Solitary wave solutions to a class of modified Green-Naghdi systems

We provide the existence and asymptotic description of solitary wave solutions to a class of modified Green-Naghdi systems, modeling the propagation of long surface or internal waves. This class was recently proposed by Duch{\^e}ne, Israwi and Talhouk (Stud. Appl. Math.,137 (2016)) in order to improve the frequency dispersion of the original Green-Naghdi system while maintaining the same precision. The solitary waves are constructed from the solutions of a constrained minimization problem. The main difficulties stem from the fact that the functional at stake involves low order non-local operators, intertwining multiplications and convolutions through Fourier multipliers.


Motivation
In this work, we study solitary traveling waves for a class of long-wave models for the propagation of surface and internal waves. Starting with the serendipitous discovery and experimental investigation by John Scott Russell, the study of solitary waves at the surface of a thin layer of water in a canal has a rich history [20]. In particular, it is well-known that the most widely used nonlinear and dispersive models for the propagation of surface gravity waves, such as the Korteweg-de Vries equation or the Boussinesq and Green-Naghdi systems, admit explicit families of solitary waves [8,40,29,41,14]. These equations can be derived as asymptotic models for the so-called water waves system, describing the motion of a two-dimensional layer of ideal, incompressible, homogeneous, irrotational fluid with a free surface and a flat impermeable bottom; we let the reader refer to [30] and references therein for a detailed account of the rigorous justification of these models. Among them, the Green-Naghdi model is the most precise, in the sense that it does not assume that the surface deformation is small. However, the validity of all these models relies on the hypothesis that the depth of the layer is thin compared with the horizontal wavelength of the flow and, as expected, the models do not describe the system accurately (for instance the dispersion relation of infinitesimally small waves) in a deep water situation. In order to tackle this issue, one of the authors has recently proposed in [23] a new family of models: where

arXiv:1706.08853v1 [math.AP] 27 Jun 2017
Here, ζ is the surface deformation, h = d + ζ the total depth (where d is the depth of the layer at rest), u the layer-averaged horizontal velocity, w = hu the horizontal momentum and g the gravitational acceleration; see Figure 1. Finally, F def = F(D) is a Fourier multiplier, i.e.
The original Green-Naghdi model is recovered when setting F(k) ≡ 1. Any other choice satisfying F(k) = 1 + O(k 2 ) enjoys the same precision (in the sense of consistency) in the shallow-water regime and the specific choice of F(k) = 3 d|k| tanh(d|k|) − 3 d 2 |k| 2 allows to obtain a model whose linearization around constant states fits exactly with the one of the water waves system. Compared with other strategies such as the Benjamin-Bona-Mahony trick [4,33] or using different choices of velocity unknowns [6,38], an additional advantage of (1.1) is that it preserves the Hamiltonian structure of the model, which turns out to play a key role since the existence of solitary waves will be deduced from a variational principle. The study of [23] is not restricted to surface propagation, but is rather dedicated to the propagation of internal waves at the interface between two immiscible fluids, confined above and below by rigid, impermeable and flat boundaries. Such a configuration appears naturally as a model for the ocean, as salinity and temperature may induce sharp density stratification, so that internal solitary waves are observed in many places [39,28,27]. Due to the weak density contrast, the observed solitary waves typically have much larger amplitude than their surface counterpart, hence the bilayer extension of the Green-Naghdi system introduced by [37,34,17], often called Miyata-Choi-Camassa model, is a very natural choice. It however suffers from strong Kelvin-Helmholtz instabilities -in fact stronger than the ones of the water waves system for large frequencies -and the work in [23] was motivated by taming these instabilities. The modified bilayer system reads Here, ζ represents the deformation of the interface, h 1 (resp. h 2 ) is the depth of the upper (resp. lower) layer, u 1 (resp. u 2 ) is the layer-averaged horizontal velocity of the upper (resp. lower) layer and w = h 1 h 2 (u 2 − γu 1 )/(h 1 + γh 2 ) is the shear momentum. In this formulation we have used dimensionless variables, so that the depth at rest of the upper layer is scaled to 1, whereas the one of the lower layer is δ −1 , in which δ is the ratio of the depth at rest of the upper layer to the depth at rest of the lower layer (see Figure 1). Similarly, γ is the ratio of the upper layer over the lower layer densities. As a consequence of our scaling, the celerity of infinitesimally small and long waves is c 0 = 1. Once again, F i (i = 1, 2) are Fourier multipliers. The choice F id i (k) ≡ 1 yields the Miyata-Choi-Camassa model while with convention δ 1 = 1, δ 2 = δ, fits the behavior of the full bilayer Euler system around constant states, and thus gives hope for an improved precision when weak nonlinearities are involved. Note that compared to equations (7)- (9) in [23] we have scaled the variables so that the shallowness parameter µ and amplitude parameter do not appear in the equations. This is for notational convenience since the parameters do not play a direct role in our results. On the other hand, we only expect the above model to be relevant for describing water waves in the regime µ 1 and the solutions that we construct in the end are found in the long-wave regime , µ 1. In the following, we study solitary waves for the bilayer system (1.2), noting that setting γ = 0 immediately yields the corresponding result for the one-layer situation, namely system (1.1). Our results are valid for a large class of parameters γ, δ and Fourier multipliers F 1 , F 2 , described hereafter. Our results are twofold: i. We prove the existence of a family of solitary wave solutions for system (1.2); ii. We provide an asymptotic description for this family in the long-wave regime.
These solitary waves are constructed from the Euler-Lagrange equation associated with a constrained minimization problem, as made possible by the Hamiltonian structure of system (1.2). There are however several difficulties compared with standard works in the literature following a similar strategy (see e.g. [1] and references therein). Our functional cannot be written as the sum of the linear dispersive contribution and the nonlinear pointwise contribution: Fourier multipliers and nonlinearities are entangled. What is more, the operators involved are typically of low order (F is a smoothing operator). In order to deal with this situation, we follow a strategy based on penalization and concentration-compactness used in a number of recent papers on the water waves problem (see e.g. [9,10,25] and references therein) and in particular, in a recent work by one of the authors on nonlocal model equations with weak dispersion, [24]. Thus we show that the strategy therein may be favorably applied to bidirectional systems of equations in addition to unidirectional scalar equations such as the Whitham equation. Roughly speaking, the strategy is the following. The minimization problem is first solved in periodic domains using a penalization argument do deal with the fact that the energy functional is not coercive. This allows to construct a special minimizing sequence for the real line problem by letting the period tend to infinity, which is essential to rule-out the dichotomy scenario in Lions' concentration-compactness principle. The long-wave description follows from precise asymptotic estimates and standard properties of the limiting (Korteweg-de Vries) model. When the Fourier multipliers F i have sufficiently high order, we can in fact avoid the penalization argument and consider the minimization problem on the real line directly, since any minimizing sequence is then also a special minimizing sequence. In particular, this is the case for the original Miyata-Choi-Camassa model (and of course also the Green-Naghdi system).
Our existence proof unfortunately gives no information about stability, since our variational formulation does not involve conserved functionals; see the discussion in Section 1.2. If sufficiently strong surface tension is included in the model, we expect that a different variational formulation could be used which also yields a conditional stability result (see [9,10,25]). A similar situation appears e.g. in the study of Boussinesq systems [15,16].

