Global Well-Posedness and Non-linear Stability of Periodic Traveling Waves for a Schrodinger-Benjamin-Ono System

The objective of this paper is two-fold: firstly, we develop a local and global (in time) well-posedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schr\"odinger-Benjamin-Ono system) for \emph{low-regularity} initial data in both periodic and continuous cases; secondly, a family of new periodic traveling waves for the Schr\"odinger-Benjamin-Ono system is given: by fixing a minimal period we obtain, via the implicit function theorem, a smooth branch of periodic solutions bifurcating a Jacobian elliptic function called {\it dnoidal}, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.


Introduction
In this paper we are interested in the study of the following Schrödinger-Benjamin-Ono (SBO) system where u is a complex-valued function, v is a real-valued function, t ∈ R, x ∈ R or T, α, β and γ are real constants such that α = 0 and β = 0, and D∂ x is a linear differential operator representing the dispersive term. Here D = H∂ x where H denotes the Hilbert transform defined as where sgn(k) = −1, k < 0, 1, k > 0.
Note that from these definitions we have that D is a linear positive Fourier operator with symbol |k|. The system (1.1) was deduced by Funakoshi and Oikawa ( [21]). It describes the motion of two fluids with different densities under capillary-gravity waves in a deep water flow. The short surface wave is usually described by a Schrödinger type equation and the long internal wave is described by some sort of wave equation accompanied with a dispersive term (which is a Benjamin-Ono type equation in this case). This system is also of interest in the sonic-Langmuir wave interaction in plasma physics [28], in the capillarygravity interaction wave [20], [26], and in the general theory of water wave interaction in a nonlinear medium [13], [14]. We note that the Hilbert transform considered in [21] for describing system (1.1) is given as −H.
When studying an initial value problem, the first step is usually to investigate in which function space well-posedness occurs. In our case, smooth solutions of the SBO system (1.1) enjoy the following conserved quantities where D 1/2 is the Fourier multiplier defined as D 1/2 v(k) = |k| 1 2 v(k). Therefore, the natural spaces to study well-posedness are the Sobolev H s -type spaces. Moreover, due to the scaling property of the SBO system (1.1) (see [9] Remark 2), we are led to investigate well-posedness in the spaces H s × H s−1/2 , s ∈ R.
In the continuous case Bekiranov, Ogawa and Ponce [10] proved local well-posedness for initial data in H s (R) × H s− 1 2 (R) when |γ| = 1 and s ≥ 0. Thus, because of the conservation laws in (1.2), the solutions extend globally in time when s ≥ 1, in the case αγ β < 0. Recently, Pecher [33] has shown local well-posedness in H s (R) × H s− 1 2 (R) when |γ| = 1 and s > 0. He also used the Fourier restriction norm method to extend the global well-posedness result when 1/3 < s < 1, always in the case αγ β < 0. Here, we improve the global well-posedness result till L 2 (R) × H − 1 2 (R) in the case γ = 0 and |γ| = 1. Indeed, we refine the bilinear estimates of Bekiranov, Ogawa and Ponce [10] in Bourgain spaces Proposition 3.1). These estimates combined with the L 2conservation law allow us to show that the size of the time interval provided by the local well-posedness theory depends only on the L 2 -norm of u 0 . It is worth to point out that this scheme applies for other dispersive systems. In fact Colliander, Holmer and Tzirakis [18] already applied this method to Zakharov and Klein-Gordon-Schrödinger systems.
They also announced the above result for the SBO system (see Remark 1.5 in [18]). However they allowed us to include it in this paper since there were not planing to write it up anymore. Note that we also prove global well-posedness in H s (R) × H s− 1 2 (R) when s > 0 in the case γ = 0 and |γ| = 1. We take the opportunity to express our gratitude to Colliander, Holmer and Tzirakis for the fruitful interaction about the Schrödinger-Benjamin-Ono system.
In the periodic setting, there does not exist, as far as we know, any result about the well-posedness of the SBO system (1.1). Bourgain [16] proved well-posedness for the cubic nonlinear Schrödinger equation (NLS) (see (1.3)) in H s (T) for s ≥ 0 using the Fourier transform restriction method. Unfortunately, this method does not apply directly for the Benjamin-Ono equation. Nevertheless, using an appropriate Gauge transformation introduced by Tao [35], Molinet [31] proved well-posedness in L 2 (T). Here we apply Bourgain's method for the SBO system to prove its local well-posedness in H s (T) × H s− 1 2 (T) when s ≥ 1/2 in the case γ = 0, |γ| = 1. The main tool is the new bilinear estimate stated in Proposition 3.3. Furthermore, by standard arguments based on the conservation laws, this leads to global well-posedness in the energy space H 1 (T)×H 1/2 (T) in the case αγ β < 0. We also show that our results are sharp in the sense that the bilinear estimates on these Bourgain spaces fail whenever s < 1/2 and |γ| = 1 or s ∈ R and |γ| = 1. In fact, we use Dirichlet's Theorem on rational approximation to locate certain plane waves whose nonlinear interactions behave badly in low regularity.
In the second part of this paper, we turn our attention to another important aspect of dispersive nonlinear evolution equations: the traveling-waves. These solutions imply a balance between the effects of nonlinearity and dispersion. Depending on the specific boundary conditions on the wave's shape, these special states of motion can arise as either solitary or periodic waves. The study of this special steady waveform is essential for the explanation of many wave phenomena observed in the practice: in surface water waves propagating in a canal, in propagation of internal waves or in the interaction between long waves and short waves as in our case. In particular, some questions such as existence and stability of these traveling waves are very important in the understanding of the dynamic of the equation under investigation.
