Keywords: rogue waves, three sisters, NLS equation, KP-I equation, rational solutions, Darboux transform ondes scélérates, trois soeurs, équation NLS, équation KPI, solutions rationnelles, transformation de Darboux. √

Je souhaite tout d'abord remercier Vladimir B. Matveev pour avoir accepté d'être mon directeur de thèse et m'avoir consacré une partie de son temps précieux. Je remercie également Christian Klein pour avoir assuré les fonctions de co-encadrant et pour ses connaissances sur l'aspect numérique des problèmes étudiés ici.

Je suis très reconnaissant à Efim Pelinovsky, Nail Akhmediev et Alexander Its pour avoir pris le temps de lire ce travail et de formuler des remarques constructives dans leur rapports.

Je tiens à remercier Hans Jauslin, Vladimir Roubtsov et John Dudley de sacrifier une précieuse journée pour participer au jury de soutenance

Solution of order 3 with all phases equal to 0 . . . 8

Solution of order 4 with all phases equal to 0 . . . 9

Solution of KP-I with ϕ 4 = 0 . . . . . . . . . . . . 10 Solution of KP-I with ϕ 4 = 5+ 1 Introduction

Notion of rogue waves in oceanography and other fields

Sailing is and always has been a dangerous activity because sea is an unpredictable element. For example here is how Philippe Lijour, first mate of the oil tanker Esso Languedoc describes the wave that hit the ship near Durban in South Africa in 1980 :

We were in a storm and the tanker was running before the sea. This amazing wave came from the aft and broke over the deck. I didn't see it until it was alongside the vessel but it was special, much bigger than the others. It took us by surprise. I never saw one again.

Another surprising and devastating phenomenon that can occur is the so called "Three sisters" wave. It consists of one huge wave followed by two others. All these waves have the particularity to be unexpectedly high and to appear suddenly before disappearing as suddenly. They are called rogue or freak. Testimonies and a list of occurrences can be found at [27]. Some descriptions are available in [START_REF] Ch | Physical mechanisms of the rogue wave phenomenon[END_REF][START_REF] Ch | Rogue waves in the Ocean[END_REF].

For a long time, sailors who survived rogue waves and told their stories weren't taken seriously. Nowadays, even if some losses of ships can be blamed on rogue waves when a human mistake is in fact the cause of the sinking, it makes no doubt that rogue waves are a reality. Lots of pictures are available on the Internet. A Google Image search for "rogue waves" returns over three millions hits including pictures in Figure 1.

But these pictures can't be the basis of scientific works. To be able to analyze and produce modelings of rogue waves we need official measurement by instruments. A good source for such recordings is oil platforms. For example Figure 2 shows the New Year wave that hit the Draupner platform in 1995 in the North sea. After that, the study of rogue waves took off.

The researches on rogue waves are very active. Several mechanisms are believed to be possible sources of appearance of rogue waves. We can cite

• Interaction with a current : Waves from one current are driven into a current of opposing direction. The wave train compresses into a rogue wave. For example in the Indian ocean the Agulhas current goes against the westerlies.

• Spatial focusing : Small waves coming from different directions (open sea, refraction from coast, etc.) interfere and energy concentration happens.

• Spatio-temporal focusing : Waves with a large group velocity overtake waves with a smaller group velocity located in front of them and a unexpectedly big wave is produced.

• Nonlinear focusing : A wave concentrates the energy from other smaller waves and becomes huge and unstable before collapsing.

This list is not exhaustive and we could add the effects of the wind or of the Benjamin-Feir instability. Current "state of art" is well reflected in the volume [START_REF] Akhmediev | Discussions and debate : Rogue waves -Towards a unifying concept ?[END_REF] and several books especially [START_REF] Ch | Rogue waves in the Ocean[END_REF].

As we pointed out, the classification of rogue waves doesn't depend as much on the height of the wave than its singularity with respect to other waves. To define rogue waves properly, we need parameters to describe the waves and the sea. First, the height of a wave H w is defined as the larger distance between the top of the considered crest and the bottom of the troughs before and after. Then, we can define the significant wave height H s of the sea as the average waves height over the higher third of the waves in a given time interval, usually between ten and thirty minutes. The considered wave is classified as rogue if If rogue waves are especially studied in oceanography, they have recently appeared in other fields like non-linear optics [START_REF] Dudley | Emergence of rogue waves from optical turbulence[END_REF][START_REF] Montina | Non-gaussian statistics and extreme waves in a nonlinear optical cavity[END_REF][START_REF] Solli | Optical rogue waves[END_REF], matter physics [START_REF] Bludov | Matter rogue waves[END_REF], physics of fluids [START_REF] Ganshin | Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium[END_REF] and even in financial markets [17,[START_REF] Yan | Financial rogue waves[END_REF]. The simplest model for deep water is the focusing non-linear Schrödinger equation. It was obtained in 1968 by Zakharov in [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF] and it reads

H w H s > 2
Even
ir t + r xx + 2|r| 2 r = 0. (1)
Let denote by σ 1 , σ 2 , σ 3 , σ + , σ -the Pauli matrices

σ 1 := 0 1 1 0 , σ 2 := 0 -i i 0 , σ 3 := 1 0 0 -1 σ + := 0 1 0 0 , σ -:= 0 0 1 0 .
Let define U := U 0 + λU 1 and V := V 0 + λV 1 + λ 2 V 2 where U i and V j satisfy

U 0 := i(rσ + + rσ -), U 1 := 1 2 σ 3 , V 0 := -i|r| 2 σ 3 + (r x σ + -r x σ -), V 1 := -iU 0 , V 2 := -iU 1 .
Using these two matrices U and V we can write a zero curvature representation for the NLS equation. Equation ( 1) is equivalent to

U t -V x + (U V -V U ) = 0. (2)
Equation ( 2) can be seen as the compatibility condition for the following overdetermined system of vectorial equations

F x = U F F t = V F (3)
Here F is a two-components vector function

F := f 1 f 2 .
The following theorem can be found in [START_REF] Matveev | Darboux Transformations and solitons[END_REF] Theorem 1.

If F = f 1 f 2
is a solution of (3) for a real λ and matrices U (0) and V (0) constructed from a solution r 0 of (1) then r 1 defined by

r 1 := r 0 + 2iλ f 2 f 1 |f 1 | 2 + |f 2 | 2
is also a solution of (1).

For example, if we set r 0 := e 2it then f 1 and f 2 defined by

f 1 := -i √ 2x + √ 2 2 -2i √ 2t e -it , f 2 := √ 2x - √ 2 2 -2i √ 2t e it
give a solution of (3) for λ = 2. If we compute r 1 given in the previous theorem we get back, up to a sign, the famous Peregrine solution obtained in [START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF] which is the first example of non-trivial quasi-rational solution of NLS. Theorem 1 can be used recursively.

Once we obtain a new solution r 1 we are theoretically able to solve the associated system (3) with new matrices U (1) and V (1) . Then we can generalize it by a determinant formula (see [START_REF] Matveev | Darboux Transformations and solitons[END_REF]) or we can use iterative application of this theorem "by hand" as Akhmediev et al. did in their articles [START_REF] Akhmediev | Rogue waves and rational solutions of the NLS equation[END_REF][START_REF] Akhmediev | Waves that appear from nowhere and disappear without a trace[END_REF][START_REF] Ankiewicz | Rogue waves, rational solutions, the patterns of their zeros and integral relations[END_REF]. In these works the machinery of Darboux transformations was used to obtain some very remarkable but isolated solutions having a rogue waves behavior. In Section 2 we present a multi-parametric class of solutions showing the existence of multiple Peregrine breather. Solutions in [START_REF] Akhmediev | Rogue waves and rational solutions of the NLS equation[END_REF] can be obtained back from those with the right choice of parameters.

