Existence and Orbital Stability of Cnoidal Waves for a 1d Boussinesq Equation

In this paper we shall study nonlinear stability of periodic solitary waves of the nonlin-ear one-dimensional Boussinesq type equation known as the one-dimensional Benney-Luke equation) Φ tt − Φ xx + aΦ xxxx − bΦ xxtt + Φ t Φ xx + 2Φ x Φ xt = 0, (1) where a and b are positive numbers such that a − b = σ − 1/3 (σ is named the Bond number). We will consider the study of periodic travelling waves for equation (1) of the form Φ(x, t) = φ c (x − ct), where satisfies 0 < c 2 < min(1, a/b). In this case, the periodic travelling wave profile φ c should satisfy the equation (c 2 − 1)φ c + (a − bc 2)φ c − 3c 2 (φ c) 2 = A, where A is an integration constant. We will show that periodic travelling wave solutions are orbitally stable when 0 < c < 1 < a/b, which corresponds to the case Bond number σ > 1/3 and when 0 < c < c * < a/b < 1, which corresponds to the case Bond number σ < 1/3. The Boussinesq type equation (1) fits into the class of abstract Hamiltonian system studied by Grillakis, Shatah and Strauss in [4]. The condition of orbital stability and instability is characterized by the convexity of the function defined by d(c) = 1 2


