Pierre Gaillard
email: pierre.gaillard@u-bourgogne.fr

Mickaël Gastineau
email: gastineau@imcce.fr

Keywords: 35Q55, 37K10, 4710A, 47.35.Fg, 47.54.Bd NLS equation, wronskians, Peregrine breather, rogue waves. 1

The Peregrine breather of order eleven (P11 breather) solution to the focusing one dimensional nonlinear Schrödinger equation (NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given : when all parameters are equal to 0 we recover the famous P11 breather. We obtain new families of quasi-rational solutions to the NLS equation in terms of explicit quotients of polynomials of degree 132 in x and t by a product of an exponential depending on t. We study these solutions by giving patterns of their modulus in the (x; t) plane, in function of the different parameters.

The eleventh Peregrine breather and twenty parameters families of solutions to the NLS equation.

Introduction

The story of the nonlinear Schrödinger equation (NLS) begins with the works of Zakharov and Shabat in 1968. It was solved in 1972 by using the inverse scattering method [START_REF] Zakharov | Stability of periodic waves of finite amplitude on a surface of a deep fluid[END_REF][START_REF] Zakharov | Exact theory of two dimensional self focusing and one dimensinal self modulation of waves in nonlinear media[END_REF]. The first quasi-rational solution to NLS equation was constructed in 1983 by Peregrine [START_REF] Peregrine | Water waves, nonlinear Schrödinger equations and their solutions[END_REF], nowadays called worldwide Peregrine breather. Akhmediev, Eleonski and Kulagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather [START_REF] Akhmediev | Generation of periodic trains of picosecond pulses in an optical fiber : exact solutions[END_REF][START_REF] Akhmediev | Exact first order solutions of the nonlinear Schrödinger equation[END_REF] in 1986; other families of higher order 3 and 4 were constructed in a series of articles by Akhmediev et al. [START_REF] Akhmediev | Rogue waves and rational solutions of nonlinear Schrödinger equation[END_REF][START_REF] Akhmediev | Rogue waves, rational solutions, the patterns of their zeros and integral relations[END_REF], using Darboux transformations.

Since the beginning of the years 2010, there was a renewed interest for this equation and many works were published using various methods. In 2011, the solutions to the NLS equation were presented as a quotient of two wronskians in [START_REF] Gaillard | Families of quasirational solutions of the NLS equation and multi-rogue waves[END_REF][START_REF] Gaillard | Wronskian representation of solutions of the NLS equation and higher Peregrine breathers[END_REF]. In 2012, Guo, Ling and Liu constructed another representation of the solutions to the NLS equation, as a quotient of two determinants [START_REF] Guo | Nonlinear Schrodinger equation: Generalized Darboux transformation and rogue wave solutions[END_REF] using the generalized Darboux transformation.

Ohta and Yang presented in [START_REF] Ohta | General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation[END_REF] solutions to the NLS equation by means of determinants, using a new approach based on the Hirota bilinear method.

Then in 2013, it was found in [START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF], solutions expressed in terms of determinants of order 2N depending on 2N -2 real parameters; the complete proof was recently given in [START_REF] Gaillard | Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves[END_REF]. A new representation has been found as a ratio of a determinant of order N + 1 by another one of order N by Ling and Zhao in [START_REF] Ling | Trajectory Characters of rogue waves[END_REF]. Very recently in 2014, another approach have been given in [START_REF] Gelash | Superregulier solitonic solutions : a novel scenario for the nonlinear stage of modulation instability[END_REF] using a dressing method where the solutions are expressed as the quotient of a determinant of order N + 1 by another one of order N .

With the method given in the present work, we construct new solutions to the focusing one dimensional nonlinear Schrödinger equation which appear as deformations of the (analogue) Peregrine breather of order 11 with 20 real parameters : when all the parameters are equal to 0, we recover the famous P 11 breather. These solutions are completely expressed as a quotient of two polynomials of degree 132 in x and t by an exponential depending on t. We do not have the space to present them here; we only present plots in the (x; t) plane to analyze the evolution of the solutions in function of the different parameters.

Determinant representation of solutions to NLS equation

We use in the following to construct deformations of the P 11 breather, solutions to the NLS equation, the results obtained in [START_REF] Gaillard | Wronskian representation of solutions of the NLS equation and higher Peregrine breathers[END_REF][START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF].

Theorem 2.1 The function v defined by v(x, t) = det((n jk ) j,k∈[1,2N ] ) det((d jk ) j,k∈[1,2N ] ) e (2it-iϕ)
is a quasi-rational solution to the NLS equation

iv t + v xx + 2|v| 2 v = 0,
where

n j1 = f j,1 (x, t, 0), n jk = ∂ 2k-2 fj,1 ∂ǫ 2k-2 (x, t, 0), n jN +1 = f j,N +1 (x, t, 0), n jN +k = ∂ 2k-2 fj,N+1 ∂ǫ 2k-2
(x, t, 0), d j1 = g j,1 (x, t, 0),

d jk = ∂ 2k-2 gj,1 ∂ǫ 2k-2 (x, t, 0), d jN +1 = g j,N +1 (x, t, 0), d jN +k = ∂ 2k-2 gj,N+1 ∂ǫ 2k-2 (x, t, 0), 2 ≤ k ≤ N, 1 ≤ j ≤ 2N
The functions f and g are defined for 1 ≤ k ≤ N by :

f 4j+1,k = γ 4j-1 k sin A k , f 4j+2,k = γ 4j k cos A k , f 4j+3,k = -γ 4j+1 k sin A k , f 4j+4,k = -γ 4j+2 k cos A k , f 4j+1,N +k = γ 2N -4j-2 k cos A N +k , f 4j+2,N +k = -γ 2N -4j-3 k sin A N +k , f 4j+3,N +k = -γ 2N -4j-4 k cos A N +k , f 4j+4,k = γ 2N -4j-5 k sin A N +k , g 4j+1,k = γ 4j-1 k sin B k , g 4j+2,k = γ 4j k cos B k , g 4j+3,k = -γ 4j+1 k sin B k , g 4j+4,k = -γ 4j+2 k cos B k , g 4j+1,N +k = γ 2N -4j-2 k cos B N +k , g 4j+2,N +k = -γ 2N -4j-3 k sin B N +k , g 4j+3,N +k = -γ 2N -4j-4 k cos B N +k , g 4j+4,N +k = γ 2N -4j-5 k sin B N +k , (1)
The arguments A ν and B ν of these functions are given for 1 ≤ ν ≤ 2N by

A ν = κ ν x/2 + iδ ν t -ix 3,ν /2 -ie ν /2, B ν = κ ν x/2 + iδ ν t -ix 1,ν /2 -ie ν /2. The terms κ ν , δ ν , γ ν are defined by 1 ≤ ν ≤ 2N κ j = 2 1 -λ 2 j , δ j = κ j λ j , γ j = 1-λj 1+λj , κ N +j = κ j , δ N +j = -δ j , γ N +j = 1/γ j , 1 ≤ j ≤ N, (2)
where λ j are given for 1 ≤ j ≤ N by :

λ j = 1 -2j 2 ǫ 2 , λ N +j = -λ j . (3)
The terms x r,ν (r = 3, 1) are defined for 1 ≤ ν ≤ 2N by :

x r,ν = (r -1) ln γν -i γν +i . (4)
The parameters e ν are given by

e j = i N -1 k=1 ãj ǫ 2k+1 j 2k+1 - N -1 k=1 bj ǫ 2k+1 j 2k+1 , e N +j = i N -1 k=1 ãj ǫ 2k+1 j 2k+1 + N -1 k=1 bj ǫ 2k+1 j 2k+1 , 1 ≤ j ≤ N, (5)
3 Quasi-rational solutions of order 11 with twenty parameters

We construct here deformations of the Peregrine breather P 11 of order 11 depending on 20 parameters. This construction is based on Theorem 2.1. This theorem in various formulations was completely shown in the articles published previously [START_REF] Gaillard | Families of quasirational solutions of the NLS equation and multi-rogue waves[END_REF][START_REF] Gaillard | Wronskian representation of solutions of the NLS equation and higher Peregrine breathers[END_REF][START_REF] Gaillard | Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves[END_REF][START_REF] Gaillard | Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves[END_REF].

To obtain the quasi rational solutions of equation NLS one uses the functions f and g defined previously by [START_REF] Zakharov | Stability of periodic waves of finite amplitude on a surface of a deep fluid[END_REF]. One carries out limited developments of these functions in ǫ and their derivatives with respect to ǫ, to order 2N if one wants the solutions with the order N . The calculation of the two determinants given in theorem 2.1 then gives the polynomials searched in the expression of the solutions.

We do not give the analytic expression of the solution to NLS equation of order 11 with twenty parameters because of the length of the expression. The computations were done using the computer algebra systems Maple and TRIP [START_REF] Gastineau | TRIP 1.3.8 : TRIP Reference manual[END_REF]. For simplicity, we denote

d 3 := det((n jk) j,k∈[1,2N ] ), d 1 := det((d jk) j,k∈[1,2N ] ).
The number of terms of the polynomials of the numerator d3 and denominator d1 of the solutions are shown in the table below (Table 1) when other a i and b i are set to 0. The computation of d 3 (a 1 , b 1 , x, t) and d 1 (a We construct figures to show deformations of the eleventh Peregrine breather. We get different types of symmetries in the plots in the (x, t) plane.

We give some examples of this fact in the following discussion. The present study follows works on order N = 3 to N = 10 given in [START_REF] Gaillard | Deformations of third order Peregrine breather solutions of the NLS equation with four parameters[END_REF][START_REF] Gaillard | Six-parameters deformations of fourth order Peregrine breather solutions of the NLS equation[END_REF][START_REF] Gaillard | The fifth order Peregrine breather and its eight-parameters deformations solutions of the NLS equation[END_REF][START_REF] Gaillard | Ten parameters deformations of the sixth order Peregrine breather solutions of the NLS equation[END_REF][START_REF] Gaillard | Higher order Peregrine breathers, their deformations and multi-rogue waves[END_REF][START_REF] Gaillard | Other 2N-2 parameters solutions to the NLS equation and 2N+1 highest amplitude of the modulus of the N-th order AP breather[END_REF][START_REF] Gaillard | The Peregrine breather of order nine and its deformations with sixteen parameters solutions of the NLS equation[END_REF][START_REF] Gaillard | Eighteen parameter deformations of the Peregrine breather of order ten solutions of the NLS equation[END_REF].

Peregrine breather of order 11

If we choose ãi = bi = 0 for 1 ≤ i ≤ 10, we obtain the classical eleventh Peregrine breather Figure 1: Solution of NLS, N=11, all parameters equal to 0, Peregrine breather P 11 .

Variation of parameters

With other choices of parameters, we obtain all types of configurations : triangles and multiple concentric rings configurations with a maximum of 66 peaks.

Conclusion

We have constructed explicitly solutions to the NLS equation of order 11 with 20 real parameters. The explicit representation in terms of polynomials of degree 132 in x and t is obtained. His expression is too large to be published in this text. It is the first time that the Peregrine breather of order eleven with its deformations with twenty parameters is presented to our knowledge. It confirms the property about the shape of the breather in the (x, t) coordinates, the maximum of amplitude equal to 2N + 1 and the degree of polynomials in x and t here equal to N (N + 1). We obtained different patterns in the (x; t) plane by different choices of these parameters. So we obtain a classification of the rogue waves at order 11.

It is fundamental to note the similar role played by a i and b i for the same given index i : one obtains exactly the same structures of the modulus of the solutions to NLS equation in the (x; t) plan.

In the cases a 1 = 0 or b 1 = 0 we obtain triangles with a maximum of 66 peaks; for a 2 = 0 or b 2 = 0 , we have 9 rings with respectively 5, 10, 10, 5, 5, 10, 5, 10, 5 peaks with in the center one peak. For a 3 = 0 or b 3 = 0, we obtain 7 rings with respectively 7, 14, 7, 14, 7, 7, 7 peaks with in the center the Peregrine P 2 . For a 4 = 0 or b 4 = 0, we have 7 rings with 9 peaks on each of them with in the center the Peregrine P 2 .

For a 5 = 0 or b 5 = 0 , we have 6 rings of 11 peaks on each of them without a central peak. For a 6 = 0 or b 6 = 0 , we have 5 rings with 13 peaks on each of them and in the center one peak. For a 7 = 0 or b 7 = 0 , we have 4 rings with 15 peaks on each of them and in the center the Peregrine breather of order 3. For a 8 = 0 or b 8 = 0, we have 3 rings with 17 peaks on each of them and in the center the Peregrine breather of order 5. For a 9 = 0 or b 9 = 0, we have 2 rings with 19 peaks and in the center the Peregrine breather of order 7. At least, for a 10 = 0 or b 10 = 0, we have only one ring with 21 peaks and in the center the Peregrine breather of order 9.