This method allows us obtain other solutions of NLS that are not quasi-rational. For example, from the same r 0 than previously and by solving the system for λ < 2 we can obtain a rational fraction in sinh(ωt) cos(ωx), cosh(ωt) cos(ωx), sinh(ωt) sin(ωx) and cosh(ωt) sin(ωx) multiplied by an exponential. It's the kind of solutions presented in [START_REF] Akhmediev | Generation of periodic trains of picosecond pulses in an optical fiber : exact solutions[END_REF] where the second order rational solutions were written down for the first time. It was then obtained by taking the rational limit of these "trigonometric-hyperbolic" solutions often called Akhmediev breather.

Darboux transform is not the only possible approach to construct solutions of NLS. Its integrability by the inverse scattering method was proved in the famous work [START_REF] Zakharov | An exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media[END_REF] by [START_REF] Zakharov | An exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media[END_REF]. A review of this topic and the Riemann-Hilbert problem approach can be found in [START_REF] Faddeev | Hamiltonian methods in the theory of solitons[END_REF][START_REF] Novikov | Theory of solitons : the inverse scattering method[END_REF]. It allowed to construct multi-soliton solutions. An another successful technics to obtain solutions of NLS is the algebro-geometric approach where solutions are expressed in terms of Riemann theta-functions of algebraic curves. Exact formulas are given in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF][START_REF] Its | Exact integration of nonlinear Schrödinger equation[END_REF].

Shallow water model : the KdV and the KP equations

In [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[END_REF], Korteweg and de Vries derived the KdV equation to represent the traveling of a wave in a channel of shallow water

u t + 6uu x + u xxx = 0. (4)
This equation can be seen at the compatibility condition of the following system

λψ = -ψ xx -uψ ψ t = -4ψ xxx -6uψ x -3u x ψ (5)
In [START_REF] Kadomtsev | On the stability of solitary waves in weakly dispersive media[END_REF], Kadomtsev and Petviashvili wrote down a 2+1 generalization of this equation the so-called KP equation

(u t + 6uu x + u xxx ) x = -3α 2 u yy . (6)
Once again, we can associate to this equation a system of linear differential equations

αψ y = -ψ xx -uψ ψ t = -4ψ xxx -6uψ x -3(u x -αv)ψ (7)
where v satisfies v x = u y . If α = i (resp. α = 1) the equation is called KP-I (resp. KP-II). The KP-I model describes waves in a very shallow water when surface tension is strong and KP-II is used for the case of weak surface tension. From now we restrict ourselves to the case α = i.

As in the case of NLS, these systems allow us to construct new solutions of ( 4) and ( 6) from simpler ones with the help of the following theorem Theorem 2. Let u be a solution of (4) (resp. ( 6)) and let ψ and ψ 1 two solutions of (5) (resp. ( 7)) for this u for different λ. Let denote by σ := ∂ x log ψ 1 . Then u defined by

u := u + 2∂ x σ
is also a solution of (4) (resp. ( 6)) and a solution of the associated system is given by ψ := ψ xσψ.

This theorem can easily be generalized in the following way Theorem 3. Let u be a solution of (4) (resp. ( 6)) and let ψ, ψ 1 , . . . , ψ n be solutions of (5) (resp. ( 7)) for this u. Then u defined by

u := u + 2∂ 2 x log W (ψ 1 , . . . , ψ n )
is also a solution of (4) (resp. ( 6)) and a solution of the associated system is given by

ψ := W (ψ 1 , . . . , ψ n , ψ) W (ψ 1 , . . . , ψ n ) .
Here W (g 1 , . . . , g n ) denotes the Wronski determinant of functions g 1 , . . . , g n with respect to x i.e. the determinant of the matrix A with entries

A ij := ∂ i-1
x g j . A proof of this theorem can be found in [START_REF] Matveev | Darboux transformation and the solutions to the KP equation depending on functional parameters[END_REF]. In Section 4 we will show that it allows to generate a family of smooth real rational solutions of the KP-I equation from the quasi-rational solutions of the focusing NLS equation.

KdV is one of the most, if not the most, studied equation in the literature. It's the first equation where soliton solutions have been found and the first equation that have been successfully integrated via the inverse scattering method. After that, it was found that other equations admit similar results including KP. Different axis of development emerged in the study of these two solitons equations like construction of algebro-geometric solutions [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF], study of solitons and rational solutions [START_REF] Ablowitz | Solitons and rational solutions of nonlinear evolution equations[END_REF][START_REF] Bordag | Two-dimensional solitons of the KP equation and their interaction[END_REF][START_REF] Krichever | Rational solutions of the Kadomtsev-Petviashvili equation and integrable systems of N particles on a line[END_REF][START_REF] Matveev | Some comments on the rational solutions of the Zakharov-Shabat equations[END_REF][START_REF] Pelinovsky | Rational solutions of the KP hierarchy and the dynamics of their poles I[END_REF][START_REF] Pelinovsky | Rational solutions of the KP hierarchy and the dynamics of their poles II[END_REF] or theory of Darboux transforms [START_REF] Liu | Vectorial Darboux tranformations for the KP hierarchy[END_REF][START_REF] Matveev | Darboux transformation and the solutions to the KP equation depending on functional parameters[END_REF][START_REF] Matveev | Darboux Transformations and solitons[END_REF][START_REF] Matveev | Darboux transformations and coherent interaction of the light pulses with twolevel media[END_REF].

Construction of solutions

Darboux transform for the non-stationary Schrödinger equation

Let consider the 1+1 non-stationary Schrödinger equation with potential u(x, t)

iψ t + ψ xx + u(x, t)ψ = 0. (8)
As stated in [START_REF] Matveev | Darboux Transformations and solitons[END_REF], this equation satisfies the following remarkable property Theorem 4. If f 1 , . . . , f n , f are solutions of (8) for a base potential u 0 (x, t) then

ψ := W (f 1 , . . . , f n , f ) W (f 1 , . . . , f n ) is also a solution of (8) for the potential u(x, t) := u 0 (x, t) + 2∂ 2 x log W (f 1 , . . . , f n ).
All these Darboux transform theorems are particular cases of a more general covariance theorem for several integrable systems obtained by Matveev.For completeness this theorem is presented in appendix A. Its use with a base potential u 0 (x, t) := 0 already gives us interesting solutions.

Let q 2n (k) be the polynomial defined by

q 2n (k) := n j=1 k 2 - ω 2m j +1 + 1 ω 2m j +1 -1 B 2 , ω := exp iπ 2n + 1 (9)
where B is a free real parameter and m j are integers satisfying

0 ≤ m j ≤ 2n -1, m l = 2n -m j (10)
for all l and j. Then, for any k,

f (k, x, t) := exp(kx + ik 2 t + Φ(k)) q 2n (k) (11)
where Φ(k) is of the form

Φ(k) := i 2n l=1 ϕ l (ik) l , ϕ j ∈ R, (12)
is a solution of ( 8) with u 0 = 0. We can construct other solutions by applying a differential operator in k to f or by evaluating in different values of k. For example, let f j be defined by

f j (x, t) := D 2j-1 k f (k, x, t) | k=B , j = 1 . . . , n, (13)
f n+j (x, t) := D 2j-1 k f (k, x, t) | k=-B , j = 1 . . . , n (14)
where

D k := k 2 k 2 + B 2 ∂ k . According to the previous theorem ψ(x, t, k) = W (f 1 , . . . , f 2n , f ) W (f 1 , . . . , f 2n )
is a solution of (8) for the potential u(x, t) = 2∂ 2 x log W (f 1 , . . . , f 2n ). Furthermore, for any constant C, s C defined by

s C (x, t) := Cψ(x, t, 0)
is a rational solution of the non-stationary Schrödinger equation for the potential u(x, t).

Reduction to the NLS equation

Let assume that, for a conveniently chosen C, the solution s

C satisfies u(x, t) = 2 |s C (x, t)| 2 -B 2 . ( 15
)
Then s C , now denoted by s, is a rational solution of the modified non-linear Schrödinger equation

is t + s xx + 2(|s| 2 -B 2 )s = 0. ( 16
)
From there, we can easily construct a non-singular quasi-rational solution of (1) given by the formula r(x, t) := s(x, t)e 2iB 2 t . Theorem 5. If C = ±q 2n (0)B 1-2n then s is a smooth rational solution of (16).