The minimization problem
We now set up the minimization problem which allows to obtain solitary waves of system (1.2). We seek traveling waves of (1.2), namely solutions of the form (abusing notation) where we denote Integrating these equations and using the assumption (since we restrict ourselves to solitary waves) lim |x|→∞ ζ(x) = lim |x|→∞ w(x) = 0 yields the system of equations We now observe that system (1.2) enjoys a Hamiltonian structure. Indeed, define the functional Under reasonable assumptions on F 1 , F 2 , and for sufficiently regular ζ, A F γ,δ [ζ] defines a well-defined, symmetric, positive definite operator [23]. We may thus introduce the variable and write It is now straightforward to check that (1.2) can be written in terms of functional derivatives of H: It is therefore tempting to seek traveling waves through the time-independent quantities Indeed any solution to (1.3) is a critical point of the functional H − cI: which, by (1.5), is the desired system of equations. However, notice that from Weyl's essential spectrum theorem, one has where H ∞ , I ∞ are the asymptotic operators as |x| → ∞, i.e.
Since the spectrum of the above operator has both negative and positive components, the desired critical point is neither a minimizer nor a maximizer, as noticed (for the Green-Naghdi system) in [31].
We will obtain solutions to (1.3) from a constrained minimization problem depending solely on the variable ζ. Notice that for each fixed c and ζ, the .
and look for critical points of H(ζ, v c,ζ ) − cI(ζ, v c,ζ ) by considering the minimization problem with c −2 acting as a Lagrange multiplier. Another way of thinking of this reduction is to solve the first equation in (1.3) for w and substitute the solution w = cζ into the second equation, yielding