Introduction
In this paper, we consider the existence of periodic travelling-waves solutions and the study of nonlinear orbital stability of these solutions for the one-dimensional Boussinesqtype equation where a and b are positive numbers.
One can see that this Boussinesq-type equation is a rescaled version of the onedimensional Benney-Luke equation 2 International Journal of Mathematics and Mathematical Sciences which is derived from evolution of two-dimensional long water waves with surface tension.In this model, Φ(x,t) represents the nondimensional velocity potential at the bottom fluid boundary, μ represents the long-wave parameter (dispersion coefficient), represents the amplitude parameter (nonlinear parameter), and a − b = σ − 1/3, with σ being named the Bond number which is associated with surface tension.An important feature is that the Benney-Luke equation (1.2) reduces to the Kortewegde-Vries equation (KdV) when we look for waves evolving slowly in time.More precisely, when we seek for a solution of the form Φ(x,t) = f (X,τ), (1.3) where X = x − t and τ = t/2.In this case, after neglecting O( ) terms, η = f X satisfies the KdV equation It was established by Angulo [1] (see also [2]) and Angulo et al. [3] that cnoidal waves solutions of mean zero for the KdV equation exist and they are orbitally stable in H 1 per [0,T 0 ].The proof of orbital stability obtained by Angulo et al. was based on the general result for stability due to Grillakis et al. [4] together with the classical arguments by Benjamin in [5], Bona [6], and Weinstein [7] (see also Maddocks and Sachs [8]).This approach is used for obtaining stability initially in the space of functions of mean zero, ᐃ 1 = q ∈ H 1 per 0,T 0 : T0 0 q(y)dy = 0 . (1.5) The reason to use the space ᐃ 1 to study stability is rather simple.Cnoidal wave solutions are not critical points of the action functional on the space H 1 per ([0,T 0 ]), however on the space ᐃ 1 cnoidal waves solutions are characterized as critical points of the action functional, as required in [4,7].The meaning of this is that the mean-zero property makes the first variation effectively zero from the point of view of the constrained variational problem, and so the theories in [4][5][6][7] can be applied.
Due to the strong relationship between the Benney-Luke equation (1.1) and the KdV equation (1.4), we are interested in establishing analogous results in terms of existence and stability of periodic travelling-waves solutions as the corresponding results obtained by Angulo et al. in the case of the KdV equation.More precisely, we want to prove existence of periodic travelling-wave solutions for the Benney-Luke equation (1.1) and to study the orbital stability of them.
In this paper, we will study travelling-waves for (1.1) of the form Φ(x,t) = φ c (x − ct) such that ψ c ≡ φ c is a periodic function with mean zero on an a priori fundamental period and for values of c such that 0 < c 2 < min{1,a/b}.So, φ c will be a periodic function.The profile φ c has to satisfy the equation where A 0 is an integration constant.So, by following the paper of Angulo et al., we obtain that ψ c is of type cnoidal and it is given by the formula . Moreover, for T 0 appropriate, this solution has minimal period T 0 and mean zero on [0,T 0 ].So, we obtain by using the Jacobian Elliptic function dnoidal, dn(•;k), that (1.6) has a periodic solution of the form for appropriate constants M and L 0 .We will show that the periodic travelling-wave solutions φ c are orbitally stable with regard to the periodic flow generated by (1.1) provided that 0 < |c| < 1 < √ a/b, which corresponds to the Bond number σ > 1/3 and when for θ small, 0 < |c| < c * + θ < √ a/b < 1, which corresponds to the Bond number σ < 1/3.Here c * is a specific positive constant (see Theorem 4.3).These conditions of stability are needed to assure the convexity of the function d defined by where ψ c = φ c and φ c is a travelling-wave solution of (1.6) of cnoidal type, with speed c and period T 0 .Unfortunately from our approach, it is not clear if our waves are stable for the full interval 0 We recall that in a recent paper, Quintero [9] established orbital stability/instability of solitons (solitary wave solutions) for the Benney-Luke equation (1.1) for 0 < c 2 < min{1,a/b} by using the variational characterization of d.Orbital stability of the soliton was obtained when 0 < c < 1 < √ a/b and orbital instability of the soliton was obtained when 0 < c 0 < c < √ a/b < 1 for some positive constant c 0 .Our result of stability of periodic travelling-wave solutions for (1.1) follows from studying the same problem to the Boussinesq system associated with (1.1), where q = Φ x , r = Φ t , and B = 1 − b∂ 2 x .More exactly, we will obtain an existence and uniqueness result for the Cauchy problem associated with system (1.10) in H 1 per ([0,T 0 ]) × H 1 per ([0,T 0 ]) and also that the periodic travelling-wave solutions (ψ c ,−cψ c ) are orbitally stable by the flow of (1.10) with periodic initial disturbances restrict to the space ᐃ 1 × H 1 per ([0,T 0 ]).In this point, we take advantage of the Grillakis et al.'s stability theory.More concretely, the stability result relies on the convexity of d defined in (1.9) and on a complete spectral analysis of the periodic eigenvalue problem of the linear operator which is related with the second variation of the action functional associated with system (1.10).We will show that ᏸ cn has exactly its three first eigenvalues simple, the eigenvalue zero being the second one with eigenfunction ψ c and the rest of the spectrum consists of a discrete set of double eigenvalues.This spectral description follows from a careful analysis of the classical Lame periodic eigenvalue problem where K = K(k) represents the complete elliptic integral of first kind defined by . (1.13) We will show here that (1.12) has the three first eigenvalues simple and the remainder of eigenvalues are double.The exact value of these eigenvalues as well as its corresponding eigenfunctions are given.We note that our stability results cannot be extended to more general periodic perturbations, for instance, by disturbances of period 2T 0 .In fact, it is well known that problem (1.12) has exactly four intervals of instability, and so when we consider the periodic problem in (1.12) but now with boundary conditions Λ(0) = Λ(4K(k)), Λ (0) = Λ (4K(k)), we obtain that the seven first eigenvalues are simple.So, it follows that the linear operator ᏸ cn with domain H 1 per ([0,2T 0 ]) will have exactly three negative eigenvalues which are simple.Hence, since the function d defined above is still convex with the integral in (1.9) defined in [0,2T 0 ], we obtain that the general stability approach in [4,10] cannot be applied in this case.
This paper is organized as follows.In Section 2, we establish the Hamiltonian structure for (1.10).In Section 3, we build periodic travelling-waves of fundamental period T 0 using Jacobian elliptic functions, named cnoidal waves, with the property of having mean zero in [0,T 0 ].We also prove the existence of a smooth curve of cnoidal wave solutions for (1.10) with a fixed period T 0 and the mean-zero property in [0, T 0 ].In Section 4, we study the periodic eigenvalue problem associated with the linear operator in (1.11).We also prove the convexity of the function d in a different fashion as it was done by Angulo et al. in [3,KdV equation (1.3)].In Section 5, we discuss the main issue regarding orbital stability for the Boussinesq system (1.10).This requires proving the existence and uniqueness results of global mild solutions for this system, and applying Grillakis, Shatah, and Strauss stability methods, as done in [3].Finally, in Section 6, we state the orbital stability of periodic wave solutions of the Benney-Luke equation, by showing the equivalence between the Cauchy problem for the Benney-Luke equation (1.1) and the Boussinesq system (1.10).

