Silva Ohannessian

Gilbert Saporta

Zero option in conjoint

Keywords: "zero" option specification, inappreciation of the products, indecision in the choice, bayesian approach, market shares

Introduction

In conjoint analysis literature, the "zero" option relates to the no choice due to the inappreciation of the scenarios presented within the frame of a survey. However, the econometric specification of this no choice, called refusal, is rarely tackled. Another type of no choice is the indecision in the choice making process resulting from the similarities of the scenarios. This second concept of no choice, labeled conflict, is presented by [START_REF] Tversky | Choice under Conflict: The Dynamics of Deferred Decision[END_REF]. The latter demonstrates that the consideration of the conflict disagrees with the utility maximisation theory or the rational theory of the choice.

Taking into account these various elements, we propose a new specification of the "zero" option that integrates the above mentioned concepts, the conflict and the refusal. To respect the unsuitability of the utility maximisation with the no choice, no utilities are associated with neither of the two concepts. Instead, we refer to the notion of ordered response models that governs the no choice. Our modeling of the "zero" option is therefore a mix of specification resulting from the utilities comparison and the ordered response models. This mix associates utilities only with the products and compares them with the "zero" option thresholds. These comparisons allow us to determine the boundaries of no choice situations. We apply our "zero" option model to the Video on Demand (VoD) market. The obtained results, by using a bayesian approach on the individual models and the likelihood maximisation on the aggregate model, show a good adequation of the model. The estimates are consistent with the reality and the significance of the refusal and the conflict demonstrates their importance in the decision making process. Moreover, the use of a bayesian approach for the parameter estimation gives better results than the widespread penalized likelihood.

The new specification of the "zero" option and its application

The specification of the model defining the choice and the two concepts of the "zero" option, i.e. the refusal and the conflict, is plotted below: where δ represent the threshold defining the refusal, δ the threshold associated with the conflict and u h , respectively u l , the utilities associated with the products h, respectively to the products l.

This illustration provides the structure of the probabilities associated with each area described above and used in the log-likelihood to estimate the model parameters.

In our application to VoD, we have been confronted with separation and non convergence problems due to the small data size in the case of individual samples. The literature deals with these separation problems by using the penalized likelihood in the case of the dichotomous logit model, later extend to multinomial logit models [START_REF] Bull | A modified score function estimator for multinomial logistic regression in small samples[END_REF]. Our specification related to a multinomial probit model addresses this issue by deriving the individual estimates from a bayesian approach, while the aggregate estimates were obtained from likelihood maximisation. In fact, the latter is also used as an a priori information in the individual estimation with the bayesian approach. The use of this method is justified in that the results better match the reality from a statistical and economic point of view than the penalized likelihood maximisation.

Conclusion

Our approach brings therefore additional accuracy to the estimation of the results and to the calculations of the market shares. Additionally, it offers the perspective to better understand the consumer behavior in the decision making process.

Figure 1 :

 1 Figure 1: Representation of the choice of the products h, l and the "zero" option


Perla Kfoury

Stefan Le Coz

Tai-Peng Tsai

ANALYSIS OF STABILITY AND INSTABILITY FOR STANDING WAVES OF THE DOUBLE POWER ONE DIMENSIONAL NONLINEAR SCHR ÖDINGER EQUATION

Keywords: Mathematics Subject Classification. 35Q55 (35B35) nonlinear Schrödinger equation, double power nonlinearity, standing waves, stability, orbital stability

For the double power one dimensional nonlinear Schrödinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.

Introduction

Consider the one dimensional nonlinear Schrödinger equation with double power nonlinearity i∂ t u + ∂ 2

x u + a p |u| p-1 u + a q |u| q-1 u = 0,

where u : R t × R x → C, a p , a q ∈ R \ {0} and 1 < p < q < ∞. When a p < 0, a q > 0, we say that the nonlinearity is defocusing-focusing, with analogous definitions for other possible signs combinations. Nonlinear Schrödinger equations appear in many areas of physics such as nonlinear optics (see e.g. [START_REF]Nonlinear fiber optics[END_REF]) or Bose-Einstein condensation. The double power nonlinearity is an important example of the possible nonlinearities appearing in soliton theory (see e.g. [START_REF] Akhmediev | Hamiltonian-versus-energy diagrams in soliton theory[END_REF]). Via gauge transformations, the double power nonlinearity is also connected with the derivative nonlinear Schrödinger equation (see e.g. [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF][START_REF] Coz | Stability of Multisolitons for the Derivative Nonlinear Schrödinger Equation[END_REF][START_REF] Van Tin | On the derivative nonlinear Schrödinger equation on the half line with Robin boundary condition[END_REF]). The double power nonlinearity is also a typical example of a nonlinearity breaking the scaling invariance of the pure power case, while still being relatively tractable, and it may be used to study phenomena in the absence of scaling symmetry (see e.g. [START_REF] Le Coz | Minimal mass blow up solutions for a double power nonlinear Schrödinger equation[END_REF] for the construction of blowing-up solutions).

The Cauchy problem for (1) is well known (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and the references therein) to be well-posed in the energy space H 1 (R): for any u 0 ∈ H 1 (R), there exists a unique maximal solution u ∈ C((-T * , T * ), H 1 (R)) ∩ C 1 ((-T * , T * ), H -1 (R)) of [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] such that u(t = 0) = u 0 . Moreover, the energy E and the mass M , defined by

E(u) = 1 2 u x 2 L 2 - a p p + 1 u p+1 L p+1 - a q q + 1 u q+1 L q+1 , M (u) = 1 2 u 2 L 2 ,
are conserved along the flow and the blow-up alternative holds (i.e. if T * < ∞ (resp. T * < ∞), then lim t→T * (resp -T * ) u(t) H 1 = ∞).

A standing wave is a solution of (1) of the form u(t, x) = e iωt φ(x) for some ω ∈ R and a profile φ ∈ C 2 (R), which then satisfies φ + ωφa p |φ| p-1 φa q |φ| q-1 φ = 0.

(

) 2
We only consider real-valued φ in this paper. Define ω * by ω * = sup ω 0 : ∃s > 0 such that ω 2 s 2 -a p p + 1 s p+1 -a q q + 1 s q+1 < 0 .

It is well known (see [START_REF] Berestycki | Nonlinear scalar field equations. I. Existence of a ground state[END_REF]) that existence of non-trivial solutions of (2) with lim |x|→∞ φ(x) = 0 holds if and only if 0 ω < ω * when a p < 0, a q > 0, 0 < ω < ω * otherwise.

In that case, the solution is positive (up to phase shift), even (up to translation) and unique. We denote it by φ ω , or simply φ when there is no ambiguity. Solitary waves are the building blocks for the nonlinear dynamics of (1), as it is expected that, generically, a solution of (1) will decompose into a dispersive linear part and a combination of nonlinear structures as solitary waves. This vague statement is usually referred to as the Soliton Resolution Conjecture.

Therefore, understanding the dynamical properties of standing waves, in particular their stability, is a key step in the analysis of the dynamics of [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. Several stability concepts are available for standing waves. The most commonly used is orbital stability, which is defined as follows. The standing wave e iωt φ(x) solution of ( 1) is said to be orbitally stable if the following holds. For any ε > 0 there exists δ > 0 such that if u 0 ∈ H 1 (R) verifies u 0φ H 1 < δ, then the associated solution u of (1) exists globally and verifies sup t∈R inf y∈R,θ∈R u(t)e iθ φ(•y) H 1 < ε.

In the rest of this paper, when we talk about stability/instability, we always mean orbital stability/instability.

The groundwork for orbital stability studies was laid down by Berestycki and Cazenave [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF], Cazenave and Lions [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] and Weinstein [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF]. Two approaches lead to stability or instability results: the variational approach of [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF], which exploits global variational characterizations combined with conservation laws or the virial identity, and the spectral approach of [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF], which exploits spectral and coercivity properties of linearized operators to construct a suitable Lyapunov functional. Later on, Grillakis, Shatah and Strauss [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] developed an abstract theory which, under certain assumptions, boils down the stability study of a branch of standing waves ω → φ ω to the study of the sign of the quantity (usually called slope) ∂ ∂ω M (φ ω ). Note that the theory of Grillakis, Shatah and Strauss has known recently a considerable revamping in the works of De Bièvre, Genoud and Rota-Nodari [START_REF] De Bièvre | Orbital stability: analysis meets geometry[END_REF][START_REF] De Bièvre | Orbital stability via the energy-momentum method: The case of higher dimensional symmetry groups[END_REF].

With the above mentioned techniques, the orbital stability of positive standing waves has been completely determined in the single power case (i.e. a q = 0) in any dimension d 1 in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF][START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF][START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF]. If a q = 0, positive standing waves exist if and only if a p > 0 and ω > 0. In this case, they are stable if 1 < p < 1 + 4 d (i.e. 1 < p < 5 in dimension d = 1), and they are unstable if 1 + 4

d p < 1 + 4 (d-2)+ (i.e. 5 p < ∞ in dimension d = 1
). Scaling properties of the single power nonlinearity play an important role in the proof and ensure in particular that stability and instability are independent of the value of the frequency ω. It turns out that there is no scaling invariance for double power nonlinearities, which makes the stability study more delicate. As a matter of fact, only very partial results are available so far in higher dimensions. In dimension 1, the situation is a bit more favorable, as one might exploit the ODE structure of the profile equation ( 2) in the analysis.

Preliminary investigations for the stability of standing waves in dimension 1 were conducted by Iliev and Kirchev [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF] in the case of a generic nonlinearity. In particular, a formula for the slope condition was obtained in [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF]. The earliest work devoted to the stability of standing waves for nonlinear Schrödinger equations with double power nonlinearity in dimension 1 is the work of Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. In this work, using the integral expression for the slope condition derived by Iliev and Kirchev [START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF], Ohta established the stability/instability of standing waves in a number of cases. Later on, Maeda [START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF] further refined the approach of Ohta and established the stability/instability in most of the situations not covered in [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. However, the stability picture was still not complete, as the following case was left partially open: a p < 0, a q > 0, 1 < p < q < 5, p + q 6 or p 7 3 .