Statement of the results
For the sake of readability, we postpone to Section 2 the definition and (standard) notations of the functional spaces used herein. The class of Fourier multipliers for which our main result is valid is the following. i. F(k) = F(|k|) and 0 < F ≤ 1; ii. F ∈ C 2 (R), F(0) = 1 and F (0) = 0; iii. There exists an integer j ≥ 2 such that iv. There exists θ ∈ [0, 1) and C F ± > 0 such that We also introduce a second class of strongly admissible Fourier multipliers which is used in our second result. Definition 1.2 (Strongly admissible class of Fourier multipliers). An admissible Fourier multipler F in the sense of Definition 1.1 is strongly admissible if F ∈ C ∞ (R) and for each j ∈ N there exists a constant C j such that |∂ j k F(k)| ≤ C j (1 + |k|) −θ−j . Notice the following. are strongly admissible, and satisfy Definition 1.1,iv with (respectively) θ = 0, 1/2. Assumption 1.4 (Admissible parameters). In the following, we fix γ ≥ 0, δ ∈ (0, ∞) such that δ 2 − γ = 0. We also fix a positive number ν such that ν ≥ 1 − θ and ν > 1/2 (the second condition is automatically satisfied if θ < 1/2). Finally, fix R an arbitrary positive constant. Remark 1.5. Our results hold for any values of the parameters (γ, δ) ∈ [0, ∞) × (0, ∞) such that δ 2 = γ, although admissible values for q 0 depend on the choice of the parameters. However, not all parameters are physically relevant in the oceanographic context. When γ > 1, the upper fluid is heavier than the lower fluid, and the system suffers from strong Rayleigh-Taylor instabilities [12]. In the bilayer setting, the use of the rigid-lid assumption is well-grounded only when the density contrast, 1 − γ, is small. In this situation, one may use the Boussinesq approximation, that is set γ = 1; see [22] in the dispersionless setting. Notice however that system (1.2) exhibits unstable modes that are reminiscent of Kelvin-Helmholtz instabilities when the Fourier multipliers F i satisfy Definition 1.1,iv with θ ∈ [0, 1); see [23]. It is therefore noteworthy that internal solitary waves in the ocean and in laboratory experiments are remarkably stable and fit very well with the Miyata-Choi-Camassa predictions [27]. The sign of the parameter δ 2 − γ is known to determine whether long solitary waves are of elevation or depression type, as corroborated by Theorem 1.7. At the critical value δ 2 = γ, the first-order model would be the modified (cubic) KdV equation, predicting that no solitary wave exists [21].
We study the constrained minimization problem arg min and q ∈ (0, q 0 ), with q 0 sufficiently small. Notice in particular that as soon as q is sufficiently small ζ L ∞ < min(1, δ −1 ) (by Lemma 2.1 thereafter and since ν > 1/2) and E(ζ) is well-defined (by Lemmata 2.3 and 2.4 and since ν ≥ 1 − θ) for any ζ ∈ V q,R . Any solution will satisfy the Euler-Lagrange equation where α is a Lagrange multiplier. Equation (1.11) is exactly (1.8) with (−α) −1 = c 2 , and therefore provides a traveling-wave solution to (1.2). Our goal is to prove the following theorems.
Theorem 1.6. Let γ, δ, ν, R satisfying Assumption 1.4 and F i , i = 1, 2 be admissible in the sense of Definition 1.1. Let D q,R be the set of minimizers of E over V q,R . Then there exists q 0 > 0 such that for all q ∈ (0, q 0 ), the following statements hold: • The set D q,R is nonempty and each element in D q,R solves the traveling wave equation (1.8), with c 2 = (−α) −1 > 1. Thus for any ζ ∈ D q,R , ζ(x ± ct), w ± = ±cζ(x ± ct) is a supercritical solitary wave solution to (1.2).
• For any minimizing sequence {ζ n } n∈N for E in V q,R such that sup n∈N ζ n H ν (R) < R, there exists a sequence {x n } n∈N of real numbers such that a subsequence of {ζ n (·+x n )} n∈N converges (strongly in H ν (R) if ν = 1 − θ > 1/2; weakly in H ν (R) and strongly in H s (R) for s ∈ [0, ν) otherwise) to an element in D q,R .
Theorem 1.7. In addition to the hypotheses of Theorem 1.6, assume that F i , i = 1, 2, are strongly admissible in the sense of Definition 1.2. Then there exists q 0 > 0 such that for any q ∈ (0, q 0 ), each ζ ∈ D q,R belongs to H s (R) for any s ≥ 0 and is the unique (up to translation) solution of the KdV equation (5.2) and In addition, the number α, defined in Theorem 1.6, satisfies uniformly over q ∈ (0, q 0 ) and ζ ∈ D q,R .
In the following, we denote C(λ 1 , λ 2 , . . . ) a positive constant depending non-decreasingly on the parameters λ 1 , λ 2 , . . . . We write A B when A ≤ CB with C a nonnegative constant whose value is of no importance. We do not display the dependence with respect to the parameters γ, δ, C Fi ± and regularity indexes.
Functional setting on the real line Here and thereafter, we denote L 2 (R) the standard Lebesgue space of square-integrable functions, endowed with the norm . We use the same notation for duality pairings which are clear from the context. The space L ∞ (R) consists of all essentially bounded, Lebesgue-measurable functions f , endowed with the norm f L ∞ = ess sup x∈R |f (x)|. For any real constant s ∈ R, H s (R) denotes the Sobolev space of all tempered is the space of functions having continuous derivatives up to order n, and C ∞ (R) = n∈N C n (R). The Schwartz space is denoted S(R) and the tempered distributions S (R). We use the following convention for the Fourier transform: We start with standard estimates in Sobolev spaces. The following interpolation estimates are standard and used without reference in our proofs.