Hamiltonian structure
The Boussinesq system (1.10) can be written as a Hamiltonian system in the new variables (q, p) ≡ q,Br + 1 2 q 2 (2.1) as with A = 1 − a∂ 2 x and B = 1 − b∂ 2 x .This system arises as the Euler-Lagrange equation for the action functional where the Lagrangian ᏽ and the Hamiltonian are given, respectively, by ( In this way, we obtain the canonical Hamiltonian form and the Hamiltonian system in the variable V = ( q p ) as We observe that the Hamiltonian in (2.4) is formally conserved in time for solutions of system (2.2), since (2.7) So, the Hamiltonian 6 International Journal of Mathematics and Mathematical Sciences associated to (1.10) is formally conserved in time.Moreover, since the Hamiltonian is translation-invariant, then by Noether's theorem there is an associated momentum functional ᏺ which is also conserved in time.This functional has the form (2.9) Next we are interested in finding periodic travelling-waves solutions for system (1.10), in other words, solutions of the form (q,r) = (ψ(x − ct),g(x − ct)).By substituting, we have that the couple (ψ,g) satisfies the nonlinear system (2.10) with A 0 and Ꮽ integration constants.Now, since our approach of stability is based on the context of the stability theory of Grillakis et al. (see proof of our Theorem 5.1), we need to show that (ψ,g) satisfies the equation with therefore it follows from (2.10) that we must have A 0 = 0.In other words, we have to solve the system On the other hand, if we look for periodic travelling-wave solutions Φ(x,t) = φ(x − ct) for (1.1), then η ≡ φ has mean zero and satisfies equation where Ꮽ 1 is an integration constant.Note that if η is a periodic solution with mean zero on [0,L], then Ꮽ 1 = 0 and φ is periodic of period L. As a consequence of this, we have to look for periodic solutions ψ with mean zero for (2.15), and so Ꮽ = 0.This simple observation shows that V c = ( ψc −cψc ) cannot be a critical point of the action functional Ᏺ.This shows the need to adapt Grillakis et al.'s stability result to the present case (see Theorem 5.1).More precisely, we need in our stability theory to have J. Angulo and J. R. Quintero 7

Existence of a smooth curve of cnoidal waves with mean zero
In this section, we are interested in building explicit travelling-wave solutions for (1.1) and (1.10).Our analysis will show that the initial profile of φ c can be taken as periodic or not, with a periodic derivative ψ c of cnoidal form.Our main interest here will be the construction of a smooth curve c → ψ c of periodic travelling-wave with a fixed fundamental period L and mean zero on [0,L], so we will have that φ c is periodic.More precisely, our main theorem is the following.Theorem 3.1.For every T 0 > 0, there are smooth curves of solutions of the equation where each ψ c has fundamental period T 0 and mean zero on [0,T 0 ].Moreover, there are smooth curves and ψ c has the cnoidal form The proof of Theorem 3.1 is based on the techniques developed by Angulo et al. in [3], so we use the implicit function theorem together with the theory of complete elliptic integrals and Jacobi elliptic functions.We divide the proof of Theorem 3.1 in several steps.The following two subsections will show the construction of cnoidal waves solutions with mean zero.Sections 3.3 and 3.4 will give the proof of the theorem.Section 3.5 gives a more careful study of the modulus function k.