In the above case, Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] established the stability of standing waves for ω large enough. The instability for small ω was obtained by Ohta [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] for p + q > 6, a condition which was later improved to (p + 3)(q + 3) > 32 by Fukaya and Hayashi [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF]. What happens in the intermediate range of ω when (p + 3)(q + 3) > 32 and p + q 6 or p 7 3 ,

was not elucidated in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF][START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF], nor what happens for small ω when (p + 3)(q + 3) 32, (except for the notable case p = 2, q = 3, where explicit calculations are possible and show that the wave is stable for any ω > 0). For convenience, we adopt the following convention. When a standing wave is stable for any ω ∈ (0, ω * ), we say that it is of type S. When there exists ω 1 ∈ (0, ω * ) such that the standing wave is unstable for ω ∈ (0, ω 1 ] and stable for ω ∈ (ω 1 , ω * ), we say that it is of type US. Other types are defined in a similar manner. Note that when instability holds the endpoint ω 1 is included in the instability range (thanks to the criterion of Comech and Pelinovsky [START_REF] Comech | Purely nonlinear instability of standing waves with minimal energy[END_REF], see ( 6)).

Our goal in this paper is to fill out the gaps left open by the previous works [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF][START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF] and to provide a complete stability picture for the standing waves of the Schrödinger equations with double power nonlinearity. Our main result is the following. Theorem 1.1. Let (φ ω ) ω∈(0,ω * ) be the family of standing waves of (1). The following gives the stability type of the family of standing waves.

(1) Assume that a p > 0 and a q > 0.

(a) If q 5, then it is of type S.

(b) If p 5, then it is of type U . (c) If p < 5 < q, then it is of type SU . (2) Assume that a p > 0 and a q < 0.

(a) If p 5, then it is of type S.

(b) If p > 5, then it is of type U S.

(3) Assume that a p < 0 and a q > 0.

(a) If q 7 -2p, then it is of type S.

(b) If 7 -2p < q < 5, then it is of type U S.

(c) If q 5, then it is of type U . In particular, in the cases 1(c), 2(b) and 3(b) with stability change, the standing wave φ ωc at the critical frequency ω c is unstable. This theorem implies in particular that stability change occurs at most once, which is conjectured in [30, p. 265], and is in contrast to NLS with triple power nonlinearity considered in [START_REF] Liu | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF].

In Theorem 1.1, cases (1), ( 2) and (3)(c) were already covered in [START_REF] Maeda | Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity[END_REF][START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. For the sake of completeness, and as the proofs are not very long, we will also cover them in our work. Cases (3)(a) and (b) were only partially solved. We provide a definitive result for these cases. Our approach relies on several ingredients. First of all, we express the slope condition in a concise, while easily tractable integral, factoring out terms which are in any case positive. Instead of working with the parameter ω, we manipulate the slope condition with the parameter φ 0 = φ ω (0) (which is in a bijective relation with ω). We are left with an integral expression F (φ 0 ) (see [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]), of which we need to determine the sign. A refactorisation allows us to introduce an auxiliary parameter γ, and differentiation with respect to φ 0 gives us an expression which we can prove to have sign, provided we have suitably chosen the parameter γ. This gives the information that F (φ 0 ) changes sign at most once. The sign for large ω (or equivalently large φ 0 ) had already been established in [START_REF] Ohta | Instability of standing waves for the generalized Davey-Stewartson system[END_REF]. On the other hand, the sign for ω close to 0 had not been computed before. Here, an astute rewriting of the slope in terms of Beta functions allows us to determine the sign for ω close to 0.

Observe that our results are not covering the zero-frequency case ω = 0. Stability or instability of the corresponding (algebraic) standing waves (when existing) can be conjectured to be the same as the one for small ω > 0 (which is consistent with the results obtained by Fukaya and Hayashi [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF]).

In the case a p < 0, a q > 0 and 1 < p < q < 5, we complement our theoretical results with numerical experiments. We first represent the critical surface {ω c (p, q)} at which the stability change occurs and discuss the different possible shapes of the surface depending on the ratio a p /a q . We then simulate the dynamics of (1) around a standing wave with the Crank-Nicolson scheme with relaxation of Besse [START_REF] Besse | A relaxation scheme for the nonlinear Schrödinger equation[END_REF]. Three types of behaviors are observed depending on the type of initial data : stability, growth followed by oscillations, and scattering.

To end this introduction, we point out that many works are devoted to standing waves of the double power nonlinear Schrödinger in higher dimension (for which our approach does not apply), and just give a small sample of the existing literature. The cubic quintic case in higher dimension was investigated in [START_REF] Carles | On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF]. Stability of standing waves in higher dimension for generic nonlinearities was considered in [START_REF] Fukuizumi | Stability and instability of standing waves for nonlinear Schrödinger equations[END_REF]. Strong instability was studied in [START_REF] Ohta | Strong instability of standing waves for nonlinear Schrödinger equations with double power nonlinearity[END_REF]. Stability results for algebraic standing waves were obtained in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF]. Uniqueness and nondegeneracy was considered in [START_REF] Lewin | The double-power nonlinear schrödinger equation and its generalizations: uniqueness, nondegeneracy and applications[END_REF]. Existence or non-existence of minimizers of the energy at fixed mass was obtained in [START_REF] Bellazzini | Ground state energy threshold and blow-up for nls with competing nonlinearities[END_REF]. Let us also mention in dimension 1 the work [START_REF] Genoud | Stable NLS solitons in a cubic-quintic medium with a delta-function potential[END_REF], which is devoted to the stability of standing waves for cubic-quintic nonlinearities in the presence of a δ potential (see [START_REF] Angulo Pava | On stability properties of the cubic-quintic Schrödinger equation with δ-point interaction[END_REF][START_REF] Angulo Pava | Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity[END_REF] for further developments).

This paper is organized as follows. We start by some preliminaries in Section 2, recalling in particular the properties of the standing wave profiles and the stability criterion. In Section 3, we reformulate the slope condition for stability, using the profile equation. In Section 4, we analyze the limit of the slope at the endpoints of the interval of admissible frequencies ω and in particular determine the sign of the slope at the endpoints. The sign of the slope on the full interval of admissible frequencies is recovered in Section 5, which shows Theorem 1.1. Finally, numerical experiments are presented in Section 6.

After the first version of this paper was posted to arXiv, Professor Hayashi kindly informed us he had an independent similar result and posted it as [START_REF] Hayashi | Sharp thresholds for stability and instability of standing waves in a double power nonlinear schrödinger equation[END_REF]. His Theorem 1.3 is similar to our Theorem 1.1 although it does not include the case 1 < p < 9/5.

Preliminaries

2.1. The profile equation. We start by some analysis around the ordinary differential equation ( 2) and its solutions (φ ω ). Apart in a few specific cases (e.g. when q = 2p -1, see e.g. [START_REF] Liu | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF]), there does not exist an explicit formula for the full standing waves profile. Note that ω * = ∞ when a q > 0, 0 < ω * < ∞ when a p > 0 and a q < 0, and ω * = -∞ (i.e. there is no solution of ( 2) in H 1 (R)) when a p , a q < 0. All along this paper, we assume that 0 < ω < ω * (excluding in particular the possibility that a p , a q < 0). Under this assumption, there exists φ 0 > 0 (depending implicitly on ω) such that

φ 0 = inf{φ > 0 : ω 2 φ 2 - a p p + 1 |φ| p+1 - a q q + 1 |φ| q+1 = 0},
and we have

φ ω (0) = φ 0 .
Observe that ω may be expressed in terms of φ 0 as follows

ω = 2a p p + 1 φ p-1 0 + 2a q q + 1 φ q-1 0 . (3)
Moreover, as ω < ω * , we have

ω -a p φ p-1 0 -a q φ q-1 0 < 0.
This implies in particular φ 0 is a C 1 -function of ω. Moreover, we always have

∂φ 0 ∂ω = 2a p (p -1) p + 1 φ p-2 0 + 2a q (q -1) q + 1 φ q-2 0 -1 > 0. ( 4
)
As a consequence, the following result holds.

Lemma 2.1. The function ω → φ 0 is a strictly increasing bijection from (0, ω * ) to (φ * , φ * ) where

φ * =    - ap aq q+1 p+1 1 q-p if a p < 0, 0 if a p > 0, φ * =    ∞ if a q > 0, - ap aq p-1 q-1 q+1 p+1 1 q-p if a q < 0. (5)
2.2. The stability criterion. As we already mentioned, stability criteria have been derived in the general case in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF][START_REF] Iliev | Stability and instability of solitary waves for one-dimensional singular Schrödinger equations[END_REF]. For the double power nonlinearity, the stability of the standing wave is determined by a slope condition (the spectral condition of [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF] being always verified in this case when ω > 0). The standing wave e iωt φ ω (x) will be stable if

∂ ∂ω M (φ ω ) > 0,
and it will be unstable if

∂ ∂ω M (φ ω ) < 0.
When ∂ ω M (φ ω ) = 0, the stability can be decided by looking at the second derivative, as was established by Comech and Pelinovsky [START_REF] Comech | Purely nonlinear instability of standing waves with minimal energy[END_REF]:

If ∂ ω M (φ ω ) = 0 and ∂ 2 ∂ω 2 M (φ ω ) = 0, (6)
then the standing wave e iωt φ ω (x) is unstable.

Reformulation of the slope

For notational convenience, we introduce the function J defined by

J(ω, p, q) = ∂ ∂ω M (φ ω ).
Hence the sign of J determines the stability of the corresponding standing wave.

The main idea in this section is to express J in terms of φ 0 instead of ω. Before doing that, we introduce some convenient notation. Let Φ p and Φ q be defined by

Φ p = 2a p p + 1 φ p+1 0 (1 -s p-1 ), Φ q = 2a q q + 1 φ q+1 0 (1 -s q-1 ), (7)
where a p , a q = 0, 1 < p < q < ∞ and 0 < s < 1.