Proof of the reduction relation

The proof presented here is a modified and simplified version of the article [START_REF] Eleonskii | Rational multisoliton solutions to the NLS equation, Soviet Doklady 1986 sect[END_REF] written by Eleonskii, Krichever and Kulagin . It consists in a residue analysis of a couple of meromorphic differentials constructed from the dual Baker-Akhiezer function introduced by Cherednik in [START_REF] Cherednik | On the dual Baker-Akhiezer functions[END_REF]. Before tackling the proof of Theorem 5 itself, we need an easy algebraic lemma. Lemma 6. The polynomial q 2n defined by (9) satisfies

2k 2 q 2n (k)q 2n (-k) = (k 2 + B 2 ) 2n+1 + (k 2 -B 2 ) 2n+1 . ( 17
)
Proof w satisfies w 2m+1 = w 4n+2-2m-1 = w 2(2n-m)+1 which leads to

2k 2 q 2n (k)q 2n (-k) = 2k 2 n j=1 k 2 - ω 2m j +1 + 1 ω 2m j +1 -1 B 2 n j=1 k 2 - ω 2(2n-m j )+1 + 1 ω 2(2n-m j )+1 -1 B 2 = 2 2n+1 l=1 k 2 - ω 2l+1 + 1 ω 2l+1 -1 B 2 .
The roots of (X + B 2 ) 2n+1 + (X -B 2 ) 2n+1 satisfy

x l + B 2 x l -B 2 2n+1 = w 2n+1 , l = 1, . . . , 2n + 1 which yields x l + B 2 x l -B 2 = w 2l+1 or equivalently x l = w 2l+1 + 1 w 2l+1 -1 B 2 .
Hence

(k 2 +B 2 ) 2n+1 +(k 2 -B 2 ) 2n+1 = 2 2n+1 l=1 k 2 - ω 2l+1 + 1 ω 2l+1 -1 B 2 = 2k 2 q 2n (k)q 2n (-k)
Let dΩ be the meromorphic differential defined by

dΩ := q 2n (k)q 2n (-k) (k 2 -B 2 ) 2n dk.
Taking into account the previous lemma, we can re-write

dΩ = (k 2 + B 2 ) 2n+1 + (k 2 -B 2 ) 2n+1 2k 2 (k 2 -B 2 ) 2n dk.
In the neighborhoods of each of the points k = ±B, we use two local parameters:

E(k) := k + B 2 k , z(k) = k - B 2 k satisfying the relation E 2 = z 2 + 4B 2 . It yields dΩ = E 2n+1 z 2n+1 + 1 dE 2 .
or equivalently:

dΩ = (z 2 + 4B 2 ) n z 2n + z 2B 1 + z 2 4B 2 -1 2 dz 2 .
This leads to the following asymptotic expansion when z → 0 for dΩ, near the points k = ±B:

dΩ = α ± 0 z 2n + α ± 1 z 2n-2 + . . . + α ± n-1 z 2 + O(1) dz (18)
It is clear that D k = ∂ z and the derivatives of ψ with respect to z are given by

∂ m z ψ = W (f 1 , . . . , f 2n , D m k f ) W (f 1 , . . . , f 2n ) which gives us ∂ 2j-1 z ψ| k=±B = 0 j = 1, . . . , n.
Therefore an expansion of ψ in the neighborhood of k = ±B when z → 0 has the form:

ψ = β ± 0 + β ± 1 z 2 + . . . + β ± n-1 z 2n-2 + O(z 2n ). (19)
Equations ( 12), ( 18) and ( 19) guarantee that dΩ 1 and dΩ 2 defined by the formulas dΩ 1 := ψ(x, t, k)ψ(x, t, -k) dΩ and

dΩ 2 := (k + B 2 k ) dΩ 1
are meromorphic differentials with vanishing residues in B and -B. If we consider these differentials defined on the compact sphere CP 1 then the sum of residues equals 0 hence res ∞ dΩ 1 = 0 and res 0 dΩ 2 = -res ∞ dΩ 2 .

When k belongs to a neighborhood of ∞, ψ and dΩ admit the following expansions:

ψ(x, t, k) = 1 + ξ 1 (x, t) k + ξ 2 (x, t) k 2 + . . . e kx+ik 2 t+φ(k) , ( 20
) dΩ = 1 + 2nB 2 k 2 + . . . dk such that dΩ 1 = 1 + ξ 1 -ξ 1 k + ξ 2 + ξ 2 -|ξ 1 | 2 + 2nB 2 k 2 + . . . dk.
From these expansions we easily see that

res ∞ dΩ 1 = ξ 1 -ξ 1
which means that ξ 1 is real and

-res ∞ dΩ 2 = ξ 2 + ξ 2 -ξ 2 1 + (2n + 1)B 2 . Lemma 7. If {m 1 < • • • < m n } ∈ F n then m 1 = 0 and either m 2 = 1 or m n = 2n -1.

Proof

Let assume m 1 > 0. Then {m 1 , . . . , m n } and {φ n (m 1 ), . . . , φ n (m n )} are disjoint hence their union is a subset of {1, . . . , 2n -1} of cardinality 2n which is absurd. Now let assume m 2 > 1 and m n < 2n-1. Then {m 2 , . . . , m n } and φ n (m 2 ), . . . , φ n (m n ) are disjoint hence their union is a subset of {2, . . . , 2n -2} of cardinality 2n -2 which is absurd. We can remark that we can't have m 2 = 1 and m n = 2n -1 simultaneously. Proposition 8. ψ defined by

ψ : {1, 2n + 1} × F n -→ F n+1 (m, {0, m 2 , . . . , m n }) -→ {0, m 2 + 1, . . . , m n + 1, m}
m i = M i . Let {0 < p 2 < • • • < p n+1 } be an element of F n+1 . If p 2 = 1 then it's the image of (1, {0, p 2 -1, . . . , p n+1 -1}) otherwise it's the image of (2n + 1, {0, p 2 -1, . . . , p n -1}). Corollary 9. card (F n ) = 2 n-1 Proof Proposition 8 yields card (F n+1 ) = 2 card (F n ) and card (F 1 ) = 1
In the rest of this thesis, we set m j = j -1.

Analysis of these solutions

These solutions apparently depend on 2n + 1 parameters but only 2n -2 of them are important. B is the well-known re-scaling parameter. If s(x, t) is a solution then

s(x, t) := Bs(Bx, B 2 t)
is also a solution. Then, we can set B = 1 without loss of generality. Using phases ϕ 1 and ϕ 2 means replacing x and t with (x-ϕ 1 ) and (tϕ 2 ) which is irrelevant for the derivatives. ϕ 1 and ϕ 2 are just translation parameters that don't change the profile of the solution.

Case n=1

Here, we obtain essentially one solution, the Peregrine solution given which reads

s(x, t) = -3 + 4x 2 + 16t 2 -16it 1 + 4x 2 + 16t 2
for ϕ 1 = 0 and ϕ 2 = √ 3 4 . A plot is given in Figure 3.

The computation of the first derivatives gives us

∂ x (|s| 2 ) = 64x(-4x 2 +48t 2 +3 (1+4x 2 +16t 2 ) 3 ∂ t (|s| 2 ) = 256t(-12x 2 +16t 2 +1 (1+4x 2 +16t 2 ) 3
which leads to three critical points in (0, 0),(

√ 3 2 , 0) and (- √ 3
2 , 0). A study of the Hessian matrix in these points shows that (0, 0) is a local maximum where |s| = 3 and the other two are local minima where s = 0. Now, we will study the energy of this solution. Of course, as the base level is 1, the energy in its strict meaning is infinite. But we can adapt the definition of energy to obtain two integral relations :

+∞ -∞ |s(x, t| 2 -1 dx = 0 (22)
and

+∞ -∞ |s(x, t) -1| 2 dx = 4π √ 1 + 16t 2 (23)
The first relation shows that the excitation below the surface compensate exactly the excitation above, and that at every moment of time. The second is the first of a list of moments that can be obtained by integration by parts.

