Local well-posedness for the Zakharov system on the background of a line soliton

We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove weak convergence to a nonlinear Schr\"odinger equation.


Introduction
In this paper, we study the initial value problem for the scalar version of the two dimensional Zakharov system i∂ t u + ∆u = nu, (1.1) 1 λ 2 ∂ 2 t n − ∆n = ∆(|u| 2 ), (1.2) where (x, y, t) ∈ R 2 × R, λ is a fixed number, u is a complex valued function, n is a real function, with initial data u(x, y, 0) = u 0 (x, y) + Q(x), n(x, y, 0) = n 0 (x, y) − Q(x) 2 , n t (x, y, 0) = n 1 (x, y). And also note that e it Q(x) is a line soliton of the cubic focusing nonlinear Schrödinger equation (NLS) i∂ t u + ∆u + |u| 2 u = 0. (1.5) The Zakharov system is introduced in [7] to describe the propagation of Langmuir waves in plasma. For more details about the derivation and the physical background of Zakharov system we refer to [6] and [7].
In (1.4) the soliton is considered as a two dimensional (constant in y) object. A natural question is that of its transverse (with respect to y) stability. When looking for localized perturbations, one is lead, as a first step, to study the Cauchy problem for the perturbed system below.
Another possibility, looking for y− periodic perturbations, would be to study (1.1)-(1.2) as a system posed on R × T, the 1-d torus case was studied in [9,8]. On the other hand, the Zakharov system in spatial space R 2 and R 3 have been studied by many authors [20,17,21,15,1,11,10,22,2,19] The local well-posedness of (1.1)-(1.2) with initial data given by (1.3) which will be proved in this paper can be viewed as the first step to study the transverse stability (or instability) of the line soliton (1.4). As far as we know, this problem is still an open problem. Concerning global issues we can quote [11,10] where the authors prove the existence of self-similar blow-up solutions and the instability by blow-up of periodic (in time) solutions (solitary wave solutions) of the Zakharov system with data in H 1 (R 2 ). Their method uses the radial symmetry of the system which is broken when one writes the system satisfied by a localized perturbation of the line solitary wave.
The transverse instability of the line solitary wave for some two dimensional models such as nonlinear Schrödinger equation, Kadomtsev -Petviashvili equation and for some general "abstract" problems have been studied extensively in [14,13,12] but the framework of those papers does not seem to include the case of the Zakharov system.
The main purpose of this paper is to prove that the Cauchy problem (1.6)-(1.7), (1.8) is locally well-posed in a suitable functional framework. The main differences between the Zakharov system and its perturbation lie in the new terms containing "Q". More precisely, i) If we reduce system (1.6)-(1.7) to a nonlinear Schrödinger equation with loss of derivative in the nonlinearity, then we can think of using the smoothing effect of Schrödinger operator as in [22]. In our case, the linear terms Qn and Q 2 u will give trouble because the function Q does not decay in y at infinity.
ii) We use the method of Bourgain (actually the techniques developed in [15,1] ) and the method of Schochet-Weinstein (see [2]) to obtain two versions of the local well-posedness result for system (1.6)-(1.7). The differences with the case of the unperturbed Zakharov system are the estimates for the new linear terms and the effects of Q in each method.
Compared to Bourgain method, Schochet-Weinstein method provides well-posednes in smaller Sobolev spaces but allows, when a suitable small parameter is included, to obtain the Schrödinger limit. For the unperturbed system see [17,16,2].
iii) There are some difficulties concerning blow-up and global existence issues. In the setting of (1.6)-(1.7), the approach in [11], [10] does not make sense because we do not have radial symmetric solutions. We also do not have L 2 conservation or at least a bound on L 2 norm that makes the usual method to extend the local solution impossible to apply.
The rest of this paper is organized as follows. In Section 2, we present the energy conservation of system (1.6)-(1.7). In Section 3, we assume that λ = 1 and use Bourgain's method to prove the local well-posedness of system (1.6)-(1.7), more precisely we have the following theorem. (1.9) Then the system (1.6)-(1.7) with initial data (u 0 , n 0 , n 1 ) ∈ H k × H l × H l−1 is locally well posed in X k,b1 for suitable b, b 1 close to 1/2 . Furthermore the solutions satisfy where T is the existence time.
(The Bourgain space X s,b j will be defined as (3.8) in section 3.) In Section 4, we present the proof by using the method of Schochet and Weinstein. We get a local (in time) uniform bound in a suitable Sobolev space that allows us to pass to limit to a nonlinear Schrödinger type equation (NLS) when λ tends to ∞. In particular we have the following theorem.
Theorem 2. Let s > 2 and consider the initial value problem (1.6)-(1.7) with initial data of the form Then (1.6)-(1.7) has a unique solution where [0, T ] is the time interval of existence, T depends only on C 1 . In addition, the solution (u, n) satisfies In Section 5, using the uniform bound in the Theorem 2 we establish a weak convergence result stating that the local solution (u(λ), n(λ)) of (1.6)-(1.7) with initial data (1.8) tends to (u, −|u| 2 ) weakly when λ tends to ∞. Where u is the unique solution of the following perturbation of the nonlinear Schrödinger equation with u(x, y, 0) = u 0 . Remark : The local well-posedness of (1.13) in a Sobolev space H s with s ≥ 1 can be obtained by using Strichartz estimate. Unfortunately, the global existence in the focusing case is still unknown to us.
Notations: F , F t , F x , F y and F −1 denote the Fourier transform of a function in spacetime, time, space variable and the inverse Fourier transform respectively. We also use " " as the short notation of the space-time Fourier transform. H s is the usual Sobolev space.
The L 2 norm in space and time. u; X or u X : The norm of function u in functional space X. C will be a general constant unless otherwise explicitly indicated. f g means that there exits a constant C such that f ≤ Cg.