Lemma 2.2 (Composition estimate). Let G be a smooth function vanishing at 0, and f ∈ H µ (R) with µ > 1/2. Then G • f ∈ H µ (R) and we have i. For any f, g ∈ L ∞ (R) ∩ H s (R) with s ≥ 0, one has f g ∈ H s (R) and ii. For any f ∈ H s (R), g ∈ H t (R) with s + t ≥ 0, and let r such that min(s, t) ≥ r and r < s + t − 1/2. Then one has f g ∈ H r (R) and iii. For any ζ ∈ L ∞ (R) such that ζ L ∞ ≤ 1 − h 0 with h 0 > 0 and any f ∈ L ∞ (R), one has iv. For any ζ ∈ H µ (R) with µ > 1/2 such that ζ L ∞ ≤ 1 − h 0 with h 0 > 0 and any f ∈ H s (R) with s ∈ [−µ, µ], one has Proof. The first two items are standard (see for instance [5,Prop. C.11 and Th. C.10]. The third item is obvious. For the last item, we use the second item and deduce that for any f ∈ H s (R), s ∈ [−µ, µ], and g ∈ H µ (R), where we have used Lemma 2.2, and the estimate is proved.
The following Lemma justifies the assumptions of admissible Fourier multipliers in Definition 1.1. i. The linear operator Moreover, for any ζ ∈ H s+1−θ , one has iii. There exists j ≥ 2 and C j such that for any ζ ∈ L 2 (R) with compact support Proof. The first result is obvious from Definition 1.1,i and the definition of Sobolev spaces. For the second, we shall first prove that the function To this aim, let us first consider G ∈ S(R) and χ a smooth cut-off function, such that χ(k) = 1 for |k| ≤ 1/2 and χ(k) = 0 for |k| ≥ 1. We decompose For the first contribution, one has and the second contribution satisfies for any j ≥ 2, , by the Cauchy-Schwarz inequality and Parseval's theorem. Thus we find, for any j ≥ 2, The same estimate applies to G(k) = kF i (k) by smooth approximation, and (2.1) follows from Definition 1.1. Now, we note that the Fourier transform of [∂ x F i , ϕ]ζ is given by By (2.1) and the mean value theorem, we find that Due to Young's inequality and Parseval's theorem, we find For the third result, let us assume at first that the kernel , remark that K i,j ∈ L 2 (R) by Definition 1.1,iii and Plancherel's theorem, and apply the Cauchy-Schwarz inequality to the convolution. If K i / ∈ L 2 (R), we obtain the result by regularizing K i (i.e. smoothly truncating F i ) and passing to the limit.
Proof. We first deal with the contribution of E(ζ) defined in (1.6). By Lemma 2.4,i we get that By Lemma 2.3,iv, one has and the triangle inequality together with Lemma 2.3,ii yields Collecting the above information, we find that . Similar estimates hold for E(ζ), and thus for E(ζ) = γE(ζ) + E(ζ).
Lemma 2.6. Let γ ≥ 0, δ > 0, µ > 1/2 and F i be admissible Fourier multipliers. Assume that, for Proof. As previously, we detail the result for E(ζ), as the similar estimate for E(ζ) is obtained in the same way. One has By Lemma 2.3, iii, and the Cauchy-Schwarz inequality, we immediately have Similarly, we find by Lemma 2.4,i and by Lemma 2.3,iv, Finally, and we conclude by Lemma 2.3, iv The result is proved.
Proof. We consider E(ζ); the corresponding expansion for E(ζ) is obtained similarly. We write Note that by the Cauchy-Schwarz inequality Applying the Cauchy-Schwarz inequality, Plancherel's theorem and the estimates (by Definition 1.1,i and ii), we deduce where the last inequality follows from Leibniz's rule and standard bilinear estimates [5, Prop. C.12].
Combining the above estimates together with similar calculations for E yields the desired estimate for |E rem |. The estimate for | dE rem (ζ), ζ | follows in the same way when decomposing and we do not detail for the sake of conciseness.
The estimate on E 3 is straightforward by the Cauchy-Schwarz inequality and applying Lemma 2.4,i and Lemma 2.3,i. We detail the estimate for |E (j) rem (ζ)|. As above, we focus on the terms involving E, the terms involving E being obtained identically. One has We estimate each bracket separately. First note that, by the Cauchy-Schwarz inequality, Lemma 2.4,i and Lemma 2.3,iii,iv one has Next consider where we used the identity ∂ x F 1 It follows from the Cauchy-Schwarz inequality, Lemma 2.4,i and Lemma 2.3,i,iv that This concludes the estimate for |E (1) rem (ζ)|. The estimate for |E (1) rem (ζ)| is obtained identically, and the one for |E (2) rem (ζ)| are obtained using similar estimates when decomposing dE(ζ), ζ given in (2.2). We do not detail for the sake of conciseness.
Periodic functional setting Given P > 0, we denote L 2 P the space of P -periodic, locally squareintegrable functions, endowed with the norm The Fourier coefficients of u ∈ L 2 P are defined bŷ We define, for s ≥ 0, The Fourier multiplier operator Λ : It maps periodic distributions to periodic distributions and we have for any s, m ∈ R. Similarly, the operators ∂ x F i extend to operators from S (R) to itself, and maps smoothly H s P into H s−1+θ P , acting on the Fourier coefficients by pointwise multiplication: For any s > 1/2, the continuous embedding holds uniformly with respect to P ≥ 1. More generally, one checks by a partition of unity argument, or repeating the proofs in the periodic setting, that Lemmata 2.1,2.2, 2.3 and as a consequence Lemmata 2.5-2.8 have immediate analogues in the periodic setting, with uniform estimates with respect to P ≥ 1, when defining