Lemma 3.1. The function J may be expressed in terms of φ 0 as follows

J(ω, p, q) = C(φ 0 )F (φ 0 ),
where

F (φ 0 ) = 1 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 sds, (8)
and C(φ 0 ) is positive and explicitly known (see [START_REF] Besse | A relaxation scheme for the nonlinear Schrödinger equation[END_REF]).

Proof. We multiply the equation ( 2) of the profile by φ x and we integrate to obtain

- 1 2 |φ x | 2 + ω 2 |φ| 2 - a p p + 1 |φ| p+1 - a q q + 1 |φ| q+1 = c.
When x → ∞, we know that φ(x) → 0 and φ (x) → 0. Therefore c = 0, and

- 1 2 |φ x | 2 + ω 2 |φ| 2 - a p p + 1 |φ| p+1 - a q q + 1 |φ| q+1 = 0. (9)
For x > 0, as φ is decreasing, from (9) we have

φ x = -ωφ 2 - 2a p p + 1 φ p+1 - 2a q q + 1 φ q+1 .
Still for x > 0, let t = φ(x), then

dx = dt φ x = - dt ωφ 2 - 2ap p+1 φ p+1 - 2aq q+1 φ q+1
.

Therefore we may perform the following change of variable:

M (φ) = 1 2 R |φ(x)| 2 dx = ∞ 0 |φ(x)| 2 dx = φ0 0 t 2 ωt 2 - 2ap p+1 t p+1 - 2aq q+1 t q+1 dt.
Changing again variable by setting t = φ 0 s, we have

M (φ) = 1 0 φ 3 0 s 2 s ωφ 2 0 - 2ap p+1 φ p+1 0 s p-1 - 2aq q+1 φ q+1 0 s q-1
ds.

Replacing ω by its value (3) in terms of φ 0 , we have

M (φ) = 1 0 φ 3 0 s 2ap p+1 φ p+1 0 + 2aq q+1 φ q+1 0 - 2ap p+1 φ p+1 0 s p-1 - 2aq q+1 φ q+1 0 s q-1
ds, which, using the notation (7) for Φ p and Φ q , gives

M (φ) = 1 0 φ 3 0 s Φ p + Φ q ds.
Differentiating with respect to ω, we have

∂ ω Φ p = (p + 1)Φ p φ -1 0 ∂ ω φ 0 , ∂ ω Φ q = (q + 1)Φ q φ -1 0 ∂ ω φ 0 .
Therefore we obtain

J(ω, p, q) = ∂ ω M (φ) = 1 0 3φ 2 0 ∂ ω φ 0 s(Φ p + Φ q ) -1 2 φ 3 0 s (p + 1)Φ p φ -1 0 + (q + 1)Φ q φ -1 0 ) ∂ ω φ 0 (Φ p + Φ q ) 3 2 ds, = ∂ ω φ 0 2 φ 2 0 1 0 6(Φ p + Φ q ) -((p + 1)Φ p + (q + 1)Φ q )) (Φ p + Φ q ) 3 2 sds, = ∂ ω φ 0 2 φ 2 0 1 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 sds, = C(φ 0 )F (φ 0 ),
where F (φ 0 ) is defined in [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF] and

C(φ 0 ) = ∂ ω φ 0 2 φ 2 0 . (10)
This concludes the proof.

We will now analyze the variations of J(ω, p, q) in terms of φ 0 . For future convenience (the reason for such a choice will appear clearly later), we introduce an auxiliary parameter γ in the following way

J(ω, p, q) = C(φ 0 )φ -γ 0 F γ (φ 0 )
, where

F γ (φ 0 ) = 1 0 φ γ 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 sds.
Denote the integrand of F γ by

I γ (φ 0 ) = φ γ 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 . ( 11
)
Observe that there is a implicit dependency in s. In the following lemma we differentiate I γ (φ 0 ) with respect to φ 0 .

Lemma 3.2. For any 0 < s < 1, the following holds:

∂I γ ∂ φ0 = 1 2 φ γ-1 0 ((5 -p)(2γ -(p + 1))Φ p + (5 -q)(2γ -(q + 1))Φ q ) (Φ p + Φ q ) -3(q -p) 2 Φ p Φ q (Φ p + Φ q ) 5 2
.

Proof. We start by differentiating the term in parenthesis in I γ (φ 0 ). We have

∂ φ0 Φ p = (p + 1)Φ p φ -1 0 , ∂ φ0 Φ q = (q + 1)Φ q φ -1 0 . Therefore, we have ∂ φ0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 , = φ -1 0 ((5 -p)(p + 1)Φ p + (5 -q)(q + 1)Φ q ) (Φ p + Φ q ) -3 2 ((5 -p)Φ p + (5 -q)Φ q ) ((p + 1)Φ p + (q + 1)Φ q ) (Φ p + Φ q ) 5 2 , = 1 2 φ -1 0 -(5 -p)(p + 1)Φ 2 p -(5 -q)(q + 1)Φ 2 q + ((5 -p)(2p -3q -1) + (5 -q)(2q -3p -1))Φ p Φ q (Φ p + Φ q ) 5 2 , = 1 2 φ -1 0 -((5 -p)(p + 1)Φ p + (5 -q)(q + 1)Φ q ) (Φ p + Φ q ) -3(q -p) 2 Φ p Φ q (Φ p + Φ q ) 5 2
.

Before going on, observe that we may rewrite the term in parentheses in I γ (φ 0 ) as

(5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 = 2 ((5 -p)Φ p + (5 -q)Φ q ) (Φ p + Φ q ) 2 (Φ p + Φ q ) 5 2
.

Finally, the full derivative of I γ (φ 0 ) is given by

∂I γ ∂φ 0 = ∂ φ0 φ γ 0 (5 -p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 3 2 , = 1 2 φ γ-1 0 ((5 -p)(2γ -(p + 1))Φ p + (5 -q)(2γ -(q + 1))Φ q ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2
.

This concludes the proof.

For future reference, we establish here the following technical lemma which we will use at several occasions.

Lemma 3.3. The function s → 1-s q-1 1-s p-1 is an increasing bijection from (0, 1) to (1, q-1 p-1 ). Proof. Let h(s) = 1-s q-1 1-s p-1 . We have h (s) = s p-2 (1 -s p-1 ) 2 l(s)
, where l(s) = (qp)s q-1 + p -1 -(q -1)s q-p . Note that l(1) = 0 and for 0 < s < 1, l (s) = (qp)(q -1)(s q-2s q-p-1 ) < 0. Hence l (s) < 0 and l(s) > 0 for 0 < s < 1. We conclude that h (s) > 0 for 0 < s < 1. As a consequence, h is increasing on the interval (0, 1). Moreover, we have h(0) = 1 and, by L'Hospital's rule,

lim s→1 h(s) = q -1 p -1 .
This concludes the proof.

The slope at the endpoints

Our goal in this section is to investigate what happens for J(ω, p, q) when ω is close to 0 and ω * .

4.1. The zero frequency case. In this section, we determine the limit of J(ω, p, q) when ω tends to zero. Let J 0 be defined by J 0 (p, q) = lim ω→0 J(ω, p, q).

We first consider the case where a p > 0.

Proposition 4.1. Let a p > 0. The following holds.

(1) If 1 < p < 7 3 , then J 0 (p, q) = 0 + . (2) If p = 7 3 , then 0 < J 0 (p, q) < ∞.

(3) If 7 3 < p < 5, then J 0 (p, q) = ∞. (4) If p = 5, then three cases have to be distinguished.

(a) If q < 9, then J 0 (p, q) =sign(a q )∞.

(b) If q = 9, then 0 <sign(a q )J 0 (p, q) < ∞.

(c) If q > 9, then J 0 (p, q) = 0 -sign(aq) . (5) If 5 < p, then J 0 (p, q) = -∞.

Proof. When a p > 0, we have lim

ω→0 φ 0 = φ * = 0.
Recall that we have shown in Lemma 3.1 that J may be written as J(ω, p, q) = C(φ 0 )F (φ 0 ). We have (recalling the definition (10) of C and the expression (4) of ∂ ω φ 0 )

C(φ 0 ) = 1 2 ∂ ω φ 0 φ 2 0 = 1 4 a p (p -1) p + 1 φ p-4 0 + a q (q -1) q + 1 φ q-4 0 -1 = φ 4-p 0 1 4 a p (p -1) p + 1 + a q (q -1) q + 1 φ q-p 0 -1 = φ 4-p 0 p + 1 4a p (p -1)
+ o(1) .

The function F (defined in ( 8)) can be written, substituting Φ p and Φ q by their expressions [START_REF] Bellazzini | Ground state energy threshold and blow-up for nls with competing nonlinearities[END_REF], as

F (φ 0 ) = 1 0 2ap(5-p) p+1 (1 -s p-1 )φ p+1 0 + 2aq(5-q) q+1 (1 -s q-1 )φ q+1 0 2ap p+1 (1 -s p-1 )φ p+1 0 + 2aq q+1 (1 -s q-1 )φ q+1 0 3 2
sds.

As we are interested in the limit φ 0 → 0, we factor out the terms in φ p+1 0 to get

F (φ 0 ) = φ -p+1 2 0 1 0 2ap(5-p) p+1 (1 -s p-1 ) + 2aq(5-q) q+1 (1 -s q-1 )φ q-p 0 2ap p+1 (1 -s p-1 ) + 2aq q+1 (1 -s q-1 )φ q-p 0 3 2 sds = (5 -p)φ -p+1 2 0 1 0 2a p p + 1 (1 -s p-1 ) -1 2 sds + o(1) .
In the particular case p = 5, we instead write

F (φ 0 ) = a q (5 -q)φ q-8 0   1 0 2 q+1 (1 -s q-1 ) ap 3 (1 -s 4 ) 3 2 sds + o(1)   .
In summary, when φ 0 → 0 (i.e. ω → 0), we have established that there exists C = C(p, q) > 0 such that when p = 5 we have

J(ω, p, q) = (5 -p)φ 7-3p 2 0 C(1 + o(1)),
and when p = 5 we have J(ω, p, q) = (5q)a q φ q-9 0

C(1 + o(1)).
This gives the desired result.