The study of the solutions to the NLS equation has been done until order N = 6 by Akhmediev et al. in [START_REF] Kedziora | Classifying the hierarchy of the nonlinear Schrödinger equation rogue waves solutions[END_REF] and extrapolated until order N = 10.

From this present study of order 11, it becomes clear that we can conjecture the structure of solutions to NLS equation.

Precisely, one can partly conjecture the structure of the rogue waves solutions to the NLS equation at the order N . Important applications for example in the fields of nonlinear optics and hydrodynamics are made recently; we can cite in particular the works of Akhmediev et al [START_REF] Chabchoub | Super rogue waves : observation of a higher-order breather in water waves[END_REF] or Kibler et al. [START_REF] Kibler | The Peregrine soliton in nonlinear fibre optics[END_REF].

Another interesting study would be to determine which initial conditions can give these types of rogue waves and to discuss the physical excitations which lead to such situations. It would be important to answer this kind of question in the future.

It would be relevant to continue this study to understand these solutions, to try to classify them in the general case of order N (N > 11) and to prove the preceding conjectures.

Figure 2 :

 2 Figure 2: Solution of NLS, N=11, ã1 = 10 3 : triangle with 66 peaks; in bottom, sight of top.

Figure 3 :

 3 Figure 3: Solution of NLS, N=11, b1 = 10 3 : triangle with 66 peaks; in bottom, sight of top.

Figure 4 :

 4 Figure 4: Solution of NLS, N=11, ã2 = 10 5 : 9 rings with 5; 10; 10; 5; 5; 10; 5 : 10; 5 peaks, with in the center one peak; in bottom, sight of top.

Figure 5 :

 5 Figure 5: Solution of NLS, N=11, b2 = 10 5 : 9 rings with 5; 10; 10; 5; 5; 10; 5 : 10; 5 peaks, with in the center one peak; in bottom, sight of top.

Figure 6 :

 6 Figure 6: Solution of NLS, N=11, ã3 = 10 7 : 7 rings with 7; 14; 7; 14; 7; 7; 7 peaks, with in the center P 2 ; in bottom, sight of top.

Figure 7 :

 7 Figure 7: Solution of NLS, N=11, b3 = 10 7 : 7 rings with 7; 14; 7; 14; 7; 7; 7 peaks, with in the center P 2 ; in bottom, sight of top.

Figure 8 :

 8 Figure 8: Solution of NLS, N=11, ã4 = 10 9 : 6 rings with 9; 9; 18; 9; 9; 9 peaks, with in the center P 2 ; in bottom, sight of top.

Figure 9 :

 9 Figure 9: Solution of NLS, N=11, b4 = 10 9 : 6 rings with 9; 9; 18; 9; 9; 9 peaks, with in the center P 2 ; in bottom, sight of top.

Figure 10 :

 10 Figure 10: Solution of NLS, N=11, ã5 = 10 11 : 6 rings of 11 peaks without a central peak; in bottom, sight of top.

Figure 11 :

 11 Figure 11: Solution of NLS, N=11, b5 = 10 11 : 6 rings of 11 peaks without a central peak; in bottom, sight of top.

Figure 12 :

 12 Figure 12: Solution of NLS, N=11, ã6 = 10 13 : 5 rings with 13 peaks and in the center one peak; in bottom, sight of top.

Figure 13 :

 13 Figure 13: Solution of NLS, N=11, b6 = 10 13 : 5 rings with 13 peaks and in the center one peak; in bottom, sight of top.

Figure 14 :

 14 Figure 14: Solution of NLS, N=11, ã7 = 10 15 : 4 rings with 15 peaks and in the center the Peregrine breather of order 3; in bottom, sight of top.

Figure 15 :

 15 Figure 15: Solution of NLS, N=11, b7 = 10 15 : 4 rings with 15 peaks and in the center the Peregrine breather of order 3; in bottom, sight of top.

Figure 16 :

 16 Figure 16: Solution of NLS, N=11, ã8 = 10 18 : 3 rings with 17 peaks and in the center the Peregrine breather of order 5; in bottom, sight of top.

Figure 17 :

 17 Figure 17: Solution of NLS, N=10, b8 = 10 18 : 3 rings with 17 peaks and in the center the Peregrine breather of order 5; in bottom, sight of top.

Figure 18 :

 18 Figure 18: Solution of NLS, N=10, ã9 = 10 20 : two rings with 19 peaks and in the center the Peregrine breather of order 7; in bottom, sight of top.

Figure 19 :

 19 Figure 19: Solution of NLS, N=10, b9 = 10 20 : two rings with 19 peaks and in the center the Peregrine breather of order 7; in bottom, sight of top.

Figure 20 :

 20 Figure 20: Solution of NLS, N=10, ã10 = 10 20 : one ring with 21 peaks and in the center the Peregrine breather of order 9; in bottom, sight of top.

Figure 21 :

 21 Figure 21: Solution of NLS, N=10, b10 = 10 20 : one ring with 21 peaks and in the center the Peregrine breather of order 9; in bottom, sight of top.

Table 1 :

 1 1 , b 1 , x, t) requires 13 days on a 32-cores computer. Number of terms for the polynomials d 3 and d 1 of the solutions to the NLS equation.

	N=11	Number of terms
	d 3 (a 1 , b 1 , x, t)	803 534
	d 1 (a 1 , b 1 , x, t)	407 850
	d 3 (a 2 , b 2 , x, t)	306 417
	d 1 (a 2 , b 2 , x, t)	155 543
	d 3 (a 3 , b 3 , x, t)	165 321
	d 1 (a 3 , b 3 , x, t)	83 925
	d 3 (a 4 , b 4 , x, t)	105 667
	d 1 (a 4 , b 4 , x, t)	53 637
	d 3 (a 5 , b 5 , x, t)	74 720
	d 1 (a 5 , b 5 , x, t)	37 930
	d 3 (a 6 , b 6 , x, t)	56 409
	d 1 (a 6 , b 6 , x, t)	28 638
	d 3 (a 7 , b 7 , x, t)	44 491
	d 1 (a 7 , b 7 , x, t)	22 590
	d 3 (a 8 , b 8 , x, t)	35 459
	d 1 (a 8 , b 8 , x, t)	17 999
	d 3 (a 9 , b 9 , x, t)	26 282
	d 1 (a 9 , b 9 , x, t)	13 332
	d 3 (a 10 , b 10 , x, t)	15 049
	d 1 (a 10 , b 10 , x, t)	7632


Frédéric Gavrel
email: fred-eric.gavrel@unicaen.fr

Thérèse Rebière
email: therese.rebiere@cnam.fr

On the Equilibrium and Welfare Consequences of Going Ahead of the Smiths

Keywords: Going ahead of the Smiths, Social interactions, Well-being, Inequalities, Efficiency. JEL Classification numbers: D3, D6, D8, I3, Z1

This paper provides an analysis of the social consequences of people seeking to go ahead of the Smiths. All individuals attempt to reach a higher rank than the Smiths, including the Smiths themselves. This attitude gives rise to an equilibrium in which all individuals have equal utilities but unequal (gross) incomes.

Due to a rat-race effect, individuals devote too much energy to climbing the social scale. However, laissez-faire equilibrium is an equal-utility constrained social optimum. Conversely, an utilitarian social planner would not choose utility equality. Unexpectedly, this social ambition theory fairly well accounts for empirical intermediate wage inequality.

Introduction

"So far as concerns the present question, the end sought by accumulation is to rank high in comparison with the rest of the community in point of pecuniary strength." - [START_REF] Veblen | The Theory of the Leisure Class. An Economic Study of Institutions[END_REF] In this paper, inspired by [START_REF] Veblen | The Theory of the Leisure Class. An Economic Study of Institutions[END_REF], we provide an analysis of the social consequences of people seeking to Go Ahead of the Smiths, GAS henceforth. In accordance with Veblen (1899) 1 , individuals' utilities not only depend on their incomes but also on their social "status". All people attempt to reach a higher social status than the Smiths, including the Smiths themselves. It emerges that the GAS attitude is capable of generating equilibrium income dispersion across ex ante homogenous agents, thus accounting for part of income inequality.

The idea that the individual well-being is to some extent relative to that of others and that we all try to keep up with wealthier than us dates back to [START_REF] Smith | An Inquiry into the Nature and Causes of the Wealth of Nations[END_REF] who noticed that happiness is not linked to the stock of acquisitions but to the progressive state of acquiring (see Book I, chapter VIII: of the wages of Labour, paragraph 42).

Smith also pointed out that prosperity makes the poorest incapable of being content with the consumption which had formerly satisfied them (see Book I, chapter VIII: of the wages of Labour, paragraph 34). [START_REF] Duesenberry | Income, Saving and the Theory of Consumer Behaviour[END_REF] was the first to empirically observe this relative income phenomenon. The well-know Easterlin paradox [START_REF] Easterlin | Does Economic Growth Improve the Human Lot? In Paul A[END_REF] tells us that while the progressive state of income acquisition is correlated with happiness, increased income does not lead itself to increased happiness. As a consequence of this transitory effect of income on life satisfaction, which acts like a focusing illusion [START_REF] Kahneman | Would You Be Happier If You Were Richer? A Focusing Illusion[END_REF], the social concerns behavior leads to a rat-race in which the individual's goal is to place her or himself higher in the income hierarchy.

Our model can be compared with James Mirrlees' path-breaking paper. Contrary to [START_REF] Mirrlees | An Exploration in the Theory of Optimum Income Taxation[END_REF], individuals are identical ex ante. But, since they wish to become VIP's, they turn out to be heterogenous ex post with VIP's consuming a great deal of energy in keeping their envied social position, common people having a low status 1 See [START_REF] Arrow | Thornstein Veblen as an Economic Theorist[END_REF] for a view of Veblen as an economic theorist.

not consuming as much energy in pursuing that end. This gives rise to an equilibrium in which all individuals have equal utilities but unequal (gross) incomes. We derive other interesting insights in this essay. First, we show that there is a single utilitarian social optimum, relative to which all effort is lower than in the case of laissez-faire. This is the consequence of individuals devoting their energy to climbing the social scale. Next, we show that this social optimum generates true inequalities, that is, inequalities in terms of utilities. The reason for this is that a social planner takes advantage of the dispersion of individuals on the ranking scale. However, we find that GAS equilibrium is an egalitarian optimum. It follows that if "you" dislike true inequalities, that is inequalities in terms of utility, then "you" should favor observed (gross) income inequalities, and "you" should firmly reject the idea of any redistributive tax policy. Finally, we show that market equilibrium is inefficient.

Pareto inefficiency is the price to pay for true equality.

We also give an assessment of the empirical potential of GAS theory. Unexpectedly, the estimated income distribution rather correctly replicates the observed distribution of wages in France, within an extensive "middle class". Our GAS model can be empirically summarized in two parameters. The first one, referred to as GAS propensity, reflects individuals' desire for advantageous income inequality, while, the other one, referred to as differentiation capability, measures their technological ability to differentiate each other by choosing different effort levels. Using our estimation as a benchmark, numerical simulations describe the effects of GAS propensity and of differentiation capability on income inequality.

The envy/pride hypothesis is indirectly confirmed in the empirical investigations by Easterlin, who finds that "growth does not buy happiness". This unfortunate outcome can be interpreted as a consequence of GAS attitude. If all incomes increase by the same rate, income hierarchy is not affected. Consequently, individuals do not perceive any improvement (in their social status). In that regard, we would like to stress that the analytical tool that we construct in the following assumes rational expectations in the sense that individuals' decisions generate the income distribution on which they base their calculations. Although rational expectations became a nec plus ultra in economic theory about forty years ago, it seems that myopic or adaptative expectations à la Milton Friedman might better account for observed facts in the so-called empirical economics of Happiness. Relative concerns have been broadly empirically tested trough well-being sample surveys and experiments since the 90s, see for instance [START_REF] Clark | Satisfaction and Comparison Income[END_REF], or more recently [START_REF] Card | Inequality at Work: The Effect of Peer Salaries on Job Satisfaction[END_REF]. Indeed, myopic individuals would realize ex post that, despite their efforts, they did not succeed in reaching a higher status since everyone else did too. In other words, while the Smiths were attempting to keep up with the Joneses or go ahead of them, the Joneses themselves moved towards the Harpers who, in turn, moved towards the Rebieres... As a result, the Smiths, Joneses, Harpers and Rebieres, and everybody else report that their happiness is either equal or even falling when asked by Happiness economists.