The periodic problem
Our first task is to construct periodic traveling-wave solutions with large periods by considering the periodic minimization problem corresponding to (1.10). We will use this in the next section to construct a special minimizing sequence for (1.10), which is useful when ν > 1 − θ. When θ < 1/2 and ν = 1 − θ, any minimizing sequence has the special property and therefore it is strictly speaking unnecessary to first consider the periodic minimization problem. Nevertheless, we consider here all possible parameters in order to highlight some interesting differences between the cases ν = 1 − θ and ν > 1 − θ.
We ensure that the hypotheses of Section 2, namely will be satisfied through a penalization argument. To this aim, we fix R > 0 and restrict ourselves to values q ∈ (0, q 0 ) sufficiently small so that ζ H ν P (R) ≤ 2R and (γ + δ) ζ 2 L 2 P = q ensures (by Lemma 2.1,i in the periodic setting) that ζ L ∞ < min(1, δ −1 ) − h 0 with some h 0 > 0, uniformly with respect to P ≥ P 0 sufficiently large (and likewise in the real line setting). We then define iii. For any a 1 ∈ (0, 1), there exists M 1 , M 2 > 0 and a 2 > 1 such that and the constraint set In the following we solve for q sufficiently small and P sufficiently large arg min Lemma 3.1. There exists q 0 > 0 such that for any q ∈ (0, q 0 ), the functional E P, : V P,q,2R → R is weakly lower semi-continuous, bounded from below and E P, (ζ) → ∞ as ζ H ν P 2R. In particular, it has a minimizer ζ P ∈ V P,q,2R , which satisfies the Euler-Lagrange equation for some Lagrange multiplier α P (ζ P ) ∈ R.
Proof. As explained above, we restrict ourselves to q ∈ (0, q 0 ) so that E P, is well-defined, and in particular sup , any minimizing sequence is bounded, and therefore weakly convergent (up to a subsequence) in the (reflexive) Hilbert space H ν P . We only need to show that for any ζ n ∈ V P,q,2R such that ζ n ζ weakly in H ν P , one has Notice first that by the weak lower semi-continuity of · H ν P , and since is non-decreasing, one has Now, by the Rellich-Kondrachov theorem, ζ n ζ in H ν P implies that ζ n → ζ in H s P , for s ∈ (1/2, ν); and in particular, ζ n → ζ in L ∞ . Moreover, one has sup n ζ n L ∞ < min(1, δ −1 ) since ζ n ∈ V P,q,2R , and therefore ζn 1−ζn → ζ 1−ζ in L ∞ , and ζ 1−ζ ∈ H ν P by Lemma 2.3,iii, iv. Since ζn 1−ζn is uniformly bounded in H ν P and converges in L ∞ , it follows that ζn In the same way, we find and The result follows from the weak lower semi-continuity of · L 2 P . Now we wish to prove that ζ P ∈ V P,q,R , and in particular satisfies the Euler-Lagrange equation From this point on, we heavily make use of the property (see Assumption 1.4) without explicit references in the statements.
Lemma 3.2. There exists m > 0 and q 0 > 0 such that for any q ∈ (0, q 0 ), and there exists P q > 0 such that for any P ≥ P q .
Proof. Let us first consider the case of the real line. Consider ψ ∈ C ∞ (R) with compact support, such that (γ + δ) ψ 2 L 2 = 1; and denote ψ λ : It follows that, for the case when γ − δ 2 < 0, one can choose ψ ≥ 0 and λ small enough so that If γ − δ 2 > 0, we instead let ψ ≤ 0 and again choose λ small enough so that the above holds. Now, consider φ q : In particular, for q sufficiently small, φ q H ν < R; and by Lemma 2.7 with l = 3, The result follows in the real-line setting.
We now deduce the result in the periodic setting. By taking P ≥ P q sufficiently large, we may ensure supp φ q ⊂ (−P/2, P/2), and define φ P,q = j∈Z φ q (x − jP ) ∈ H n P (n ∈ N).
One has (γ + δ) φ P,q and φ P,q H ν P < R for q sufficiently small, so that as well as E P, (φ P,q ) = E P (φ P,q ) = (γ + δ) The result is proved.
There exists q 0 > 0 such that for any q ∈ (0, q 0 ), one has where α P is defined in Lemma 3.1 and P q in Lemma 3.2.
Proof. We use the Euler-Lagrange equation satisfied by ζ P , namely This equation is well-defined in (H ν P ) and testing against ζ P ∈ H ν P yields Using the decompositions in Lemma 2.8 (changing the domain of integration to [−P/2, P/2]) yields so that one obtains the identity Let us now use Lemma 3.2, which asserts where = ν −1/2 > 0 and we use in the last estimate that ζ P ), by the interpolation estimate Lemma 2.1,i in the periodic-setting.
Combining with (3.5) yields Hence, by (3.1) Now, by Lemma 2.5 and using once again (3.5), one has Thus by Lemma 2.8, one has and |E (2) rem,P (ζ P )| + |E Plugging the above estimates into (3.4) yields and the proof is complete.
Lemma 3.4. Let q 0 , q ∈ (0, q 0 ) and P q be as in Lemma 3.3. There exists M > 0 such that one has uniformly over q ∈ (0, q 0 ), P ≥ P q and ζ in the set of minimizers of E P, over V P,q,2R .
Proof. Recall the Euler-Lagrange equation: It follows from the proof of Lemma 3.3 that for q 0 sufficiently small Thus the result is proved if ν = 1−θ, and we focus below on the situation ν > 1−θ.
In this case we obtain the desired estimate in a similar fashion after finite induction. Indeed, define r n = min(ν − (1 − θ), n(1 − θ)), n ∈ N, and assume that ζ P 2 H rn P q. Note that this is satisfied for n = 0 by assumption. We will show below that (3.7) Since 1 − θ > 0, the desired result follows by finite induction.
The first term in (3.10) is estimated by the Cauchy-Schwarz inequality and Lemma 2.2: It is clear that the second term can be estimated in the same way. Next we see that and Lemma 2.3,ii, Lemma 2.4,i and Lemma 2.1,ii yield for any 0 < < min(ν + θ − 1, 1 − θ, ν − 1/2), We therefore have that We next consider the remaining term in (3.10) and note that (3.13) First we see that, by Lemma 2.4,i, (3.14) We estimate I in (3.13) proceeding as previously: where we choose 0 < < min{ν − 1/2, 1 − θ}. The remaining terms in I are of higher order and can be estimated in the same way. Next we estimate II: where we used Lemma 2.3,i and Lemma 2.1,i. Finally consider III: proceeding as above, with > 0 sufficiently small. It is clear that the same estimate holds for dE(ζ P ), Λ 2rn ζ P , and if we use these in (3.9), we obtain Thus one may choose q sufficiently small, so that (3.7) holds. This concludes the proof.
We now collect the preceding results and deduce solutions of the non-penalized periodic problem.
Proof. From Lemma 3.4, any minimizer of E P, over V P,q,2R satisfies, for q 0 sufficiently small and P ≥ P q sufficiently large, Thus the Euler-Lagrange equation (3.3) becomes (3.21), and the control on α P is stated in Lemma 3.3. Moreover, since E P, = E P over V P,q,R , ζ P minimizes E P over V P,q,R . The theorem is proved.
Remark 3.6. If θ ∈ [0, 1/2) and ν = 1 − θ, then the functional E P is coercive on V P,q,R by Lemma 2.5, and it isn't necessary to consider the penalized functional E P, to construct periodic minimizers. Indeed, one can minimize E P over V P,q,R directly, noting that any minimizing sequence satisfies (up to subsequences) sup n ζ P,n 2 H ν P ≤ M q < R 2 if q 0 is sufficiently small.