Building periodic solution.
One can see directly that travelling-waves solutions for (1.1), that is, solutions of the form Φ(x,t) = φ(x − ct), have to satisfy the equation Integrating over [0,x], we find that φ satisfies equation and so ψ ≡ φ satisfies equation where A 0 is an integration constant.Note that for periodic travelling-wave solution φ with a specific period L, we have that ψ has mean zero on [0,L], therefore A 0 needs to be nonzero.Moreover, if ψ is a periodic solution with mean zero on [0, L], then A 0 = 0 and φ is periodic of period L.
Next we scale function ψ.Defining we have that ϕ satisfies the ordinary differential equation with A ϕ = −3cA 0 .For 0 < c 2 < 1, a class of periodic solutions to (3.9) called cnoidal waves was found already in the 19th century work of Boussinesq [11,12] and Korteweg and de Vries [13].It may be written in terms of the Jacobi elliptic function as where Here is a classical argument leading exactly to these formulas.Fix c ∈ (−1,1) and multiply (3.9) by the integrating factor ϕ , a second exact integration is possible, yielding the first-order equation where B ϕ is another constant of integration.Suppose ϕ to be a nonconstant, smooth, periodic solution of (3.12).The formula (3.12) may be written as If F ϕ has only one real root β, say, then ϕ (z) can vanish only when ϕ(z) = β.This means that the maximum value of ϕ which takes on its period domain [0, T] is the same, with T = T/θ, as its minimum value there, and so ϕ is constant, contrary to presumption.Therefore F ϕ must have three real roots, say β 1 < β 2 < β 3 (the degenerate cases will be considered presently).Note J. Angulo and J. R. Quintero 9 that for the existence of these different zeros, it is necessary to have that (1 − c 2 ) 2 + 2A ϕ > 0. So, we have where we have incorporated the minus sign into the third factor.Of course, we must have (3.15) It follows immediately from (3.13)-(3.14) that ϕ must take values in the range where . By translation of the spatial coordinates, we may locate a maximum value of ρ at x = 0.As the only critical points of ρ for values of ρ in [η 2 ,1] are when ρ = η 2 < 1 and when ρ = 1, it must be the case that ρ(0) = 1.One checks that ρ > 0 when ρ = η 2 and ρ < 0 when ρ = 1.Thus it is clear that our putative periodic solution must oscillate monotonically between the values ρ = η 2 and ρ = 1.A simple analysis would now allow us to conclude that such periodic solutions exist, but we are pursuing the formula (3.10), not just existence.Change variables again by letting with ρ(0) = 0 and ρ continuous.Substituting into (3.16)yields the equation To put this in standard form, define Of course 0 ≤ k 2 ≤ 1 and > 0. We may solve for ρ implicitly to obtain The left-hand side of (3.20) is just the standard elliptic integral of the first kind (see [14]).Moreover, the elliptic function sn(z;k) is, for fixed k, defined in terms of the inverse of the mapping ρ → F(ρ;k).Hence, (3.20) implies that sinρ = sn( x;k), (3.21) and therefore As sn 2 +cn 2 = 1, it transpires that ρ = η 2 + (1 − η 2 )cn 2 ( √ x;k), which, when properly unwrapped, is exactly the cnoidal wave solution (3.10), or ψ c has the form Next we consider the degenerate cases.First, fix the value of c and consider whether or not periodic solutions can persist if As ϕ can only take values in the interval [β 2 ,β 3 ], we conclude that the second case leads only to the constant solution ϕ(x) ≡ β 2 = β 3 .Indeed, the limit of (3.10) as β 2 → β 3 is uniform in x and is exactly this constant solution.If, on the other hand, c and β 1 are fixed, say, β 2 ↓ β 1 and , the elliptic function cn converges, uniformly on compact sets, to the hyperbolic function sech and (3.10) becomes, in this limit, So, by returning to the original function ψ, we obtain the standard solitary-wave solution of speed 0 < c 2 < min{1,a/b} of the Benney-Luke equation (see [9]).Next, by returning to original variable φ c , we obtain after integration and using the formula (see [14]) where M is an integration constant and physically amounts to demanding that the wavetrain has the same mean depth as does the undisturbed free surface (this is a very good presumption for waves generated by an oscillating wavemaker in a channel, e.g., as no mass is added in such a configuration).
Wavetrains with non-zero mean are readily derived from this special case as will be remarked presently.Let a phase speed c 0 be given with 0 < c 2 0 < min{1,a/b}, and consider four constants β 1 , β 2 , β 3 and k as in (3.10).The complete elliptic integral of the first kind (see [1,Chapter 2], or [14]) is the function K(k) defined by the formula (3.31) The fundamental period of the cnoidal wave ϕ c0 in (3.10) is T c0 = T ϕc 0 , with K as in (3.31).The period of cn is 4K(k) and cn is antisymmetric about its half period, from which (3.32) follows.
The condition of mean zero of ϕ c0 over a period [0,T c0 ] is easily determined to be Simple manipulations with elliptic functions put (3.33) into a more useful form, namely where k = (1 − k 2 ) 1/2 and E(k) is the complete elliptical integral of the second kind defined by the formula Thus the zero-mean value condition is exactly (see [14]), the relation (3.36) has the equivalent form We note that by replacing K(k) and E(k), we have that (3.38) is equivalent to have A(β 2 ,β 3 ) = 0, where ).Now we are in a good position to prove that under some consideration, ϕ c0 has mean zero.
Proof.We proceed as by Angulo et al. in (see [3]).Let Ω ⊂ R 4 be the set defined by , and let Φ : Ω → R 3 be the function defined by where (3.56) From Theorem 3.2, Φ(β 1 ,β 2 ,β 3 ,c 0 ) = 0.The first observation is that J. Angulo and J. R. Quintero 15 On the other hand, a direct computation shows that and that If we assume that α 1 K(k) + (α 3 − α 1 )E(k) = 0, then using that we obtain the following formulas: (3.61) Similarly, under such assumptions, we conclude that (3.62) Replacing these in previous equalities, we have, for ( − → α ,c) satisfying and that (3.65) In particular, for ( − → β ,c 0 ), we obtain that λ(c 0 ) = 1, so it follows that and that with k 2 1 = (β 3 − β 2 )/(β 3 − β 1 ).The properties of the cnoidal wave ϕ c0 (•, Using previous calculation, the Jacobian determinant of Φ(•,•,•,c) at ( As a consequence of this, the implicit function theorem implies the existence of a function Π from a neighborhood of I(c 0 ) of c 0 to a neighborhood of (β 1 ,β 2 ,β 3 ) satisfying the first part of the lemma.The second part of the lemma is immediate.

