On the Cauchy problem for the Zakharov-Rubenchik/Benney-Roskes system

We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system (known as the Benney-Roskes system in water waves theory), which models the interaction of short and long waves in many physical situations. Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Cauchy problem in the background of a line solitary wave.


Introduction
This paper is concerned with various issues concerning the Cauchy problem for the two or three-dimensional Zakharov-Rubenchik (or Benney-Roskes) system and its perturbation by a line soliton. The Zakharov-Rubenchik system is by no doubts a fundamental one, being a "generic" asymptotic system in the so-called modulation regime (slowly varying envelope of a fast oscillating train) and it was actually derived in various physical contexts. Moreover it contains in various limits the classical (scalar) Zakharov system (coupling a nonlinear Schrödinger equation and a wave equation, see (1.3) below) and the Davey-Stewartson systems (coupling a nonlinear Schrödinger equation and an elliptic equation). We refer to [52] for more details on the formal derivation of those systems and on the physical background.
The Davey-Stewartson system was first derived formally in the context of water waves in [14,1,15] (see also [11,12] for a derivation of Davey-Stewartson systems in a different context). However, as noticed in [25] it is less general than the Benney-Roskes system (1.6) below in the sense that the initial conditions for the acoustic type components have to be prepared to obtain an approximation of the full water waves system. We refer to [13] for a rigorous justification of the Zakharov limit of the Zakharov-Rubenchik system and to [36] for the Schrödinger limit of the Zakharov-Rubenchik system in the one-dimensional case and for well-prepared initial data.
The Zakharov-Rubenchik/Benney-Roskes system is thus richer than those simpler models and should capture more of the original dynamics. It was introduced in [53] (see also the survey article [52]) to describe the interaction of spectrally narrow high-frequency wave packet of small amplitude with a low-frequency acoustic type oscillations. The analysis is general and carried out in the Hamiltonian formalism and yields the following universal system where v g , ω", k, q, β, α, ρ 0 , c are parameters .The two last equations describe the acoustic type waves and ∆ ⊥ = ∂ 2 x + ∂ 2 y or ∂ 2 x , ∆ = ∆ ⊥ + ∂ 2 x . In two space dimensions a more specific (formal) derivation in the context of surface water waves is displayed in [6] and rigorously justified in [25], see below for a more precise description.
In the notations of [41] (see also [40] where it is used in the context of Alfvén waves in dispersive MHD), the Zakharov-Rubenchik system has the form M 2 ρ + |ψ| 2 = 0, where ψ : R × R d → C, ρ, φ : R × R d → R, d = 2, 3 describe the fast oscillating and respectively acoustic type waves.
Here σ 1 , σ 2 , σ 3 = ±1, W > 0 measures the strength of the coupling with acoustic type waves, M > 0 is a Mach number, D ∈ R is associated to the Doppler shift due to the medium velocity and δ ∈ R is a nondimensional dispersion coefficient.
When α = 0 (resp. D = 0) in (1.1) (resp. (1.2)) the Zakharov-Rubenchik system reduces to the classical (scalar) Zakharov system (see eg Chapter V in [46]). More precisely, in the framework of (1.1), one gets which is a form of the 2 or 3D Zakharov system. Note however that the second order operator in the first equation is not necessarily elliptic. [41] by using the local smoothing property of the free Schrödinger operator after reducing the system to a quasilinear (non local) Schrödinger equation. Since it uses dispersive properties of the free Schrödinger group that are valid only in the whole space the proof does not extend to the Cauchy problem posed in T d or R d−1 × T, the later situation being relevant for transverse stability issues. On the other hand, when applied to the Benney-Roskes system (1.5) below, it provides an existence time of order O(1) 1 while an existence time of order O(1/ǫ) is needed to fully justify the Benney-Roskes as a water wave model on the correct time scales (see [24]).
Local well-posedness of the Zakharov-Rubenchik/Benney-Roskes system was also obtained in [31], for s > 2 with the additional condition δσ 1 > 0 (that is the second order operator in the first equation of (1.2), (1.1) is elliptic) by using an energy method inspired by the work of Schochet-Weinstein in [45] on the nonlinear Schrödinger limit of the Zakharov system. The method used in [31] and [45] consists in rewriting the Zakharov system (or the Zakharov-Rubenchik system) as a dispersive (skew-adjoint) perturbation of a symmetric nonlinear hyperbolic system and it uses only the algebraic structure of the system. A shortcoming of the method is that one has to prepare the initial data.