The real line problem
The construction of a minimizer for the real line problem, (1.10), will follow from Lions' concentration compactness principle. The main difficulty consists in excluding the "dichotomy" scenario. To this aim, we shall use a special minimizing sequence (satisfying the additional estimate ζ n 2 H ν q) to show that the function q → I q is strictly subhomogeneous (see Proposition 4.2), which implies that it is also strictly subadditive (Corollary 4.3). This special subsequence is constructed from the solutions of the periodic problem, obtained in Theorem 3.5, with period P n → ∞.

A special minimizing sequence
Proof. The estimate on I q was proved in Lemma 3.2; thus we only need to construct a minimizing sequence satisfying ζ n 2 H ν ≤ M q. If ν = 1−θ, then any minimizing sequence satisfies this property as a consequence of Lemma 2.5, so we assume in the following that ν > 1 − θ. Let q 0 be sufficiently small so that Theorem 3.5 holds. By the construction of [24, p. 2918 and proof of Theorem 3.8], one obtains, for any P n sufficiently large, x n ∈ R, ζ Pn ∈ H ν Pn and ζ n ∈ H ν (R) such that where ζ Pn is defined by Theorem 3.5, supp ζ n ⊂ (−P n /2 + P 1/2 n , P n /2 − P 1/2 n ) and ζ Pn = l∈Z ζ n (· + lP n ). uniformly with respect to P n sufficiently large. By (4.4) and Theorem 3.5, one has ζ n 2 H ν ≤ M q < R 2 provided that P n is sufficiently large and q 0 is sufficiently small; and ζ n ∈ V q,R by (4.3). Thus there only remains to prove that ζ n is a minimizing sequence.
One has

Using Lemma 2.4,i, (4.4) and the Cauchy-Schwarz inequality, we deduce
Notice now that, by uniqueness of the Fourier decomposition in L 2 Pn , one has the identity and therefore, by Lemma 2.4,iii and (4.2), one has The component E satisfies the same bounds, thus we proved Now by Lemma 2.6 (which holds in the periodic setting and uniformly with respect to P > 0) with ν replaced by some ν ∈ (1/2, ν) and (4.5), one has Thus we found that There remains to prove the converse inequality. For any > 0, there exists ζ ∈ V q,R such that By the same argument as above, we construct by smoothly truncating and rescaling,ζ ∈ V q,R such that suppζ ∈ (−P , P ), and E(ζ) ≤ E(ζ) + 3 .
This concludes the proof.
The following proposition is essential to rule out the "dichotomy" scenario in Lions' concentrationcompactness principle (see below).
Proof. Let us consider ζ n the special minimizing sequence defined in Theorem 4.1. We first fix a 0 > 1, and restrict q 0 > 0 if necessary, so that for any a ∈ (1, a 0 ] and q ∈ (0, q 0 ) such that aq < q 0 , one has a Moreover, by Theorem 4.1, one has It follows that one has for q ∈ (0, q 0 ) with q 0 sufficiently small and n sufficiently large, Thus we find for n sufficiently large, We now estimate the last contribution, treating separately E The series are absolutely convergent provided q 0 is sufficiently small, and start at index k = 4, as pointed out in the proof of Lemma 2.8. We now subtract the contributions of aE (1) rem (ζ n ) and by the triangle and Cauchy-Schwarz inequalities, Using that |a 2 , Lemma 2.4,i, that H ν is a Banach algebra as well as the continuous embedding H ν ⊂ L ∞ , we find that one can restrict q 0 > 0 such that the above series is convergent and yields uniformly over q ∈ (0, q 0 ) and a ∈ (1, a 0 ] such that aq < q 0 . Plugging this estimate in (4.6) and restricting q 0 if necessary, we deduce I aq < aI q for 0 < q < aq < q 0 , a ∈ (1, a 0 ].
Consider now the case when a ∈ (1, a p 0 ] for an integer p ≥ 2. Then a 1 p ∈ (1, a 0 ] and so The result is proved.
By a standard argument, Proposition 4.2 induces the subadditivity of the map q → I q .

Concentration-compactness; proof of Theorem 1.6
We now prove Theorem 1.6. Let us first recall Lions' concentration compactness principle [32]. We shall apply Theorem 4.4 to e n = γ ζ 2 where ζ n is a minimizing sequence of E over V q,R with sup n ζ n 2 H ν < R 2 . Such a sequence is known to exist provided that q ∈ (0, q 0 ) is sufficiently small, by Theorem 4.1 (and any minimizing sequence is valid when ν = 1 − θ, by Lemma 2.5; see Remark 3.6). The choice of density is inspired by the recent paper [2], and allows (contrarily to the more evident choice e n = ζ 2 n ) to show, when ν = 1−θ, that the constructed limit satisfies E(η) = I q and is therefore a solution to the constrained minimization problem (1.10). Notice that R e n dx = E(ζ n ) → I q (n → ∞) and that there exists a constant C such that for any interval J ⊆ R. We exclude the two first scenarii in Lemmata 4.5 and 4.6, below. Thus the concentration scenario holds and, using (4.7), we find that there exists {x n } n∈N ⊂ R such that for any > 0, there exists r > 0 with η n L 2 (|x|>r) < , where {η n } n∈N def = {ζ n (· + x n )} n∈N . Since sup n∈N η n H ν (R) < R, there exists η ∈ H ν (R) satisfying η H ν (R) < R and η n η weakly in H ν (R) (up to the extraction of a subsequence). Increasing r if necessary, we have η L 2 (|x|>r) < .
In either case we have that E(η) = I q .
The constructed function η ∈ H ν (R) is therefore a solution to the constrained minimization problem (1.10). In particular, it solves the Euler-Lagrange equation (1.11) with α < 0 provided that q ∈ (0, q 0 ) is sufficiently small (proceeding as in Lemma 3.3), and therefore satisfies (1.8) with This proves the first item of Theorem 1.6, as well as the second item -except for the strong convergence in H ν (R) when ν = 1−θ > 1/2. This result follows from the fact that weak convergence together with convergence of the norm implies strong convergence in a Hilbert space (applied to . There remains to prove the estimates of the third item. Proceeding as in Lemma 3.4, we find uniformly over the minimizers of E over V q,R . Moreover, by Lemma 2.8, one has where E 2 (ζ) ≥ q and and the result follows. Theorem 1.6 is proved.
Hence one has for n sufficiently large from which, using (4.7), it follows that "vanishing" cannot occur.
Proof. We denote by χ ∈ C ∞ (R + ) a non-increasing function with and such that where χ 1 and χ 2 are smooth. For instance, set χ(r) = 1 − (1 − χ 2 (r)) 2 with χ ∈ C ∞ (R + ) nonincreasing and satisfying (4.8). Define η n = ζ n (· + x n ), and noting that After possibly extracting a subsequence, we can assume that with q * ∈ [0, q]. For n sufficiently large, one has N n > N n /2 > 2M n > M n and therefore We claim that E(η (1) n ) → I * . To show this, note that We next find that Noting that by Lemma 2.3 i and Lemma 2.1,i, and using η On the other hand, as the remainder term is bounded by a constant times Mn≤|x−xn|≤Nn e n dx. An analogous argument for E reveals that We next claim that q * > 0. Indeed, if q * = 0, we set By (4.10) and since q * = 0, one has c n → 1. Thus we note n H ν → 0 by Lemma 2.6 and lim sup n→∞ η (2) n H ν < R, resulting in the contradiction I q ≤ E( η (2) n ) → I q − I * < I q as n → ∞. We obtain a similar contradiction involving η (1) n and (4.9) if we assume that q * = q. Hence, 0 < q * < q. In view of the above, we can rescale so that (γ + δ) η (1) n 2 L 2 = q and (γ + δ) η (2) n 2 L 2 = q − q * for any n ∈ N. One easily checks that Thus we arrive at the following contradiction to Corollary 4.3: This concludes the proof of Lemma 4.6.