We now discuss the case a p < 0 and a q > 0.

Proposition 4.2. Let a p < 0 and a q > 0.

(1) Assume that p < 7 3 . Then J 0 (p, q) ∈ R and the following holds. (a) If 2p + q < 7, then J 0 (p, q) > 0. (b) If 2p + q = 7, then J 0 (p, q) = 0. (c) If 2p + q > 7, then J 0 (p, q) < 0.

(2) Assume that p 7 3 . Then J 0 (p, q) = -∞. We start with some preliminaries. To establish the first part of Proposition 4.2, we will calculate J 0 in terms of the Beta function. Recall that the Beta function, also called Euler integral of the first kind, is a special function closely related to the Gamma function. It is defined for x > 0 and y > 0 by the integral

B(x, y) = 1 0 t x-1 (1 -t) y-1 dt. ( 12
)
The relation between the Beta function and the Gamma function is given by (see e.g [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF])

B(x, y) = Γ(x)Γ(y) Γ(x + y) .
We introduce the function H defined for x > 0 and y > 0 by

H(x, y) = 1 0 t x-1 (1 -t y ) (1 -t) 3 2 dt. (13)
The relation between H and B is given in the following lemma.

Lemma 4.3. For x > 0 and y > 0, we have

H(x, y) = -(2x -1)B x, 1 2 + (2x + 2y -1)B x + y, 1 2 . ( 14
)
Proof. Let u(t) = t x-1 (1t y ).

Rewrite 1

(1t)

3 2 = v (t) - 1 (1 -t) 1 2
, where

v(t) = 2 (1 -t) 1 2 -2(1 -t) 1 2 = 2t (1 -t) 1 2
.

We have

H(x, y) = 1 0 u(t) v (t) - 1 (1 -t) 1 2 dt, =u(1 -)v(1 -) -u(0 + )v(0 + ) - 1 0 u (t)v(t)dt - 1 0 u(t) (1 -t) 1 2 dt, =0 - 1 0 2tu (t) + u(t) (1 -t) 1 2
dt.

Above we have used u(1) = 0 of order 1 to cancel the singularity of v(1 -) of order 1 2 , and v(0) = 0 with order 1 to cancel the singularity of u(0 + ) of order x -1. Note that

2tu (t) + u(t) = (2x -1)t x-1 -(2x + 2y -1)t x+y-1 .
Therefore, using the definition of B given in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] with y = 1 2 , we have

H(x, y) = -(2x -1)B x, 1 2 + (2x + 2y -1)B x + y, 1 2 .
This concludes the proof.

The value J 0 (p, q) may be expressed using B as follows.

Lemma 4.4. Let a p < 0 and a q > 0. Assume that 1 < p < 7/3. Then

J 0 (p, q) = (7 -2p -q)C 0 B 7 -3p 2(q -p) , 1 2 ,
where C 0 is a positive constant explicitly known (given by ( 16)).

The first part of Proposition 4.2 is a direct consequence of Lemma 4.4.

Proof of Lemma 4.4. Let 1 < p < 7/3. Recall that J(ω, p, q) = C(φ 0 )F (φ 0 ), with C(φ 0 ) > 0 and F given by [START_REF] Berestycki | Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires[END_REF]. Observe that, using the value of φ * given in (5), we may introduce the constant

C * = 2a q q + 1 φ q+1 * = - 2a p (p + 1) φ p+1 * .
Using the definition (10) of C(φ 0 ) and the expression (4) of ∂ ω φ 0 , we have

lim φ0→φ * C(φ 0 ) = C(φ * ) = φ 5 * 2C * (q -p) > 0, lim φ0→φ * Φ p = 2a p (p + 1) φ p+1 * (1 -s p-1 ) = -C * (1 -s p-1 ), lim φ0→φ * Φ q = 2a q (q + 1) φ q+1 * (1 -s q-1 ) = C * (1 -s q-1 ).
As a consequence, we get lim φ0→φ *

(Φ p + Φ q ) = C * (s p-1 -s q-1 ), lim φ0→φ * ((5 -p)Φ p + (5 -q)Φ q ) = C * -(5 -p)(1 -s p-1 ) + (5 -q)(1 -s q-1 ) .
As a consequence,

F (φ * ) = C -1 2 * 1 0 -(5 -p)(1 -s p-1 ) + (5 -q)(1 -s q-1 ) (s p-1 -s q-1 ) 3 2 sds = C -1 2 * 1 0 -(q -p)(1 -s p-1 ) + (5 -q)(s p-1 -s q-1 ) (1 -s q-p ) 3 2 s 5-3p 2 ds = C -1 2 * -(q -p) 1 0 (1 -s p-1 )s 5-3p 2 (1 -s q-p ) 3 2 ds + (5 -q) 1 0 s 3-p 2 1 -s q-p -1 2 ds . (15)
Changing variable t = s q-p , we obtain

F (φ * ) = C -1 2 * - 1 0 (1 -t p-1 q-p )t 7-p-2q 2(q-p) (1 -t) 3 2 ds + 5 -q q -p 1 0 t 5+p-2q 2(q-p) (1 -t) - 1
2 ds .

We now use B and H to express the above quantity. Setting

(x 1 , y 1 ) = 7 -p -2q 2(q -p) + 1, p -1 q -p = 7 -3p 2(q -p) , p -1 q -p , (x 2 , y 2 ) = 5 + p -2q 2(q -p) + 1, 1 2 = 5 -p 2(q -p) , 1 2 , we get
F (φ * ) = C -1 2 * -H(x 1 , y 1 ) + 5 -q q -p B(x 2 , y 2 ) .
Observe that we have assumed p < 7 3 , p < q, which ensures that x 1 , x 2 , y 1 , y 2 are positive. This a posteriori justifies the fact that J 0 (p, q) is finite. The formula [START_REF] Comech | Purely nonlinear instability of standing waves with minimal energy[END_REF] allows us to express H(x 1 , y 1 ) in the following way (using y 2 = 1/2):

H(x 1 , y 1 ) = -(2x 1 -1)B (x 1 , y 2 ) + (2x 1 + 2y 1 -1)B (x 1 + y 1 , y 2 ) .

It turns out that

-(2x 1 -1) = - 7 -2p -q q -p , (2x 1 + 2y 1 -1) = 5 -q q -p , x 1 + y 1 = 5 -p 2(q -p) = x 2 .
As a consequence, there is a simplification in the expression of F (φ * ), which becomes

F (φ * ) = C -1 2 * 7 -2p -q q -p B (x 1 , y 2 ) .
Setting

C 0 = C -1 2 * q -p C(φ * ) > 0 ( 16
)
gives the desired result.

Lemma 4.5. Assume that a p < 0 and a q > 0. For p 7/3 and 1 < p < q, we have

lim ω→0 J(ω, p, q) = -∞.
The second part of Proposition 4.2 is a direct consequence of Lemma 4.5.

Proof. Coming back to the expression (15) of F (φ * ) in the proof of Lemma 4.4, we observe that if 5-3p 2 -1, i.e. p 7 3 , then F (φ * ) = -∞, and, since lim φ0→φ * C(φ 0 ) = C(φ * ) > 0, we also have J 0 (p, q) = -∞ when p 7 3 .

4.2.

The large frequency case. In this section, we determine the limit of J(ω, p, q) when ω tends to ω * . Let J * be defined by J * (p, q) = lim ω→ω * J(ω, p, q). We first consider the case where a q > 0.

Proposition 4.6. Let a q > 0. The following holds.

(1) If 1 < q < 7 3 , then J * (p, q) = 0 + . ( 2) If q = 7 3 , then 0 < J * (p, q) < ∞.

(3) If 7 3 < q < 5, then J * (p, q) = ∞. (4) If q = 5, then J * (p, q) = 0 sign(ap) . ( 5) If 5 < q, then J * (p, q) = -∞.

Proof. Since a q > 0, we have ω * = ∞ and therefore φ * = ∞. Following similar arguments as in the proof of Proposition 4.1, as φ 0 → ∞, for q = 5, we have

C(φ 0 ) = φ 4-q 0 q + 1 4a q (q -1) + o(1) , F (φ 0 ) = (5 -q)φ -q+1 2 0 1 0 2a q q + 1 (1 -s q-1 ) -1 2 sds + o(1) .
As a consequence, for q = 5, when φ 0 → ∞ (i.e. ω → ∞), there exists C = C(a q , q) > 0 such that

J(ω, p, q) = (5 -q)φ 7-3q 2 0 C(1 + o(1)).
In the particular case q = 5, we instead write

F (φ 0 ) = a p (5 -p)φ p-8 0   1 0 2 p+1 (1 -s p-1 ) aq 3 (1 -s 4 ) 3 2 sds + o(1)   .
and therefore we get

J(ω, p, q) = a p (5 -p)φ p-9 0 C(1 + o(1)).
The two estimates on J lead to the desired result.

Then we consider the case where a q < 0 (and thus a p > 0 to ensure existence of standing waves).

Proposition 4.7. Let a p > 0, a q < 0 and 5 p. Then

J * (p, q) = ∞.
Proposition 4.7 does not cover the whole possible range of p and q. As it was not necessary in our analysis, we did not try to cover the remaining cases.

Proof of Proposition 4.7. By construction, ω * = ω(φ * ) is the value of ω at which ∂ φ0 ω(φ * ) = 0. As a consequence, we have lim

φ0→φ * ∂φ 0 ∂ω = ∞,
which, given the value (10) of C(φ 0 ), readily implies

lim φ0→φ * C(φ 0 ) = ∞.
Using the expressions of (7) of Φ p and Φ q and the expression (8) of F we have

F (φ * ) = s 0 2a p (5-p)(φ * ) p+1 (1-s p-1 ) p+1 + 2a q (5-q)(φ * ) q+1 (1-s q-1 ) q+1 2a p (φ * ) p+1 (1-s p-1 ) p+1 + 2a q (φ * ) q+1 (1-s q-1 ) q+1 3 2

sds.