To position our contribution in the literature2 , it is convenient to clearly distinguish between two behavioral issues3 . The first issue is: what motivates wealth accumulation? As clearly expressed in the introductory quote, the end sought by accumulation is to rank high in comparison with the rest of the community. This is the social competition argument which we here develop. The second issue is: why do we observe so large expenses in apparently useless items? An answer is that these expenses make wealth visible. We refer to this answer as the signalling argument.

Papers like Hopkins and Korienko (2004) or [START_REF] Ireland | Optimal Income Tax in the Presence of Status Effects[END_REF] retain the "ordinal" version of social concerns. As in the present paper, individuals are affected by their rank in the income hierarchy. In our terminology, they are driven by the GAS attitude.

In these papers, contrary to ours, incomes are private information. These papers thus belong to the signalling branch in which agents must be heterogenous. Relative to this strand of literature, our specification of preferences is more restrictive. The reason is that, in an empirical perspective, we have to derive an explicit solution to our GAS model.

Contrary to our contribution, many papers follow the "cardinal" tradition. Similar to Duesenberry, individuals' utilities are affected by the gap between their own "conspicuous" consumption and the average in the population. The cardinal version of social concerns is also referred to as the KUJ (Keeping Up with the Joneses) attitude. To our knowledge, in these papers, information is always perfect. Conse-quently, these papers belong to the social competition branch, in which agents can be homogenous. [START_REF] Friedman | Conspicuous Consumption Dynamics[END_REF] develop a very nice study of the dynamics of conspicuous consumption. They also show that social competition may generate ex post heterogeneities among ex ante homogenous agents. When individuals only care about people whose conspicuous consumption is lower -the "pride" hypothesis -, the underlying static model possesses a stable dispersed equilibrium. In general, KUJ applies to the consumption of a positional good but KUJ can also apply to income (models have the same structure) or to real assets as in [START_REF] Cahuc | Social Status and the Overworked Consumer. Annales d[END_REF] who develop a behavioral growth model. In a very recent paper where agents are ex ante heterogeneous, [START_REF] Ulph | Keeping up with the Joneses: Who Loses out?[END_REF] shows that well-being could decrease with wage rate for those workers who would rather have chosen not to work in the absence of a KUJ behavior.

To summarize, to our best knowledge, earlier papers either develop the social competition argument in the KUJ environment or the signalling argument in the GAS environment. We develop a competition model in the GAS environment. Although both versions are admissible, the GAS assumption sounds more attractive than KUJ whose behavioral foundations are not so clear. In addition, relative to [START_REF] Friedman | Conspicuous Consumption Dynamics[END_REF] in which ex post heterogeneities result from the "pride" version of KUJ hypothesis, we find that the GAS attitude necessarily gives rise to income inequality, even though agents are ex ante homogenous.

This insight is close in spirit to [START_REF] Burdett | Equilibrium Price Dispersion[END_REF] for example where equilibrium price dispersion does not result from some ex ante heterogeneity of agents but only from non-sequential search. Another example is [START_REF] Burdett | Wage Differentials, Employer Size, and Unemployment[END_REF] where the same holds for the equilibrium dispersion of wages which entirely stems from on-the-job search. In other words, we exhibit an income dispersion (i.e inequality) equilibrium which entirely results from the GAS behavior.

Section 2 builds our modeling of GAS equilibrium. The welfare implications of social ambition are studied in section 3 while section 4 examines the empirical potential of GAS model. The conclusion summarizes our findings and sets a non-exhaustive agenda for further investigations.

Environment and GAS equilibrium

Let us consider an environment à la James Mirrlees. Following this author, individuals' income, y, is an increasing function of their effort, k. But, contrary to Mirrlees

(K 1 < ... < K i < ... < K n )
where the positive integer n can be unbounded. The

corresponding frequencies are (φ(K 1 ), ..., φ(K n )). If n is zero, Π(.) is continuous on [0, ∞[.
These people are envious. They are all willing to go ahead of the Smiths, implying that they are affected by their location in the income scale. In this context, the utility of an individual, U (k, .), is decreasing in her effort, while increasing in her rank in the income hierarchy as well as in her output. In an empirical perspective, we need to derive an explicit solution. In this spirit, we retain the following specification

U (k, .) = -k + J(Π(k))H(k) (1)
In the previous expression, the term J(.), referred to as the "GAS multiplier", is a strictly increasing (continuous) function of the share, Π(k), of strictly lower ranked participants.5 It is worth noting that, according to our definition, Π(k

) = Π(k -). Thus, if K i is a mass point, Π(K + i ) = Π(K i ) + Φ(K i ).
With no loss of generality, the top, J(1), can be set to 1 while the bottom, J(0), assumed to be strictly positive, is denoted by µ (0 < µ < 1). This specification of utility U (.) nicely captures the ambition of Going Ahead of the Smiths. It is not additive6 , implying that the downgrading (or upgrading) effect is all the higher as individuals are rich. This is in conformity with Thorstein Veblen's message (1899). It is worth noting that individuals only care about lower ranked people. The reason is that, contrary to the cardinal version of social concerns, the ordinal version does not allow for distinguishing between higher and lower ranked players. More rigorously, this distinction can be made but has not any formal incidence. Indeed, suppose that instead of Π(k), individuals care about the weighted

average αΠ(k) -(1 -α)(1 -Π(k)) with (0 ≤ α ≤ 1).
In the terminology of [START_REF] Friedman | Conspicuous Consumption Dynamics[END_REF], the term αΠ(k) translates a "pride" feeling while a "frustration"

feeling is expressed through the term -(1-α)(1-Π(k). As αΠ(k)-(1-α)(1-Π(k)) = Π(k) -(1 -α),
Π(k). It is also assumed that function V (k) = -k + H(k) goes to -∞ as k goes to
+∞ and that the derivative of V (k) is strictly positive for k = 0. This implies that

V (.) has a single maximum k > 0 with V ( k) = Ṽ . On the other hand, function W (k) denotes -k + µH(k).
In this environment an equilibrium can be defined as below.

Definition 1 A GAS equilibrium is a distribution function Π * (.) such that, for all k in its support, U (k) is maximized with respect to k ≥ 0.

It can be noted that this definition implies that U (k) is a constant U * ≥ U (0) > 0, for all k in the support of an equilibrium distribution. Let us study this equilibrium.

In Appendix A, it is proved that the equilibrium distribution is continuous and that its support, denoted by

S * , is a bounded interval [A, B] with 0 ≤ A < B < ∞.
For simplicity, the analysis focus on the case in which the lower bound, A, is strictly positive. In S * , U (k) is a constant U * > 0. Consider the behavior of the GAS multiplier J(k) = J(Π(k)) in the interval [A, B]. We have J(A) = µ. In the interval [A, B], J(k) is strictly increasing and equal to one for k = B. In this interval, J(k) satisfies (see equation (1))

J(k) = k + U * H(k) (2)
So J(k) has a derivative J (k) such that

J (k) = H(k) -(k + U * )H (k) H(k) 2 One can check that if J (A) ≥ 0, then J (k) > 0 in ]A, B]. As U (A) should be equal to U * , it follows that J (A)H(A) = 1 -µH (A) ≥ 0. Consider the behavior of U (k) in [0, A[. In this interval, U (k) = W (k) should be lower than U (A). The concavity of H(.) then imposes that -1 + µH (A) ≥ 0. See Figure 1. It results that -1 + µH (A) = 0. Consequently, S * is an interval [A, B] whose lower bound is determined by -1 + µH (A) = 0. The latter equation has a solution A * (> 0) if and only if -1 + µH (0) > 0. From A * , we deduce U * = U (A * ).
Let us now turn to the upper bound. Since Π *

(B) = 1, B * satisfies -B + H(B) = U (A * ). For k > B * , U (k) = V (k) = -k + H(k). This implies that B * ≥ k. If not, U (k) = V (k) would be strictly increasing for B * < k < k. Consequently, B * is the highest solution to -B + H(B) = U (A * ). See Figure 1.
Results are summarized in the proposition below.

Proposition 1 If and only if -1 + µH (0) > 0, the support of Π * (.) is an interval

[A * , B * ] with 0 < A * < B * . A * is the solution to -1 + µH (A) = 0 while B * is the highest solution to -B + H(B) = U * = U (A * ). In this interval, GAS equilibrium satisfies Π * (k) = J -1 ((k + U * )/H(k)). 7
In fact, the derivation of a GAS equilibrium is simple. The investment of individuals at the bottom of hierarchy does not depend on the investment distribution. And the same holds for individuals at the top. Between the two extremes, Π * (k) is set so as to ensure that utilities are constant and equal to U (A * ). From Π * (k) = 1, hence, J(k) = 1, one deduces the upper bound of the support (which should be greater than the lower bound).

Figure 1 illustrates the determination of an equilibrium. As

W (k) is maximized in A * , U (k) coincides with W (k) for 0 ≤ k ≤ A * , is equal to U * (= U (A * )), for
A * ≤ k ≤ B * , and coincides with V (k), beyond B * .

Figure 1: GAS Equilibrium, W (0) > 0.

-

6 k U * Ṽ A * B * k V (.) W (.)
Individuals are identical ex ante. But, since they wish to become VIP's, they turn out to be heterogenous ex post, with VIP's consuming a great deal of energy in keeping their envied social position, common people having a low status not consuming as much energy to pursuing that end. This gives rise to an equilibrium in which all individuals have equal utilities, but unequal (gross) incomes. Income inequality is not very surprising. Indeed, starting from a situation of income equality, a (very) small increase in effort is sufficient to reach the top of the social hierarchy, implying that such a symmetric outcome cannot be an equilibrium. On the other hand, one could predict that all individuals will be prompted to "do the maximum". This is what can occur when beyond some ceiling, the marginal return to effort is reduced to zero. It can be noted that equilibrium existence is ensured for all strictly increasing GAS multiplier, J(Π(k)). Numerical simulations given in section 4 throw some light on the comparative statics.

Welfare analysis

In this section, we study the welfare properties of laissez-faire equilibrium in relation to different criteria. We find that a decentralized equilibrium is not an utilitarian optimum. The reason is that a social planner would take advantage of the income hierarchy. Interestingly, laissez-faire generates an egalitarian optimum. Which means that market equilibrium maximizes the utilitarian criterion subject to the constraint of utility equality. To conclude, it is proved that decentralized equilibrium is inefficient in the sense of Pareto. This stresses the price to pay for true equality (equality in terms of utilities).

Utilitarian criterion

Following the utilitarian tradition, the welfare criterion is the weighted average of utilities. Let S(Π(.)) denote the support of Π(.). Social welfare is Σ(Π(.)) = -S(Π(.))

kdΠ(k) + S(Π(.)) J(Π(k))H(k)dΠ(k) (3)
To characterize the social optimum, we rank the individuals uniformly (and continuously) on the segment [0, 1]. As the ranking is uniform, there is a share r of individuals whose ranks are lower than r (i.e. whose ranks lie in the interval [0, r]).

Making use of this ranking model of effort dispersion, [k(r)] 1 0 , denotes the effort of an individual whose rank in the income scale is r (0 ≤ r ≤ 1). The effort function, k(.), is the reciprocal of the distribution function Π(.) (when it exists). Accordingly, the welfare criterion can be rewritten as

Σ([k(r)] 1 0 ) = - 1 0 k(r)dr + 1 0 J(r)H(k(r)dr (4)
An utilitarian social optimum, denoted by [k S (r)] 1 0 , maximizes the welfare criterion with respect to [k(r)] 1 0 . Let k * (r) denote the reciprocal of laissez-faire equilibrium, Π * (k). We can prove the following Proposition 2 (i) According to an utilitarian planner, GAS model has a single social optimum, [k S (r)] 1 0 , such that -1

+ J(r)(dH(k(r))/dk(r)) = 0 for all r in [0, 1]. (ii) k * (r) > k S (r) for all r in [0, 1].
From the ranking optimum k S (r), we deduce the optimum distribution Π S (k) which is the reciprocal of k S (r).

Statement (i) results from the fact that the two models of effort dispersion, either the distribution Π(k), or the ranking function k(r), are isomorphic. See Appendix B.