Long-wave asymptotics
In this section we prove that the solutions of (1.8) obtained in Theorem 1.6 are approximated by solutions of the corresponding KdV equation in the long-wave regime, i.e. letting q → 0 in the constrained minimization problem (1.10). Indeed, if we introduce the scaling in (1.11) and denote α + 1 = α 0 q 2 3 , then we find that the leading order part of the equation as q → 0 is Recall (see e.g. [1]) that ξ ∈ L 2 (R) satisfying (5.2) uniquely defines (up to spatial translation) a solitary-wave solution of the KdV equation, with explicit formula Equation (5.2) can also be obtained as the Euler-Lagrange equation associated with the minimizer of the scalar functional E KdV (consistently with Lemma 2.7) which is (5.2) with α 0 the Lagrange multiplier. Testing the constraint (γ + δ) ξ 2 L 2 = 1 with the above explicit formula, we find that (5.4)

Additional computations show that
We aim at proving that the variational characterization of (5.2), and therefore its explicit solutions, approximate (after suitable rescaling) the corresponding one of (1.8), namely (1.10), in the limit q → 0.

Refined estimates
We start by establishing estimates on ζ ∈ D q,R the set of minimizers of E over V q,R , as provided by Theorem 1.6. Here and below, we rely on extra assumptions on the Fourier multipliers, which are assumed to be strongly admissible, in the sense of Definition 1.2.
Lemma 5.1. There exists q 0 > 0 such that ζ ∈ H s for any s ≥ 0, and there exists M s > 0 such that ζ 2 H s ≤ M s q uniformly for q ∈ (0, q 0 ) and ζ ∈ D q,R .
Proof. Once the regularity property ζ ∈ H s has been established, the corresponding estimate is obtained as in the proof of Lemma 3.4, thus we focus only on the regularity issue. This follows from the Euler-Lagrange equation (1.11) and elliptic estimates. However, the ellipticity property is not straightforward to ascertain when γ = 0, and we will make use of paradifferential calculus. These tools are recalled in Appendix A.
The following lemma shows that the minimizers of E over V q,R , as provided by Theorem 1.6, scale as (5.1). Lemma 5.3. There exists q 0 > 0 and C > 0 such that the estimates hold uniformly for q ∈ (0, q 0 ) and ζ ∈ D q,R , the set of minimizers of E over V q,R .
Proof. Let ζ be minimizer over V q,R . Since 2(γ + δ)αζ + dE(ζ) = 0, we get from Lemma 2.8 that We also have that In frequency space equation (5.18) becomes By using the estimate for α in Theorem 1.6, we deduce .
The estimates follow from (5.19) and a suitable decomposition into high-and low-frequency components. In order to estimate the right-hand-side, we heavily make use of Lemma 5.1: ζ 2 H n q for all n ∈ N. This will be used again throughout the proof without reference.
We first deduce from Lemma 2.3 that By the definition of admissible Fourier multipliers in (1.1), there exists c 0 , k 0 > 0 such that We also assume that F(k) > 0, and therefore there exists c 0 > 0 such that As a consequence, we have (5.20) into high-and low-frequency components and estimate each part. For the low frequency part we have For the high-frequency part q.
Combining the above estimates in (5.20) gives us the inequality (5.15).
Let us now turn to (5.16). By (5.19) we have We estimate the low-frequency part as above: As for the high frequency part, we notice that where we used (5.15). It follows that Combining the high and low frequency estimates into (5.21) gives us hence, for q 0 sufficiently small we get (5.16).
We conclude with the proof of (5.17). By (5.19) we have Now we remark that by the Cauchy-Schwarz inequality and integration by parts, one has Since ζ L ∞ and ∂ x ζ L 2 satisfy similar estimates by (5.15) and (5.16), we have Thus, proceeding as above, we find k0 −k0 Plugging these estimates into (5.22) and restricting q ∈ (0, q 0 ) if necessary yields (5.17), and the proof is complete.