Conservation law
The system (1.6)-(1.7) can be rewritten as follows where Proof. To obtain (2.4) we proceed formally. A rigorous proof can be obtained by smoothing the initial data and passing to the appropriate limit. We multiply (2.1) by ∂ tū , integrate and take its real part to get We also multiply (2.3) by ∇ −1 n t or λ 2 v and integrate to get Combining (2.6) and (2.7) we obtain (2.4).
Remark : The energy space of system (1.6)-(1.7) is H 1 × L 2 × H −1 and the Theorem 1 gives the local well-posedness of (1.6)-(1.7) in the energy space. It is expected to get a global solution with small initial data in the energy space, but in our case, the difficulty is the lack of L 2 conservation.

Bourgain method
3.1. Linear estimate. Throughout this Section we will assume that λ = 1 and split the function n in (1.6)-(1.7) into two parts n = n + + n − , where n ± = n±iω −1 ∂ t n, ω = (−∆) 1/2 , then the system (1.6)-(1.7) can be rewritten as The equations (3.1) and (3.2) have the form where φ is a real function defined in R 2 and f some nonlinear function . The Cauchy problem for (3.3) with initial data u 0 is rewritten as the integral equation where U (t) = e −itφ(−i∇) and * R denotes the retarded convolution in time.
Let ψ 1 ∈ C ∞ (R, R + ) be even with 0 ≤ ψ 1 ≤ 1, ψ 1 (t) = 1 for |t| < 1, ψ 1 (t) = 0 for |t| ≥ 2 and let ψ T = ψ 1 (t/T ) for 0 < T ≤ 1. One then replaces equation (3.4) by the cut off equation (3.5) Then, we have the cut off integral equations associated with (3.1) and (3.2) namely where U (t) = e it∆ and V ± (t) = e ∓iωt . We use a standard contraction method on the two operators on the right hand side of (3.6)-(3.7) with u ∈ X k,b1 1 and n ± ∈ X l,b 2 . X k,b1 1 and X l,b 2 are the Bourgain spaces associated to two operators with symbols φ 1 (ξ) = |ξ| 2 and φ 2 (ξ) = ±|ξ|, which are given by the following definition where φ j (ξ) is the symbol of the associated differential operator. We will also use the following definition of space time Sobolev space, The linear estimate is given by the following lemma (3.10) For b > 1/2, it is clear that X s,b ⊂ C (R, H s ). This is no longer true if b ≤ 1/2 and we shall need the following Lemma for that result.
Remark : 1. In the case: k < l + 1 we shall be able to take (3.10) with b, b 1 < 1/2 and suitable 0 < c, c 1 < 1/2 to estimate the following terms i) n ± u, n ± Q, Q 2 u and u in X k,−c1 , we also need to estimate the terms on Y k 1 and Y l 2 respectively. (Y j corresponds to the symbol φ j , j = 1, 2) 2. In the limit case: k = l + 1 we shall be forced to take b 1 = 1/2. The estimates for Q 2 u , Qn ± , n ± u and u in Y k 1 is also needed for the proof of the local well-posedness. 3. We are allowed to assume that u and n ± have compact support in t by using additional cutoffs inside f in (3.5) and consider the equation.
For the effect of those factors in the spaces X s,b , we refer [1] Lemma 2.5.
The rest of this Section will be organized as follows: In section 3.2, we will prove the estimates for linear terms which are new in this context. The estimates for nonlinear terms were proved completely in [1] and we will recall them in section 3.3. Finally, we give the final step of the proof of Theorem 1.