Proof. Using the formula (8) of F (φ 0 ) and replacing in the numerator of the integrand Φ p and Φ q by their expressions [START_REF] Bellazzini | Ground state energy threshold and blow-up for nls with competing nonlinearities[END_REF], we obtain

F (φ 0 ) = 1 0 (5 -p) 2ap p+1 φ p+1 0 (s -s p ) + (5 -q) 2aq q+1 φ q+1 0 (s -s q ) (Φ p + Φ q ) 3 2 ds. Let l(s) = (5 -p) 2a p p + 1 φ p+1 0 (s -s p ) + (5 -q) 2a q q + 1 φ q+1 0 (s -s q ), and
k(s) = (Φ p + Φ q ) 3 2
. We may reformulate l(s) in the following way:

l(s) = (5 -p) 2a p p + 1 φ p+1 0 + (5 -q) 2a q q + 1 φ q+1 0 1 -s q-1 1 -s p-1 (s -s p ).
From Lemma 3.3, we know that the function s → 1-s q-1 1-s p-1 is increasing from 1 to q-1 p-1 when s goes from 0 to 1. Let φ 0,0 be given by φ q-p 0,0 = -a p (5p)(q + 1)(p -1) a q (5q)(p + 1)(q -1) ,

and assume from now on that φ 0 > φ 0,0 . Then lim s→1 l(s) ss p < 0, and there exists s * ∈ [0, 1) such that l(s) > 0 for s ∈ (0, s * ) and l(s) < 0 for s ∈ (s * , 1).

Define k by k(s) = k(s) k(s * ) . Then k(s * ) = 1. As k and therefore k is a positive decreasing function of s, for all s ∈ (0, 1) we have l(s) k(s) < l(s).

Integrating over (0, 1), we obtain

F (φ 0 ) < 1 k(s * ) 1 0 l(s)ds,
and F (φ 0 ) will be negative if the integral in the right member is. Define φ 0,1 > φ 0,0 by

φ q-p 0,1 = - a p (5 -p)(p -1)(q + 1) 2 a q (5 -q)(q -1)(p + 1) 2 . ( 17
)
If φ 0 > φ 0,1 , then

1 0 l(s)ds = (5 -p) a p p + 1 φ p+1 0 p -1 p + 1 + (5 -q) a q q + 1 φ q+1 0 q -1 q + 1 < 0.
Hence for any φ 0 > φ 0,1 we have F (φ 0 ) < 0. This concludes the proof.

Lemma 5.4. Let a p > 0, a q > 0 and p < 5 < q. Let γ = p+1 2 . There exists φ 0,2 (explicitly given in [START_REF] Grillakis | Stability theory of solitary waves in the presence of symmetry[END_REF]) such that the integrand I γ of F γ defined in [START_REF] Carles | On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF] verifies ∂I γ ∂φ 0 < 0 for all φ 0 ∈ (0, φ 0,2 ).

Proof. As γ = p+1 2 , from Lemma 3.2 we have

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -q)(p -q)Φ q ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2 , = 1 2 φ γ-1 0 Φ q (p -q) (5 + 2q -3p)Φ p + (5 -q)Φ q (Φ p + Φ q ) 5 2
.

Proof. From Lemma 3.2 with γ = p+1 2 , we have

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -q)(p -q)Φ q ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2
.

Since 5q 0, pq < 0 and Φ q < 0, we have ∂Iγ ∂φ0 > 0 for any φ 0 ∈ (0, φ * ).

Lemma 5.9. Let a p > 0, a q < 0 and 5 < p < q. Let γ = pq + 3. Then the integrand I γ of F γ defined in (11) verifies ∂I γ ∂φ 0 > 0 for all φ 0 ∈ (0, φ * ).

Proof. Let γ = pq + 3.

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -p)(p -2q + 5)Φ p + (5 -q)(2p -3q + 5)Φ q ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2
.

The sign of ∂Iγ ∂φ0 is the same as the sign of the numerator of the fraction. Factoring out Φ 2 p , the sign is the same as the one of the second order polynomial in Φq Φp given by

(5 -q)(2p -3q + 5) Φ q Φ p 2 + 2(5 -p)(2p -3q + 5) Φ q Φ p + (5 -p)(p -2q + 5).
As 2p -3q + 5 < 0 and 5q < 0, the coefficient of the term of order 2 is positive. Therefore to show that the polynomial is positive, it is sufficient to show that the discriminant ∆, given by ∆ = 4(5p)(2p -3q + 5)((5p)(2p -3q + 5) -(5q)(p -2q + 5)), = -8(5p)(2p -3q + 5)(pq) 2 , is negative. We have 2p -3q + 5 < 0 and 5p < 0, therefore ∆ < 0. This concludes the proof.

Lemma 5.10. Let a p > 0, a q < 0.

• Let p < q 5. Then for any ω ∈ (0, ω * ), we have J(ω, p, q) > 0, and the family of standing waves is of type S. • Let 5 < p < q. Then there exist ω 1 ∈ (0, ∞) such that J(ω, p, q) < 0 for ω < ω 1 , J(ω 1 , p, q) = 0, J(ω, p, q) > 0 for ω > ω 1 , and the family of standing waves is of type US.

Proof. In both cases, we infer from Lemmas 5.8 and 5.9 that for any ω ∈ (0, ω * ), the function ω → J(ω, p, q) changes sign (from negative to positive) at most once on ω ∈ (0, ω * ).

To establish the desired conclusion, we consider the values of J close to the endpoints. As ω → 0, we have established in Proposition 4.1 that for ω close to 0, we have J(ω, p, q) > 0 for p 5, J(ω, p, q) < 0 for p > 5.

As J is increasing, this gives the conclusion for the first part of the Lemma.

For the second part of the Lemma, we look at the limit ω → ω * (i.e. φ 0 → φ * ). From Proposition 4.7, for 5 < p < q and for ω close to ω * we have J(ω, p, q) > 0, which gives the second part of the Lemma. 5.3. The defocusing-focusing case. In this section, we consider the case a p < 0, a q > 0. In this case Φ p < 0 and Φ q > 0.

Lemma 5.11. Let a p < 0, a q > 0 and p < q < 5. Let γ = q+1

2 . If 3q 2p + 5, then the integrand I γ of F γ defined in [START_REF] Carles | On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF] verifies

∂I γ ∂φ 0 > 0 for all φ 0 ∈ (φ * , ∞).
Proof. As γ = q+1 2 , from Lemma 3.2 we have

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -p)(q -p)Φ p ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2 , = 1 2 φ γ-1 0 Φ p (q -p) (5 -p)Φ p + (5 -p)Φ q + 3(p -q)Φ q (Φ p + Φ q ) 5 2 , = 1 2 φ γ-1 0 Φ p (q -p) (5 -p)Φ p + (5 + 2p -3q)Φ q (Φ p + Φ q ) 5 2
.

As 0 < 5p and 2p + 5 -3q 0 we have

(5 -p)Φ p + (5 + 2p -3q)Φ q < 0.
As a consequence ∂Iγ φ0 > 0 for all φ 0 ∈ (φ * , ∞) when 5 + 2p -3q 0, which is the desired conclusion. Lemma 5.12. Let a p < 0, a q > 0 and p < q < 5. Let γ = pq + 3. If 3q < 2p + 5 then the integrand I γ of F γ defined in [START_REF] Carles | On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF] verifies ∂I γ ∂φ 0 > 0 for all φ 0 ∈ (φ * , ∞).

Proof. As γ = pq + 3, from Lemma 3.2 we have

∂I γ ∂φ 0 = 1 2 φ γ-1 0 ((5 -p)(p -2q + 5)Φ p + (5 -q)(2p -3q + 5)Φ q ) (Φ p + Φ q ) -3(p -q) 2 Φ p Φ q (Φ p + Φ q ) 5 2
.

If the numerator of the fraction is positive then the derivative is positive. Factorizing out Φ 2 p , the sign of the numerator is the same as the one of the quadratic polynomial in Φq Φp given by (5q)(2p -3q + 5)

Φ q Φ p 2 + 2(5 -p)(2p -3q + 5) Φ q Φ p + (5 -p)(p -2q + 5).
As 2p -3q + 5 > 0 and 5q > 0, the coefficient of the term of order 2 is positive. Therefore to show that the polynomial is positive, it is sufficient to show that the discriminant ∆, given by ∆ = 4(5p)(2p -3q + 5)((5p)(2p -3q + 5) -(5q)(p -2q + 5)), = -8(5p)(2p -3q + 5)(pq) 2 , is negative. We have 2p -3q + 5 > 0 and 5p > 0, therefore ∆ < 0. This concludes the proof.

Lemma 5.13. Let a p < 0, a q > 0 and p < q < 5.

• If q 7 -2p, then for any ω ∈ (0, ∞), we have J(ω, p, q) > 0, and the family of standing waves is of type S. • If q > 7 -2p, then there exist ω 1 ∈ (0, ∞) such that J(ω, p, q) < 0 for ω < ω 1 , J(ω 1 , p, q) = 0, J(ω, p, q) > 0 for ω > ω 1 , and the family of standing waves is of type US.

We have represented the critical surface {(p, q, ω c (p, q))} for a p = -1 and three different values of a q = 1/2, 1, 2 in Figure 1. Critical surface {(p, q, ω c (p, q))} for a p = -1 and a q = 1/2, 1, 2. The white lines represent q = 7 -2p and q = p, where the transition from ω c = 0 to ω c > 0 occurs.

Several observations can be made on the critical surface. As (p, q) approaches the line q = 5, we have ω c (p, q) → ∞, which is consistent with the fact that standing waves are all unstable on this line.