Existence of curve of solutions.
In this subsection as a consequence of the previous results, we establish the proof of Theorem 3.1.
Proof of Theorem 3.1.We start by proving the existence of a smooth curve of cnoidal waves solutions for (3.7) with a fixed period a − bc 2 0 T c0 and with mean zero on [0, a − bc 2 0 T c0 ].In fact, let ϕ c0 (•,α i ) be the cnoidal wave determined by Lemma 3.3 with 2 , and mean zero on [0, T ϕc ].Moreover, it is not hard to see that ϕ c (•,α i ) satisfies the differential equation where Next, we obtain a smooth curve of solution for (3.7).From (3.8), define Then it is easy to see that ψ c has period T 0 = a − bc 2 0 T c0 and mean zero on [0,T 0 ].On the other hand, from (3.71) and (3.72), it follows that ψ c satisfies the differential equation (3.2) with A ψc given by (3.3).Now, the regularity of the map c → ψ c follows from the properties of ϕ c0 and α i .Moreover, from Theorem 3.2, we obtain that T c0 = T c0 (β 3 ) can be taken arbitrarily in the interval (0, +∞), and so the solution ψ c can be taken with an arbitrary period T 0 with T 0 ∈ (0,+∞).Finally, by uniqueness of the map Π in Lemma 3.3 and by c 0 being arbitrary with 0 < c 2 0 < min{1,a/b}, we can conclude that the map Π can be extended such that we obtain the following smooth curves of cnoidal wave solutions to (3.7): with an arbitrary period T c0 and mean zero on [0,T c0 ].This finishes the proof of Theorem 3.1.

Monotonicity of the modulus k.
In this subsection, we show some properties of the modulus k determined in Lemma 3.3.We start by recalling that for every c ∈ I(c 0 ), Φ(Π(c),c) = (0,0,0), where Π(c) = (α 1 (c),α 2 (c),α 3 (c)).As done above, from formulas (3.57), (3.64), and (3.65) it is not hard to see that As a consequence of this, we conclude that with α = (α 1 ,α 2 ,α 3 ).Next by finding the inverse matrix in (3.76), we obtain that Using this fact, we are able to establish that k is a monotone function, depending on the wave speed.

Spectral analysis and convexity
In this section, attention is turned to set the main tools to be used in order to establish stability of the cnoidal-wave solutions (ψ c ,−cψ c ) determined by Theorem 3.1 for system (1.10).

