ON THREE-WAVE INTERACTION SCHR¨ODINGER SYSTEMS WITH QUADRATIC NONLINEARITIES: GLOBAL WELL-POSEDNESS AND STANDING WAVES

. Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1- dimensional three-component systems of nonlinear Schr¨odinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.


1.
Introduction. The propagation of optical beams in a nonlinear dispersive medium with quadratic response has attracted the attention of a broad community of physicists in recent years. Particularly because for very short pulses, the nonlinear cubic Schrödinger (NLS) equation, which is good enough to describe long-distance propagation, should be corrected to include additional terms that take into account effects as higher-order dispersion and Raman scattering (see [15] for a brief explanation). As a result, models generalizing the cubic NLS equation should be derived. An important processes in such a direction is the so-called multistep cascading mechanism. In particular, multistep cascading can be achieved by second-order nonlinear processes such as second harmonic generation (SHG) and sum-frequency mixing (SFM) (see [18]). These procedures has gained attention due to their potential ability to produce stronger nonlinearities.
Here, we are mostly interested in two particular models: three-step cascading models. For the first one, we consider the fundamental beam with frequency ω entering a nonlinear medium with a quadratic response. Let A and B be two orthogonal polarization components of the fundamental wave. Also, denote by S and T two orthogonal polarizations of the second harmonic wave with frequency 2ω. According to [16] (see also [15]) a multistep cascading process consists of the following. The fundamental harmonic wave generates the second harmonic wave S via SHG process. Then by a down-conversion process the orthogonal fundamental wave B is generated and, finally, the fundamental wave A can be reconstructed. The reduced amplitude equations derived in the slowly varying envelope approximation with the assumption of zero absorption of all interacting waves is given, in dimensionless variables, by the following coupled nonlinear Schrödinger system where x, t ∈ R, w = w(x, t), v = v(x, t), and u = u(x, t) are complex-valued functions representing the S, B, and A polarizations, respectively. The positive real constants β and β 1 are dimensionless parameters that characterize the nonlinear phase matching between the parametrically interacting waves. The parameter χ > 0 depends on the type of phase matching and can take any positive value. Here and throughout, z denotes the complex conjugate of z. For the second model, the procedure is similar but now, as a second step, the third-harmonic wave is generated. The reduced amplitude equations is now given by (see [15]) where again w = w(x, t), v = v(x, t), and u = u(x, t) are complex-valued functions representing (in dimensionless variables) the complex electric fields envelopes of the fundamental harmonic, second harmonic, and third harmonic, respectively. The constants β, β 1 , and χ are, as before, dimensionless parameters and has similar physical meaning. The system (2) presents a fundamental model for three-wave multistep cascading solitons in the absence of walk-off (see [15]). From the mathematical point of view, the study of nonlinear quadratic Schrödinger systems with two or three-wave interaction, in one or higher dimension, has attracted the attention of many researchers in recent years (see [1,3,4,5,9,17,22,23,26,33,34], and references therein). Since such systems appear in physical applications, the topics of study are mainly focused on local and global well-posedness of the Cauchy problem, existence and nonlinear/spectral stability of standing or traveling waves, asymptotic behaviour of global solutions and blow up. In general, even when a result is known for the corresponding scalar equation, in view of the wave interactions, the extension of such a result for systems demands extra efforts.
Our first concern in this paper is to establish the global well-posedness of (1) and (2) in L 2 and in H 1 (the usual L 2 -based Sobolev space of order 1), the energy space. The local-in-time existence of solutions follows from the standard contraction mapping principle, applied to the equivalent integral formulation. The global wellposedness in L 2 follows directly in view of the conservation of the total power. In a similar fashion, the global well-posedness in H 1 follows from de conservation of the energy.
Our second concern consists in establishing the existence of standing-wave solutions. Here, we will pay particular attention to the case χ = 1. This case remains physically important because according to [15], when sum-frequency mixing is due to the third-order down-conversion frequency mixing process one should take χ = 1. The mathematical reason for this is that under such an assumption the systems can be viewed as Hamiltonian systems.