It can be observed that on the line q = 7 -2p the transition is continuous, no matter the value of a q . To the contrary, the transition is continuous on the line p = q when a q 1, whereas it becomes discontinuous when a q < 1, in which case ω c (p, q) → ∞ as q → p.

To investigate more the transition close to the lines q = 7 -2p and q = p, we plot slices of the critical surface for a fixed value of q in Figure 2. We chose to present the results when q = 4, but similar results are obtained with other values of q. On Figure 2, we observe that when a q = |a p | = 1, the transition between ω c (p, q) = 0 and ω c (p, q) > 0 at q = p and q = 7 -2p is Lipschitz. When a q = 2 > |a p | = 1, the transition seems smoother (but closer observations will reveal otherwise) when q = p, whereas it remains Lipchitz when q = 7 -2p. To the contrary, when a q = 1/2 < |a p | = 1, the transition is discontinuous when q = p, whereas it seems smoother at q = 7 -2p. To confirm our previous observations, we zoomed on the slices of Figure 2 and obtained the results presented in Figure 3. Observing closer the transition from ω c > 0 to ω c = 0 on fixed q slices of Figure 3, we realize that the transition on the left (q = 7 -2p) seems to be always only Lipschitz, contrary to what could be inferred from the previous observation. On the other hand, the previous observation when p = q is confirmed: the transition seems smooth when a q = 2, Lipschitz when a q = 1, and discontinuous when a q = 1/2. This is reflecting the fact that when p → q, the family of soliton profiles has a different behavior for different values of a q . When when ap = -1 and aq = 2 ωc(p, q = 4.00) Figure 3. Zoom on slices of the critical surface for fixed value of q = 4 a q > |a p |, soliton profiles for p = q exist and are stable (hence ω c (p, q = p) = 0), whereas for a q = |a p | the two nonlinearities exactly compensate and for |a p | > a q the defocusing nonlinearity becomes the dominant one (and solitary waves do not even exist).

From the previous observations, we know that at fixed q the map p → ω c (p, q) has a unique maximum if a q = 1 or a q = 2 (if a q = 1/2, we have seen that the map increases towards infinity as p approaches q). Denote by p max (q) the value realizing this maximum, i.e.

ω c (p max (q), q) = max 1<p<5 ω c (p, q).
The line {(p max (q), q), q > 7/3} is represented in Picture 4. When a q = 1, we observe that the line is tangent when a p = -1 and a q = 1 p = 7/3 p = q q = 7 -2p p max (q) when a p = -1 and a q = 2 p = 7/3 p = q q = 7 -2p p max (q) Figure 4. Curve of the argument of max p ω c (p, q) in terms of q to the line p = q when q is close to 7/3 or close to 5. On the other hand, when a q = 2, the line seems to be tangent to the line p = 7/3 when q is close to 7/3. It approaches the point [START_REF] Angulo Pava | Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity[END_REF][START_REF] Angulo Pava | Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity[END_REF] as q goes to 5, but does not seem to be tangent to the line p = q (it was however not possible to obtain numerically a relevant picture closer to q = 5, which leaves open the question of the behavior when q is close to 5). 6.2. Evolution for initial data close to standing waves. We now turn to numerical experiments for the stability/instability of solitary waves for the flow of (1). For the experiments, we have used the Crank-Nicolson scheme with relaxation presented in [START_REF] Besse | A relaxation scheme for the nonlinear Schrödinger equation[END_REF] which has been proved to be efficient for the numerical simulation of the Schrödinger flow (see e.g. [START_REF] Antoine | Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations[END_REF] for the comparison of various schemes used for the dynamical simulations of the nonlinear Schrödinger flow).

For a time discretization step δ t (typically δ t = 10 -3 ), denote by u n the approximation of u at time t n = nδ t . The semi-discrete (in time) relaxation scheme is then given by 

  φ n+ 1 2 +φ n-1 2 2 = a p |u n | p + a q |u n | q , i u n+1 -u n δt + ∂ xx u n+1 +u n 2 = -u n+1 +u n 2 φ n+ 1 2 ,
with the understanding that u 0 = u 0 and φ -1 2 = a p |u 0 | p + a q |u 0 | q . For the implementation, the scheme is further discretized in space with second order finite differences for the second derivative operator, with Dirichlet boundary conditions.

We have performed simulations for (p, q) on the line q = 2p -1, as for this range of exponents explicit formulas are available for solitary wave profiles (see e.g. [START_REF] Liu | Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities[END_REF]) and can be used easily to construct initial data. Considering other ranges of (p, q) would have been possible, to the extend of additional computations to first obtain numerically solitary waves. As we do not expect different behavior to occur for other values of (p, q), the restriction to the line q = 2p -1 is harmless.

The initial data that we construct are all based on a solitary wave profile φ ω . They are of the form

u 0 = φ ω + εψ,
where 0 < ε 1 is used to adjust the size of the perturbation and ψ is the direction of perturbation, which can be for example ψ = φ ω , ψ = φ ω cos, ψ = φ ω tanh, ψ = φ ω (• -3). As our numerical scheme uses Dirichlet conditions at the bounds of the space interval, we have chosen to work with well-localized perturbation in order to avoid possible numerical reflections due to the boundary conditions. Our experiments consisted in taking one of the previous possibility as initial data, running the simulation of the nonlinear Schrödinger flow, and observe the pattern of the outcome. It turns out that after running numerous simulations, we have observed only three possible types of behavior:

• Stability;

• Growth followed by slightly decreasing oscillations;

• Dispersion. Observe that our numerical results are in part similar to the ones obtained and discussed in further details in [START_REF] Carles | On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations[END_REF]Section 4] in the case of the 2d cubic-quintic (focusing-defocusing) nonlinear Schrödinger equation.

Stability means that the solution does not leave the neighborhood of φ ω (up to phase shift and translations). We obviously expect to see this behavior in the cases where the values of the parameters p, q, and ω ensure that the solitary wave will be stable. However, one thing which is not easily decided by the theory is the size of the basin of stability of the solitary wave. In other words, finding a perturbation of the solitary wave sufficiently large to be visible, but small enough so that the corresponding solution remains in the vicinity of the solitary wave requires delicate adjustments.

An example of a stable behavior is provided in Figure 5. Observe that while on the global scale the solution seems to be behave exactly as a solitary wave (left picture), when getting a closer look at the maximum value (right picture) we observe small oscillations (with an amplitude of order 0.03).

The second behavior consists in a first phase of focusing growth of the profile, which is similar to what can be observed when instability of solitons is by blow-up (e.g. for power-type supercritical nonlinearities. However, after a certain time, the focusing phase stops and is followed by a phase in which the solution seems to oscillate around another profile. The size of the oscillation is decaying, but at a slow pace, and we have not run the simulation long enough to observe convergence toward a final state. An example of such a behavior is presented in Figure 6.

Finally, the third behavior that we have observed could be characterized as scattering, as the profile of the solution is simultaneously decreasing in height while spreading over the whole line. As before, the decay is rather slow and we have not run the simulation long enough for the solution to converge to 0. An example of such a behavior is presented in Figure 7. Observe that the domain of calculation is [-50, 50], but the solution is represented only on [-20, 20], which explains the non-zero values observed at the boundaries on the left figure. a p = -1, a q = 1, p = 1.25, q = 1.50, ω = 1.00, ω c = 0.00, init=1 a p = -1, a q = 1, p = 2.67, q = 4.33, ω = 1.00, ω c = 2.14, init=2

1010

  Critical surface when ap = -1 and aq = 0.5 Critical surface when ap = -1 and aq = 2

Figure 1 .

 1 Figure1. Critical surface {(p, q, ω c (p, q))} for a p = -1 and a q = 1/2, 1, 2. The white lines represent q = 7 -2p and q = p, where the transition from ω c = 0 to ω c > 0 occurs.

whenFigure 2 .

 2 Figure 2. Slices of the critical surface for fixed value of q = 4

Figure 5 .a

 5 Figure 5. Example of a stable numerical solution. The initial data is u0 = (1 + ε)φ ω , ε = 10 -2 .

Figure 6 .

 6 Figure 6. Example of a growing/oscillating numerical solution. The initial data is u 0 = (1 + ε)φ ω , ε = 10 -2 .

Figure 7 .

 7 Figure 7. Example of a scattering numerical solution. The initial data is u 0 = (1ε)φ ω , ε = 10 -2 .

The work of P. K. and S. L. C. is partially supported by ANR-11-LABX-0040-CIMI and CNRS IEA 296038. The work of T.-P. T. is partially supported by the NSERC grant RGPIN-2018-04137.

1

If p = 5, then we have F (φ * ) > 0 and the conclusion follows. From now on, assume that p > 5. Recalling the value of φ * given in [START_REF] Angulo Pava | Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity[END_REF], we infer that 2a p

(5p)(φ * ) p+1 (1s p-1 ) p + 1 + 2a q (5q)(φ * ) q+1 (1s q-1 ) q + 1 = 2a q (φ * ) p+1 (1s q-1 ) (5q) q + 1 a p (5p)(q + 1) a q (5q)(p + 1) (1s p-1 ) (1s q-1 ) + (φ * ) q-p = 2a p (φ * ) p+1 (1s q-1 ) (5q) q + 1 (5p)(q + 1)

(5q)(p + 1) (1s p-1 ) (1s q-1 ) -p -1 q -1 q + 1 p + 1 = 2a p (φ * ) p+1 (1s q-1 ) (5q) p + 1 (5p)

where we have used in particular Lemma 3.3 for the last inequality. This implies that F (φ * ) > 0 which, since J(ω, p, q) = C(φ 0 )F (φ 0 ), finishes the proof.