The solitary waves are in general a single crested, symmetric, localized traveling waves, with sech-profiles (see Ono [32] and Benjamin [11] for the existence of solitary waves of algebraic type or with a finite number of oscillations). The study of the nonlinear stability or instability of solitary waves has had a big development and refinement in recent years. The proofs have been simplified and sufficient conditions have been obtained to insure the stability to small localized perturbations in the waveform. Those conditions have showed to be effective in a variety of circumstances, see for example [1], [2], [3], [12], [15], [25], [36].
The situation regarding to the study of periodic traveling waves is very different. The stability and the existence of explicit formulas of these progressive wave trains have re-ceived comparatively little attention. Recently many research papers about this issue have appeared for specify dispersive equations, such as the existence and stability of cnoidal waves for the Korteweg-de Vries equation [5] and the stability of dnoidal waves for the one-dimensional cubic nonlinear Schrödinger equation where u = u(t, x) ∈ C and x, t ∈ R (Angulo [4], see also Angulo&Linares [6] and Gallay&Hȃrȃgus [22], [23]).
In this paper we are also interested in giving a stability theory of periodic traveling waves solutions for the nonlinear dispersive system SBO (1.1). The periodic traveling waves solutions considered here will be of the general form where φ, ψ : R → R are smooth, L-periodic functions (with a prescribed period L), c > 0, ω ∈ R and we will suppose that there is a q ∈ N such that 4qπ/c = L.
So, by replacing these permanent waves form into (1.1) we obtain the pseudo-differential system where σ = ω − c 2 4 and A φ,ψ is an integration constant which we will set equal zero in our theory. Existence of analytic solutions of system (1.5) for γ = 0 is a difficult task. In the framework of traveling waves of type solitary waves, namely, the profiles φ, ψ satisfy φ(ξ), ψ(ξ) → 0 as |ξ| → ∞, it is well known the existence of solutions for (1.5) in the form when γ = 0, σ > 0, and αβ > 0. For γ = 0 a theory of even solutions of these permanent waves solutions has been established in [7] (see also [8]) by using the concentrationcompactness method. For γ = 0 and σ > 2π 2 /L 2 we prove (along the lines of Angulo [4] with regard to (1.3)) the existence of a smooth curve of even periodic traveling wave solutions for (1.5) with α = 1, β = 1/2; note that this restriction does not imply loss of generality. This construction is based on the dnoidal Jacobian elliptic function , namely, where η 1 and k are positive smooth functions depending of the parameter σ. We observe that the solution in (1.7) gives us in " the limit " the solitary waves solutions (1.6) when η 1 → √ 4cσ and k → 1 − , because in this case the elliptic function dn converges, uniformly on compacts sets, to the hyperbolic function sech.
In the case of our main interest, γ = 0, the existence of periodic solutions is a delicate issue. Our approach for the existence of these solutions uses the implicit function theorem together with the explicit formulas in (1.7) and a detailed study of the periodic eigenvalue problem associated to the Jacobian form of Lame's equation where sn(·; k) is the Jacobi elliptic function of type snoidal and K = K(k) represents the complete elliptic integral of the first kind and defined for k ∈ (0, 1) as So, by fixing a period L, and choosing c and ω such that σ ≡ ω − c 2 4 satisfies σ > 2π 2 /L 2 , we obtain a smooth branch γ ∈ (−δ, δ) → (φ γ , ψ γ ) of periodic traveling wave solutions of (1.5) with a fundamental period L and bifurcating from (φ 0 , ψ 0 ) in (1.7). Moreover, we obtain that for γ near zero φ γ (x) > 0 for all x ∈ R and ψ γ (x) < 0 for γ < 0 and x ∈ R.
Furthermore, concerning the non-linear stability of this branch of periodic solutions, we extend the classical approach developed by Benjamin [12], Bona [15] and Weinstein [36] to the periodic case. In particular, using the conservation laws (1.2), we prove that per ([0, L]) at least when γ is negative near zero. We use essentially the Benjamin&Bona&Weinstein's stability ideas because it gives us an easy form of manipulating with the required spectral conditions and the positivity property of the quantity d dσ φ 2 γ (x)dx, which are basic information in our stability analysis.
However, we do not use the abstract stability theory of Grillakis et al. in our approach basically because of the two circumstances above. We recall that Grillakis et al. theory in general requires a study of the Hessian for the function with γ = γ(c, ω), and a specific spectrum information of the matrix linear operator H c,ω = L ′′ (e icξ/2 φ γ , ψ γ ). In our case, these facts do not seem to be easily obtained.
So, for γ < 0 we reduce the required spectral information (see formula (5.6)) to the study of the self-adjoint operator L γ , where K −1 γ is the inverse operator of K γ = −γD + c. Hence we obtain via the min-max principle that L γ has a simple negative eigenvalue and zero is a simple eigenvalue with eigenfunction d dx φ γ provide that γ is small enough. Finally, we close this introduction with the organization of this paper: in Section 2, we introduce some notations to be used throughout the whole article; in Section 3, we prove the global well-posedness results in the periodic and continuous settings via some appropriate bilinear estimates; in Section 4, we show the existence of periodic traveling waves by the implicit function theorem; then, in Section 5, we derive the stability of these waves based on the ideas of Benjamin and Weinstein, that is, to manipulate the information from the spectral theory of certain self-adjoint operators and the positivity of some relevant quantities.