Convergence results
; proof of Theorem 1.7 We are now in position to relate the minimizers of E in D q,R with the corresponding solution of the KdV equation. We first compare Lemma 5.4. There exists q 0 > 0 such that the quantities I q and I KdV satisfy , uniformly over minimizers of E in V q,R , (5.23) uniformly over q ∈ (0, q 0 ).
Proof. Recall that, from Lemma 2.7, one has for any ζ ∈ H 2 , . Let ζ be a minimizer of E in V q,R and note that ζ ∈ H 2 by Lemma 5.1. Using Lemma 5.3 and (5.25), we obtain By Theorem 5.5 there exists x ζ such that S −1 and therefore dE(ζ), ζ = 2q + q

Numerical study
In this section, we provide numerical illustrations of our results as well as some numerical experiments for situations which are not covered by our results. We first describe our numerical scheme, before discussing the outcome of these simulations.
Description of the numerical scheme Our numerical scheme computes solutions for (1.8) for given value of c (and hence does not not follow the minimization strategy developed in this work). Because we seek smooth localized solutions and our operators involve Fourier multipliers, it is very natural to discretize the problem through spectral methods [43]. We are thus left with the problem of finding a root for a nonlinear function defined in a finite (but large) dimensional space. To this aim, we employ the Matlab script fsolve which implements the so-called trust-region dogleg algorithm [19] based on Newton's method. For an efficient and successful outcome of the method, it is important to have a fairly precise initial guess. To this aim, we use the exact solution of the Green-Naghdi model, which is either explicit (in the one-layer situation [41]) or obtained as the solution of an ordinary differential equation (in the bi-layer situation [37,17]) that we solve numerically. Our solutions are compared with the corresponding ones of the full Euler system. To compute the latter, we use the Matlab script developed by Per-Olav Rusås and documented in [26] in the bilayer configuration while in the one-layer case, the Matlab script of Clamond and Dutykh [18] offer faster and more accurate results (although limited to relatively small velocities).
As the velocity approaches c max , the solitary waves broadens and its mass keeps increasing. These type of profiles or often refered to as "table-top" profiles, and lead to bore profiles in the limit c → c max . When the velocity is small the numerically computed solitary wave solutions of the bilayer original (F i = 1) and full dispersion (F i = F imp i ) Green-Naghdi systems and the one of the water waves systems (and to a lesser extent the KdV model) agree, so that the curves corresponding to the  three former models are indistinguishable in see Figure 2(a). For larger velocities, as in Figure 2(b), the numerically computed solitary wave solutions of the Green-Naghdi and water waves systems is very different from the sech 2 profile of the solitary wave solution to the Korteweg-de Vries equation. It is interesting to see that both the original and full dispersion Green-Naghdi models offer good approximations, even in this "large velocity" limit (the normalized l 2 difference of the computed solutions is ≈ 2.10 −3 in both cases). This means that the internal solitary wave keeps a long-wave feature even for large velocities. These observations were already documented and corroborated by laboratory experiments in [36,26,11].
One-layer setting In the one-layer setting, the script by Clamond and Dutykh [18] allows to have a very precise numerical computation of the water waves solitary solution, from which the    numerical solutions of the Green-Naghdi models can be compared. In this setting, namely γ = 0 and δ = 1, we have an explicit solution for the Green-Naghdi model [41]: In Figure 3, we compute the solitary waves for our models with different (small) values of the velocity, rescaled by S −1 KdV . One clearly sees, as predicted by Theorem 1.7 and the above formula, that the solitary waves converge towards ξ KdV after rescaling, as c 1. One also sees that the water waves solution is closer to the one predicted by the model with full dispersion than the original Green-Naghdi model. Figure 4 shows that the convergence rate is indeed quadratic for the full dispersion model whereas it is only linear for the original Green-Naghdi model (and therefore only qualitatively better than the KdV model).

A Paradifferential calculus
The definitions and properties below are collected from [35]; see also [13,5] for relevant references.
Below, we use an admissible cut-off function ψ in the sense of [35, Definition 5.1.4] and define paradifferential operators as follows (the constant factor depends on the choice of convention for the Fourier transform). where ψ(D, ξ) is the Fourier multiplier associated with ψ(η, ξ) (here, ξ is a parameter). The operator is defined for u ∈ H s (R) by density and continuous linear extension.
The following lemma is a direct application of the above definitions [35,Theorem 5.1.15].
Lemma A.3. For any r ≥ 0 and a ∈ Γ m r ⊂ Γ m 0 , and for all s ∈ R, the operator T a extends in a unique way to a bounded operator from H s+m to H s .
If a(ξ) is a symbol independent of x, then T a = a(D), the corresponding Fourier multiplier.
The main tool we use is the following composition property [35, Theorem 6.1.1] Lemma A.4. Let a ∈ Γ m r and b ∈ Γ m r where 0 < r ≤ 1. Then ab ∈ Γ m+m r and T a T b − T ab is a bounded operator from H s+m+m −r to H s , for any s ∈ R.
Of particular interest is the case when the symbol a(x) ∈ L ∞ is independent of ξ. The admissible cut-off function can be constructed so that the paraproduct T a u corresponds to a standard Littlewood-Paley decomposition of the product au. This allows to show that au−T a u is a smoothing operator provided that a is sufficiently regular.
Lemma A.5. Let v ∈ H s and u ∈ H t , and r ≥ 0. Then uv−T v u ∈ H r provided that s+t ≥ 0, s ≥ r and s + t > r + 1/2.
The definitions of the paraproduct in [13] and [35] differ slightly but it is not hard to show that [13, Theorem 2.4.1] still holds for the paraproduct as it is defined in [35], and Lemma A.5 follows directly from this theorem.