial and most relevant part of simulation, i.e., in which the prediction error is nonzero. We neglect the remaining part since the estimated state follows exactly the real one.

The plots show that, even with a discontinuous step reference on both the outputs, the proposed method preserve the stability of the closed loop system.

In the initial part of Fig. 1b, we can notice that, since the initial velocity is zero, the selector chooses the wrong observer. Nevertheless, after just some milliseconds, at time t 1 , since the controller makes the elevation velocity nonzero, the correct observer is selected and never changed again. Since the elevation trajectory lives in the third and forth quadrant, we can imagine the robot attached to the roof. Moreover, since we are requiring compression, the stabilization point Marco Tognon and Antonio Franchi are with LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse CEDEX 4, France.

mtognon@laas.fr,antonio.franchi@laas.fr is not trivial since the problem is similar to the inverted pendulum.

For the step reference passing from tension to compression showed in Fig. 2, we notice that, instead of inverting the sign of the thrust, the vehicle changes its attitude from up facing to down facing.

B. Smooth Trajectory

The Figs 3,4,5,6 show the behavior of the system following smooth trajectory from an initial to a final output configuration. The plots of the tracking errors show that the proposed controller is able, after a short transient, to perfectly follow the time varying smooth trajectories of class C 3 and C 1 for the elevation and stress, respectively.

In Fig. 5 it is interesting to notice that to pass from tension to compression the vehicle turns upside-down and keeping the thrust always positive. On the other side, in the simulation of Fig. 6 the transition from tension to compression is obtained with the thrust that passes through zero and inverts its sign in order to obtain the same final compressing force of the simulation of Fig. 5. This happens because in the second case the desired trajectory requires zero thrust at a certain moment. Then, since it is not possible to instantaneously turn the vehicle the controller inverts the sign of the thrust in order to provide compression. In the case of vehicles able to provide also negative thrust this is not a problem. While, in the case of robots providing only positive in order to generate feasible trajectories.

C. Sinusoidal Trajectory

In the third groups of simulations we want to show the ability of the presented method to follow a continuously changing trajectory as a sine curve, on both elevation and tension. As we can see in Fig. 7 and 8, the system shows an initial tracking error due to the transient in the observer. However, as soon as the estimation error goes to zero, the tracking error quickly goes to zero as well, i.e., the elevation and stress of the link follow exactly the desired sinusoidal trajectories.

[1] M. Tognon and A. Franchi, "Nonlinear observer-based tracking control of link stress and elevation for a tethered aerial robot using inertial-only measurements," in 2015 IEEE Int. Conf. on Robotics and Automation, Seattle, WA, 2015. [s] t1 ẽ (b) Observer Results: the vertical line, labeled t 1 , indicates the instant on which the selector changes observer, from the incorrect to the correct one. After this instant the 'good' observer is never changed again.

-

2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -3 -2.5 -2 -1.5 -1 -0.5 0 x [m] z [m] Time Begin End fL = -20 -20 -20 -20[N]
(c) Trajectory visualization.

  Controller Results: notice that after the transient due to the estimation error, the stress is kept constant even if a step in the elevation is required.

Fig. 1 :

 1 Fig. 1: The desired elevation trajectory is a discontinuous step from the initial value ϕ d 0 = 5π/4 [rad] to the final ϕ d f = 7π/4 [rad]. While the desired stress is a constant compression, f d L = -20 [N].

  Controller Results: asking an instantaneous transition from tension to compression we can notice a peak on the torque needed to rotate the vehicle as fast as possible in order to provide the compression.

Fig. 2 :

 2 Fig. 2: The desired elevation trajectory is a discontinuous step from the initial value ϕ d 0 = π/4 [rad] to the final ϕ d f = 3π/4 [rad]. The desired stress trajectory is a step from the initial tension f d L 0 = 20 [N] to the final compression f d L f = -20 [N].

Fig. 3 :

 3 Fig. 3: The desired smooth elevation trajectory of class C 3 goes from the initial value ϕ d 0 = π/4 [rad] to the final ϕ d f = 3π/4 [rad]. The desired smooth stress trajectory of class C 1 goes from the initial tension f d L 0 = 20 [N] to the final tension f d L f = 40 [N].

Fig. 4 :

 4 Fig. 4: The desired smooth elevation trajectory of class C 3 goes from the initial value ϕ d 0 = 5π/4, [rad] to the final ϕ d f = 7π/4, [rad]. The desired smooth stress trajectory of class C 1 goes from the initial compression f d L 0 = -20 [N] to the final compression f d L f = -40 [N].

Fig. 5 :

 5 Fig. 5: The desired smooth elevation trajectory of class C 3 goes from the initial value ϕ d 0 = π/4 [rad] to the final ϕ d f = 3π/4 [rad]. The desired smooth stress trajectory of class C 1 goes from the initial tension f d L 0 = 20 [N] to the final compression f d L f = -20 [N].

Fig. 6 :

 6 Fig. 6: The desired smooth elevation trajectory of class C 3 goes from the initial value ϕ d 0 = 5π/4 [rad] to the final ϕ d f = 7π/4 [rad]. The desired smooth stress trajectory of class C 1 goes from the initial tension f d L 0 = 20 [N] to the final compression f d L f = -20 [N].

Fig. 7 :

 7 Fig. 7: The desired elevation trajectory is a sine curve from the minimum value ϕ d min = π/8 [rad] to the maximum ϕ d max = 7π/8 [rad] with period T ϕ = 3 [s]. The desired stress is sinusoidal from the minimum tension f d L min = 20 [N] to the maximum f d L min = 40 [N] with period T f L = 3 [s].

Fig. 8 :

 8 Fig. 8: The desired elevation trajectory is a sine curve from the minimum value ϕ d min = π/4 [rad] to the maximum ϕ d max = 3π/4 [rad] with period T ϕ = 3 [s]. The desired stress trajectory is sinusoidal from the minimum compression f d L min = 20 [N] to the maximum tension f d L min = 40 [N] with period T f L = 3 [s].