Notation
For any positive numbers a and b, the notation a b means that there exists a positive constant θ such that a ≤ θb. Here, θ may depend only on certain parameters related to the equation (1.1) such as γ, α, β. Also, we denote a ∼ b when, a b and b a.
For a ∈ R, we denote by a+ and a− a number slightly larger and smaller than a, respectively.
Let L > 0, the inner product of two functions in L 2 ([0, L]) is given by where ( f (n)) n∈Z denote the Fourier series of f (for further information see Iorio&Iorio [27]). Sometimes we also write H s (T) to denote the space H s per ([0, L]) when the period L does not play a fundamental role.
Similarly, when s ∈ R, we denote by H s (R) the set of all f ∈ S ′ (R) such that where S ′ (R) is the set of tempered distributions and f is the Fourier transform of f . When the function u is of the two time-space variables (t, x) ∈ R × R, periodic in space of period L, we define its Fourier transform by and similarly, when u : Next, we introduce the Bourgain spaces related to the Schrödinger-Benjamin-Ono system in the periodic case: and the continuous case: where x := 1 + |x|. The relevance of these spaces are related to the fact that they are well-adapted to the linear part of the system and, after some time-localization, the coupling terms of (1.1) verifies particularly nice bilinear estimates. Consequently, it will be a standard matter to conclude our global well-posedness results (via Picard fixed point method).

Global Well-Posedness of the Schrödinger-Benjamin-Ono System
This section is devoted to the proof of our well-posedness results for (1.1) in both continuous and periodic settings.