3.2.
Estimates for linear terms. We may assume that n ± ∈ X l,b 2 and u ∈ X k,b1 1 then they can be rewritten in the form where v, w ∈ L 2 (R 2 ). Then Furthermore, in order to estimate ω(QRe(u)) we will rewrite it as ω Q u+ū 2 , then we also need the following form In order to estimate n ± Q in X k,−c1 1 , we take its scalar product with a generic function in X −k,c1 then takes the form where with the notation ξ = (ξ 1 , ξ 2 ). In order to estimate Q 2 u in X k,−c1 1 , the required estimate takes the form where with v 2 ∈ L 2 . The required estimate for u in X k,−c1 1 is simpler than (3.13) since we do not have the term Q and the variable ξ ′ 1 . In order to estimate ω(QRe(u)) in X l,−c 2 , we will estimate ω(Qu) and ω(Qū) in X l,−c 2 . We take its scalar product with a generic function in X −l,c 2 with Fourier transform ξ l τ ± |ξ| −c v 2 and v 2 ∈ L 2 . The required estimates for ωRe(Qu) then takes the form (3.14) where We now consider the estimates for the linear terms in Y k 1 and Y l 2 . The estimates for Q 2 u and u are similar, then we will only estimate Q 2 u in Y k 1 . Similarly to the previous part, the required estimates for Q 2 u in Y k 1 will take the form : where Estimate for Qn ± in Y k 1 : where where where Because of the effect of cutoff function, in (3.12)- (3.19) we are allowed to assume that the following functions will be supported in a region |t| < CT . Preparing for the proofs of (3.12)-(3.17), we first recall the Strichartz estimate and some elementary inequalities which we will need.
Let 0 < η ≤ 1 and define q and r by Then with θ ≥ 0. Note that θ = 0 if and only if a = 0 or γ = 0.
Remark : In particular, in this paper, in order to estimate the linear terms we shall use only the special case q = r = 2, γ = 1 for both Schrödinger equation and wave equation.
By using Plancherel identity, Hölder inequality, we havē The first term on the right hand side is rewritten as follow We used the fact that F We used the fact that D x s Q(x) ∈ L ∞ x for any s ≥ 0, and θ 1 > 0 comes from the Strichartz estimate (3.22) with α > 0. Thus , (3.12) holds even in the limit case c 1 = 1/2.
ii) The proof of (3.13) is easier than (3.12), because we only need to use the estimate (3.25) to remove ξ k in the numerator and using: x for any s ≥ 0. iii) Proof of (3.14) and (3.15). We only prove (3.14), (3.15) is treated similarly, because, with a > 0 By using symbol inequalities (3.24), (3.25) and the fact that k = l + ǫ, we have Subtracting 1−ǫ 2 from one of (b 1 , c, 1/2) and using the fact that b 1 > 1−ǫ 2 , we only need to consider the terms of the following form with α > 0, β ≥ 0 and s > 0.
Remark : In the case ǫ = 1, we take b > 1/2 since we need a positive power of T on (3.12) and also on (3.17).

3.3.
Estimates for nonlinear terms. Remark : In the following Lemmas, we are allowed to assume that n ± and u have compact support in time. Then holds, with θ > 0. Then holds, with θ > 0.