Case n=2

The solutions obtained here depend on two parameters ϕ 3 and ϕ 4 . For convenience, we choose

ϕ 1 := 3ϕ 3 , ϕ 2 := 2ϕ 4 + 3 + √ 5 16 10 -2 √ 5.
We obtain

s(x, t) = 1 -12 G(2x, 4t) + iH(2x, 4t) Q(2x, 4t) (24)
where

G(x, t) := x 4 + 6g 2 (t)x 2 + 2βx + g 0 (t) H(x, t) := tx 4 + 2h 2 (t)x 2 + 2βtx + h 0 (t) Q(x, t) := x 6 + 3g 2 (t)x 4 -2βx 3 + 3q 2 (t)x 2 + 6βg 2 (t)x + q 0 (t)
= t 2 + 1 g 0 (t) := 5t 4 + 18t 2 -4αt -3 h 2 (t) := t 3 -3t + α h 0 (t) := t 5 + 2t 3 -2αt 2 -15t + 2α q 2 (t) := t 4 -6t 2 + 4αt + 9 q 0 (t) := t 6 + 27t 4 -4αt 3 + 99t 2 -36αt + β 2 + 4α 2 + 9
In these formulas parameters α and β are given by We can remark that the profile shown in Figure 6 is different of the other two. The system of three peaks of similar height is replaced by one giant peak surrounded by four peaks of smaller amplitude.

α := (5 + √ 5) 10 -2 √ 5 -96ϕ 4 , β := 96ϕ 3 .
In this case the solution ( 24) is the second order solution obtained by Akhmediev et al. in [START_REF] Akhmediev | Rogue waves and rational solutions of the NLS equation[END_REF].

Higher order solutions

In this subsection conjectures about solutions obtained for n > 2 are presented. We have to work from plots of solutions. Two of them can be found in figures 7 and 8. It represents third and fourth order solutions when all phases are equal to zero. It's believed to be the general profile. The solutions seem to have in general n(n+1) 2 maxima of comparable amplitude and n(n+1) minima.

But once again for exceptional solutions we should observe the appearance of a "super-peak" surrounded by n(n + 1) -2 peaks of significantly smaller amplitudes. t with y and ϕ 3 with -4t then f defined by [START_REF] Bludov | Matter rogue waves[END_REF] satisfy

if y = -f xx f t = -4f xxx
and so do the functions f j defined by ( 13) and ( 14).

Proof

The fact that u is a solution comes from Theorem 3 and condition [START_REF] Eleonskii | Rational multisoliton solutions to the NLS equation, Soviet Doklady 1986 sect[END_REF]. The integral relation is derived from the Wronskian representation u(x, y, t) = 2∂ 2

x log W (f 1 , . . . , f 2n ).

It means that relation [START_REF] Ch | Physical mechanisms of the rogue wave phenomenon[END_REF] holds for every solution of NLS constructed in Section 2.

These solutions are clusters of peaks traveling to infinity, possibly on a line. Two series of plots for different values of ϕ 4 are presented in figures 9 and 10. We set ϕ 1 = ϕ 2 = 0.

Solutions of the Johnson equation

In [START_REF] Johnson | Water waves and Korteweg-de-Vries equations[END_REF] the author introduces a two dimensional generalization of the cylindrical KdV equation called the Johnson equation (or cylindrical KP equation) that reads

v t + v xxx + 6vv x + v 2t x = 3v yy t 2 . ( 25
)
As stated in [START_REF] Klein | Cylindrical Kadomtsev-Petviashvili equation : old and new results[END_REF], the Johnson equation is closely related to the KP-I equation.

Proposition 11. If u is a solution of ( 6) then v defined by v(x, y, t)

:= u(x + y 2 t 12 , yt, t)
is a solution of [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[END_REF].

We can use this change of variables to construct solutions of (25) from the solutions of KP-I presented above. Some of these solutions are represented in figures 11 and 12 for different values of ϕ 4 and ϕ 1 = ϕ 2 = 0. We can remark that this time the influence of ϕ 2 on these solutions is more complicated than a translation.

Conclusion and open questions

The solutions presented here were obtained for the first time in the article [START_REF] Eleonskii | Rational multisoliton solutions to the NLS equation, Soviet Doklady 1986 sect[END_REF] by Eleonskii, Krichever and Kulagin in 1986 up to some inaccuracies. Surprisingly this work has been overlooked in the integrable systems community and the rogue waves community until now. I'm inclined to believe that the work presented here have several advantages over the original exposition :

• The quasi-rational solutions of the NLS equation are described in a simpler and more elegant way using Wronski determinants.

• Using a Darboux transformation approach for the non-stationary Schrödinger equation gives a simplified version of the proof.

• It also allows us to construct for "free" a large family of real non-singular rational localized solution of the KP-I equation which might be associated with shallow water rogue waves.

• Maybe more than everything else it contains a qualitative study of these solutions making their multi-rogue waves behavior obvious.

• Finally the approach used here can be extended to many other equations of physical interest, for example the vectorial NLS equation. This work is in progress.

Nonetheless, several questions stay open.

First, a better understanding of the location and the dynamics of peaks and holes would be interesting. It would also provide a better understanding of the propagation of the peaks of the associated solution of KP-I.

Then, we can tackle the problem of higher values of n. If, theoretically, the same scheme can be applied for solutions obtained with n ≥ 3, the amount of calculations would be overwhelming and the results not very satisfactory in the long term. To obtain similar results for generic n, we need to work from the Wronskian formula. Maybe, a re-writing of this formula in terms of theta functions or Fredholm determinants would help. See [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF][START_REF] Gaillard | From finite-gap solutions of KdV in terms of theta functions to solitons and positons[END_REF][START_REF] Its | Exact integration of nonlinear Schrödinger equation[END_REF] for more information.

We can also wonder about the generality of these solutions. Can we choose the different parameters in the definition of f j more generally and still obtain rational solutions with a system of peaks ? What is the position of these solutions among all rational solutions of NLS ? Finally, other well-known equations admit rogue rational solutions. For example, some special rational solutions of the Hirota equation having the behavior of higher order Peregrine breathers were obtained in [START_REF] Ankiewicz | Rogue waves and rational solutions of the Hirota equation[END_REF]. It could be interesting to use the same method as here to obtain the more general family of rational solutions.

A Covariance theorem

In this section the subscript denotes the derivative with respect to the evolution variable z and the prime or power notation is used for the derivative with respect to x. Theorem 12. Let f be a solution of

f z = n m=0 u m (x, z)f (m) (26)
and let denote by φ another solution of ( 26) and σ = φ φ . Then ψ defined by

ψ = (∂ x -σ)f (27)
satisfies

ψ z = n m=0 u m (x, z)ψ (m) (28)
where u m are defined by .

As φ is solution of (26) the terms for j = 0 in the right-hand side of the last equation cancels out the term φ z x x 0 φ -1 ψ dτ in the left-hand side. Hence

Multiplying both sides of this relation by φ -1 and differentiating by x we get the following formula :

(φ -1 ψ) z = φ -1 n k=1 u k k j=1 C j k φ (k-j) (φ -1 ψ) (j-1)
x .

We can easily check that (31) yields, up to a factor φ -1 , equation [START_REF] Liu | Vectorial Darboux tranformations for the KP hierarchy[END_REF] with the u m defined in theorem 12.

This result can be generalized in the case of a matricial equation (see [START_REF] Matveev | Darboux Transformations, Covariance Theorems and Integrable systems[END_REF]). It can also be used iteratively. In this case the formula is simpler than expected.

Theorem 13. Let φ 1 , . . . , φ N be linearly independent solutions of [START_REF] Krichever | Rational solutions of the Kadomtsev-Petviashvili equation and integrable systems of N particles on a line[END_REF]. The result of a N -step Darboux transformation applied to f is given by D N is a differential operator of order N of the form

D N = ∂ N x + N -1 m=0 a m ∂ m x ( 32
)
such that D N f = 0 when f = φ i . The coefficients a m are solutions of a system of N equations and can be written down with Cramer's formula. Then we can see that the right-hand side of ( 32) is the development along the last column of W (φ 1 , . . . , φ N , .) W (φ 1 , . . . , φ N ) .