Global well-posedness on R
The bulk of this subsection is the proof of the following theorem: Theorem 3.1. Let 0 < |γ| = 1. Then, the SBO system is globally well-posed for initial data In the rest of this section, we will denote by U(t) := e it∂ 2 x and V γ (t) := e −γtH∂ 2 x the unitary groups associated to the linear part of (1.1). The proof of Theorem 3.1 follows the lines of [18]. Let us first state the linear estimates:
Proof of Lemma 3.1. Estimate (3.1) is proved in [18] Lemma 2.1 (a). Next we combine Estimate (3.1) and the fact that where Once these linear estimates are established, our task is to prove the following bilinear estimates: Proposition 3.1. Let γ ∈ R such that |γ| = 1 and γ = 0. Then, we have for any where the implicit constants depend on γ.
For the proof of these bilinear estimates, we need the following standard Bourgain-Strichartz estimates: Finally, we recall the two following technical lemmas proved in [24]: Assume that f and g are nonnegative, even and nonincreasing for positive argument. Then f * g enjoys the same property. In particular f * g takes its maximum at zero.
The proof of Estimate (3.9) is actually identical to that of Estimate (3.8). Indeed, The algebraic relation associated to this integral is given by Then, we note that Estimate (3.21) is exactly the same as Estimate (3.12), replacing c by We now slightly modify the bilinear estimates of Proposition 3.1.
Finally, we conclude this subsection with the proof of theorem 3.1: Proof of Theorem 3.1 Case s = 0. The system (1.1) is, at least formally, equivalent to the integral system , we want to use a contraction argument to solve (3.25) in a product of balls (3.8) and (3.9) imply that whenever T satisfies (3.27) and (3.28). Since the L 2 -norm of u is a conserved quantity by the SBO flow, we can suppose that v 0 H − 1 2 ≫ u 0 L 2 , otherwise we can repeat the above argument and extend the solution globally in time. Hence Condition (3.28) is automatically satisfied and Condition (3.27) implies that the iteration time T must be . Then we deduce from (3.7), (3.9), (3.25) and (3.29) that there exists a positive constant C such that Since ∆T only depends on u 0 L 2 , we can repeat the above argument and extend the solution (u, v) of (1.1) globally in time. Moreover, we deduce that there exists c > 0 such This time we want to solve the integral system (3.25) in a space of the type . Moreover we can always suppose that T satisfies (3.27) and (3.28), so that Estimate (3.29) holds. We also observe from the third conservation law in (1.2) and a priori Estimate Therefore we deduce taking Using the conservation laws (1.2) as in [33], our local existence result implies The fundamental technical points in the proof of Theorem 3.2 are the following bilinear estimates. The rest of the proof follows by standard arguments, as in [19].
where the implicit constants depend on γ.
These estimates are sharp in the following sense  (ii) The estimate (3.3) fails for any s ∈ R.
The following Bourgain-Strichartz estimates will be used in the proof of Proposition 3.3:
Proof. The first estimate of (3.4) was proved by Bourgain in [16] and the second one is a simple consequence of the first one (see for example [31]).
In the proof of Proposition 3.4, we will use the following lemma which is a direct consequence of the Dirichlet theorem.
Lemma 3.5. Let γ ∈ R such that γ = 0 and |γ| < 1 and Q γ defined as in (3.12). Then, there exists a sequence of positive integers {N j } j∈N such that and [x] denotes the closest integer to x. Theorem 3.4 (Dirichlet). Let α ∈ R \ Q. Then, the inequality has infinitely many rational solutions p q . Proof of Lemma 3.5. Fix γ ∈ R such that |γ| < 1. Let N a positive integer, N ≥ 2, α = 2 1+γ and N 0 = [αN]. Then, from the definition in (3.12), we deduce that When α ∈ Q, α = p q , it is clear that we can find an infinity of positive integer N satisfying the right-hand side of (3.21) choosing N j = jq, j ∈ N. When α ∈ R \Q, this is guaranteed by the Dirichlet theorem.

Existence of Periodic Traveling-Wave Solutions
The goal of this section is to show the existence of a smooth branch of periodic travelingwave solutions for (1.5). Initially we show a novel smooth branch of dnoidal waves solutions for (1.5) in the case γ = 0. After that, by using the implicit function theorem, we construct (in the case γ = 0) a smooth curve of periodic solutions bifurcating from these dnoidal waves.

Dnoidal Waves Solutions
We start by finding solutions for the case γ = 0 and σ > 0 in (1.5). Henceforth, without loss of generality, we will assume that α = 1 and β = 1 2 . Hence, we need to solve the system Then, by replacing the second equation of (4.1) into the first one, we obtain that φ 0 satisfies Equation (4.2) can be solved in a similar fashion to the method used by Angulo [4] (in the context of periodic traveling-wave solutions for the nonlinear Schrödinger equation (1.3)).
Next, we show that Λ is a strictly decreasing function. We know that Ψ(Λ(σ), σ) = L for every σ ∈ I(σ 0 ), then Thus, using the relation we obtain the following formal equivalences This completes the proof of the Theorem.
Remark 4.1. In the case that the polynomial F (t) = −t 4 + 4cσt 2 + 4cB φ 0 has a pure imaginary root and the other two roots are real we can show the existence of two smooth curves of periodic solutions for (4.2) of cnoidal type, more precisely we have where a, b, k are smooth functions of ω.
The following result will be used in our stability theory.
Proof. By (4.4), (4.10), and the formula . So, since k → K(k)E(k) and σ → k(σ) are strictly increasing functions we have that This finishes the Corollary.