Proof of Theorem 1
Proof. We finish the proof of Theorem 1 by combining the estimates for the linear and nonlinear terms in Sections 3.2, 3.3 and the contraction argument described in Section 3.1. If k < l + 1, by the Lemmas 3.1, 3.2, 3.6 (i) and 3.7 (i), it is sufficient to find 1/2 > c, c 1 ≥ 0 such that the estimates (3.35)-(3.38) hold. In the Lemmas 3.8, 3.9, 3.10 and 3.11 we choose Up to now we proved the well-posedness of the cut off integral equation (3.6)-(3.7), for the proof of the independence of the cut off function we refer to [1].
In the next section, we will use the above argument to rewrite (1.6)-(1.7) in a similar form.
4.2. Perturbed Zakharov system as a dispersive perturbation of a symmetric hyperbolic system. In Section 3, we already considered the perturbation of (1.1)-(1.2) by Q(x) = 2 √ 2/(e x + e −x ) that is (1.6)-(1.7). In this section, we will consider the perturbation of (1.1)-(1.2) by e it Q(x). This trick will make the calculation simpler by using the previous calculation. Furthermore, in order to make computation transparent, we will denote (e it Q(x), −|Q(x)| 2 ) by (φ, φ 1 ) 1 and change the notation of spatial variables from (x, y) to (x 1 , x 2 ). Then, the perturbed system will have the form then (φ, Q r , V r ) is also a solution of (4.3)-(4.5). We denote thatṼ then because of (4.15)-(4.16), (u + φ,P ,Ṽ ) is a solution of (4.3)-(4.5). Now applying the argument in the previous section, if we denote then (P r , V r , F r , G r , H r , L r ) T and (P ,Ṽ ,F ,G,H,L) T are the solutions of (4.8)-(4.13). Therefore, we can get a system solved by (P, V, F, G, H, L) T by eliminating the terms only depend on (P r , V r , F r , G r , H r , L r ), that is the following system Where the residuals are Therefore, we now can rewrite (4.15)-(4.16) as a dispersive perturbation of a symmetric hyperbolic system where U = (P, V, F, G, H, L) T , A j , B j , C j are symmetric 9 × 9 matrices, K is an anti-symmetric 9 × 9 matrix.
Now, instead of studying the Cauchy problem for the perturbed Zakharov system (4.15)-(4.16) we will study the Cauchy problem for (4.23).
Theorem 3. Let s > 2 and U 0 ∈ (H s (R 2 )) 9 , then there exists T = T ( U 0 (H s (R 2 )) 9 ) such that the equation (4.23) has a unique solution Proof. The proof of the existence proceeds via a classical iteration scheme. We first regularize the initial data by taking a family of self-adjoint regularization operators J ǫ as following.
We construct a local solution of (4.23) by considering the iteration scheme : H s we will estimate J s x1 U k+1 L 2 and J s x2 U k+1 L 2 separately. Estimate for J s x1 U k+1 L 2 . Applying J s x1 to (4.25) , we get We multiply (4.28) by J s x1 U k+1 and integrating in R 2 , with integration by parts and using the symmetry of A j and B j we obtain Note that ·, · denotes L 2 (R 2 ) scalar product and ·, · 1 denotes L 2 (R) scalar product with respect to x 1 .
Definition 4.1. We say that a Fourier multiplier σ(D) is of order s (s ∈ R) and write σ ∈ S s if ξ ∈ R d → σ(ξ) ∈ C is smooth and satisfies where C(σ) depends only on σ.
First, using (4.28) with d = 2, σ(D) = J s x1 , Sobolev embedding inequality and note that D 1 (U ) contains linear and quadratic terms on U and A j (U ) depends linearly on U , we get Now using (4.28) with d = 1, σ(D) = J s x1 and Sobolev embedding inequality, we have By the Aubin-Lions theorem (Lemma 5.1), there is a subsequence of {U kn } (with the same notation) converging strongly in L ∞ ([0, T ]; (H s−δ loc ) 9 ) with 0 < δ < 2. This allows to take the limit in the nonlinear terms of (4.23).
The uniqueness of solution of (4.23) is done by using energy method for the difference of two solutions since the dispersive part contributes nothing to the energy.
Proof of Theorem 2.

A weak convergence result
We consider the following family of system labeled by the parameter λ with initial data given by We study the behaviour of the solution (u λ , n λ ) when λ tends to ∞, that is the theorem We use a classical compactness method and we will follow the ideas of H. Added and S. Added in [17]. Before giving the proof of theorem 4 we want to recall the two following lemma in [18].
W is a Banach space with norm Then the embedding W ֒→ L p0 (0, T, B) is compact. When p 0 = ∞, p 1 > 1, the above statement is also true, see [23].
Let Ω be an open set of R n and let g, g ε ∈ L p (R n ), 1 < p < ∞, such that g ε → g a.e in Ω and g ε L p (Ω) ≤ C.
Then g ε → g weakly in L p (Ω).
Conclusions: 1) We proved the local well-posedness in the usual Sobolev spaces for the 2-d Zakharov system perturbed by its 1-d soliton solution. This is the preliminary step to studying transverse stability (or instability) of that 1-d soliton under the 2-d Zakharov flow. Moreover, Theorem 4 shows that one obtains an NLS type equation in an appropriate limit.
It is also interesting to consider the same problem for the general vectorial Zakharov system. 2) Schochet-Weinstein method is also interesting if we consider a perturbation of a 3-d Zakharov system by its 2-d soliton. We think the same method should work there, too, since we only used the algebraic structure of the system and the fact that the soliton is smooth and bounded.