In words, deciding on a non-decreasing function k(r) is not different from deciding on a distribution function Π(k).8 So the proof of statement (i) amounts to observing that, due to the concavity of H(.), k S (r) is strictly increasing in r. At first glance this result might look counter-intuitive, however. Indeed, one could object that, if defined with a c.d.f. Π(k), a social optimum might have mass points. This can be excluded. The reason is that any mass point K i of Π(k) corresponds to a subinterval of [0, 1] in which k(r) is constant and equal to K i . Consequently, any mass point generates a constraint on [k(r)] 1 0 . Since J(r) is strictly increasing, such constraints would be binding for the social problem. Symmetrically, one can easily exclude the case in which the social optimum defined as a distribution Π S (k) would be constant in a subinterval of [A S = k S (0), B S = k S (1)]. This is because the ranking of individuals on [0, 1] would create a discontinuity of k S (r) in such a case. Hence, the social optimum, Π S (k), which is unique, is also continuous and strictly increasing in the interval [A S , B S ]. See Appendix B and C. Statement (ii) results from the strict concavity of -k(r) + J(r)H(k(r)) (in k(r)).

In words, this result means that, relative to the social optimum, laissez-faire generates too much investment. In a precise sense, that is, for all r in [0, 1]. As state-ment (i) makes clear, this comes from the ranking effect, measured by the term dJ(Π(k)/dk)H(k). In GAS language, this result is the consequence of individuals devoting their energy k to climb up the social scale, aiming at having more people below them, or, equivalently, fewer people above them. This generates a rat-race effect which makes individuals' efforts excessive. On the contrary, a social planner would decide on investments [k(r)] 1 0 for a given ranking of individuals.

One implication of statement (i) is interesting per se. It turns out that, in the social optimum, the utility U (r) = U (k S (r)) (whose derivative reduces to U (r) = J (r)H(k S (r)) > 0 ) is strictly increasing. By this path, we have thus reached the result that, in this context, although individuals are homogenous ex ante, utility inequality is better for welfare. This is an unexpected outcome, since a priori, one could surmise that, due to the concavity of H(.), the social optimum should be symmetric. As the ranking model makes clear, a social planner takes advantage of the dispersion of individuals across the ranking scale. The reason is that the marginal product of effort grows with the individuals' position in the hierarchy. In other words, rich people profit more from an increase in their (gross of effort) revenue.

Or, equivalently, people are increasingly sensitive to social degrees when their income is higher. Consequently, utilitarian welfare is higher with a hierarchy of individuals.

It is worth noticing that Proposition 2 dramatically depends on the definition of aggregate welfare. A (virtual) situation in which true (i.e. utility) inequality would be favored because "rich people should be very rich" is a bit shocking. Extending the analysis to other welfare criteria is an interesting line for further investigations.

At this stage, we can ask the following question. Which income distribution would an egalitarian planner select? We can prove that GAS equilibrium maximizes the utilitarian criterion subject to the constraint of utility equality. In other words Proposition 3. According to an egalitarian planner, GAS equilibrium is a social optimum.

The analysis can be restricted to a continuous distribution whose support is connected. 9 In this case, the proof of Proposition 4 consists in observing that the equal-9 Mass points of probability (strictly) lower than one are clearly incompatible with the equal-utility constraint U (r) = 0 implies that, for all r in [0, 1],

-k (r) + J(r)H (k(r))k (r) + J (r)H(k(r)) = 0 or, -1 + J(k(r))H (k(r)) + dJ(Π(k(r))) dk(r) H(k(r)) = 0
The latter condition is satisfied by GAS equilibrium Π * (k). It follows that this (strictly) egalitarian social planner only has a single degree of freedom which is used to make U (0) = -k(0) + µH(k(0)) as high as possible. This maximum is reached for k(0) = A * . Since k(r) cannot be decreasing, it results that k(1) = B * . 10 This shows that GAS equilibrium is an equal-utility constrained social optimum.

The implications of Proposition 3 can be expressed in the following maxima: Whoever dislikes true inequalities, that is inequalities in terms of utility, should like observed (gross) income inequalities, and should then fight against any redistributive tax policy. It is worth noting that "true" equality (in terms of utilities) is particularly relevant in an economy where individuals are identical.

Pareto criterion

Let us finally prove that an equilibrium is Pareto inefficient: all participants invest too much in the neighborhood of GAS equilibrium.

In market equilibrium Π * (.), the effort of r-workers satisfies r = Π * (k) or k = k * (r), with k * (.) being the reciprocal of Π * (.). The equilibrium utility of an individual with rank r is

-k * (r) + J(r)H(k * (r) = -k + J(Π * (k))H(k)
Making use of this ranking of individuals, let us study the effect of a small change dK in all efforts in the neighborhood of equilibrium. Since all investments vary utility constraint. The same holds for holes in the support which are mapped into a discontinuity of k(r) (see Appendix B). Conversely, one can see that the equal-income social optimum is another equal-utility constrained social optimum.

10 Notice that the derivative dJ(Π(k(r)))/dk(r) is equal to zero for r = 0 but J (r) > 0. This is because, for r = 0, dJ(Π(k(r)))/dk(r) is zero but, since r = Π(k(r)), (dΠ(k(r))/dk(r))k (r) = 1 for all r in [0, 1].

by the same amount, the rank (r) associated with (k * (r) + dK) is left unchanged.

Consequently, the change in the utility of an agent of rank r is

dU (r) = [-1 + J(r) dH(k * (r)) dk(r) ]dK
On the other hand, in GAS equilibrium, all individuals have the same utility U * , implying that

-1 + J(r) dH(k * (r)) dk(r) = - dJ(Π * (k * (r))) dk(r) H(k * (r))
.

We then obtain

dU (r) dK = - dJ(Π * (k * (r))) dk(r) H(k * (r)) < 0
This proves that11

Proposition 4 According to a Paretian planner, the neighborhood of GAS equilibrium, lowering individuals' efforts improves welfare.

This result clearly comes from the GAS hypothesis. A decrease in all investments (i.e. efforts), which does not affect the position of agents, increase all utilities. The same ranks are reached at a lower "price ". Cooperation could enhance welfare in the sense of Pareto.12 This insight stresses the cost to pay for egalitarian efficiency. Lowering all efforts is Pareto-improving but individuals enjoy different utility gains. Assuming that GAS multiplier J(r) is convex, one can see that the higher the rank, the higher the marginal utility gain (J (r)H(k(r)). These disparities are not compatible with egalitarian efficiency.

4 Empirical potential and predictions of GAS theory

Can GAS equilibrium replicate observed wage dispersion? This section shows that GAS equilibrium fairly well accounts for the empirical wage distribution in France.

How does a change in GAS attitude affect income inequality? We also carry out numerical simulations aimed at throwing some light on different issues.

The specification for the GAS multiplier is borrowed to urn-ball matching with recruitment selection (see [START_REF] Gavrel | On the Inefficiency of Matching Models of Unemployment with Heterogeneous Workers and Jobs when Firms Rank their Applicants[END_REF][START_REF] Gavrel | Participation, Recruitment Selection, and the Minimum Wage[END_REF]. That is

J(r) = exp (-λ(1 -r)) (5)
In the previous expression for the GAS multiplier, the positive parameter λ can be regarded as a measure of individuals' GAS propensity, also referred to as their "social ambition". The higher this parameter, the higher the effect of social rank on individuals' utilities. From section 2 (Proposition 1), we know that the bottom and the top of (gross) income distribution are respectively determined by H(A * )

with -1 + J(0)H (A * ) = 0, and by H(B * ) with B * being the highest solution to

U (B) = U * = U (A), or -B + H(B) = -A * + J(0)H(A * ).
We also know that the reciprocal of the equilibrium distribution of investments, k * (r) (0 ≤ r ≤ 1) can be deduced from

J(r) = k -U * H(k)
or, in terms of income y,

J(r) = H -1 (y) + U * y
In the following calculations, the production function, H(.) is specified as H(k) = a + bk 1/2 with a and b being strictly positive. Notice that H (0) = ∞, implying that A * > 0, as in GAS equilibrium (W (0) > 0). GAS propensity, λ, is strictly positive.

The equilibrium value of utility, U * = U (A * ) is deduced from

A * = (bJ(0)) 2 4
It results that

U * = (bJ(0)) 2 4 + aJ(0)
Let q denote q = b 2 /a. The reciprocal of the cumulative distribution functionthe income function y * (r) -should satisfy the following second degree equation:

(y/a) 2 -[2 + qJ(r)](y/a) + [1 + qJ(0) + (qJ(0)) 2 4 ] = 0 (6)
It is easy to prove that y * (r) is the highest solution to the former equation. This is because y(r) should be increasing. We obtain the following econometric model:

y(r) = a 2 + qJ(r) + [qJ(r)(4 + qJ(r)) -qJ(0)(4 + qJ(0))] 1/2 2 + (7)
where J(r) is set by equation ( 5).

It should be noted that results depend on the units in which wages are measured.

In other words, a is a scale parameter. Only coefficient q has an economic meaning.

This coefficient measures the extent to which individuals can differentiate each other by deciding on different effort levels. The random variable is the error term. This GAS model of income dispersion is estimated by applying non linear least squares.

Notice that estimating the reciprocal of the cumulative distribution function allows us to treat the income y(r) associated with rank r as a random variable. In addition, the bounds z = H(A * ) and Z = H(B * ) do not necessarily coincide with the bounds of the observed (truncated) distribution. showing similar behavior should belong to the same "community". For this reason, we limit our study on the one hand to the poverty threshold, that is to say to 60% of the median income 13 , and on the other hand to the 95th percentile so that monthly gross incomes vary between 1 317 and 5 586 euros. The Theory of Leisure Class, which here extends to a large "middle class", the actual truncated wage distribution is rather better replicated by the estimated distribution (see the observed and estimated reciprocal of the income cumulative distribution presented on Figure 3). Table 1 reports the econometric results.

Empirical potential

13 This amounts to dropping about 10% of the bottom of the distribution.

Predictions

Let us first study the impact of GAS propensity on income inequality and on poverty.

Figures 5 shows the simulated Lorenz curves for different values of GAS propensity, λ, using the estimated value of the differentiation parameter, that is q = 1.925017. Lorenz curve simulation: sensitivity to parameter lambda GAS attitude gives rise to inequalities. The intuition behind this is simple. Individuals are homogenous, implying that, in the absence of "social ambition" (λ = 0), they all earn the same income. In the presence of social ambition, equality becomes impossible, as explained in the comment on Proposition 1. Notice that beyond some threshold, curves intersect: an increase in GAS parameter would tend to lower income inequalities. This is because all efforts go to zero as parameter λ goes to infinity. This point can be made clearer by making use of Gini coefficient for inequality. The Gini coefficient of the empirical truncated income distribution is 0.2023696 while the simulation leads to pretty close coefficient of 0.2011106. Figure 6 depicts the Gini coefficient as a function of GAS propensity. Above about λ = 6, an increase in GAS propensity reduces income inequalities. Besides, one can show that an increase in GAS propensity lowers individuals' utilities14 . But for any given rank in the income scale, an increase in GAS propensity does not invariably raise (or lower) the corresponding (gross) income (see Figure 7).

In other words, the effect on "perceived" wealth (or poverty) is ambiguous.

Figure 7: Simulated income distribution sensitivity to GAS propensity

Income cumulative distribution functions

Note: z and Z are the inferior bound (H(A * )) and superior bound (H(B * )).

Another interesting issue is the way in which individuals' ability to differentiate each other -their differentiation capability -affects income inequalities. Holding parameter a as a constant, an increase of parameter q can be seen as a consequence of economic development. 15 Figure 8 describes the effect of such improvement of labor efficiency on "perceived" inequality. In conformity with Veblen's intuition, an increase in labor efficiency would tend to expand income inequality. Indeed, individuals cannot differentiate when labor efficiency is very low.

It can be noted that these comparative statics exercises have very different meanings. In the second case, wealth is affected by a "material" phenomenon whereas, in the first case, the change in utilities and incomes results from a social phenomenon.

This highlights the role of social values (like social ambition) in the economic sphere as well as the practical importance that value-oriented policies may have in the real world. Contrary to K. Marx but in accordance with J.M. Keynes, "ideas" may have economic consequences. They may exert an influence on the so-called economic "infra-structures". This is a domain where "free lunches" could be found, although a marxian economist would probably object that GAS behavior, like individualism, is determined by the economic environment.