Standing waves or optical solitons are, roughly speaking, waves that propagate without changing their profile. For system (1) they are special solutions of the form and for system (2) they are of the form where γ is a real parameter and ψ, φ, and ϕ are real-valued functions decaying to zero as x → ±∞. Substituting (3) into (1), we obtain the following nonlinear system of ordinary differential equations Hence, (1) possesses standing waves of the form (3), if one shows that (5) admits at least one solution decaying to zero at infinity. We are first interested in proportional waves. Under suitable assumptions (see Section 3), we can show that (5) admits a solution with φ = aψ, ϕ = bψ, and Similarly, substituting (4) in (2), we see that ψ, φ, and ϕ must satisfy the system Also in this situation, assuming suitable balance between the parameters, we are able to show that (8) has a solution fulfilling relation (6) with With the above standing waves in hand, their stability is at issue. Our main goal here is to show that the standing waves (3) and (4) are spectrally stable. The approach employed here was established by Grillakis [6,7] (see also [24,25,32] where the theory was already been used for coupled systems) and, roughly speaking, it consists in counting the number of real eigenvalues of the linearized problem around the standing wave. We emphasize that for both systems, we are taking the advantage of their Hamiltonian structures.
It is to be observed that the above proportional standing wave solutions are obtained in a very special regime. Next, we will be concerned with non-proportional waves. To this end, after renaming variables, we note that (5) and (8) may be written in the form with F (w, v, u) = 1 2 w(u 2 + v 2 ) and F (w, v, u) = wvu + 1 2 w 2 v, respectively. This permits us writing such systems in a variational form. The method we use here allows us to consider more general quadratic nonlinearities. So, we can see (5) and (8) as particular cases of the more general system (10).
From this point on, we consider system (10) and assume the following concerning the nonlinearity F : with a ijk ≥ 0, and Under assumption (F), we follow close the arguments in [34], which employs the Mountain Pass Theorem, to prove that the variational functional associated with (10) has at least one nontrivial critical point in the functional space H 1 (R) × H 1 (R) × H 1 (R). From elliptic regularity, such a critical point turns out to be a smooth solution of (10).
Besides this introduction, the paper is organized as follows. In Section 2 we observe that the local Cauchy problems associated with (1) and (2) can be solved in a standard way and then establish the global well-posedness. In Sections 3 and 4 we show the existence and stability of the proportional standing waves. In Section 5, we introduce the basic framework to study (10) by employing a variational approach and prove our main theorem concerning non-proportional standing waves. Finally, in Section 6, it is proved some uniqueness results for the particular systems (5) and (8).
Notation. Let us give some notation. By C we denote several constants that may vary from line to line. By L p (R), 1 ≤ p ≤ ∞ we denote the standard Lebesgue space with norm · L p . For s ∈ R, we let H s (R) be the usual L 2 -based Sobolev space of order s. To simplify the notation, we set endowed with their standard norms. In particular, for real-valued functions, in H 1 it is defined the inner product Thus, H 1 becomes a real Hilbert space with induced norm It should be clear that in Section 2 the functions are taken to be complex-valued. We still use the above notations with the standard modifications and believe this will not cause a confusion. The product of a 3-by-3 matrix A by a column vector 2. Well-posedness results. This section is devoted to establish the global wellposedness of systems (1) and (2). Attention is mainly turned to the energy spaces L 2 and H 1 , because these are the natural spaces to study the soliton solutions.
2.1. Local and global well-posedness in L 2 . Our first result establishes the local and global well-posedness in L 2 . More precisely, Then, there is a unique global solution (w, v, u) of (1) (or (2)) such that (w(0), v(0), u(0)) = (w 0 , v 0 , u 0 ) and, for any T > 0, (w, v, u) ∈ C([0, T ]; L 2 ). In addition, for any T < ∞, and the map data-solution is Lipschitz from a neighborhood of Proof. As usual, first we need to show the local well-posedness, that is, the solution exists at least on a time interval, say, [0, T ]. This is quite standard by now. Indeed, the existence of a unique local solution with and T depending only upon the L 2 norm of the initial data follows from an application of the contraction mapping principle, applied to the equivalent integral formulation, taking into account the well-known Strichartz estimates. This is a consequence of the fact that quadratic nonlinearities are "subcritical" in one dimension (see, for instance, [20,Chapter 5]). The global existence then follows because (1) and (2) conserve the total power given, respectively, by and This completes the proof of the theorem.