Periodic Traveling Waves Solutions for Eq. (1.5)
In this subsection we show the existence of a branch of periodic traveling waves solutions of (1.5) for γ close to zero such that these solutions bifurcate the dnoidal waves solutions found in Theorem 4.1.
We start our analysis by studying the periodic eigenvalue problem considered on [0, L], where for σ > 2π 2 /L 2 , φ 0 is given by Theorem 4.1 and satisfies (4.2). , has its first three eigenvalues simple with zero being its second eigenvalue (with eigenfunction d dx φ 0 ). Moreover, the remainder of the spectrum is constituted by a discrete set of eigenvalues which are double and converging to infinity. Theorem 4.2 is a consequence of the Floquet theory (Magnus&Winkler [30]). For convenience of the readers, we will give some basic results of this theory. From the classical theory of compact symmetric linear operator we have that problem (4.14) determines a countable infinity set of eigenvalues {λ n |n = 0, 1, 2, ...} with λ 0 ≤ λ 1 ≤ λ 2 ≤ λ 3 ≤ λ 4 ≤ ..., where double eigenvalue is counted twice and λ n → ∞ as n → ∞. We shall denote by χ n the eigenfunction associated to the eigenvalue λ n . By the conditions χ n (0) = χ n (L), χ ′ n (0) = χ ′ n (L), χ n can be extended to the whole of (−∞, ∞) as a continuously differentiable function with period L.
We know that with the periodic eigenvalue problem (4.14) there is an associated semiperiodic eigenvalue problem in [0, L], namely, As in the periodic case, there is a countable infinity set of eigenvalues {µ n |n = 0, 1, 2, 3, ...}, where double eigenvalue is counted twice and µ n → ∞ as n → ∞. We shall denote by ξ n the eigenfunction associated to the eigenvalue µ n . So, we have that the equation has a solution of period L if and only if γ = λ n , n = 0, 1, 2, · · · , while the only periodic solutions of period 2L are either those associated with γ = λ n , but viewed on [0, 2L], or those corresponding to γ = µ n , but extended in form ξ n (L+ x) = ξ n (L−x) for 0 ≤ x ≤ L, n = 0, 1, 2, · · · . If all solutions of (4.16) are bounded we say that they are stable; otherwise we say that they are unstable. From the Oscillation Theorem of the Floquet theory (see [30]) we have that The intervals (λ 0 , µ 0 ), (µ 1 , λ 1 ), ···, are called intervals of stability. At the endpoints of these intervals the solutions of (4.16) are unstable in general. This is always true for γ = λ 0 (λ 0 is always simple). The intervals, (−∞, λ 0 ), (µ 0 , µ 1 ), (λ 1 , λ 2 ), (µ 2 , µ 3 ), · · · , are called intervals of instability 1 . The interval (−∞, λ 0 ) of instability will always be present. We note that the absence of an instability interval means that there is a value of γ for which all solutions of (4.16) have either period L or semi-period L, in other words, coexistence of solutions of (4.16) with period L or period 2L occurs for that value of γ.

Stability of Periodic Traveling-Wave Solutions
We begin this section defining the type of stability of our interest. For any c ∈ R + define the functions Φ(ξ) = e icξ/2 φ(ξ) and Ψ(ξ) = ψ(ξ), where (φ, ψ) is a solution of (1.5). Then we say that the orbit generated by (Φ, Ψ), namely, The main result to be proved in this section is that the periodic traveling waves solutions of (1.1) determined by Theorem 4.3 are stable for σ > 2π 2 /L 2 and γ negative close to 0.
Proof of Theorem 5.1. Consider the perturbation of the periodic traveling wave (φ γ , ψ γ ) Hence, by the property of minimum of (θ, x 0 ) = (θ(t), x 0 (t)), we obtain from (5.4) that p(x, t) = Re(ξ(x, t)) and q(x, t) = Im(ξ(x, t)) satisfy the compatibility relations Now we take the continuous functional L defined on H 1 where E, G, H are defined by (1.2). Then, from (5.4) and (1.5), we have where, for γ < 0 we define K −1 γ as which is the inverse operator of are the positive roots of K γ and K −1 γ respectively. Now, we need to find a lower bound for ∆L(t). The first step will be to obtain a suitable lower bound of the last term on the right-hand side of (5.6). In fact, since K where C 1 and C 2 are positive constants.
(2) L γ has a simple eigenvalue at zero with eigenfunction d dx φ γ .
γ is a non-negative operator which has zero as its first eigenvalue with eigenfunction φ γ . The remainder of the spectrum is constituted by a discrete set of eigenvalues.
Moreover, we have that L γ d dx φ γ = 0. Next, for f ∈ H 1 per ([0, L]) and f = 1, we have where the last inequality is due to that γ < 0 and DK −1 γ is a positive operator. So, since we have from Theorem 4.3 that for γ near 0 − and ǫ small, L γ f, f ≥ L 0 f, f − ǫ.