Lorenz curve simulation: sensitivity to parameter q

To sum up, these empirical exercises permit us predict that the GAS model does have empirical potential. We acknowledge that, to some extent, GAS theory is difficult to refute, since the plausibility of parameters values cannot be easily assessed. A natural way of circumventing this problem would be to assume that, excepting GAS intensity λ, other parameters are common to different countries. In that regard, notice also that the inference method to apply in the estimation of inequalities is not obvious.

One could argue that the distance between observations and estimations should be linked to a measure of inequalities.

We would like to add that we see the empirical relevance of GAS model as an intriguing but encouraging outcome. It is both intriguing and encouraging for the same reason which is that, apparently, usual explanations of wage inequality are not required.

Should GAS hypothesis be treated as an "as if" explanation? This is possible though individuals' efforts may reflect their investments in human capital, learning by doing, networking etc...

Concluding remarks

Perhaps the main insight of this essay is that income inequality can result from the desire for (advantageous) inequality. This insight calls into question the precise meaning of studies showing that income inequalities "explain" life expectancy, mental health, toxicomania, obesity... 16 It cannot be ruled out that all these social issues, including income inequalities, are driven by one common cause: the GAS attitude.

We also showed that the GAS hypothesis has an empirical potential in replicating the observed wage dispersion.

In this conclusion we would like to note that our GAS model could be helpful in many other contexts where, whatever its form may be, a rat-race prevails, as, for instance, informative advertising17 with endogenously differentiated goods, or in the population density of cities where workers' on-the-job performance increases with their proximity to firms. Indeed, the proximity to a place of work can be regarded as a means of dealing with shocks, as with human capital in the Schultz/Nelson-Phelps view.18

Our analysis of GAS attitude raises different issues. As noted in the introduction, building a dynamic model with adaptative expectations would make the analysis more expressive. A limitation to our analysis is that incomes are assumed to be common knowledge. What is going on when incomes are private information? Why are tax returns public in some countries -like Norway where daily local newspapers report your neighbors' incomes -whereas they are secret in many others? And, still assuming imperfect information, what are the social consequences of the poor mimicking the rich, or hiding their poverty, and of the rich seeking to demonstrate to the poor that they are rich. Which goods are produced to help rich in their ostentation strategy?

In France, Hermes offers a handbag which it advertises as reserved to women with the necessary age and elegance. Symmetrically, which goods do poor people use in their (paltry) attempt at cheating. Allowing for imperfect information would lead to build a two-stage game which would combine the social competition argument with the signalling argument. 19 We leave these different questions for further investigation.

(

  1971), all individuals have the same efficiency at work, implying that they are perfectly identical ex ante. In other words, there is a single technology y = H(k), with H(k) being an increasing (strictly) concave function which satisfies H(0) > 0. 4 The set of individuals is a continuum whose measure is normalized to 1. Individuals' investments generate an endogenous c.d.f., Π(k), which represents the proportion of individuals whose investments are strictly lower than k, i.e. Π(k) = Prob[Effort < k]. This definition of Π(.) makes the exposition easier. A priori this function is assumed to be piecewise continuous. Its mass points are denoted by

  this distinction, which clearly has a behavioral content, has not any analytical content.Each individual maximizes U (k) with respect to her effort k for a given c.d.f.

Figure 2

 2 Figure 2 reports the actual distribution of wages in France in 2010. It can be noted that the lower tail of the actual distribution is quite flat. In addition, individuals

Figure 2 :

 2 Figure 2: Income distribution for France, full distribution

Figure 5 :

 5 Figure 5: Lorenz curve sensitivity to GAS propensity

Figure 6 :

 6 Figure 6: Gini sensitivity to GAS propensity λ

Table 1 :

 1 Figure 3: Actual and estimated reciprocal of the income cumulative distribution for France OLS estimation of income function

				Income function
	Monthly gross earnings	1000 2000 3000 4000 5000 6000		France 2010
		0	.2	.4	.6	.8	1
				Cumulative population frequence
				Observations		Simulation
		Source: Eurostat Anonymized SES Microdata, 2010.
		Note: starting from 60% of median income; top 5% has been dropped.
		a=1393.882, q=1.925017, lambda=4.239692
				Coef. Std. Err.	Number of obs = 184236
	Differentiation capability q 1.925017 * * *	0.001256	R-squared =	0.999
	GAS propensity	λ 4.239692 * * * 0.0029779
	Scale parameter	a 1393.882 * * * 0.5622717
	Figure 4 depicts the empirical Lorenz curve together with the estimated Lorenz
	curve (deduced from the estimated cumulative distribution). One can notice that
	the two curves are very close to each other. GAS theory fairly well accounts for
	intermediate wage inequality.

See Hopkins (2008) for a survey.

We thank Ed Hopkins for very useful comments on the pros and cons of this behavioral literature.

This (reasonable) assumption makes the analysis simpler.

Results extend to the case in which the multiplier associated with a mass point K i is higher than J(Π(K i )) but lower than J(Π(K i ) + Φ(K i )).

Additive relative concerns would make technological progress incompatible with steady income inequality.

One can see that if -1 + µH (0) ≤ 0, then S * is an interval [0, B * ], with B * being the highest solution to -B + H(B) = µH(0).

If X is a random variable whose c.d.f. F(.) is continuous and strictly increasing, then the random variable Y = F(X) is uniformly distributed on (0,1). The proof of Proposition 3 just generalizes this property to a piecewise continuous non-decreasing function F(.), then allowing for mass points.

Provided that they keep in the same position, all individuals enjoy a utility gain.

Notice that Proposition 4 obviously implies that market equilibrium is not an utilitarian optimum. On the contrary, Pareto efficiency does not imply utilitarian efficiency.

Knowing that all individuals have the same utility in GAS equilibrium, the proof amounts to see that an increase in social ambition (parameter λ) reduces the utility U * = U (A * ) = -A * +J(0)H(A * of individuals at the lower end of the social scale. GAS multiplier J(0) = exp (-λ) falls.

Holding parameter q as a constant, an increase in parameter a leaves income inequality unchanged. In this sense, our specification of preferences makes that technological progress is compatible with steady income inequality.

See Wilkinson and Pickett (2010). Whatever their interpretation may be, the empirical results given by the authors are puzzling.

[START_REF] Butters | Equilibrium Distribution of Sales and Advertising Prices[END_REF] and[START_REF] Grossman | Informative Advertising with Differentiated Products[END_REF] are seminal papers.

See Acemoglu and Autor (2012).

See Danau and Gavrel (2015) for a first pass.

We also assign to a companion paper the task of appraising the extent to which international disparities in income inequalities can be imputed to cultural specificities of GAS social rules.

Appendix

A. Equilibrium distribution Π * (.)

This appendix shows that Π * (.) is continuous and that its support is connected and bounded above. In a preliminary step, we show that Π * (.) is continuous for k ≥ 0. Suppose it is not. This means that Π * (.) has at least one mass point K i ≥ 0 of probability φ(K i ) > 0.

Suppose that a participant, D, who invests K i initially, deviates and decide on a

, we can deduce that, due to the continuity of J(.) and H(.), her utility will jump upwards, implying that K i is not an optimum. This proves by contradiction that Π * (.) is continuous.

Next step shows that if the support of Π * (.), denoted by S * , contain two intervals

We can now show that the support of Π * (.) is bounded above. Suppose it is not. Namely, we show that any piecewise continuous c.d.f., Π(k) translates into a piecewise continuous non-decreasing "ranking function" k(r).

Consequently, this support is an

Let us first consider an interval

Let us now consider the case of a mass point, K, of Π(.) such that Π(K + ) = Π(K) + Φ(K), with 0 < Φ(K) ≤ 1. Remember that, according to our definition, Π

One can see that in this case, k(r) is a constant equal to K for all r in the interval

In this interval, the GAS multiplier, J(Π(k(r))), is constant and equal to J(r 3 ). This

Finally, we have to deal with holes in the support of Π(k).

Suppose Π(.) is constant in an interval [k 5 , k 6 ] while increasing in the left hand neighborhood of k 5 as well as in the right hand neighborhood of k 6 . One can see that the rank associated with this interval is R = Π(k 5 ) = Π(k 6 ). In addition, k(R) = k 5 and k(R + ) = k 6 . In words, the ranking function k(.) is discontinuous at R and jumps upwards.

C. Social optimum

Due to the isomorphism between Π(.) and k(.), the social optimum can be defined as a ranking function k S (r) on [0, 1] which maximizes

It follows that, assuming that -1 + µH (0) > 0, as in Proposition 1, k S (r) is determined by -1 + J(r)[dH(k(r))/dk(r)] = 0 for all r in [0, 1].

Since H(.) is strictly concave and J(.) is strictly increasing, k S (r) is strictly increasing.

Consequently, the case in which the social optimum k S (r) would be constant on a subinterval of [0, 1] is clearly excluded. Indeed, since J(r) is strictly increasing, the maximum of r 4 r 3 [-k(r) + J(r)H(k(r))]dr with respect to [k(r)] r 4 r 3 is higher than the maximum of (r 4 -r 3 )[-K + J(r 3 )H(K)] with respect to K. In words, assuming that k(r) is constant on an interval [r 3 , r 4 ] with (0 ≤ r 3 < r 4 ≤ 1) would generate a constraint on the planner's problem. The same holds for a discontinuity of k(r).

Since k S (r) is strictly increasing, the optimal distribution, Π S (k) is the reciprocal of k S (r). This implies that Π S (k) is strictly increasing in its (connected) support [A S = k S (0), B S = k S (1)]. Mass points and holes in the support of Π S (k) are not compatible with optimality.


R Émi

Anne Nouri
email: anne.nouri@univ-amu.fr

D Lax

MONOKINETIC SOLUTIONS TO A SINGULAR VLASOV EQUATION FROM A SEMICLASSICAL PERSPECTIVE

Solutions to a singular one-dimensional Vlasov equation are obtained as the semiclassical limit of the Wigner transform associated to a logarithmic Schrödinger equation. Two frameworks are considered, regarding in particular the initial position density: Gaussian initial density, or smooth initial density away from vacuum. For Gaussian initial densities, the analysis also yields global solutions to the isothermal Euler system that do not enter the frame of regular solutions to hyperbolic systems by P.

INTRODUCTION AND MAIN RESULTS

This paper is concerned by the Cauchy problem for the Vlasov equation

(1.1) ∂ t f + ξ∂ x f -λ ∂ x ln(ρ) ∂ ξ f = 0, f (0, x, ξ) = f 0 (x, ξ), t > 0, (x, ξ) ∈ R 2 ,
where λ = 0 and ρ(t, x) = R f (t, x, ξ)dξ. For λ > 0 it arises in plasma physics, e.g. for quasineutral plasmas in the core or tokamaks when one focuses on the direction of the magnetic lines. There, f denotes the ionic distribution function and the electrons of the plasma are assumed adiabatic. Due to the derivative of the density ρ with respect to space in the force term ∂ x ln(ρ), this equation is highly singular. The Cauchy problem can in particular be proven to be wellposed for very specific initial data like mono-kinetic distribution functions of the form f 0 (x, ξ) = ρ 0 (x)dx ⊗ δ ξ=v0(x) , with time-dependent mono-kinetic solutions of the form f (t, x, ξ) = ρ(t, x)dx ⊗ δ ξ=v(t,x) .

From a fluid dynamics perspective for λ > 0, it is well-known that f (t, x, ξ) = ρ(t, x)dx⊗ δ ξ=v(t,x) is a distributional solution of (1.1) if and only if its moments ρ(t, x) = f (x, ξ)dξ and ρ(t, x)v(t, x) = ξf (t, x, ξ)dξ are solutions of the isothermal Euler system (1.2)

∂ t ρ + ∂ x (ρv) = 0, ∂ t (ρv) + ∂ x ρv 2 + λρ = 0.
This paper addresses the existence of mono-kinetic solutions to (1.1), as limits of the Wigner transform of solutions to a logarithmic Schrödinger equation introduced in [START_REF] Mycielski | Nonlinear wave mechanics[END_REF] in the context of wave mechancis. As noticed there, the evolution of initial Gaussian data can be computed rather explicitly, a remark which yields our first main result:

RC was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BECASIM (ANR-12-MONU-0007-04). AN was supported by the French A*MIDEX project (Nr. ANR-11-IDEX-0001-02).

Theorem 1.1. Let ρ * , σ 0 > 0 and ω 0 , p 0 ∈ R. Set ρ 0 (x) = ρ * e -σ0x 2 , v 0 (x) = ω 0 x + p 0 , and consider the ordinary differential equation (1.3) γ = 2λσ 0 γ , γ(0) = 1, γ(0) = ω 0 .