We will see that, when the small parameter ǫ is included, this method provides also the existence on the time scale O(1) in the context of water waves (see the Benney-Roskes system (1.6) below) and moreover that it can be applied to the system obtained from (1.2) which is satisfied by a (localized) perturbation of a line soliton. Also, since it does not use any dispersive property of the Schrödinger group, it applies to the Cauchy problem in T d or R d−1 × T, a situation that has not been addressed before (see on the other hand [7,8] for the periodic Zakharov system) .
Thus, none of the two aforementioned methods seems to give the expected existence time scale for the Benney-Roskes system. Nevertheless they provide different results for Zakharov-Rubenchik type systems. The "dispersive method" used in [41] works only in R d but does not need the Schrödinger part of the system to be "elliptic" (that is it does not need the condition δσ 1 > 0). Also it lowers the regularity on the initial data (an effect of the dispersive smoothing effect) and could be applied as well to (possibly non physical) nonlinear perturbations of the system.
On the other hand, the Schochet-Weinstein type, "hyperbolic like" methods allow to deal with the periodic or semi-periodic cases, but are relatively rigid (they rely on the algebraic structure of the system) and require initial data in the "hyperbolic space" H s (R d ), s > d 2 + 1. . The situation is better understood in spatial dimension one. Oliveira [35] proved the local (thus global using the conservation laws below) well-posedness in H 2 (R)× H 1 (R) × H 1 (R). This result was improved in [26] where in particular global wellposedness was established in the energy space It is worth noticing that (1.2) possesses two conserved quantities, the L 2 norm 1 Roughly speaking, the idea in [41] is to reduce the system to a (nonlocal) quasilinear Schrödinger equation. When ǫ is taken into account, the crucial dispersive smoothing estimate on the Schrödinger group has a 1/ǫ factor while the nonlinear term has a ǫ factor.
|ψ(x, y ⊥ , t)| 2 = |ψ(x, y ⊥ , 0)| 2 where y ⊥ = y or (y, z), and, after the change of variable (x, t) → (x + σ 3 t, t), the Hamiltonian The conservation laws are used in [41] to obtain global weak solutions under suitable assumptions on the coefficients. We will use them in Section 5 to prove the global existence of weak solutions of the systems obtained by perturbing a line (dark) soliton. Note also that the conservation laws can be used to get the global well-posedness of the Zakharov-Rubenchik, Benney-Roskes system in space dimension one (see [35]).
As aforementioned, in the context of water waves, the Zakharov-Rubenchik system is known as the Benney-Roskes sytem and it was formally derived in [6]. We follow here the notations in [25], where a rigorous derivation is performed.
where ω(ξ) = g + σ ρ |ξ| 2 |ξ| tanh(µ|ξ|) is the dispersion relation of water waves and where |k| is a fixed wave number, g is the gravity, σ ≥ 0 is a surface tension coefficient, ρ is the density of the water and µ is the shallowness parameter (square of the typical fluid depth over a typical horizontal scale) which is large or infinite in the deep water models and The small parameter ǫ is the wave steepness that is the ratio of a typical amplitude of the wave over a typical horizontal scale. Recall ( [25]) that the typical time scale for the solutions of (1.6) below is 1/ǫ and so it is crucial to establish the well-posedness on those time scales.
The Benney-Roskes equations can then be written in 2 dimensions as follows It is known (see eg [25] Chapter 8) that ω ′ > 0, while for purely gravity waves (σ = 0) ω is a concave function, thus ω ′′ < 0 and the Schrödinger equation in the Benney-Roskes system is "non elliptic".
On the other hand, in presence of surface tension, the condition ω ′′ > 0 is possible as shown in the following computation.
In order to apply Schochet-Weinstein method we will need the condition δσ 1 > 0 and we will only consider the Zakharov-Rubenchik (or Benney-Roskes) system of the form of (1.2) satisfying this condition.