2.2.
Local and global well-posedness in H 1 . This subsection is devoted to study local and global well-posedness in H 1 . As we show below, with no much efforts we can indeed establish the local well-posedness in H s , s > 1/2. So, we start with the following result.
Then, there are T > 0, depending only on the norm of the initial data, and a unique local solution (w, v, u) of In addition, the map data-solution is Lipschitz from a neighborhood of Proof. In this case, the existence of a local solution follows directly from the contraction mapping principle, applied to the equivalent integral formulation, just taking into consideration that H s (R) is a Banach algebra for s > 1/2. Of course, the existence time T depends on the H s norm of the initial data.

Remark 2.
As is well known, in addition to the conclusions of Lemma 2.2, there is a blow up alternative, that is, there exists 0 < T * ≤ ∞ such that the solution Concerning global well-posedness, our result writes as follows. Proof. In order to extend the solution globally by using the standard extension principle, it suffices to establish a uniform bound for the local solution.
Initially, we turn our attention to system (1). In what follows we proceed formally and deduce some suitable quantities. This procedure can be made rigorous by taking sufficient regular solutions and then passing to the limit. Multiplying the first equation in (1) by w t , summing with its complex conjugate, and integrating on R, we deduce Similarly, and Now let us define the quantity, By using (14)- (16), it is easily seen that dE dt = 0.
This means that E is a conserved quantity of (1), that is, E(t) = E(0) as long as the local solution exists. This is enough to get the uniform bound of the solutions. Indeed, since β, β 1 , χ > 0, the quantity where we used Sobolev and Young's inequalities. Combining (19) with (12), we easily see that (w, v, u) H 1 is bounded by a universal constant.
Next, we turn attention to system (2). Define, Following the same steps as before, we deduce Again, we see that E is conserved and, consequently, the norm (w, v, u) H 1 is uniformly bounded, in any finite-length interval, as long as the local solution exists. The proof of Theorem 2.3 is thus completed.
3. Spectral stability of standing waves for system (1). In this section, we will first prove that under suitable conditions on the parameters, system (5) possesses an explicit solution. The spectral stability/instability of such a solution then becomes the main issue of study.
In order to solve (5), we look for proportional solutions φ = aψ and ϕ = bψ, where a and b are real constants. By assuming 4γ + β = γ + β 1 = γ + 1, that is, Finally, by assuming that a and b belong to the circle of radius √ 2, that is, system (21) reduces to the single equation For γ + 1 > 0, which implies β < 4, (23) has the solution Next, we intent to study the spectral stability of the above solution. As we will see below, our approach is based on the Hamiltonian structure of (1). In order to write (1) as a real Hamiltonian system we set w = P + iQ, v = M + iN , and u = R + iS. Separating real and imaginary parts, we see that it reads as where U = (P, M, R, Q, N, S), J is the skew-symmetric matrix

ADEMIR PASTOR
and E is the energy functional (17). In (25), E denotes the Fréchet derivative of E. To simplify notation we set We now define the constrained energy G by In view of (5), it is easily seen that Ψ is a critical point of G, that is, G (Ψ) = 0. As a result, it is expected that the stability of the waves in question is determined by the Hessian G (Ψ). Next we observe that G (Ψ) is given by a 6-by-6 matrix operator, more precisely, where and 3.1. Spectral analysis. Here we will study the spectrum of the operator L γ . Due to the interaction between the components, the spectrum of matrix-type operators are, in general, hard to be described in details (see [29] for additional discussions). However, by using a diagonalization argument we are able to count the number of nonpositive eigenvalues of L γ . First of all, by noting that L γ is a "diagonal" operator, a complex number λ is an eigenvalues of L γ if and only if λ is an eigenvalue of either L R or L I . Thus, it suffices to study the spectrum of the operators L R and L I .
3.1.1. The spectrum of L R . By recalling that β + 4γ = γ + 1 and φ = aψ, ϕ = bψ, we see that (32) It then suffices to diagonalize the constant-coefficient matrix in the second term of (32), which we shall call E R . Since E R is a symmetric matrix, it is equivalent to a diagonal matrix for which the principal diagonal entries are the eigenvalues of E R . It is easy to see that E R has r 1 = 1, r 2 = −1, and r 3 = −2 as eigenvalues. Let A R be the matrix whose columns are normalized eigenvectors associated with the eigenvalues r 1 , r 2 , and r 3 , respectively. More precisely, Note that With the matrix A R in hand, we are able to diagonalize the operator L R by defining In the sequel we shall study the spectrum of L RD . Since L RD is a diagonal operator, it is sufficient to describe the spectrum of the operators appearing in the diagonal entries. In what follows we set Since L j is a compact perturbation of −∂ 2 x + γ + 1, Weyl' theorem implies that the essential spectrum of L j is the interval [γ + 1, ∞). (i) The operator L 2 defined in L 2 (R) with domain H 2 (R) has a unique negative eigenvalue, which is simple. Zero is a simple eigenvalue with associated eigenfunction ψ . Moreover, the rest of the spectrum is positive and bounded away from zero.
Zero is a simple eigenvalue with associated eigenfunction ψ. Moreover, the rest of the spectrum is positive and bounded away from zero. (iii) The operator L −1 defined in L 2 (R) with domain H 2 (R) is strictly positive.
Proof. The proof is essentially an application of the Sturm-Liouville theory combined with the comparison theorem. Indeed, by taking the derivative with respect to x in (23), we immediately see that zero is an eigenvalue of L 2 with associated eigenfunction ψ . Since ψ has exactly one zero on the whole line, it follows that the eigenvalue zero is the second one. This proves part (i).
Also from (23), we promptly see that zero is the first eigenvalue of L 1 with eigenfunction ψ. Finally, to establish part (iii) it suffices to apply the comparison theorem taking into account part (ii) and the fact that ψ > −ψ.
As an immediate consequence we have. Corollary 1. Let ψ be the function in (24). The operator L RD defined in L 2 with domain H 2 has a unique negative eigenvalue, which is simple. Zero is a double eigenvalue with associated eigenfunctions (0, ψ, 0) and (0, 0, ψ ). Moreover, the rest of the spectrum is positive, bounded away from zero and the essential spectrum is the interval [γ + 1, +∞).
Proof. This is a consequence of the facts that if L RD u = λu then L R A R u = λA R u and, conversely, if L R u = λu then L DR A −1 R u = λA −1 R u.