Suppose that (1.3) has a solution γ ∈ C 2 ([0, T [). Set ρ(t, x) = ρ * γ(t) e -σ0(x-p0t) 2 /γ(t) 2 , v(t, x) = γ(t) γ(t)

x + p 0 .

Then µ = ρ(t, x)dx ⊗ δ ξ=v(t,x)
is a measure solution to (1.1) on [0, T [, with µ |t=0 = ρ 0 dx ⊗ δ ξ=v0 .

In the case λ > 0, the solution γ is globally defined and smooth, so T can be taken arbitrarily large. In addition, it safisfies γ(t) ∼ t→∞ 2t λσ 0 ln t , γ(t) ∼ t→∞ 2 λσ 0 ln t.

On the other hand, for λ < 0, the solution γ becomes singular in finite time.

Remark 1.2. In the case λ > 0 the global smooth solutions to the isothermal Euler system that are obtained as part of the result, do not enter the frame of regular solutions developed by Lax [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF]. Indeed the Lax solutions require bounded initial data, whereas the initial velocity in Theorem 1.1 is unbounded.

Remark 1.3. To our knowledge, the case λ < 0 does not correspond to a physical model related to (1.1). However, the logarithmic nonlinear Schrödinger presented in Section 3 was introduced initially exactly in the case λ < 0 ( [START_REF] Mycielski | Nonlinear wave mechanics[END_REF]). We will see that the analysis of this case requires very little extra effort.

Our second result deals with more general solutions to (1.1) such that ρ remains bounded away from zero. To do so, we do not consider ρ ∈ H s (R) the standard Sobolev space, or ρ ∈ ρ * + H s (R) for some ρ * > 0, but rather Zhidkov spaces, as introduced in [START_REF] Vries | and nonlinear Schrödinger equations: qualitative theory[END_REF][START_REF] Zhidkov | The Cauchy problem for a nonlinear Schrödinger equation[END_REF], and further analyzed in [START_REF] Gallo | Schrödinger group on Zhidkov spaces[END_REF] in the case of Schrödinger equations. For s 1, we set

X s (R) = f ∈ L ∞ (R), f ′ ∈ H s-1 (R) .
Note that being in dimension one, H s (R) ⊂ X s (R) holds for all s 1, and X s (R) is an algebra. The second result of the paper is the following theorem.

Theorem 1.4. Let λ > 0 and s 2. Suppose that (ρ 0 , Φ 0 ) ∈ X s (R) × C(R) with Φ ′ 0 ∈ X s (R) and ρ 0 (x) ρ 0 * for some positive constant ρ 0 * . There are T > 0, ρ ∈ C([0, T [; X s (R)), Φ ∈ C([0, T [×R), with ∂ x Φ ∈ C([0, T [; X s (R)), such that µ = ρ(t, x)dx ⊗ δ ξ=∂xΦ(t,x) with (ρ, Φ) t=0 = (ρ 0 , Φ 0 ),
is a measure solution to (1.1). Moreover, ρ(t, •) is bounded from below by a positive decreasing function of time. Remark 1.5. In (1.1), the term ∂ x ln µdξ ∂ ξ µ can be considered in a weak sense as

∂ ξ ∂ x ln µdξ µ , since µdξ = ρ ∈ C([0, T ]; C 1 (R)) and ρ > 0.
Remark 1.6. Although the Cauchy problem for the isothermal Euler (1.2) has global in time entropy weak solutions (ρ, u) ∈ L ∞ (R + × R) 2 (see [START_REF] Chen | Euler equations and related hyperbolic conservation laws[END_REF][START_REF] Chen | Compressible Euler equations with general pressure law[END_REF]), we cannot use them directly for our purpose. Indeed, the momentum equation in (1.2) is obtained from the kinetic equation (1.1) by multiplying (1.1) by ξ and integrating the resulting equation with respect to ξ. This leads to the product ρ ∂ x ln(ρ).

Since it is not under a conservative form, it is well known that there is no rigorous way to give a sense to this product for a general ρ ∈ L ∞ . It is why we have recourse to regular solutions to the isothermal Euler system, thus restricting for general initial bounded data to local in time solutions far from vacuum.

Remark 1.7. For (ρ, v) with values in ]0, +∞[×R, (1.2) is a strictly hyperbolic system. Consequently, for (ρ 0 , v 0 ) ∈ (X 2 (R)) 2 with ρ 0 α for some α > 0, there are T > 0 and

(ρ, v) ∈ C 1 ([0, T ] × R)
2 solution to the Cauchy problem associated to (1.2) and the initial datum (ρ 0 , v 0 ). This theorem makes an extra connection with a logarithmic Schrödinger equation, for which the above system corresponds to a limit system in the semiclassical regime.

Remark 1.8. The assumption on the initial density ρ 0 is more general than merely ρ 0 ∈ ρ 0 * + H s (R) for some s 2, since for instance, ρ 0 may have different limits as x → -∞ and x → +∞, or, even, no limit at all.

The plan of the paper is the following. Section 2 recalls the main steps of the derivation of the model for λ > 0 and related mathematical results. In Section 3, we show how (1.1) can be obtained formally from a nonlinear Schrödinger equation through the semiclassical limit. Section 4 establishes Theorem 1.1. In Section 5 , we prove Theorem 1.4. Section 6 adapts the proof of the boundedness from below of the density in the isotropic case by [START_REF] Chen | Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations[END_REF] to the isothermal case.

DERIVATION OF THE MODEL AND RELATED RESULTS

In this section, we recall the main lines of the derivation of the model with λ > 0, used for studying fusion plasmas ( [START_REF] Ghendrih | Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution[END_REF]). The evolution of the ions in the core of such plasmas is well described by the Vlasov equation

∂ t f + v • ∇ x f + Ze m i (-∇ x Φ + v ∧ B) • ∇ v f = 0,
where f is the ionic distribution function depending on time, position (in the domain Ω of the plasma) and velocity (in R 3 ), Ze and m i are the ion charge and mass respectively. The electric potential Φ and the magnetic field B should be governed by the Maxwell equations. But a finite Larmor radius approximation is derived in the limit of a large and uniform external magnetic field. This leads to the following equation for the ionic distribution function f in gyro coordinates, (2.1)

∂ t f + v ∂ x f -J 0 ρL (∂ x Φ)∂ v f -J 0 ρL ∇ x ⊥ Φ ⊥ • ∇ x ⊥ f = 0.
Here, the index (resp. ⊥) refers to the direction parallel (resp. orthogonal) to the external magnetic field. For any vector u = (u i ) ∈ R 3 , u ⊥ denotes the vector (u 2 , -u 1 , 0). The operator J 0 ρL is a Bessel operator performing averages on circles of Larmor radius ρ L in planes orthogonal to the magnetic field. Since it is not used in this paper, we do not enter into more details about it. The electrons move quite more quickly than the ions, so that their density n e is given in terms of the electric potential Φ by the Maxwell-Boltzmann equation Here, e (resp. T e ) is the electronic charge (resp. temperature), and < Φ > is the average of the potential on a magnetic field line. Due to the electroneutrality of the plasma, the Poisson equation is replaced by the electroneutrality equation

n e = ρ,
where ρ is the ionic density. The operator J 0 ρL induces some regularity in the orthogonal direction, but none in the parallel direction. The two-dimensional dynamics in the direction perpendicular to the magnetic field is studied in [START_REF] Hauray | Well-posedness of a diffusive gyro-kinetic model[END_REF]. In order to analyze the difficulty coming from the highly singular term J 0 ρL (∂ x Φ)∂ v f , we restrict to a one-dimensional spatial setting, e.g. by considering ionic distribution functions written in the form

f (t, x, v) = f (t, x , v )f ⊥ (|v ⊥ |), with +∞ 0 f ⊥ (|v ⊥ |)2π|v ⊥ |d|v ⊥ | = 1.
Then the term f ⊥ has no incidence in equation (2.1) and can be factorized. The equation that f should solve is

∂ t f + v ∂ x f -λ ∂ x ln(ρ ) ∂ v f = 0, t > 0, (x , v ) ∈ R 2 ,
i.e. the partial differential equation in (1.1) for f (resp. ρ , x , v ) denoted by f (resp. ρ, x, ξ), and λ = Te e .

Mathematical results related to (1.1) have been obtained for a system close to equilibrium, i.e. in the case where the departure of the electric potential Φ from its average along the magnetic lines < Φ > is small. Equation (2.2) simplifies into

n e = n 0 1 + e T e (Φ-< Φ >) , so that (1.1) is replaced by (2.3) ∂ t f + ξ∂ x f -λ ∂ x ρ ∂ ξ f = 0, t > 0, (x, ξ) ∈ R 2 .
The Cauchy problem for (2.3) is locally well-posed either for initial analytic data [START_REF] Jabin | Analytic solutions to a strongly nonlinear Vlasov equation[END_REF] or in Sobolev spaces and satisfying a Penrose stability condition [START_REF] Rousset | Quasineutral limit for Vlasov-Poisson with Penrose stable data[END_REF], but is ill-posed in the sense of Hadamard for regular initial data in Sobolev spaces and arbitrarily small time [START_REF] Bardos | A Vlasov equation with Dirac potential used in fusion plasmas[END_REF].

FROM SCHR ÖDINGER TO VLASOV VIA EULER

In this section, we show how a logarithmic Schrödinger equation can be formally related to the isothermal Euler system (1.2), in the semiclassical limit. For ε > 0, consider the Cauchy problem for the Schrödinger equation

(3.1) iε∂ t u ε + ε 2 2 ∂ 2 xx u ε = λ ln(|u ε | 2 )u ε , u ε (0, x) = ρ 0 (x)e iΦ0(x)/ε .
Following the idea from [START_REF] Grenier | Semiclassical limit of the nonlinear Schrödinger equation in small time[END_REF], any function

u ε = a ε e iΦ ε /ε , with (t, x) → a ε (t, x) ∈ C and (t, x) → Φ ε (t, x) ∈ R solutions to the quasilinear problem (3.2) ∂ t Φ ε + (∂ x Φ ε ) 2 2 + λ ln |a ε | 2 = 0, Φ ε (0, x) = Φ 0 (x), (3.3) ∂ t a ε + ∂ x Φ ε ∂ x a ε + a ε 2 ∂ 2 xx Φ ε = i ε 2 ∂ 2 xx a ε , a ε (0, x) = ρ 0 (x) =: a 0 (x),
is a solution to (3.1). An important remark is that by allowing a ε to be complex-valued (even though its initial datum is real-valued), one gains a degree of freedom to dispatch terms from (3.1) into (3.2)-(3.3), and the choice introduced by Grenier is much more robust than the Madelung transform when semiclassical limit is considered (see [6]).

Determining Φ ε solution to (3.2) turns out to be equivalent to determining

v ε = ∂ x Φ ε and a ε solution to (3.4)    ∂ t v ε + v ε ∂ x v ε + λ ∂ x ln |a ε | 2 = 0, v ε (0, x) = Φ ′ 0 (x), ∂ t a ε + v ε ∂ x a ε + a ε 2 ∂ x v ε = i ε 2 ∂ 2 xx a ε , a ε (0, x) = a 0 (x).
Indeed, given (v ε , a ε ) solution to (3.4), we can define Φ ε by

(3.5) Φ ε (t, x) = Φ 0 (x) - t 0 1 2 |v ε (τ, x)| 2 + λ ln |a ε (τ, x)| 2 dτ.
We check that

∂ t (∂ x Φ ε -v ε ) = ∂ x ∂ t Φ ε -∂ t v ε = 0, so that v ε = ∂ x Φ ε and Φ ε solves (3.
2), and a ε solves (3.3). Passing formally to the limit ε → 0 in (3.4), we get the system

(3.6)    ∂ t v + v∂ x v + λ∂ x ln(|a| 2 ) = 0, v |t=0 = Φ ′ 0 , ∂ t a + v∂ x a + a 2 ∂ x v = 0, a |t=0 = √ ρ 0 ,
which turns out to be the symmetrized version of (1.2), with ρ = |a| 2 (see [START_REF] Chemin | Dynamique des gaz à masse totale finie[END_REF][START_REF] Makino | Sur la solution à support compact de l'équation d'Euler compressible[END_REF]). As noticed in the introduction, we then formally obtain a solution to (1.1) by setting

f (t, x, ξ) = |a(t, x)| 2 dx ⊗ δ ξ=v(t,x) .
A more direct link from (3.1) to (1.1) is provided by the notion of Wigner measure. The Wigner transform of u ε , solution to (4.2)-(4.3), is defined by (see e.g. [START_REF] Burq | Mesures semi-classiques et mesures de défaut[END_REF][START_REF] Érard | Homogenization limits and Wigner transforms[END_REF][START_REF] Lions | Sur les mesures de Wigner[END_REF])

(3.7) W ε (t, x, ξ) = R e iyξ u ε t, x - ε 2 y u ε t, x + ε 2 y dy, (t, x, ξ) ∈ [0, T ] × R 2 .
Up to the extraction of a subsequence, W ε converges to a non-negative measure on the phase space. In general, several limits may exist (see the above references). We will see that in the framework of this paper, the limit is unique, and solves (1.1).