We can also generalize the formulas for the coefficients of the equation. For example

B Maple code for plots

Here an example of the code used to produce the plots. This corresponds to Figure 4.

1 . 1 3 = 1 and ϕ 4

 1134 Notion of rogue waves in oceanography and other fields . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Deep water model : the NLS equation . . . . . . . 1.3 Shallow water model : the KdV and the KP equations 2 Construction of solutions 2.1 Darboux transform for the non-stationary Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . 2.2 Reduction to the NLS equation . . . . . . . . . . . 2.3 Proof of the reduction relation . . . . . . . . . . . 2.4 Further remarks on condition (10) . . . . . . . . . 3 Analysis of these solutions 3.1 Case n=1 . . . . . . . . . . . . . . . . . . . . . . 3.2 Case n=2 . . . . . . . . . . . . . . . . . . . . . . 3.3 Higher order solutions . . . . . . . . . . . . . . . . 4 Solutions of the KP-I and the CKP equations 4.1 Solutions of the KP-I equation . . . . . . . . . . . 4.2 Solutions of the Johnson equation . . . . . . . . . 5 Conclusion and open questions A Covariance theorem B Maple code for plots List of Figures 1 Some pictures of rogue waves . . . . . . . . . . . . 2 The Draupner wave . . . . . . . . . . . . . . . . . 3 The Peregrine solution . . . . . . . . . . . . . . . 4 ϕ 3 = 0 and ϕ 4 = 0 . . . . . . . . . . . . . . . . . . 5 ϕ

Figure 1 :

 1 Figure 1: Some pictures of rogue waves

Figure 2 : 10 1. 2

 2102 Figure 2: The Draupner wave

  is a bijection. Proof Let assume that (m, {0, m 2 , . . . , m n }) and (M, {0, M 2 , . . . , M n }) have the same image by ψ. If m = M then 1 and 2n + 1 are both in the set which is impossible. Then m = M and it's straightforward that

Figure 3 :

 3 Figure 3: The Peregrine solution

Figures 4 to 6

 6 Figures 4 to 6 show this solution for different value of ϕ 3 and ϕ 4 .We can remark that the profile shown in Figure6is different of the other two. The system of three peaks of similar height is replaced by one giant peak surrounded by four peaks of smaller amplitude. In this case the solution[START_REF] Klein | Cylindrical Kadomtsev-Petviashvili equation : old and new results[END_REF] is the second order solution obtained by Akhmediev et al. in[START_REF] Akhmediev | Rogue waves and rational solutions of the NLS equation[END_REF].

Figure 4 :Figure 5 : 5 Figure 6 :

 4556 Figure 4: ϕ 3 = 0 and ϕ 4 = 0

Figure 7 :Figure 8 :

 78 Figure 7: Solution of order 3 with all phases equal to 0

Theorem 10 .

 10 u(x, y, t) := 2 |s| 2 -B 2 is a smooth real solution of KP-I that satisfies ∀ t u(x, y, t) -→ 0 when x 2 + y 2 -→ +∞ and ∀ y, t +∞ -∞ u(x, y, t) dx = 0.

Figure 9 :Figure 10 :

 910 Figure 9: Solution of KP-I with ϕ 4 = 0

Figure 11 :Figure 12 :

 1112 Figure 11: Solution of CKP with ϕ 4 = 0

u n = u n , 1 = φ x x 0 φ - 1

 101 (u kσu k )φ (k-j) + u k φ (k-j+1) φ -1 (j-1-m) j u k φ (k-j) φ -1 (j-m) 0 ≤ m ≤ n -1.Proof Considering (27) as a differential equation for f we can write f (τ, z)ψ(τ, z) dτ.

  φ (k-j) φ -1 ψ (j-1) .

ψ

  N := D N f = W (φ 1 , . . . , φ N , f ) W (φ 1 , . . . , φ N )ProofWe prove this by induction on N . For N = 1 it's the previous theorem. Let assume thatψ N -1 = W (φ 1 , . . . , φ N -1 , f ) W (φ 1 , . . . , φ N -1) .Thenφ[N ] = W (φ 1 , . . . , φ N -1 , φ N ) W (φ 1 , . . . , φ N -1 )is a solution of the equation satisfied by ψ N -1 . If we denote by σ[N ] the logarithmic derivative of φ[N ] then ψ N = ψ N -1σ[N ]ψ N -1 .

u n- 2

 2 [N ] = u n-2 +nu n N m=1 σ[m] = u n-2 +nu n ∂ 2x log(W (φ 1 , . . . , φ N )).

Remerciements

Substituting [START_REF] Johnson | Water waves and Korteweg-de-Vries equations[END_REF] into [START_REF] Ankiewicz | Rogue waves, rational solutions, the patterns of their zeros and integral relations[END_REF] we obtain the formulas u = -2∂ x ξ 1

The real part of ( 21) combined with the reality of ξ 1 yields

Therefore

Since res

Comparing the behaviors when |x| → ∞ in this identity, we get A(t) = -2nB 2 which completes the proof of the crucial relation [START_REF] Eleonskii | Rational multisoliton solutions to the NLS equation, Soviet Doklady 1986 sect[END_REF]. Now, we can use the remark in [START_REF] Its | Exact integration of nonlinear Schrödinger equation[END_REF] that any meromorphic solution of the focusing NLS equation can't have real poles x j (t). [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF] In this subsection we denote by F n the set

Further remarks on condition


PhD Adam Fletcher
email: adam.fletcher@lshtm.ac.uk

PhD Chris Bonell

MSc Annik Sorhaindo

Keywords: Adolescent, Social Networks, Obesity, Eating Disorders, Review. [ Adolescent, Social Networks, Obesity, Eating Disorders, Review. [MeSH terms]

You are what your friends eat: systematic review of social network analyses of young people's eating behaviours and bodyweight.