3.1.2.
The spectrum of L I . Here we will study the spectrum of L I . As above, there exists a matrix A I , satisfying A −1 I = A * I = A T I , such that the operator L ID := A −1 I L I A I is diagonal. More precisely, The operators in the diagonal of L ID , except L −2 , were studied in Lemma 3.1.
Lemma 3.2. Let ψ be the function in (24). The operator L −2 defined in L 2 (R) with domain H 2 (R) is strictly positive.
Proof. Because 2ψ > −ψ, the proof is a consequence of the comparison theorem and Lemma 3.1 (ii).

Corollary 3.
Let ψ be the function in (24). The operator L ID defined in L 2 with domain H 2 has no negative eigenvalues. Zero is a simple eigenvalue with associated eigenfunction (0, 0, ψ). Moreover, the rest of the spectrum is positive, bounded away from zero and the essential spectrum is the interval [γ + 1, +∞).

Corollary 4.
Let ψ be the function in (24). The operator L I defined in L 2 with domain H 2 has no negative eigenvalues. Zero is a simple eigenvalue with associated eigenfunctions (2ψ, aψ, bψ) ≡ (2ψ, φ, ϕ). Moreover, the rest of the spectrum is positive and bounded away from zero.

Spectral stability.
To start with, we observe that if (w, v, u) is a solution of (1) so is (e 2is w, e is v, e is u), for any s ∈ R. We denote this symmetry by T p (s). In terms of the real coordinate U , this action is represented by Next, we define V = V (t) by By using the group properties of T p (s) together with the fact that Ψ is a critical point of the functional G, it is easy to see, from Taylor's expansion and (25), that where J and L γ are the operators defined in (26) and (29), respectively. The precise definition of stability we are interested in is given below. First of all, due to the Hamiltonian structure of the problem in hand, we point out that JL γ has finitely many eigenvalues with strictly positive real part (see, for instance, Lemma 5.6 and Theorem 5.8 in [8]). Thus, studying spectral stability amounts in locating the eigenvalues off the imaginary axis. The theory for counting such eigenvalues has gained substantially attention in recent decades (see, for instance, [6,7,8,11,13,14,27], and references therein).
Our main theorem in this section reads as follows.
Theorem 3.4. Let Ψ be as in (27). Then the standing wave (3), with γ = (1−β)/3, is spectrally stable Proof. In order to show Ψ is spectrally stable we need to prove that JL γ has no eigenvalues with strictly positive real part. According to Corollary 4, the number of negative eigenvalues of L I is zero. Hence, we deduce that the unstable eigenvalues of JL γ , that is, those with a strictly positive real part, may occur only as real positive eigenvalues (see e.g. [6] or [27]). Thus our task is to show when JL γ has or not one positive real eigenvalue. We will use the theory put forward in [7]. Define where the restriction operators are understood to act from Y to Y . With these notation, Theorem 2.6 in [7] states that JL γ has exactly ± pairs of real eigenvalues, where C(L) = {y ∈ Y ; Ly, y < 0} denotes the negative cone of the operator L, d(C(L)) denotes the dimension of a maximal linear subspace that is contained in C(L) and n(L) denotes the number of negative eigenvalues. Thus, we need to decide if the number in (40) is positive or zero. If it is zero we have spectral stability while if it is positive we have spectral instability. Because n(L I ) = 0 (see Corollary 4), we deduce that it reduces to n( L R ). As a consequence, we have spectral stability if n( L R ) = 0 (and spectral instability if n( L R ) > 0). To study the quantity n( L R ) we make use of information on the spectrum of L R , paying particular attention to how projection onto the subspace Y might affect the negative index. As L R is self-adjoint, its negative eigenspace is orthogonal to its kernel, and hence is influenced only by the projection off the kernel of L I . Consequently, we are faced the problem of locating the negative eigenvalues of a constrained self-adjoint operator. Here, we will take the advantage of the theory put forward in [11] (see also [12]). In fact, we will use the following result. Now, turning back to our problem, we apply Proposition 