GAUSSIAN INITIAL DATA

Let us first study the ordinary differential equation (1.3). Local existence and uniqueness of a C 1 solution stem from Cauchy-Lipschitz Theorem. Indeed, the nonlinearity in the above equation is locally Lipschitzean away from {γ = 0}. We now address the global existence issue.

In the case λ > 0, multiplying (1.3) by γ and integrating yields (4.1) ( γ) 2 = ω 2 0 + 4λσ 0 ln γ. This readily shows that γ is bounded from below away from zero, so the flow is global. In the case λ < 0, suppose that γ is bounded away from zero, i.e. there is δ > 0 such that γ(t) δ. Then (1.3) yields

γ 2λσ 0 δ , hence γ(t) λσ0 δ t 2 + ω 0 t + 1
, and a contradiction for t sufficiently large. Now suppose that γ ∈ C 2 (0, ∞) with γ > 0: from the above argument, there exists a sequence t n along which γ(t n ) → 0 + . From (1.3), γ(t n ) → -∞, hence a contradiction.

To prove Theorem 1.1, we start from a semi-classically scaled logarithmic Schrödinger equation, (4.2)

iε∂ t u ε + ε 2 2 ∂ 2 xx u ε = λ ln(|u ε | 2 )u ε ,
together with the initial value

(4.3) u ε (0, x) = √ ρ * e -σ0x 2 /2 e iω0x 2 /(2ε) e ip0x/ε .
Such an initial datum does not fit into the framework of [START_REF] Carles | WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity[END_REF], since it goes to zero at infinity. However, as noted in [START_REF] Mycielski | Nonlinear wave mechanics[END_REF], for fixed ε > 0, the solution to (4.2) with such an initial datum can be computed rather explicitly. Indeed, bearing in mind the propagation of coherent states in the semi-classical regime ( [START_REF] Hagedorn | Semiclassical quantum mechanics. I. The → 0 limit for coherent states[END_REF], see also [START_REF] Combescure | Coherent states and applications in mathematical physics[END_REF]) it is consistent to look for a solution of the form (4.4) u ε (t, x) = b ε (t)e -Ω ε (t)(x-q(t)) 2 /2+ip(t)(x-q(t))/ε+iS(t)/ε , with b ε , Ω ε ∈ C, and q, p, S ∈ R. Plugging this ansatz into (3.1), we find:

iε ḃε -iε Ωε (x -q) 2 2 b ε + iεΩ ε q(x -q)b ε -ṗ(x -q)b ε + p qb ε - p 2 2 b ε -Ṡb ε -ε 2 Ω ε 2 b ε + ε 2 (Ω ε ) 2 (x -q) 2 2 b ε -ipεΩ ε (x -q)b ε = λb ε ln |b ε | 2 -λb ε (x -q) 2 Re Ω ε .
Cancelling the polynomial in x -q, sufficient conditions for the previous equation to hold are

q = p, ṗ = 0, Ṡ = p q - p 2 2 = p 2 2 , (4.5) iε ḃε = ε 2 2 Ω ε b ε + λb ε ln |b ε | 2 , (4.6) iε Ωε = ε 2 (Ω ε ) 2 + 2λ Re Ω ε . (4.7)
Equation (4.5) corresponds to the classical Hamiltonian flow in the absence of external force, and the associated classical action. We compute exactly

p(t) = p 0 , q(t) = p 0 t, S(t) = p 2 0 2 t.
Since we are eventually interested only in the modulus of b ε , we infer from (4.6):

d dt |b ε | 2 = 2 Re bε ḃε = ε|b ε | 2 Im Ω ε , hence (4.8) |b ε (t)| 2 = ρ * e ε t 0 Im Ω ε (s)ds .
For fixed ε > 0 and Re Ω ε (0) > 0, (4.7) has a unique, global solution Ω ε ∈ C(R), whose large time behavior depends on the sign of λ. Indeed, in a similar fashion as in [START_REF] Li | Blowup phenomena of solutions to the Euler equations for compressible fluid flow[END_REF], we seek Ω ε of the form

Ω ε = -i ηε εη ε . Then (4.7) becomes ηε = 2λ ε η ε Im ηε η ε .
Introducing the scaled polar decomposition η ε = γ ε e iεα ε , Ω ε is given by (4.9

)

Ω ε = αε - i ε γε γ ε ,
and the above equation reads:

γε -ε 2 γ ε ( αε ) 2 = 2λγ ε αε , αε γ ε + 2 αε γε = 0.
The second equation yields

d dt (γ ε ) 2 αε = 0.
In view of (4.9), we have a degree of freedom to set γ ε (0). Setting γ ε (0) = 1, we have

αε (0) = Re Ω ε (0) = σ 0 , γε (0) = -ε Im Ω ε (0) = ω 0 .
Therefore, we have

(4.10) (γ ε ) 2 αε = σ 0 .
The equation for γ ε boils down to

γε = ε 2 σ 2 0 (γ ε ) 3 + 2λσ 0 γ ε ,
along with the initial data that we recall:

γ ε (0) = 1, γε (0) = ω 0 .
Local existence and uniqueness of a C 1 solution stem from Cauchy-Lipschitz Theorem. Indeed, the nonlinearity in the above equation is locally Lipschitzean away from {γ ε = 0}. We now address the global existence issue. Multiplying the above equation by γε and integrating in time, we infer:

( γε ) 2 = 4λσ 0 ln γ ε - ε 2 σ 2 0 (γ ε ) 2 + ε 2 σ 2 0 + ω 2 0 .
This shows that for fixed ε > 0, γ ε remains bounded away from zero, for if we had γ ε (t n ) → 0 for some sequence t n , then the above right hand side would become negative for n large enough, hence a contradiction. Thus, for fixed ε > 0, (4.7) has a unique, global solution

Ω ε ∈ C 1 (R).
In view of (4.10), (4.9) also reads (4.11)

Ω ε = σ 0 (γ ε ) 2 - i ε γε γ ε .
Given (4.8) and (4.11), the Wigner transform (3.7) of u ε is given by (4.12)

W ε (t, x, ξ) = ρ * γ ε (t) e - σ 0 (x-p 0 t) 2 (γ ε (t)) 2 e - σ 0 ε 2 y 4(γ ε (t)
) 2 e iy(ξ- γε γ ε (x-p0t)-p0) dy.

On every time interval such that γ is bounded away from zero, the Gronwall lemma shows that

γ ε -γ = O(ε), γε -γ = O(ε).
Consequently, when ε → 0, the Wigner transform W ε of u ε weakly converges to the bounded measure

(4.13) µ(t, dx, dξ) = ρ * γ(t) e - σ 0 (x-p 0 t) 2 (γ(t)) 2 dx ⊗ δ ξ= γ γ (x-p0t)+p0 . Straightforward computations show that (ρ(t, x), v(t, x)) := ρ * γ(t) e - σ 0 (x-p 0 t) 2 (γ(t)) 2 , γ(t) γ(t) (x -p 0 t) + p 0
is a solution to the isothermal Euler system (1.2) on the time interval of the existence of γ.

To conclude, we consider the large time behavior of γ for λ > 0. If γ was bounded from above, γ M , then (1.3) would yield

γ 2λσ 0 M > 0,
hence a contradiction. Therefore, for t sufficiently large, γ(t) 1 and γ(t) > 0. Since γ 0, integration then shows (see e.g. [START_REF] Lax | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]), we infer that γ(t) ∼ t→+∞ 2t λσ 0 ln t.

γ(t) γ(t)(t -t) + 1 -→ t→∞ ∞,
Since γ = ω 2 0 + 4λσ 0 ln γ, we conclude γ(t) ∼ t→+∞ 2 λσ 0 ln t.

In the case λ < 0, suppose that γ is bounded away from zero, γ(t) δ > 0. Then (1.3) yields γ 2λσ 0 δ , hence γ(t) λσ0 δ t 2 + ω 0 t + 1, and a contradiction for t sufficiently large. Now suppose that γ ∈ C 2 (0, ∞) with γ > 0: from the above argument, there exists a sequence t n along which γ(t n ) → 0 + . From (1.3), γ(t n ) → -∞, hence a contradiction. This ends the proof of Theorem 1.1.

WKB ANALYSIS

In this section, we justify the formal approach presented in Section 3 in the framework of Theorem 1.4, thus proving this result. 5.1. Constructing the solution. Note that for fixed ε > 0, the Cauchy problem for (3.1) has been considered in [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF] (see also [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Section 9.1]), for initial data in the class

W = f ∈ H 1 (R), R |f (x)| 2 ln |f (x)| 2 dx < ∞ .
This class is not compatible with the assumption ρ(x)

ρ 0 * > 0 from Theorem 1.1, which is equivalent to |u ε (0, x)| 2 = |a 0 (x)| 2
ρ 0 * > 0 in the approach that we follow. Therefore, we choose to rather work in Zhidkov spaces X s (R). The system (3.4) has a unique smooth solution as stated in the following proposition, which includes the case ε = 0. Proposition 5.1. Let s > 5/2 and λ > 0. Suppose that ρ 0 , Φ ′ 0 ∈ X s (R), with ρ 0 (x) ρ 0 * > 0.

Then there exists T independent of s > 5/2 and ε ∈ [0, 1], and a unique solution

(a ε , v ε ) ∈ C([0, T ]; X s × X s ) to (3.4).
Proof. This result is a rather direct consequence of [2, Proposition 2.1], whose proof we recall the main idea. Separate real and imaginary parts of a ε , a ε = a ε 1 + ia ε 2 , and introduce

u ε =   a ε 1 a ε 2 v ε   , u 0 =   √ ρ 0 0 Φ ′ 0   , L =   0 -∂ 2 xx 0 ∂ 2 xx 0 0 0 0 0   and A(u) =   v 0 a1 2 0 v a2 2 2λa1 a 2 1 +a 2 2 2λa2 a 2 1 +a 2 2 v   .
We now have the system:

(5.1)

∂ t u ε + A(u ε )∂ x u ε = ε 2 Lu ε ; u ε |t=0 = u 0 .
Since ρ 0 is bounded away from zero, its square root is also in X s , so that u 0 ∈ X s (R) 3 . The matrix A is symmetrized by the matrix

S = I 2 0 0 a 2 1 +a 2 2 4λ
, which is symmetric positive if and only if a 2 1 + a 2 2 > 0, that is, so long as no vacuum appears. By assumption,

(a ε 1 ) 2 + (a ε 2 ) 2 t=0 ρ 0 * > 0.
Then the main idea is that the operator L is skew-symmetric, and so does not appear in L 2 -based energy estimates. Standard tame estimates (see e.g. [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF][START_REF] Taylor | Partial differential equations[END_REF]) do not involve the L 2 norm of u ε , and so the only aspect remaining is that L ∞ -estimates can be obtained rather directly. So long as, say,

(5.2) (a ε 1 (t, x)) 2 + (a ε 2 (t, x)) 2 ρ 0 * 2 , ∀x ∈ R,
we have:

u ε (t) L ∞ u 0 L ∞ + t 0 A(u ε (τ ))∂ x u ε (τ ) L ∞ dτ + t 0 ∂ 2 xx a ε (τ ) L ∞ dτ u 0 L ∞ + C t 0 u ε (τ ) L ∞ ∂ x u ε (τ ) L ∞ dτ + t 0 ∂ 2 xx a ε (τ ) L ∞ dτ u 0 L ∞ + C t 0 u ε (τ ) 2 X s dτ + t 0 u ε (τ ) X s dτ,
where we have used Sobolev embedding under the assumption s > 5/2. Indeed, by definition, X s ⊂ L ∞ , and for

u ε ∈ X s , ∂ x u ε ∈ H s-1 ⊂ L ∞ , provided that s > 3/2, and similarly, ∂ 2 xx u ε ∈ L ∞ for s > 5/2. Now we set P = (I -∂ 2 xx ) (s-1)/2 ∂ x , so that f X s ≈ f L ∞ + P f L 2 , and denote by f, g = ∞ -∞ f (x)g(x)dx,
the scalar product in L 2 . Since L is skew-symmetric and S is real-valued,

d dt SP u ε (t), P u ε (t) = (∂ t S)P u ε (t), P u ε (t) + 2 Re S∂ t P u ε (t), P u ε (t) = (∂ t S)P u ε (t), P u ε (t) + ε Re SLP u ε (t), P u ε (t) -2 Re SP A(u ε (t))∂ x u ε (t) , P u ε (t) .
So long as (5.2) holds, we have the following set of estimates. First,