Spectral analysis of the operator
. By Theorem 3.1, we consider for L = T 0 > 0 the smooth curve of cnoidal wave c → ψ c ∈ H 1 per ([0,L]) with fundamental period L. As it is well known, the study of the periodic eigenvalue problem for the linear operator ᏸ cn considered on [0,L] is required in the stability theory.The spectral problem in question is where c is fixed such that 0 < c 2 < min{1,a/b}.The following result is obtained in this context.
Theorem 4.1.Let ψ c be the cnoidal wave solution given by Theorem 3.1.Then the linear operator defined on H 2 per ([0,L]) has exactly its three first eigenvalues λ 0 < λ 1 < λ 2 simple.Moreover, ψ c is an eigenfunction with eigenvalue λ 1 = 0.The rest of the spectrum is a discrete set of eigenvalues which is double.The eigenvalues only accumulate at +∞.
Proof.From the theory of compact symmetric operators applied to the periodic eigenvalue problem (4.1), it is known that the spectrum of ᏸ cn is a countable infinity set of eigenvalues with where double eigenvalue is counted twice and λ n → ∞ as n → ∞.Now, from the Floquet theory [15], with the eigenvalue periodic problem, there is an associated eigenvalue problem, named semiperiodic problem in [0, L], As in the periodic case, there is a sequence of eigenvalues where double eigenvalue is counted twice and μ n → ∞ as n → ∞.So, for the equation we have that the only periodic solutions, f , of period L correspond to γ = λ j for some j whilst the only periodic solutions of period 2L are either those associated with γ = λ j , but viewed on [0,2L], or those corresponding γ = μ j , but extended as Next, from oscillation theory [15], we have that the sequences of eigenvalues (4.3) and (4.5) have the following property: Now, for a given value γ, if all solutions of (4.6) are bounded, then γ is called a stable value, whereas if there is an unbounded solution, γ is called unstable.The open intervals (λ 0 ,μ 0 ),(μ 1 ,λ 1 ),(λ 2 ,μ 2 ),(μ 3 ,λ 3 ),... are called intervals of stability.The endpoints of these intervals are generally unstable.This is always so for γ = λ 0 as λ 0 is always simple.The intervals (−∞,λ 0 ),(μ 0 ,μ 1 ),(λ 1 ,λ 2 ),(μ 2 ,μ 3 ),..., and so on are called intervals of instability.
Of course, at a double eigenvalue, the interval is empty and omitted from the discussion.
On the other hand, ( Since γ 0 < γ 1 = 4(1 + k 2 ), we have that λ 0 < 0 and it is the first negative eigenvalue of ᏸ cn with eigenfunction χ 0 (x) = Λ 0 (x/η) which has no zeros.We also have that γ 1 < γ 2 for any k ∈ (0,1), then we get from (4.9) that This implies that λ 2 is the third eigenvalue of ᏸ cn with eigenfunction χ 2 (x) = Λ 2 (x/η), which has exactly 2 zeros in [0, L).On the other hand, it can be shown that the first two eigenvalues of Lame's equation in the semiperiodic case are with corresponding eigenfunctions with exactly one zero in [0,2K), (4.17) But we have that μ 0 < μ 1 < γ 1 = 4 + 4k 2 and that As a consequence of this, we found that the first three intervals of instability of ᏸ cn are We also have, eigenvalues with corresponding eigenfunctions respectively, with exactly three zeros in [0,2K).Finally, we conclude from (4.18) that the last interval of instability for ᏸ cn is (μ 2 ,μ 3 ).