The one-dimensional Zakharov-system possesses solitary wave solutions and in [35], Serra de Oliveira proved their orbital stability. One motivation of the present paper was the study of their transverse stability. The transverse instability of the line solitary wave for some two dimensional models such as the nonlinear Schrödinger equation (NLS), the Kadomtsev-Petviashvili equation (KP) and some general "abstract" Hamiltonian systems have been carried out extensively in [42,43,44,29,30].
It is thus of interest to study the transverse stability of the line soliton for the two dimensional model (1.2) and the first step is to study the Cauchy problem of a localized perturbation of (1.2) by a line soliton. Another possibility is to consider y or (y, z) -periodic perturbations of the line solitary wave, a first step being to establish the well-posedness of the Cauchy problem for the Zakharov-Rubenchik, Benney-Roskes system in R d ×T, d = 1, 2, which could not result from the methods used in [41] but we achieve here and also in the pure periodic case T d+1 .
In order to unify the notation we will rewrite the Benney-Roskes system (1.6) in the form of (1.2). We replace (ψ 01 , ζ 10 ) by (ψ, ρ, φ) and after calculating the corresponding coefficients, we have: The paper is organized as follows. In the next section we reformulate the existence of one-dimensional solitary waves (bright and dark) in our framework. In section 3 we use the Schochet-Weinstein method to prove a local existence for the Benney-Roskes/ Zakharov-Rubenchik system, keeping the small parameter ǫ which is relevant for deep water waves. In section 4 we consider the case of a localized perturbation of a line solitary wave. Finally we prove in Section 5 the global existence of weak solutions perturbing a dark solitary wave.
We conclude the paper by a list of open questions. Notations.
• ∂ x or (·) x will be used to denote the derivative with respect to variable x.
• H s (D), s ∈ R denotes the classical Sobolev space in the domain D.
• · X : The norm in a functional space X.
• F and F −1 denote the Fourier and inverse Fourier transform respectively.
• ξ = 1 + |ξ| 2 for ξ ∈ R n and σ(D) denotes the Fourier multiplier with the symbol σ(ξ). • ℜ and ℑ denote the real part and imaginary part of a complex number respectively.

Existence of one dimensional solitary waves
In this section, we reframe the proof of the existence of 1-d solitary waves in [35] in our setting. The 1-d Zakharov-Rubenchik system has the form Let c ≥ 0, we look for solutions of the system (2.2) of the form From the last two equations of (2.2) we deduce that Then the first equation of (2.2) is equivalent to The equation (2.4) has a unique positive solution if : We see that if c → (1/M ) − and λ is large enough then (2.5) holds assuming that W > 0 and δ > 0 which holds true in both models (1.2) and (1.6) . In this case, In the context of water waves ( . Therefore if µ is large enough (which occurs in the context of deep water waves) then the above conditions hold. In this case, Then the system (2.2) has two kind of solitary waves corresponding to the two conditions (2.5) and (2.7): Recalling thatφ = φ x , the solutions of system (2.1) should have thus the form Where in the case R(x) is given by (2.6). And Remark : Similarly to the case of the cubic nonlinear Schrödinger equation, we will call the 1-d solitary wave corresponding to the condition (2.5) and (2.7) the "bright", "dark" soliton respectively.

The Z-R/B-R system
As aforementioned the asymptotic model (1.6) is a good approximation of the full water wave system on a time scale O(1/ǫ) (see [25] page 233). It is thus crucial to prove the well-posedness of the Cauchy problem on time scales of order 1/ǫ. However, the existence time obtained by using the method in [41] does not reach the O(1/ǫ) time scale (as we already mentionned it is of order O(1)) In this section, we give the proof of the local well-posedness for (1.2) by using Schochet-Weinstein method in [31] but keeping the parameter ǫ in (1.6) to estimate the existence time obtained by this method. It turns out, however, that one does not improve upon the previously known O(1) result (see however the comments in the Introduction).
We consider the following system M 2 ρ + |ψ| 2 = 0, with initial conditions (ψ 0 , ρ 0 , φ 0 ) for which we obtain a local existence result : Then there exist T > 0 independent of ǫ and a unique solution Remark 3.1. With some minor changes, one obtains the same result in the threedimensional case, that is ψ yy replaced by ∇ ⊥ ψ. Proof. We follow closely the proof in [31], Section 3.3, but we keep track of the parameter ǫ.