BACKGROUND

In high-income countries, obesity among children and young people has increased rapidly since the latter part of the twentieth century. [START_REF] Strauss | Epidemic increase in childhood overweight, 1986-1998[END_REF] For example, a third of children and young people in the USA are considered to be overweight or obese, and prevalence is now also as high as 35% in parts of Europe with rates of increase accelerating worldwide. [START_REF] Lobstein | International Comparisons of Obesity Trends, Determinants and Responses: Evidence Review[END_REF] In addition to the long-term health risks associated with obesity in adulthood, obesity is associated with type-2 diabetes among children and adolescents, and overweight adolescents have higher rates of depressive symptoms and lower self-esteem. [START_REF] Strauss | Personal and interpersonal characteristics associated with childhood obesity[END_REF] Conversely, trends in young people's dieting/weightcontrol behaviours and eating disorders are also of public health concern: 2-3% of adolescent females are thought to have a clinical eating disorder. [START_REF] Favaro | The spectrum of eating disorders in young women: a prevalence study in the general population sample[END_REF] These young people are at risk of longterm physical harms and early mortality due to vitamin deficiencies and rapid weight loss. [START_REF] Steinhausen | An analysis of the outcome of anorexia nervosa and bulimia nervosa[END_REF] Recognition of these trends and harms has led to an increasing emphasis on prevention and early intervention, and a need to understand better the determinants of such problems. [START_REF] Pearson | Prevention of eating disorders: challenges and opportunities[END_REF][START_REF] Jacobi | Coming to terms with the risk factors for eating disorders: application of risk terminology and suggestions for a general taxonomy[END_REF][START_REF] Koplan | Preventing Childhood Obesity: Health in the Balance[END_REF] Social network analysis is one approach, examining the influence of social relationships on these outcomes. Sociometric studies examine social relationships in terms of nodes (individuals) and ties (relationships), drawing on reciprocal reports from each individual in a putative tie. Network analysis based on these socio-metric data allows examination of how an individual's relationships and the characteristics of the broader social network are associated with, and can influence, outcomes. Such analyses can overcome problems with earlier studies of peer effects on health-related behaviours which only examined immediate friendships rather than the wider peer group [START_REF] West | Smoking and peer influence[END_REF] and also generally relied on data from a single informants' reports of others' actions, which tends to over-estimate both the reciprocity of ties and the sharing of friends' behaviours and charateristics. [START_REF] Alexander | Peers, schools and adolescent smoking[END_REF] Recent longitudinal studies of adults' social networks suggests that both obesity and eating disorders diffuse through social ties and may be 'socially contagious'. [START_REF] Christakis | The Spread of Obesity in a Large Social Network Over 32 Years[END_REF][START_REF] Crandall | The social contagion of binge eating[END_REF] Although these studies focus on adults not young people, theoretical models to understand body-image concern and eating disorders have included peers as an important socio-cultural determinant [e.g. 13] and empirical studies report that young people perceive their close friends and peers to influence their eating behaviours. [START_REF] Wertheim | Why do adolescent girls watch their weight? An interview study examining sociocultural pressures to be thin[END_REF][START_REF] Lieberman | Interpersonal influence and disordered eating behaviors in adolescent girls: the role of peer modeling, social reinforcement, and body-related teasing[END_REF][START_REF] Ludvigsen | Burger boy and sporty girl: children and young people's attitudes towards food in school[END_REF][START_REF] Milner | Freaks, geeks and cool kids: American teenagers, schools and the culture of consumption[END_REF] This suggests social networks may influence young people's eating behaviours and adolescent bodyweight as they do in adults. However, the evidence regarding associations and causal influences between young people's social networks, and eating behaviours, calorific intake and/or bodyweight has not been synthesized. Determining whether social networks causally influence these outcomes requires more than evidence of association. [START_REF] Christakis | The Spread of Obesity in a Large Social Network Over 32 Years[END_REF] As well as peer influence, associations may reflect reverse causality, whereby young people select friends as based on shared behaviours and/or bodyweight (homophily) [START_REF] Mcpherson | Birds of a Feather: Homophily in Social Networks[END_REF], or confounding due to structural equivalence, whereby friends share characteristics other than their peer group which shape their eating behaviours and weight, such as family income and neighbourhood context. [19] This systematic review aimed to examine whether: [START_REF] Strauss | Epidemic increase in childhood overweight, 1986-1998[END_REF] there are associations between measures of individual social position (e.g. popularity) or a measure of the overall structure of social networks (e.g. density) and young people's eating behaviours, calorific intake and/or bodyweight; and (2) explore how these associations vary according to the setting and participants' socio-demographic characteristics. To do this we examined cross-sectional and longitudinal studies based on analyses of socio-metric data. Although longitudinal data is needed to examine the direction of potential network effects and processes over time, crosssectional studies can provide further evidence regarding the nature and extent of associations, informing further research and new interventions. [START_REF] Valente | Opinion leader interventions in social networks[END_REF]

METHODS

Search strategy

Major bibliographic databases (Social Science Citation Index, PubMed) and specialist registers (ADOLESC, ERIC) were searched in March 2009 using appropriate free-text and thesaurus terms relating to the population (e.g. "adolescent" or "youth"), social networks (e.g. "network" or "peer group") and health-related outcomes (e.g. "diet" or "obesity"). A full list of terms is available on request. Keywords, titles and abstracts were then screened according to the inclusion criteria specified below. The bibliographies of included studies were also searched and key investigators were contacted to identify 'grey literature', PhD thesis and forthcoming publications.

Inclusion criteria

There were no restrictions according to language or publication date. Cross-sectional and longitudinal observational studies examining the association between measures of young people's social networks based on reciprocal socio-metric data and eating behaviours or bodyweight were identified, including studies of dyadic relationships nested within larger sociometric datasets. Those studies which described non-reciprocal ego-centric reports of relations were not included. Studies were therefore eligible for inclusion if they: (a) collected 'reciprocal' data from a defined group of young people in the age range [START_REF] Christakis | The Spread of Obesity in a Large Social Network Over 32 Years[END_REF][START_REF] Crandall | The social contagion of binge eating[END_REF][13][START_REF] Wertheim | Why do adolescent girls watch their weight? An interview study examining sociocultural pressures to be thin[END_REF][START_REF] Lieberman | Interpersonal influence and disordered eating behaviors in adolescent girls: the role of peer modeling, social reinforcement, and body-related teasing[END_REF][START_REF] Ludvigsen | Burger boy and sporty girl: children and young people's attitudes towards food in school[END_REF][START_REF] Milner | Freaks, geeks and cool kids: American teenagers, schools and the culture of consumption[END_REF][START_REF] Mcpherson | Birds of a Feather: Homophily in Social Networks[END_REF] in order to be able to describe the pattern and/or nature of friendship connections (ties) between individual young people (nodes);

(b) provided summaries of social-network characteristics at the level of the individual (e.g. popularity) or network (e.g. density);

(c) reported on measures of eating-related behaviours, caloric intake or bodyweight at the individual or network level; and

(d) reported one or more measures of association between (b) and (c).

Data extraction and synthesis

A standardised framework was used by two independent reviewers to extract data from studies (available on request). Neither outcomes nor summaries of social-network characteristics were sufficiently homogenous to undertake statistical meta-analysis and therefore findings are synthesised narratively.

RESULTS

After screening 2,242 titles and abstracts, we identified ten studies (published between 1995 and 2010) which met our inclusion criteria. All of these focused on school social networks and five used data collected in the US National Longitudinal Study of Adolescent Health (Add Health) surveys (but undertook different analyses of these data). [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF][START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF][START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF][START_REF] Trogdon | Peer effects in adolescent overweight[END_REF][START_REF] Halliday | Weight gain in adolescents and their peers[END_REF] A further two were based on other US studies [26,[START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF] and three used social network data collected in Australian high schools. [START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF][START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF][START_REF] De La Haye | Obesity-related behaviours in adolescent friendship networks[END_REF] Sample sizes ranged from 385 to 17,557 school students (see Table 1 for further details of study design, sample size and characteristics, methods for collecting and analysing social network data, and key findings reported by authors). Three studies reported on longitudinal data, [START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF][START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF][START_REF] Trogdon | Peer effects in adolescent overweight[END_REF] while seven drew on cross-sectional data. One study reported only on associations between individual-level measures of young people's school social networks -such as the number of nominations received/given (in-degree/out-degree), reciprocity, and centrality -and bodyweight. [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF] Eight used sociometric data to examine network-level associations with eating behaviours/bodyweight. [START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF][START_REF] Trogdon | Peer effects in adolescent overweight[END_REF][START_REF] Halliday | Weight gain in adolescents and their peers[END_REF][26][START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF][START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF][START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF][START_REF] De La Haye | Obesity-related behaviours in adolescent friendship networks[END_REF] Two of these also reported analyses of individual-level measures of young people's school social networks. [START_REF] Trogdon | Peer effects in adolescent overweight[END_REF][START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF] One study also reported associations among dyads constructed within the Add Health dataset. [START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF] The studies reported on a range of outcome measures. One examined high calorie food consumption. [START_REF] De La Haye | Obesity-related behaviours in adolescent friendship networks[END_REF] Eight studies examined body mass index (BMI), used to define 'healthy', overweight, obese and underweight individuals. [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF][START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF][START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF][START_REF] Trogdon | Peer effects in adolescent overweight[END_REF][START_REF] Halliday | Weight gain in adolescents and their peers[END_REF][START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF][START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF][START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF] Two examined dieting, extreme weight-loss behaviours, binge eating and body-image concern as well as BMI score. [START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF][START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF] One focused on bulimic symptoms. [26] School social networks and high calorie food consumption One cross-sectional study focused on secondary school students' high calorie food consumption and found that male friends were significantly alike in their consumption of fast food but no similarities were identified between girls' friends and fast food consumption, or other measures of high calorie food consumption (e.g. high calorie drink consumption). [START_REF] De La Haye | Obesity-related behaviours in adolescent friendship networks[END_REF] Odds that an ego becomes obese if his or her 'best friend' is obese:

Unadjusted = 0.8
After adjusting for school-specific trends = 0.51

After adjusting for school-specific trends and individual fixed effects = no-sig. effect de la Haye et al [START_REF] De La Haye | Obesity-related behaviours in adolescent friendship networks[END_REF] Australia Crosssectional High-calorie food consumption

Body image concern

No significant differences Paxton et al [START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF] Australia Crosssectional

Strauss and

Pollack [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF] Add Health In-degree (mean no. of nominations) Trogdon et al [START_REF] Trogdon | Peer effects in adolescent overweight[END_REF] Add Health

Frequency named as a friend

No association with being overweight idea of social influence comes from Fowler and Christakis finding that non-reciprocal 'named friends' were not influenced by 'namers' over time. [START_REF] Halliday | Weight gain in adolescents and their peers[END_REF] Conversely, Cohen-Cole and Fletcher suggest that observed similarities in friends BMI over time in the Add Health study are explained by homophily. [START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF] They constructed friendship dyads based on the first friend named by respondents in the Add Health survey and found that, after controlling for potential confounding due to shared school environment, the odds that an ego becomes obese increase by 51% if his or her 'alter' (friend) was also obese. [START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF] After controlling for both contextual factors and individual-level 'fixed effects' the relationship between friendship dyads and BMI disappeared, which the authors suggest indicates that young people select friends similar to themselves.

