On coupled nonlinear Schrödinger systems

A class of coupled Schrödinger equations is investigated. First, in the stationary case, the existence of ground states is obtained and a sharp Gagliardo–Nirenberg inequality is discussed. Second, in the energy critical radial case, global well-posedness and scattering for small data are proved.


Introduction
Consider the Cauchy problem for a fractional Schrödinger system with power-type coupled non-linearities ⎧ ⎪ ⎨ ⎪ ⎩ iu j − (− ) s u j = γ m k=1 a jk |u k | p |u j | p−2 u j ; u j (0, .) = ψ j . (1.1) Here and hereafter, s ∈ (0, 1), γ = ±1, u j : R × R N → C, for j ∈ [1, m] and a jk = a k j are positive real numbers. The fractional Laplacian operator stands for where F denotes the Fourier transform. This system of fractional partial differential equations arises in quantum mechanics. It describes how the quantum state of some physical system changes with time [15].
A solution u := (u 1 , . . . , u m ) to (1.1) satisfies (formally) conservation of the mass M(u j (t)) := 1 2 R N |u j (t, x)| 2 dx = M(u j (0)); and the energy is denoted by If γ = 1, the energy is always positive and the problem (1.1) is said to be defocusing, otherwise a control of a solution to (1.1) with the energy is no longer possible and a local solution may blow-up in finite time, we speak about focusing problem. In the classical case s = 1, the m-component coupled nonlinear Schrödinger system with power-type non-linearities arises in many physical problems in which the field has more than one component such as the interactions of M-wave packets, the nonlinear waveguides and the optical pulse propagation in birefringent fibers. In nonlinear optics [2] u j denotes the jth component of the beam in Kerr-like photo-refractive media. The coupling constant a jk acts as the interaction between the jth and the kth components of the beam. This system arises also in plasma physics, multispecies, spinor Bose-Einstein condensates, biophysics and nonlinear Rossby waves. Readers are referred, for instance, to [5,7,16,28,29]. For mathematical point of view, wellposedness issues were investigated by many authors. Indeed, global existence of solutions and scattering hold [3,22,[24][25][26][27].
When m = 1, the problem (1.1) is a non-local model known as nonlinear fractional Schrödinger equation which has also attracted much attention recently [9][10][11][12][13][14]. It is a fundamental equation of fractional quantum mechanics, which was derived by Laskin [17,18] as a result of extending the Feynman path integral, from the Brownian-like to Levy-like quantum mechanical paths. It is proved that the Cauchy problem is well-posed and scatters in the radial energy space [10,11], see also [27].
The purpose of this paper is twofold. First, the stationary problem associated to (1.1) is investigated, where the existence of ground states is obtained and a sharp Gagliardo-Nirenberg type inequality is discussed. Second, global well-posedness and scattering for small data are proved in the radial case with energy critical non-linearity.
It is the contribution of this work to extend known results in the case of the scalar fractional Schrödinger equation [4,23] about the potential well theory, global well-posedness and scattering, to the coupled fractional system. The main difficulty is to overcome the presence of a non-local operator and a combined non-linearity. Indeed, to use Strichartz estimate without loss of regularity, the radial energy space is used. Moreover, in well-posedness, we are restricted to space dimensions because of the combined non-linearity.
The rest of this paper is organized as follows. The next section contains the main results and some technical tools needed in the sequel. In Sects. 3 and 4, the stationary problem associated to (1.1) is investigated, precisely the existence of ground state and a sharp Gagliardo-Nirenberg type inequality are obtained. The goal of the fifth and sixth sections is to prove well-posedness and scattering of (1.1) in the radial energy space. The two last sections deal with global well-posedness via potential-well method.
We end this section with some definitions. We mention that C will denote a constant which may vary from line to line and if A and B are non negative real numbers, A B means that A ≤ C B. For 1 ≤ r ≤ ∞, we denote the Lebesgue space L r := L r (R N ) with the usual norm . r := . L r and . := . 2 . For simplicity, we denote the usual Sobolev Space W s, p := W s, p (R N ) and H s := W s,2 . If X is an abstract space C T (X ) := C([0, T ], X ) stands for the set of continuous functions valued in X and X rd is the set of radial elements in X , moreover for an eventual solution to (1.1), T * > 0 denotes its lifespan.

Main results and background
In what follows, the main results and some estimates needed in the sequel are given.
and it minimizes the problem Moreover, in such a case is called vector ground state if at least, two components are nonzero.

Remark 2.2
If ∈ H is a solution to (2.2), then e it is a global solution of the focusing problem (1.1), called standing wave.
For α, β ∈ R, define the sets This minimal constant in the previous inequality is determined by the equation Finally, let us give some properties of the free fractional Schrödinger kernel.

Proposition 2.4
Denoting the free operator associated to the fractional Schrödinger system by In the next sub-section, the main contribution of this note is given.

Main results
First, we deal with the stationary problem associated to (1.1). The existence of a ground states of (1.1) is claimed.
Next, a sharp vector-valued Gagliardo-Nirenberg inequality is studied. Moreover, Now, we are interested on the evolution problem (1.1). First, local well-posedness is claimed.
Then, there exist T * > 0 and a unique maximal solution to (1.1), Moreover, 2. u satisfies conservation of the energy and the mass; 3. T * = ∞ in the defocusing case (γ = 1).
In the energy critical case, global well-posedness and scattering of (1.1) hold for small data.
Remark 2.9 Some technical difficulty imposes the condition p ≥ 2 which requires the restriction N = 2 in the two previous results.
Finally, using the potential well method [23], a global well-posedness result about the focusing problem (1.1), is obtained.
In what follows, some intermediate estimates are collected.