Convexity of d(c).
As we showed in Section 2, cnoidal wave solutions ψ c are characterized in such a way that the couple with Ꮽ being a nonzero number.Now we consider the study of the convexity of the function d defined by where the cnoidal wave solution ψ c is given by Theorem 3.1.If we differentiate (4.23) with respect to c, we get that where •, • represents the pairing between H 1 per ([0,T 0 ]) and H −1 per ([0,T 0 ]).Using that T0 0 (d/dc)ψ c (x)dx = 0, we have Ꮽ,(d/dc)ψ c = 0.As a consequence, we obtain Next we obtain the following expression to d in (4.25): where A ϕc is given by (3.72) and from (3.52), (3.12), (3.15), and (3.70), we have that where In order to establish the convexity of d, we have to compute The following result is obtained in this context.
Lemma 4.2.Let α 1 , α 2 , and α 3 be as in Lemma 3.3.Then, where α 0 = 3(1 − c 2 0 ), M(a,b,c) ≡ 2λ (c)/λ(c), and F = M/ i< j α i α j .Proof.First we note that for d/dc = " " that Using expressions for α i obtained in Lemma 3.3, we have that for i, j,k ∈ {1, 2,3} with j = i, j = k, and k = i that To get the second part, we note that and for i, j,k ∈ {1, 2,3} with j = i, j = k, and k = i, J. Angulo and J. R. Quintero 25 Thus, we get (4.45) Proof.From our previous computations and using formulas in Lemma 4.2 for the derivatives of i< j α i α j and α 1 α 2 α 3 with respect to c, we obtain that (4.46) Using that we obtain by a direct computation that Now, let us define the following polynomial: The first observation is that Therefore we have from Lemma 3.3 that α 1 α 2 α 3 > 0 and from (3.72) that i< j α i α j < 0, so we can conclude initially from (4.46) and (4.49) that (1) d is a strictly convex function in 0 Next, note that d (c * ) > 0, and so from continuity we can choose θ small such that (2) d is a strictly convex function in 0

Stability theory for the Boussinesq-type system (1.10)
In this section, we establish a theory of stability for the branch of cnoidal waves solutions We first note from (2.12) that ψ c = (ψ c ,−cψ c ) t is not a critical point to the action functional Ᏺ in (2.13) indicating that the general theory of Grillakis, Shatah, and Strauss cannot be applied directly to the problem at hand over all H 1 per ([0,T 0 ]) × H 1 per ([0,T 0 ]).To overcome this, we will proceed as in the proof of orbital stability of cnoidal wave solutions with respect to the periodic flow of solutions with mean zero for the initial value problem associated with the KdV equation (see [3]).In other words, we consider the following spaces: where per .We will see below that Grillakis et al.'s approach in [4] can be used to obtain the stability of ψ c by perturbations belonging to ᐄ.In fact, in this case we consider Ᏺ defined on the space ᐄ and so the cnoidal waves ψ c is a critical point, namely, (5.3) More exactly, we obtain the following stability result associated to system (1.10).
The proof of Theorem 5.1 needs some preliminary results.First of all, we have to establish the existence and uniqueness of global mild periodic solutions for the periodic Cauchy problem associated with system (1.10), and second, we need to study the periodic eigenvalue problem associated with the operator Ᏺ (ψ c ,−cψ c ).

3 . 3 .
Fundamental period.The first step to establish the existence of a curve of periodic wave solutions to the Benney-Luke equation with a given period is based on proving the existence of an interval of speed waves for cnoidal waves ϕ c in (3.10).

( 4 .
44) Now we are in position to prove the convexity of function d.Theorem 4.3.d is a strictly convex function for 0 < |c| < 1 < √ a/b, and for 0 < |c| < c * + θ < √ a/b < 1, where θ is small and c * is the unique positive root of the polynomial .51) If we assume that a > b and c 2 < 1, we have that b 2 c 6 − a 2 < 0. Thus we conclude that P(c) < 0, for a > b and c 2 < 1.On the other hand, if we assume that a < b and 0 < |c| < √ a/b, we have that P(± √ a/b) = 16a 2 (b − a)/b > 0, but P(0) = −6a 2 < 0. As a consequence of this, there exists a unique c * with P(c * ) = 0, 0 < c * < √ a/b < 1, such that P(c) < 0 for 0 < |c| < c * in case of having b > a.Note that P (c) > 0 for c > 0.
1 rΦ xx + 2Φ x r x .(6.15) Again, applying the operator ( ∂ −1 1 rΦ xx + 2Φ x r x .(6.5)It is not hard to verify that associated with the linear operator M 0 , there exists a 0 group defined in ᐂ × H 1 per , whose Fourier symbols are given by In this paper, we are going to say that a mild solution of the Benney-Luke equation (1.1) with initial data (u 0 ,u 1 ) is a couple (Φ,r) such that (Φ,r) ∈ C R t ;ᐂ × H 1