We first rewrite (3.1) as a dispersive perturbation of a symmetric hyperbolic system. We take the time derivative of the second and the third equation of the system (1.2). This allows to decouple the linear parts of those equations We then get a coupled system for ψ and U We can rewrite the second equation in (3.5) as Integrating with respect to time between 0 and t and using (3.2) and the two last equations in (3.1), we infer that U t + 2DW (|ψ| 2 ) x is a divergence. Then, in order to reduce to a first order system, following the idea in [45] for the Zakharov system, we define the following auxiliary (vector valued) function V Plugging this expression into the second equation of (3.5) and assuming that U, V, ψ tend to zero at infinity together with their derivatives we infer that so that we obtain the equivalent first order system Remark : By combining (3.2), (3.4), (3.7) and the two last equations of (3.1), we have that V 0 is actually the value of of V at t = 0.. Recall that the initial data of U is defined by ρ 0 and ∂ x φ 0 . We now set D = U + σ 1 2 |ψ| 2 then the system (3.8) becomes For simplicity of notation we set Furthermore, we split the ψ in real and imaginary part Multiplying the first equation of (3.9) byψ and taking the real part, we deduce that We insert (3.10) into (3.9) and then separate the real and imaginary parts of the first equation of the system. We furthermore apply the spatial gradient to the first equation of (3.9) to get the equations satisfied by H and L. This leads to the following system Since σ 1 δ > 0 , we can perform the following change of variables and we then set U = (H * , L * , F, G, D, V ) T . Therefore, (3.1) is rewritten as a dispersive (skew adjoint) perturbation of a symmetric hyperbolic system given by where A 1 , A 2 , B 1 and B 2 are symmetric matrices, K 1 , K 2 are skew symmetric matrices.
And note that C(U ) contains the term that is independent of ǫ which is Next,we prove that if the initial data U (0, x, y) = U 0 ∈ (H s (R 2 )) 9 , s > 2, then there exists T = T ( U 0 (H s (R 2 )) 9 ) such that equation (3.17) has a unique solution in L ∞ ([0, T ], (H s (R 2 )) 9 ).
The proof of the existence of solution is standard and proceeds via a classical iteration scheme for symmetric hyperbolic system (see [28,23]). The presence of C 1 (U ), unfortunately, leads to the existence time of order 0(1).
The uniqueness of solution of (3.17) is classically obtained by estimating the difference of two solutions since the dispersive part does not contribute to the L 2 energy estimate.
The last step will be the recovering the solution of (3.1) from the solution of (3.17).
Next, we are going to recover (ρ, φ). From (3.5) we already know ψ and U. Let φ be the unique solution of the following linear wave equation Then we get that (ψ, ρ, φ) solves (3.3) uniquely with respect to the given initial data.

The perturbed Z-R/B-R system
In this section, we consider the Cauchy problem for (1.2) when it is perturbed by the line solitary wave Q given by (2.9). That means, we will find solutions of (1.2) of the form (ψ Remark 4.1. A natural way to solve the Cauchy problem for (4.1) would be to use the "dispersive method" in [41]. However the fact that the line soliton does not decay to 0 in the transverse direction leads to a difficulty when dealing with a new nonlocal linear term and seems to preclude to extend this method in a straightforward way. Therefore we will apply the method of the previous section to this case.
In the first step, we need to rewrite (4.1) in the form of a skew-adjoint perturbation of a symmetric hyperbolic system.