Determination of the sign of the slope

In this section, we determine for each possible values of a p , a q , p and q the sign of J(ω, p, q). Combined with the stability criteria of Section 2.2, this will prove Theorem 1.1. The general strategy of our proofs is the following. Recall from Lemma 3.1 that J(ω, p, q) = C(φ 0 )F (φ 0 ), where

sds,

and C(φ 0 ) > 0. Moreover, ω and φ 0 are in an increasing one to one correspondence. Hence, to determine the sign of J, it is sufficient to determine the sign of F (φ 0 ). To do this, we have two ingredients at our disposal. First, it is usually not difficult to establish that F has a constant sign on intervals of the type (φ * , φ 0,1 ) or (φ 0,1 , φ * ). On the other hand, the expression for ∂ φ0 F (φ 0 ) given in Lemma 3.2 allows us to show that ∂ φ0 F (φ 0 ) has a constant sign on intervals of the type (φ * , φ 0,2 ) or (φ 0,2 , φ * ). If the intervals of the two ingredients overlap and if the signs are matching, the conclusion will follow. For example, if F (φ 0 ) > 0 on (φ * , φ 0,1 ), and ∂ φ0 F (φ 0 ) > 0 on (φ 0,2 , φ * ) and φ 0,1 > φ 0,2 , then F (φ 0 ) > 0 on (φ * , φ * ). The detail of each case is given in the following sections.

5.1. The focusing-focusing case. In this section, we consider the case a p > 0, a q > 0. In this case we have Φ p > 0 and Φ q > 0.

Lemma 5.1. Let a p > 0, a q > 0 and q 5. Then for all ω ∈ (0, ∞) we have J(ω, p, q) > 0, and the family of standing waves is of type S.

Proof. If q 5, then 5p > 0 and 5q 0. Therefore for any φ 0 ∈ (0, ∞) we have

which gives the desired conclusion. Lemma 5.2. Let a p > 0, a q > 0 and p 5. Then for all ω ∈ (0, ∞) we have J(ω, p, q) < 0, and the family of standing waves is of type U.

Proof. If p 5, then 5p 0 and 5q < 0. Therefore for any φ 0 ∈ (0, ∞) we have F (φ 0 ) < 0, which gives the desired conclusion.

The remaining case p < 5 < q is a bit more involved to consider. Lemma 5.3. Let a p > 0, a q > 0 and p < 5 < q. There exists φ 0,1 (explicitly given in [START_REF] Fukaya | Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities[END_REF]) such that if φ q-p 0 > φ q-p 0,1 then

Replacing Φ p and Φ q by their expressions [START_REF] Bellazzini | Ground state energy threshold and blow-up for nls with competing nonlinearities[END_REF], this is equivalent to

Since p < 5 < q, we have 5 + 2q -3p > 0, and therefore (18) becomes

We know from Lemma 3.

If φ 0 < φ 0,2 then ( 19) is verified, which concludes the proof.

Lemma 5.5. Let a p > 0, a q > 0 and p < 5 < q. The function F γ (φ 0 ) has at most one zero in (0, ∞).

Proof. As p < q, we have 3(p -1)(pq) < 0, hence (5p)(q + 1) < (p + 1)(5 + 2q -3p).

It implies

.

Therefore, we have φ q-p 0,1 < φ q-p 0,2 . We know from Lemma 5.3 that F γ (φ 0 ) < 0 if φ 0 ∈ (φ 0,1 , ∞), and from Lemma 5.4 that F γ (φ 0 ) is decreasing for all φ 0 ∈ (0, φ 0,2 ). As φ 0,1 < φ 0,2 , this implies that F γ (φ 0 ) has at most one zero.

Lemma 5.6. Let a p > 0, a q > 0 and p < 5 < q. There exists ω 1 ∈ (0, ∞) such that J(ω, p, q) > 0 for ω < ω 1 , J(ω 1 , p, q) = 0, J(ω, p, q) < 0 for ω > ω 1 , and the family of standing waves is of type SU.

Proof. From Proposition 4.1, we know that J(ω, p, q) > 0 for ω close to 0. Combined with Lemmas 5.3 and 5.5, this implies the desired result.

5.2.

The focusing-defocusing case. In this section, we consider the case a p > 0, a q < 0. In this case Φ p > 0 and Φ q < 0. Lemma 5.7. Let a p > 0, a q < 0 and p 5 < q. For any ω ∈ (0, ω * ), we have J(ω, p, q) > 0, and the family of standing waves is of type S.

Proof. We have 5p 0, and 5q < 0. Therefore F (φ 0 ) > 0 for any φ 0 ∈ (0, φ * ), which gives the desired result.

Lemma 5.8. Let a p > 0, a q < 0 and p < q 5. Let γ = p+1 2 . Then the integrand I γ of F γ defined in (11) verifies ∂I γ ∂φ 0 > 0 for all φ 0 ∈ (0, φ * ).

Proof. Lemmas 5.11 and 5.12 implies F γ changes sign only once on (0, ω * ). From Proposition 4.2, we know that as ω → 0, we have J(ω, p, q) > 0 when q 7 -2p, which gives the conclusion for the first part of the Lemma. When q > 7 -2p, from Proposition 4.2, we know that as ω → 0, we have J(ω, p, q) < 0. Since, from Proposition 4.6, we know that J(ω, p, q) > 0 for large ω, the conclusion follows for the second part of the Lemma.

Lemma 5.14. Let a p < 0, a q > 0 and p < 5 q. For any ω ∈ (0, ∞), we have J(ω, p, q) < 0, and the family of standing waves is of type U .

Proof. We have 5p > 0, 5q 0, Φ p < 0 and Φ q > 0. Therefore we directly see on the expression (8) of F (φ 0 ) that F (φ 0 ) < 0, which gives the desired result.

Lemma 5.15. Let a p < 0, a q > 0 and 5 p < q. For any ω ∈ (0, ∞), we have J(ω, p, q) < 0, and the family of standing waves is of type U .

Proof. We know that φ * < φ 0 , therefore -ap aq q+1 p+1 < φ q-p 0 . As 5-p 5-q < 1, we have -

which implies (5p)Φ p + (5q)Φ q < 0. This implies that F (φ 0 ) < 0 which gives the desired result. 5.4. The critical frequency. Observe that, as a by-product of the analysis of the previous sections, we always have instability at the critical frequency when there is a stability change. Indeed, we have

. At the stability change, we have F (φ 0 ) = 0. Therefore, at the stability change,

). As we have shown that in this case ∂ φ0 F γ (φ 0 ) = 0, the criterion (6) holds.

Numerical experiments

To explore further the stability/instability of standing waves, we have performed a series of numerical experiments in the case a p < 0, a q > 0, 1 < p < q < 5.

The Python language and the specific libraries Numpy, Scipy and Matplotlib have been used to perform the experiments. The code is made available in [START_REF] Kfoury | Stability-of-standing-waves-of-the-double-power-1D-NLS[END_REF].

6.1. The critical surface for stability/instability. We first analyzed the critical surface in (ω, p, q) separating instability from stability. To this aim, we first have implemented the calculation of J(ω, p, q). The function integrate.quad has been used to perform the integration. While the results are overall satisfactory, in some cases the function returned incorrect results, with problems increasing as ω was taken closer to 0.

To estimate the critical ω at given (p, q), we have used the classical bisection method, which has the advantage of being very robust. The algorithm is divided into two parts.

First, we find an initial interval [ω 0 , ω 1 ] in which we are sure that ω → J(ω, p, q) changes sign. A natural choice for ω 0 is 0. To find a suitable ω 1 , we simply start with ω 1 = 1 and test if J(ω 1 , p, q) > 0. If not, we replace ω 1 by 2ω 1 and repeat until J(ω 1 , p, q) > 0. To avoid running an infinite loop, we break it when ω 1 > 10 10 and do not search for ω c in these cases. Second, we apply the bisection method to search for a root of J(ω, p, q) inside [ω 0 , ω 1 ]. As this approach, while being efficient, is also relatively slow, we took advantage of the computer power of our department to run computations in parallel on the (p, q) ∈ [1, 5] × [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF][START_REF] Angulo Pava | Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity[END_REF] grid with dp = dq = 0.01.


O L Cruz-González

R Rodríguez-Ramos

F Lebon

Evaluation of the effective viscoelastic properties of short fibers reinforced composites

Keywords: Short fibers reinforcement, Two-scale asymptotic homogenization method, Viscoelastic composites

In the present work, we address the calculation of the effective viscoelastic properties of short fibers reinforced composites via a semi-analytical approach that combines the two-scale Asymptotic Homogenization Method (AHM) and numerical computations based on Finite Elements (FE). As starting point and considering the elastic-viscoelastic correspondence principle, we propose the solution of the heterogeneous problem as a formal series expansion and then, we derive the associated local and homogenized problems along with the effective coefficients, all in the Laplace-Carson domain. Moreover, three-dimensional finite element simulations are performed to find the numerical solution of the local problems. Then, by inverting the Laplace-Carson transform, we obtain the effective coefficients in the time domain. We present the results for a viscoelastic composite reinforced by perfectly aligned short fibers. The influence of square or hexagonal periodic cells, as well as different fiber orientations, is analyzed. Finally, we deal with the case of random orientations.

Introduction

Innovative materials with improved mechanical properties are continuously demanded by high-performance applications in structural engineering. Fiber-reinforced composites are one of the most widely used and several examples consist of viscoelastic constituents in which their phases generally involve both instant elastic and time-dependent viscous behavior (see [START_REF] Ornaghi | Viscoelastic characteristics of carbon fiber-reinforced epoxy filament wound laminates[END_REF]). Short and discontinuous fibers usually act as fillers and, although the mechanical properties in terms of stiffness and strength of these composites are inferior to those reinforced with continuous fibers, the former are preferred in manufacturing processes due to the complexity associated with the use of long continuous fibers. For example, in injection molding, short fibers present a great advantage as allowing very short cycle times, which are mandatory in industries involving high volume [START_REF] Burgarella | Effective viscoelastic behavior of short fibers composites using virtual DMA experiments[END_REF].