Trogdon and colleagues found that overweight adolescents were less likely to be nominated as a friend, [START_REF] Trogdon | Peer effects in adolescent overweight[END_REF] as did Strauss and Pollack who studied overweight students' social position and report that for all network measures overweight adolescents were more likely to receive no or few friendship nominations and report fewer reciprocal friendships. [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF] After adjusting for socio-demographic factors, participation in physical activity and clustering by school, the average 'social penalty' was minus 1.25 friendship nominations for girls and minus 1.03 for boys (p < 0.001). While Valente and colleagues found that overweight girls were less likely to be named as a friend, there was no such association among boys. [START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF] At the network level, the two Australian cross-sectional studies suggests that female highschools students' friendship groups resemble each other with respect to BMI scores, and that differences in BMI are significantly greater between rather than within friendship groups. [START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF][START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF]

School social networks, body image concerns and eating disorders

There is evidence that among girls behaviours such as dieting and extreme weight loss behaviours are clustered in certain groups within schools. In a cross-sectional US study, Pike found that bulimic symptoms were strongly predicted by having female high-school friends who also reported bulimic symptoms [26], while Hutchinson and Rapee also found significant withingroup similarity regarding students' dieting, extreme weight loss behaviours and binge eating among girls at Australian high schools, even after controlling for BMI, self-esteem and 'negative' emotions. Around 25% of the variance in these behaviours could be explained by group membership. [START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF] Friendship groups which reported high levels of dieting and extreme weight loss behaviours were also associated with high mean group reports of: friends' concern with thinness and dieting; peer pressure to lose weight and to be thin; and, peer teasing. [START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF] In contrast, another cross-sectional study of female high school students in Australia by Paxton and colleagues found that friendship groups among girls in Australian high schools shared similar levels of body image concern, dietary restraint and, use of extreme weight loss behaviours after adjusting for individual physical and psychological factors, [START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF] However, within this sample, binge eating was not associated with social network once individual-level sociodemographic and physical factors were included in multivariate analysis. Further analyses suggested that perceptions of friends' views and actions with respect to body image concern and dieting, and the extent to which a student reported comparing her body with others, were influential in these behaviours clustering. [START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF] How do these associations vary according to the setting and socio-demographic characteristics?

The study by de le Haye and colleagues only found associations between boys' friendship groups and fast-food consumption within two out of three school networks. [START_REF] De La Haye | Obesity-related behaviours in adolescent friendship networks[END_REF] Within one network the consumption of fast food was positively associated with popularity (receiving nominations), suggesting that the underlying social processes may vary across different school cultures and contexts. Although fast-food consumption appeared to cluster among groups of male school-friends, both the propensity to be overweight when friends were overweight and the risk of social isolation due to being overweight appeared to be greatest among females. [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF][START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF][START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF] There is also some evidence that the importance of friendships on having eating disorders might be greatest in early adolescence, for example among 12-14 year olds, and this diminishes later in females' high school careers.

[26]

DISCUSSION

The evidence reviewed above suggests that, in high-income countries, school friendships may be critical in shaping young people's eating behaviours and bodyweight and/or vice versa. There is consistent evidence that school friends are significantly similar in terms of their BMI, and adolescents with the highest BMI appear to be most similar. Overweight youth are also less likely to be popular students and are at risk of being socially isolated at school. Frequency of fast-food consumption has also been found to cluster within groups of boys, as have body image concerns, dieting and eating disorders among girls.

Although there is consistent evidence about associations, the studies reviewed do not provide definitive evidence about the relative importance of peer influence, peer selection, and structural equivalence/confounding. It is plausible that all three occur and partly explain the patterns reported. There is some evidence that peer influence may underlie diffusion of obesity [START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF][START_REF] Halliday | Weight gain in adolescents and their peers[END_REF] and eating disorders [START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF][START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF], and that homophily may explain observed similarities in friends BMI over time. [START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF] Although structural equivalence may partly explain within-group similarities, these may decline once potential confounders, such as school environment, are adjusted for. However, none of the studies found that such external, shared characteristics fully explained significant relationships between students' social networks, on the one hand, and their eating behaviours or bodyweight, on the other.

Limitations 20

Only a limited number of studies were identified (n=10), all of which drew on data from the USA or Australia, so that these findings may not be generalizable elsewhere. Furthermore, the majority were cross-sectional and cannot determine causal direction. The three longitudinal studies drew different conclusions due to their different methodological approaches. [START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF][START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF] Furthermore, Cohen-Cole and Fletcher's findings are limited because they only use friendship dyads (rather than all observed friendships) that are static over time, which leaves students' 'random' ex-friends in the model and thus reduces the chance of finding any effects due to friendship. [START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF] Although all the included studies aimed to control for confounding by age, gender, ethnicity and/or socio-economic status, there may have been residual confounding from factors such as parenting styles or baseline non-health risk behaviours. However, over-adjustment is also a potential limitation, and some studies may therefore have under-estimated the true association between students' social networks and eating behaviours/bodyweight. For example,

Hutchinson and Rapee report that around 25% of the variance in the eating behaviours they studied could be explained by group membership after controlling for individuals' BMI, selfesteem and 'negative' emotions [START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF], although factors such as self-esteem may lie on the causal pathway. More generally, the statistical methods applied in the field of social network analysis are constantly evolving, and results may differ according to the whether logistic regression models or ERGM is applied. [START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF][START_REF] De La Haye | Obesity-related behaviours in adolescent friendship networks[END_REF] All these analyses examined young people's social networks in school, ignoring out-of-school friendships which may be influential. [START_REF] Kirke | Collecting peer data and delineating peer networks in a complete network[END_REF] While schools provide a unique institutional context in which to collect and construct 'complete' (rather than 'ego-centric') network data, [START_REF] Scott | Social Network Analysis: a handbook[END_REF] young people have other peer groups and different friends outside school so the identified networks are incomplete. [START_REF] Kirke | Collecting peer data and delineating peer networks in a complete network[END_REF] For example, other friends who live in the same neighbourhood but go to different school, or have left school, could be influential in determining eating behaviours. A further limitation with the studies identified is that they do not examine how a school or community's macro-social network structure can influence aggregate health outcomes over and above the effects of friendship networks. For example, do schools with denser networks and/or larger components facilitate the diffusion of obesity or 'crash' dieting?

Finally, a limitation of all systematic reviews is their reliance on authors' descriptions of the methods and exposures under study, and how the findings are presented in published reports and papers. For example, authors do not always define concepts such as obesity in exactly the same way or report how they measured this. Furthermore, measures of effect sizes were reported inconsistently across studies, and in some cases the effect sizes were not reported. It is also important to note that in some cases multiple tests of significance within papers may have resulted in some apparent associations having arisen due to chance.