Using the fact that Q = (φ 1 , φ 2 , φ 3 ) is also a solution of (3.1) with ǫ = 1 then by the same calculations as in the previous section, we obtain that (H r , L r , F r , G r , D r , V r ) T is a solution of (3.11)-(3.16) with ǫ = 1, where Similarly, if (ψ, ρ, φ) is a solution of (4.1) or if (ψ + φ 1 , ρ + φ 2 , φ + φ 3 ) is a solution of (3.1) (with ǫ = 1), then (H,L,F ,G,D,Ṽ ) T is a solution of (3.11)-(3.16) with ǫ = 1, where We now set (H, L, F, G, D, Remark : In order to define the initial data for V , we use the following expression 1 Similarly to the last Section, if ρ 1 δ > 0 , we can change variables as follows H * = ( δσ 1 H 1 , σ 1 H 2 ) T and L * = ( δσ 1 L 1 , σ 1 L 2 ) T , and then we set U = (H * , L * , F, G, D, V ) T . Therefore, the perturbation of (1.2) by the line solitary wave Q is rewritten as a dispersive perturbation of a symmetric hyperbolic given by Where, with j ∈ {1, 2}, A j , B j , C j are symmetric matrices, K j are skew symmetric and C j are constant matrices. A j have the same form as in the proof of Theorem 3.1, C(U ) contains quadratic and linear elements, and B j have the form Furthermore, note that the matrixC(Q) depends on (φ 1 , φ 2 , ∂ x φ 3 ) making the following analysis holds for both cases when Q is the bright or the dark soliton.
We have written the perturbation of (1.2) by the line solitary wave Q to the form of a symmetric hyperbolic system. Applying the same method as in the proof of Theorem 3.1 we obtain the following result.
Then there exists T > 0 and a unique solution Proof. The proof of Theorem 4.1 and Theorem 3.1 are essentially the same except the estimates for the terms B j (φ 1 ) andC(Q). In the proof of Theorem 3.1, in order to estimate the derivative of order s, we use the commutator estimate and the Bessel potential J s = (1 − ∆) s/2 . Although, in this case, the 1-D soliton solution Q does not decay in "y" direction that make that argument is not true. Therefore, we use the fact that J s ∼ J s x + J s y , where J x = F −1 ξ 1 F , J y = F −1 ξ 2 F . Hence, we only need to estimate J s x U and J s y U instead of J s U . Since Q is independent of y, J s y is harmless and since J s , we can apply the commutator estimate in one dimensional case.
The rest of the proof proceeds exactly as in the proof of Theorem 3.1. Again we emphasize that the same result holds true mutatis mutandi in a R × T setting, a framework that would be needed to study the stability of the line soliton with respect to periodic transverse perturbations.

Global solution
In this section we will establish the conservation of energy for the perturbation of (1.2) by the line soliton Q given in Section 2 and as a consequence, the existence of a global weak solution when Q is the dark soliton.
In order to make the calculation easier, we will consider the solution of the form (e iλt e i σ 3 2δ x ψ(x, y, t), ρ(x, y, t), φ(x, y, t)) then the 1-d solitary wave will have the following form with R(x), P (x) are given in Section 2 (note that this trick will not affect the analysis in section 4).
Then the system (4.1) becomes In this section we will establish the energy conservation for (5.1) when it is perturbed by a line soliton Q and the existence of a global weak solution when Q is the dark soliton.
ii) We shall use a classical compactness method (see for instance [27]). The estimates in (i) prove that regular solutions of (5.1) are uniformly bounded in H 1 × L 2 × H 1 . To obtain global weak solutions, as in [47], we first implement a Galerkin approximation process (possibly after smoothing the initial data) , yielding a sequence of approximate solutions (ψ m , ρ m , φ m ) of (5.1) Using part i), we have that for any T > 0, (ψ η , ρ η , φ η ) is bounded, independently of η, in the space Hence, up to a subsequence one can assume that By Aubin-Lions lemma one can furthermore assume that up to a subsequence (5.13) ψ m → ψ in L p loc ([0, T ] : L q loc (R 2 )), for any 2 ≤ p, q < ∞. Similar convergence results hold true for φ m , ρ m .
These convergences allow to pass to the limit in the distribution sense in (5.1) for (ψ m , ρ m , φ m ), proving that (ψ, ρ, φ) satisfies
Various stability and instability results of solutions to (6.1) have been obtained in [10,32,33,34] in the context of the Davey-Stewartson systems but no similar results seemed to be known when they are viewed as solutions to the Zakharov-Rubenchik systems. In particular one does not know if the solutions of (6.1) are constrainded minimizers of the Zakharov-Rubenchik system.
According to the Davey-Stewartson case, one could conjecture that those localized solitary waves are unstable.