(∂ t S)P u ε (t), P u ε (t) ∂ t S L ∞ P u ε (t) 2 L 2 C ( u ε (t) L ∞ ) ∂ t u ε (t) L ∞ u ε (t) 2 X s .
Directly from (5.1), we have:

∂ t u ε (t) L ∞ C ( u ε (t) L ∞ ) ∂ x u ε (t) L ∞ + ∂ 2 xx a ε (t) L ∞ C ( u ε (t) X s ) u ε (t) X s .
Since SL is skew-symmetric, we have Re SLP u ε (t), P u ε (t) = 0, which prevents any loss of regularity in the estimates. For the quasi-linear term involving the matrix A, we note that since SA is symmetric, commutator estimates (see [START_REF] Lannes | Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators[END_REF]) yield:

SP (A(u ε )∂ x u ε ) , P u ε (t) C ( u ε (t) L ∞ ) P u ε (t) 2 L 2 ∂ x u ε (t) L ∞ C ( u ε (t) X s ) P u ε (t) 2 L 2 .
Finally, we have:

d dt SP u ε (t), P u ε (t) C ( u ε (t) X s ) u ε (t) 2 X s .
This estimate, along with the L ∞ -estimate, shows that on a sufficiently small time interval [0, T ], with T > 0 independent of ε ∈ [0, 1], (5.

Φ ′ 0 ∈ L 2 (R), then (3.1) has a unique solution u ε ∈ C([0, T ]; X s (R))
, where T is given by Proposition 5.1.

Proof. From Proposition 5.1, (3.4) has a solution (v ε , a ε ) ∈ C([0, T ]; X s ×X s ). Plugging this information into (3.4), we infer

v ε (t) L 2 Φ ′ 0 L 2 + t 0 v ε (τ ) L ∞ ∂ x v ε (τ ) L 2 dτ + C t 0 1 a ε (τ ) L ∞ ∂ x a ε (τ ) L 2 dτ.
Therefore, v ε ∈ C([0, T ]; L 2 ), and Φ ε , stemming from v ε via the formula (3.5), satisfies

Φ ε ∈ C([0, T ]; X s+1
). The existence part follows readily, since X s (R) is an algebra, by setting u ε = a ε e iΦ ε /ε . For the uniqueness property, consider two such solutions u ε , ũε ∈ C([0, T ]; X s ), and set w ε = u ε -ũε . It satisfies

(5.3) iε∂ t w ε + ε 2 2 ∂ 2 xx w ε = λ ln(|u ε | 2 )u ε -ln(|ũ ε | 2 )ũ ε , w ε |t=0 = 0.
Recall the pointwise estimate from [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF] (see also [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Lemma 9.3.5]),

Im ln(|u ε | 2 )u ε -ln(|ũ ε | 2 )ũ ε (u ε -ũε ) 4|u ε -ũε | 2 .
Multiply (5.3) by w ε , integrate on an interval I = [M -, M + ], and take the imaginary part. This yields, along with the above estimate, ε 2

d dt I |w ε (t, x)| 2 dx + ε 2 2 Im I w ε ∂ 2 xx w ε 4λ I |w ε (t, x)| 2 dx.
We have, by integration by parts,

Im I w ε ∂ 2 xx w ε = Im w ε (t, M + )∂ x w ε (t, M + ) -Im w ε (t, M -)∂ x w ε (t, M -).
Since w ε ∈ C([0, T ]; X 1 ), we can choose sequences M n ± → ±∞ along which the above term goes to zero, and the Gronwall lemma implies w ε (t) L 2 ≡ 0. W ε (t, x, ξ) = e iξv a ε t, x -ε 2 y a ε t, x + ε 2 y e iϕ ε (t,x,y)/ε dy, where

C independent of ε ∈ [0, 1] such that ∂ x (Φ ε -Φ) L ∞ ([0,T ];X s-2 ) + a ε -a L ∞ ([0,T ];X s-2 ) Cε.
ϕ ε (t, x, y) = Φ ε t, x - ε 2 y -Φ ε t, x + ε 2 y .
Theorem 5.4. When ε → 0, the Wigner transform W ε of u ε weakly converges to the bounded measure

µ(t, dx, dξ) = |a(t, x)| 2 dx ⊗ δ ξ=∂xΦ(t,x) ,
where (∂ x Φ, a) is a solution of (3.6). Moreover, µ is a solution to (1.1).

Proof. In view of Proposition 5.3,

a ε = a + r ε a and ∂ x Φ ε = ∂ x Φ + r ε v , with r ε a L ∞ ([0,T ];X s-2 ) + r ε v L ∞ ([0,T ];X s-2 )
Cε.

Therefore,

W ε (t, x, ξ) = e iy(ξ-∂xΦ(t,x)) a t, x - ε 2 y ā t, x + ε 2 y dy + R ε 1 + R ε 2 + R ε 3 ,
where R ε j (t, x, ξ) = e iy(ξ-∂xΦ(t,x)) r ε j (t, x, y)dy, 1 j 3,

r ε 1 (t, x, y) = a ε t, x - ε 2 y a ε t, x + ε 2 y e i(ϕ ε (t,x,y)+ε∂xΦ(t,x))/ε -1 , r ε 2 (t, x, y) = ā t, x + ε 2 y r ε a t, x - ε 2 y + a t, x - ε 2 y r ε a t, x + ε 2 y , r ε 3 (t, x, y) = r ε a t, x - ε 2 y r ε a t, x + ε 2 y .
Propositions 5.1 and 5.3 yield, along with Taylor's formula for the term r ε 1 ,

r ε j L ∞ ([0,T ]×R 2 )
Cε, 1 j 3.

Consequently, W ε tends to |a| 2 dx ⊗ δ ξ=∂xΦ in M b ([0, T ] × R 2 )
) ∈ C 1 with compact support in [0, T ] × R 2 , it holds µ, ∂ t α + ξ∂ x α -λ ∂ x ln|a| 2 ∂ ξ α = |a(t, x)| 2 (∂ t α(t, x, ∂ x Φ(t, x)) + ∂ x Φ(t, x)∂ x α(t, x, ∂ x Φ(t, x))) dxdt -λ |a(t, x)| 2 ∂ x ln|a| 2 ∂ ξ α(t, x, ∂ x Φ(t, x)) dxdt = |a| 2 (∂ t (α(t, x, v(t, x))) -∂ T v∂ ξ α(t, x, v(t, x))) dxdt + |a| 2 (∂ x (v α(t, x, v(t, x))) -∂ x v α(t, x, v) -v∂ x v ∂ ξ α(t, x, v)) dxdt -λ |a| 2 ∂ x ln|a| 2 ∂ ξ α(t, x, v(t, x)) dxdt = -α(t, x, v) ∂ t |a| 2 + v∂ x |a| 2 + |a| 2 ∂ x v dxdt -|a| 2 ∂ ξ α(t, x, v) ∂ t v + v∂ x v + λ∂ x ln|a| 2 dxdt.
This is zero, in view of (3.6), since

∂ t |a| 2 + ∂ x Φ∂ x |a| 2 + |a| 2 ∂ 2 xx Φ = 2 Re a ∂ t a + ∂ x Φ∂ x a + a 2 ∂ 2 xx Φ = 0.
Therefore, any solution to (3.6) yields a solution to (1.1).

THE BOUND FROM BELOW OF THE DENSITY

It remains to prove that the density is bounded from below to complete the proof of Theorem 1.4. The result from [START_REF] Chen | Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations[END_REF] Proof. In [START_REF] Chen | Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations[END_REF], the proof is presented for the pressure law p(ρ) = ρ in (1.2) replaced by the isentropic one, p(ρ) = ρ γ , with γ > 1. We simply perform the standard modification to adapt the proof to the isothermal case (see e.g. [START_REF] Chen | Euler equations and related hyperbolic conservation laws[END_REF][START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]). Note that Proposition 6.1 is also a consequence of [10, Remark 3.3]: we sketch the proof in the isothermal case for λ = 1 for the convenience of the reader. First, the isothermal Euler equation in Lagrangian coordinates reads (see e.g. [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]) (6.1) τ t -u x = 0, u t -τ x τ 2 = 0, with τ = 1/ρ. Setting s = u -ln τ, r = u + ln τ, we find

∂ + s = ∂ -r = 0,
where

∂ ± = ∂ t ± 1 τ ∂ x .
Following [START_REF] Chen | Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations[END_REF], introduce α = s x , β = r x .

By assumption, we know that u, τ , α and β are uniformly bounded at time t = 0: there exists M > 0 such that sup x∈R (α(0, x), β(0, x)) < M.

We check

∂ + α = 1 2τ
α(β -α), (6.2)

∂ -β = 1 2τ β(α -β). (6.3)
As in [START_REF] Chen | Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations[END_REF], (6.4) sup (t,x)∈[0,T )×R (α(t, x), β(t, x)) < M, where T > 0 is such that (6.1) has a C 1 solution on [0, T ). (6.4) is established by contradiction: assume for instance that α(t 0 , x 0 ) = M . Because the wave speed 1 τ is bounded from above, we can consider the characteristic triangle with vertex (t 0 , x 0 ) and lower boundary at time t = 0, denoted by Ω. Let t 1 ( t 0 ) be the first time such that α(t 1 , x 1 ) = M or β(t 1 , x 1 ) = M in Ω. Assume α(t 1 , x 1 ) = M for instance (the other case is similar), and let Ω 1 denote the characteristic triangle with vertex (t 1 , x 1 ): sup (t,x)∈Ω1, t<t1 (α(t, x), β(t, x)) < M.

Since α(t 1 , x 1 ) = M , there exists t 2 ∈ [0, t 1 ) such that inf (t,x)∈Ω1, t t2 α(t, x) > 0.

Let t 2 < t < t 1 . On [t 2 , t], (6.2) yields

∂ + α K(M α -α 2 ).
Integrating along some charateristic between t 2 and t, we find 1 M ln α(t) M -α(t)

1 M ln α(t 2 ) M -α(t 2 )
+ K(t -t 2 ).

Letting t → t 1 then leads to a contradiction. We infer that u x is bounded from above, hence τ t is bounded from above, in view of (6.1): τ grows at most linearly in time, and since ρ = 1/τ , the proposition follows.

(2. 2 )

 2 n e = n 0 e e Te (Φ-<Φ>) .

and γ > 0

 0 for t sufficiently large. The asymptotic behavior announced in Theorem 1.4 then follows by integrating the identity dγ ω 2 0 + 4λσ 0 ln γ = dt, to obtain the asymptotic behavior of γ. 2 -ω 2 0 )/(4λσ0) dy, where we have changed the variable as y = 4λσ 0 ln γ + ω 2 0 . Recall that the Dawson function, defined by F (x) = e -x 2 x

5. 2 .

 2 Asymptotic expansion. Proposition 5.1 with ε = 0 yields the existence of a unique solution (v, a) ∈ C([0, T ]; (X s (R)) 2 ) to (3.6) As a direct consequence of Proposition 5.1 and [2, Proposition 3.1], we have: Proposition 5.3. Under the assumptions of Proposition 5.1, there exists

5. 3 .

 3 Convergence of the Wigner transform. For u ε = a ε e iΦ ε /ε , with (Φ ε , a ε ) solution to (3.2)-(3.3), or equivalently (∂ x Φ ε , a ε ) solution to(3.4), the Wigner transform defined in (3.7) is equal to(5.4)

yields: Proposition 6 . 1 .

 61 Under the assumptions of Theorem 1.4, the density ρ solution to (1.2) satisfies ρ(t) ρ 0 * 1 + Ct , t ∈ [0, +∞[, for some constant C > 0.

  2) holds, hence the existence of a unique solution. The fact that the local existence time does not depend on s > 5/2 follows from the continuation principle based on Moser's calculus and tame estimates (see e.g.[27, Section 2.2] or [29, Section 16.1]).

	Corollary 5.2. Under the assumptions of Proposition 5.1, if we suppose in addition that