Implications for policy and practice 'multiplier' effect [START_REF] Christakis | Social Networks and Collateral Health Effects -Have Been Ignored in Medical Care and Clinical Trials, But Need to Be Studied[END_REF] whereby their initial impact spreads more widely via social diffusion. This would, in turn, imply that those school-based interventions which have previously been found to be effective, such as whole-school healthy eating programmes [START_REF] Shepherd | Young people and healthy eating: a systematic review of research on barriers and facilitators[END_REF], may have a greater 'reach', wider benefits, and be more cost-effective than previously thought.

Implications for research

Well-conducted longitudinal studies are necessary to determine the relative importance of selection, influence and structural equivalence. Such studies need to occur in a wider range of contexts, and in countries other than the USA and Australia, and should also aim to capture non-school friends via community and on-line surveys. Future studies should also examine the effects of macro-as well as micro-social networks, and the processes through which any effects occur (e.g. through the establishment of BMI norms and a reference for body image).

Additionally, little attention has been paid to how and why social-network influences can vary according to contextual and socio-demographic characteristics and this should be a priority for future analyses of social network studies of young people's eating and weight. For example, does the relative importance of selection versus influence vary according to school context, and if so how? Qualitative studies are also needed to understand the meaning and importance of network structures within school hierarchies [START_REF] Michell | Girls, Pecking Order and Smoking[END_REF], and explore how school-environmental factors vary, such as the canteen peer-group environment. [START_REF] Neumark-Sztainer | Factors influencing food choices of adolescents: Findings from focus-group discussions with adolescents[END_REF] Finally, the current evidence ignores peer group influences on eating disorders among boys, despite increasing concerns [START_REF] Dominé | Disordered Eating Behaviors: What About Boys?[END_REF],

who could also be subject of future social network studies on the issue.

CONCLUSION

Social network analysis is a rapidly emerging area within the research on adolescent eating behaviours and bodyweight, with seven out of the ten studies in this review published since 2007. In the field of public health, network analysis can be applied to understand how social relationships determine health-related knowledge, attitudes and behaviours and to inform new health promotion interventions. [START_REF] Valente | Opinion leader interventions in social networks[END_REF] There is consistent evidence that school friends are significantly similar in terms of their eating behaviours and bodyweight, particularly the most overweight young people. Further longitudinal research is needed to examine the processes via which this clustering occurs, and how it varies according to school context. There is also scope What is already known on this subject?

Recent evidence suggests that obesity and eating disorders can spread via social networks.

Various theories and empirical studies suggest young people's friends may influence their eating behaviours, bodyweight and body image concerns.

No systematic review has synthesized the evidence about the association of young people's social networks and eating behaviours or bodyweight, and whether associations reflect influence, selection or confounding.

What does this study add?

There is consistent evidence that school friends are significantly similar in terms of eating behaviours and body mass index (BMI), with the strongest associations for BMI found among the most overweight. Eating disorders have also been found to cluster within groups of female friends. Overweight youth are less likely to be popular and more likely to be socially isolated at school.

  inter-clique variability BMI Significantly higher between-than within-group variance Body image concerns Significantly higher between-than within-group variance (p=.002) Dieting Significantly higher between-than within-group variance (p=.024) Extreme weight loss Significantly higher between-than within-group variance (p=.001)

  to promote healthy eating and reduce body image concerns.

Table 1 :

 1 Summary of included studies (N=10)

	Study	Country Study design Sample size and	Data collection	Method(s) for	Method(s) for	Eating	Key findings
			characteristics	methods	collecting and	analysing social	behaviour
					summarising	network data	and/or
					social network		bodyweight
					data		measures
	Cohen-Cole	USA	Longitudinal Schools	School-based	Respondents	Logistic and OLS	Obesity
	and		N/S	and in-home	asked to list up	regression
	Fletcher			surveys.	to 10 of their
	[22]		Students		closest friends
			N=1988		(5 males, 5
	National Longitudinal Study of Adolescent Health (Add Health)				females). Dyads constructed based on 'first friend named'
					by each subject

  The evidence synthesised here suggests the potential of social-network-based health promotion interventions in schools. One such intervention is the AHEAD project, an intervention which aims to promote healthy eating via the ASSIST model of peer education and positive influence[START_REF] Campbell | An informal school-based peer-led intervention for smoking prevention in adolescence (ASSIST): a cluster randomised trial[END_REF], currently being piloted by researchers at Bristol University.

	Youth is a period of dramatic physical, cognitive, social and emotional development and offers
	health promoters a special opportunity to encourage healthy lifestyles. Although peer
	education has been widely used to address adolescent substance use and sexual health [33,
	34], school-based peer education addressing young people's eating behaviours and bodyweight
	has received relatively little attention. Interventions addressing parental influences, food prices
	and marketing, and school nutrition policies continue to receive greatest attention and
	investment.[8] There may also be scope for developing peer-group-based counselling and motivational
	interviewing interventions targeting those groups of young women most at risk of eating
	disorders in a supportive, non-stigmatising group-setting. It would also be appropriate to pilot
	interventions aiming to support overweight students in making new friendships in school in
	order to avoid isolation or confinement to obesogenic cliques.
	Furthermore, if school friendships are influential in determining young people's eating
	behaviours and bodyweight, then interventions such as those above could have a social
	22

  This evidence suggests the potential of social-network-based health promotion interventions in schools to complement existing school-based initiatives addressing young people's diet.The relative importance of influence, selection and confounding is uncertain because of the inconsistency of findings from the small number of longitudinal studies. (http://jech.bmj.com/site/about/licence.pdf).

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	in our licence

Acknowledgements

This research was funded by the UK Medical Research Council.

School social networks, BMI and obesity

All eight studies which used body mass index (BMI) scores to calculate bodyweight and/or identify overweight and underweight individuals found evidence that school friends tend to be similar in terms of their BMI. [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF][START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF][START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF][START_REF] Trogdon | Peer effects in adolescent overweight[END_REF][START_REF] Halliday | Weight gain in adolescents and their peers[END_REF][START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF][START_REF] Paxton | Friendship Clique and Peer Influences on Body Image Concerns, Dietary Restraint, Extreme Weight-Loss Behaviors, and Binge Eating in Adolescent Girls[END_REF][START_REF] Hutchinson | Do friends share similar body image and eating problems? The role of social networks and peer influences in early adolescence[END_REF] Five of these studies were based on analyses of Add Health surveys, which include questions about friendship networks and height/weight. [START_REF] Strauss | Social Marginalisation of Overweight Children[END_REF][START_REF] Cohen-Cole | Is obesity contagious? Social network vs. environmental factors in the obesity epidemic[END_REF][START_REF] Fowler | Estimating peer effects in social networks: A response to Cohen-Cole and Fletcher; and Trogdon, Nonemaker, and Pais[END_REF][START_REF] Trogdon | Peer effects in adolescent overweight[END_REF][START_REF] Halliday | Weight gain in adolescents and their peers[END_REF] Trogdon and colleagues' cross-sectional analyses of the Add Health data found that friends' BMI is positively associated with a student's own BMI. When an adolescent's friend's mean BMI is 1 unit higher (approximately 6 pounds for a 16-year-old of average height), the adolescent's own weight is higher by 0.3 BMI units (approximately 2 pounds for a 16-year-old of average height). [START_REF] Trogdon | Peer effects in adolescent overweight[END_REF] These associations were strongest among those young people at the higher end of the BMI spectrum (i.e. overweight). Similarly, Valente and colleagues analysed cross-sectional data collected in four schools in the Los Angeles area and found that overweight youth were twice as likely to have overweight friends. [START_REF] Valente | Adolescent Affiliations and Adiposity: A Social Network Analysis of Friendships and Obesity[END_REF] Drawing on longitudinal data from two waves of the Add Health surveys, Halliday and Kwak also found that friends' average BMI was positively associated with a student's own BMI at followup, and that those students who have friends who are overweight are also more likely to subsequently become overweight themselves. [START_REF] Halliday | Weight gain in adolescents and their peers[END_REF] This suggests the importance of social influence. Fowler and Christakis similarly found that risk of becoming obese increased when friends were obese, even after controlling for overall school trends. [START_REF] Halliday | Weight gain in adolescents and their peers[END_REF]