The modeling of this kind of composites has been increasingly addressed in recent contributions. For instance, in [3], the Authors investigate the effective viscoelastic properties of short fiber-reinforced composites using Scott Blair-Rabotnov fractional-exponential operators. In [4], a variational model is proposed for the description of the failure mechanisms observed in short reinforcements subjected to tensile loading. Moreover, the Authors in [5] developed a micromechanics-based Artificial Neural Networks model to predict the elastic properties of short fiber composites.

The aim of this work is to calculate the effective viscoelastic properties of short fiber reinforced composite by means of a semi-analytical technique that combines the theoretical strengths of AHM with the advantages of developing three-dimensional finite element simulations. Specifically, AHM takes advantage of the well-separated length scales to decouple the spatial variables into a local and a global one. Thus, the analysis of the original problem at the macroscale is reduced to solving the local problems on the periodic cell [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF]. Due to the lack of knowledge of an analytical solution, we perform a numerical approach based on the finite element method (FEM) providing a suitable alternative to solve the local problems. For this purposes, we use the finite element software COMSOL Multiphysics® in conjunction with LiveLink TM for Matlab® scripting.

It worth mentioning that we employ the modeling approach introduced in previous works [7, 8] and here we exploit it to perform the calculation in the case of short fiber reinforcements according to square or hexagonal arraignments, and different orientations. In addition, we investigate the effective coefficients for a particular case of random orientation.

Model Description and Statement of the Problem

Let us denote by 3



a linear viscoelastic composite material with periodic microstructure and well-separated length scales and L . In particular, we consider  as a two-phase composite made of a viscoelastic matrix reinforced by square or hexagonal arrays of elastic short fibers (see Fig. 1).

We define a dimensionless scaling parameter  by the expression 1 L   . Additionally, in our framework,

x and y respectively refer to the macroscopic and the microscopic spatial variables, related through

x y

  .
In what follows, we consider the notation ( , ) ( , , )

x t x y t   
, where   is assumed to be periodic in y . Thus, since the constitutive response of all constituents is linear viscoelastic, ignoring inertia and external volume forces, the problem in  reads, (

.

xt x t x t            0 00 σ u σn (Eq.1)
As observed in Eq. 1, we further impose continuity conditions for displacements and tractions on   . The term n stands for outward unit vectors to the surfaces   and the operator   denotes the jump of   across the interface between the constituents. More details about the boundary and initial conditions are reported in [START_REF] Cruz-González | On the effective behavior of viscoelastic composites in three dimensions[END_REF].

Regarding Eq. 1, the notation  σ represents the second-order stress tensor and the scale-dependent constitutive law for viscoelastic materials is stated,

0 ( ( , )) ( , ) ( , ) :
.

t x x t x t d          ξu σ (Eq.2)
Here, the term  u stands for the displacement field and the fourth-order tensor  refers to the relaxation modulus, where the following symmetry properties are assumed

ijkl jikl ijlk klij        .
Additionally, ξ represents the second-order strain tensor and it is determined by the formula

    1 ( , ) ( , ) ( ( , )) . 2 T x t x t x t        ξ u u u (Eq.3)
Due to the scale-dependent constitutive law in Eq. 2 corresponds to the special form of a non-aging linear viscoelastic materials, we rely on the elastic-viscoelastic correspondence principle and the Laplace-Carson transform, and we rewrite Eq. 1 as follows,

            ˆ, : , in ( \ ) [0, ), ( , ) , , : , , on [0,
)

x s x s x s x s x s                         0 00 ξu u ξ u n
(Eq.4) with the respective boundary and initial conditions. In Eq. 4, the "hat" symbol and variable s denote the Laplace-Carson domain.

Overview of the Semi-Analytical Approach

In this section, we present a brief description of the semi-analytical approach. For this purpose, we keep track of the methodology described in [7, 8] and highlight the main results of the asymptotic homogenization process and the required three-dimensional computational study.

The AHM [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF] proposes the solution of the viscoelastic heterogeneous problem Eq. 4 as a formal series expansion in powers of  , namely

          0 1 ˆˆ, , , , , . i i
i x s x y s x y s       u u u (Eq.5)
After substitution in Eq. 4, using the chain rule and equating the result in the same powers of  , we obtain that, in the limit 0

  ,               0 (1) ˆˆˆ, , , , , , , , ,
m klm kl x y s x s u x y s y s x s   u v v (Eq.6)
where v is a smooth vector function of x and s , and ˆklm  is a third order tensor periodic in y . Moreover, by following the standard procedure, we derive the homogenized problems at the macroscale, In addition, ˆklm  represents the solution of the local problems and a remarkable simplification emerges from assuming a y -constant nature of the relaxation modulus ( ( , ) ˆijkl ys) along each phase in the periodic cell, namely

    (*) (*) ( ) ( ) ( , ) ( ) ( , ) ( ˆ 0, [ , ) ( , 0,
(1) 1 2 (2) ( ), if ( , ) ( ), ˆ ˆif ijkl ijkl ijkl s y Y ys s y Y         (Eq.9)
wherein superscripts (1), ( 2) indicate the corresponding constituents, i.e. the matrix and the inclusion. Considering the above, the local problems are stated,

    ( ) ( ) ( ) (1) (2) (1) ( ) (1) ( ) (2) ( ) (2) ( ) ( )
ˆ( ) ( ( , )) 0 in [0, ) , ˆ( , ) ( , ) on [0, ), ( ) ( ( , )) ( ) ( ( , ) ( 1, 2)
) , , k l k l  when discontinuities in the coefficients Eq. 9, between the matrix and inclusions arise.

y
                    χ χχ   on [0, ).
In summary, the semi-analytical technique (AHMFE) to calculate the effective viscoelastic properties consists, firstly, in numerically solving the set of elastic local problems Eq. 10 in Laplace-Carson space employing the finite element software COMSOL Multiphysics® and LiveLink TM for Matlab® scripting. So that, we are able to calculate the effective relaxation modulus (*) ˆ() ijkl s by means of Eq. 7.2. Finally, we applied the inversion of the Laplace-Carson transform to recover the original temporal space. We refer to Fig. 2 of [8] for more details about the methodology.

Results and Discussion

In this section, we study the two extreme cases that arise in the geometrical configurations of short fibers, i.e. perfectly aligned and random orientations.

The former is developed within the framework set up in section 5.2.3 of [8]. In particular, we consider a viscoelastic composite material with square or hexagonal arrangement of perfectly aligned short fibers (see Fig. 1). We deal with the pair T300 graphite fiber/ 934 epoxy matrix at room temperature. The properties of the transversely isotropic fibers are shown in Table 1 whereas the matrix is well modeled by the power-law and the properties for a constant bulk modulus are given in Table 2. In addition, we define the parameters 1

1 1 :/ hH   and 2 2 2 :/ hH  
relating the measurement of fiber and matrix length in the square and hexagonal periodic cell, respectively (see Fig. 2 (d)).

Table 1. Elastic moduli of the transversely isotropic fibers at room temperature.

 

() GPa f A E   () GPa f T E   () GPa f A  () f A  () f T  202.82
25.30 44.12 0.443 0.05

The fibers are modeled as cylindrical and non-overlapped inclusions, centered in the periodic cell and which can uniformly rotate a counterclockwise angle  about 2 X -axis for 0  .  . It should be noted that, by using a refined mesh, we obtain more accurate results than those shown in [START_REF] Cruz-González | Effective behavior of long and short fiber-reinforced viscoelastic composites[END_REF] for the hexagonal cell.

In regards to the random orientation case, we take inspiration from [9] and we model the randomness by means of a new periodic cell made of 27 similar short fibers embedded in the matrix. We consider all the 27 fibers to behave as T300 graphite and still use the 934 epoxy matrix. Now, by assuming a continuous uniform distribution in the interval   0º ,180º for the fiber orientations, we generate 27 random angles and rotate respectively each fiber about 2 X -axis. The frequency of occurrence is shown in the histogram of Fig. 2 (d). It is worth mentioning that as the inclusions are transversely isotropic, the relaxation modulus should be rotated according to each fiber orientation. In Fig. 2 (a), (b) and (c), we highlight in red color the effective moduli for the random orientation case.

Conclusions

In this work, we applied a semi-analytical approach to calculate the effective viscoelastic properties of short fiber reinforced composites. The model focused on three-dimensional geometrical configurations. We studied square and hexagonal fiber arrangements along with different fiber orientations. In addition, the fiber orientation analysis comprised two different cases, namely perfectly aligned and randomly oriented. The results indicated that these geometrical features influence the macroscopic effective response of the viscoelastic composite.

A further generalization of the present framework includes the consideration of fiber orientations with two degrees of freedom allowing us to achieve any orientation in three-dimensional space. Another extension could be to take into account different aspect ratios of the fibers.

Figure 1 .

 1 Figure 1. The composite material is made up of a viscoelastic matrix reinforced by square or hexagonal arrangement of elastic short fibers with cylindrical shape. The fibers do not intersect the boundaries.

  stress jump condition Eq. 10.3 and represent the driving force to obtain nontrivial solutions of the six elastic-type local problems

1 E 1 E

 11 Fig.2 (a), (b) and (c), we show the results for the effective moduli (*) fibers, wherein the square and hexagonal periodic cells are highlighted in blue and black colors, respectively. We set a ratio  as in relation to what was observed in [8] and also confirm a decreasing behavior for (*) and (*) 12

Figure 2 .

 2 Figure 2. Chart (a), (b) and (c) shows the calculation of the effective moduli. Chart (d) offers general information.

  boundary and initial conditions. The expression for the effective relaxation modulus, in Laplace-Carson space, is shown in Eq. 7.2, where the notation y stands for the cell average operator. Furthermore, we have introduced the notation

	with the respective     () ˆ,.     ˆ,, 1 2 klp klq y pq kl qp χ y s χ y s ys yy         χ	(Eq.8)
	ijkl	s	x  	j		kl	v	x	s		  	)	(Eq.7)
	ˆ) ˆ, ˆĥ y s y s y s y s   χ
	ijkl				ijkl				ijpq	pq	kl
													y

Table 2 .

 2 Material properties of the viscoelastic matrix at room temperature.