Tools
A standard tool to study the Schrödinger problem is the so-called Strichartz estimate [14].

Remark 2.13
Taking μ = 0 in the previous result, one obtains the classical Strichartz estimate.

Proposition 2.14 ∈ H is a solution to (2.2) if and only if S
The following fractional chain rule [6] will be useful.
The following Sobolev injections [1,20] give a meaning to the energy and several computations done in this note.

The stationary problem
The goal of this section is to prove that the elliptic problem (2.2) has a ground state solution which is a vector one in some cases.
Proof With the definition, Since μ ≥ 0 and p > p * , we obtain, if β < 0, Hence, H α,β (u) > 0. Moreover, by a direct computation we find Arguing as previously, it follows that £H α,β (u) > 0. The last point is a consequence of the equality The next intermediate result is the following. Then, there exists n 0 ∈ N such that K (u n 1 , . . . , u n m ) > 0 for all n ≥ n 0 . Proof We have Thus, Next, we present an auxiliary result.

Lemma 3.4 Let (α, β) ∈ A.
Then, Proof Denoting by a the right hand side of the previous equality, it is sufficient to prove that m α,β ≤ a. Take u∈ H such that K (u) < 0. Because lim λ→−∞ K Q (u λ ) = 0, by the previous Lemma, there exists some λ < 0 such that K (u λ ) > 0. With a continuity argument there exists λ 0 ≤ 0 such that K (u λ 0 ) = 0. Then, since λ → H (u λ ) is increasing, we get This finishes the proof.
Proof of Theorem 2.5 Let (φ n ) := (φ n 1 , . . . , φ n m ) be a minimizing sequence, namely Therefore, the following sequences are bounded Thus, for any real number a, the following sequence is also bounded Choosing a ∈ (1, p), it follows that (φ n ) is bounded in H.

Existence of vector ground states
Now, we present a proof of the last part of Theorem 2.5, which deals with the existence of a more than one non zero component ground state for large μ. Take φ := (φ 1 , . . . , φ m ) such that (0, . . . , φ j , . . . , 0) is a ground state solution to (2.2). So, φ j satisfies Moreover, by Pohozaev identity it follows that Collecting the previous identities, we may write Setting, for t > 0, the real variable function γ (t) Thanks to (3.8), g(t) < 0 for large t. Then, since g(0) ≥ 0, the maximum of g(t) for t ≥ 0 is achieved at t > 0. Precisely g(t) = max t≥0 g(t). Moreover, Thus, the maximum value of g is . Now, from the previous equality via the fact that K 0,1 (γ (t)) < 0, for large μ it follows that This contradiction completes the proof.

Well-posedness
In what follows, we prove Theorem 2.7, therefore, in all this section we take N = 2. The proof contains two steps. First, we prove the existence of a unique local solution to (1.1), second, we establish the global existence.
Since the sign of the non-linearity has no local effect, we take γ = 1.

Local existence and uniqueness
We use a standard fixed point argument. For T, ρ > 0, denote the space E T,ρ : ) ≤ ρ} and with the complete distance   d (h 1 , . . . , h m ), (g 1 , . . . , g m We prove the existence of some small T, ρ > 0 such that φ is a contraction of E T,ρ . Taking u, v ∈ E T , applying the Strichartz estimate (2.6), we get .
To derive the contraction, consider the function Since p ≥ 2, by the mean value Theorem we see that Using Hölder inequality, Sobolev embedding and denoting the quantity , we compute via a symmetry argument Thus, for T > 0 small enough, φ is a contraction satisfying , v). (4.9) Taking in the last inequality v = 0, yields It remains to estimate Using the fractional chain rule via Strichartz estimate and Hölder inequality, we get for θ := 4sp( p− 1) 2sp−N ( p−1) and 2C H s < ρ, φ is a contraction of E T,ρ for some T > 0 small enough. Using (4.9), uniqueness follows for small time and then for all time with a translation argument.

Global existence
In the sub-critical defocusing case, the global existence is a consequence of energy conservation and previous calculations. Let u ∈ C([0, T * ), H) be the unique maximal solution of (1.1). We prove that u is global. By contradiction, suppose that T * < ∞. Consider for 0 < s < T * , the problem By the same arguments used in the local existence, we can prove the existence of a real τ > 0 and a solution v = (v 1 , . . . , v m ) to (P s ) on C [s, s + τ ], H). Using the conservation of energy, we see that τ does not depend on s. Thus, if we let s be close to T * such that T * < s + τ, this fact contradicts the maximality of T * .

Global existence and scattering in the critical case
In this section, we establish the global existence of a solution to (1.1) in the critical case p = p * for small data as claimed in Theorem 2.8, therefore, in all this section we take N = 2. Define the set where a > 0 is sufficiently small to fix later. Using Strichartz estimate, we get The Hölder inequality and Sobolev embedding yield Then, Using the fractional chain rule via Strichartz estimate and Hölder inequality, we find By a classical Picard argument, for small a > 0, there exists u ∈ X a , a solution to (1.1) satisfying We are ready to prove Theorem 2.8. it suffices to prove that u Ḣ remains small on the whole interval of existence of u. Letting the functional ξ be defined for u ∈ H by we write using the conservation identities and Lemma 2.16, Scattering is proved.

Invariant sets and applications
This section is devoted to obtaining global existence of solutions to the focusing system (1.1). Precisely, we prove Theorem 2.10. Let us start with a classical result about stable sets under the flow of (1.1).