1 with L = L R and S = ker(L I ) 1 to obtain But using (22) we easily see that Thus, from (41) and (35), we infer Let ζ 1 and ζ 3 be such that L −1 ζ 1 = f 1 and L 2 ζ 3 = f 3 . From the Fredholm alternative it follows that such functions do exist and are unique. Taking into account the explicit expression of L 2 it is not difficult to see that and (solving the integral for ψ) Next, let us inspect the quantity L −1 −1 f 1 , f 1 L 2 = ζ 1 , f 1 L 2 . Although we are not able to give the exact value, we can obtain a suitable estimate and compare it with L −1 2 f 3 , f 3 L 2 . Indeed, by definition, ζ 1 satisfies Integrating both side of (44) on R and taking into account that ζ 1 together with its first derivative must go to zero at infinity, we obtain Now, since the right-hand side of (44) is negative, the maximum principle implies that ζ 1 must be negative. So, and Finally, from (42), (43), and (46), we deduce that This completes the proof of the Theorem.
Remark 3. Note we have solved system (5) only for γ = (1 − β)/3. Thus, in particular, we do not have the existence of a smooth curve of critical points of G, say, γ → Ψ γ , for γ in some open interval. The existence of such a curve plays a crucial role in many situations in order to determine the spectral/orbital stability of standing waves for Hamiltonian systems (see e.g., [8,13,14,27] and subsequent results). Note also that the kernel of L γ is 3-dimensional. Hence, here we cannot directly apply the abstract criterion established in these cited works to prove Theorem 3.4 or the orbital stability.
4. Spectral stability of standing waves for system (2). In this section we will show the existence and spectral stability of proportional standing wave solutions for (2). Since the analysis is similar to that in Section 3 we only give the main steps. Substituting (4) in (2), we see that ψ, φ, and ϕ must satisfy system (8). Assume that 4γ + β = γ + 1 = 9γ + β 1 > 0 (47) and φ = aψ, ϕ = bψ, where a and b are nonzero real constants. Note that (47) implies that β and β 1 must satisfy the relation 8(1 − β) = 3(1 − β 1 ). In this case, we must have With these assumptions, system (8) reduces to By assuming that b and a are given by the relations we see that system (49) reduces to the single equation which has the solution Remark 4. Note that each one of the equations in (50) has two real roots. In particular, we have Contrary to the analysis for system (1), where a and b may belong to the circle a 2 + b 2 = 2, here there are only two possibilities for b and two possibilities for a (according to the sign of b). In addition, in the present case, the function ψ may be negative.
Hence the Hessian of G at Ψ takes the form with and 4.1. Spectral analysis. We will now study the spectrum of the operator T γ . As discussed in Section 3, it suffices to study the spectra of T R and T I .
where | · | stands for the Euclidean norm in R 3 . It is easy to see that By defining T RD : The spectrum of T RD is studied next. (i) The operator has a unique negative eigenvalue, which is simple. Zero is a simple eigenvalue with associated eigenfunction ψ . Moreover, the rest of the spectrum is positive, bounded away from zero and the essential spectrum is the interval [γ + 1, +∞).
Proof. The proof is similar to that of Lemma 3.1. For the second part, observe that Therefore, the potential functions of T 0 and T 1 are positive (even for b negative).
As a consequence of the above result, the following facts become clear.

Corollary 5.
Let ψ be the function in (51). The operator T RD defined in L 2 with domain H 2 has a unique negative eigenvalue, which is simple. Zero is a simple eigenvalue with associated eigenfunction (0, 0, ψ ). Moreover, the rest of the spectrum is positive, bounded away from zero and the essential spectrum is the interval [γ + 1, +∞).

Corollary 6.
Let ψ be the function in (51). The operator T R defined in L 2 with domain H 2 has a unique negative eigenvalue, which is simple. Zero is a simple eigenvalue with associated eigenfunction A R (0, 0, ψ ) ≡ (ψ , φ , ϕ ). Moreover, the rest of the spectrum is positive and bounded away from zero.

4.1.2.
The spectrum of T I . Here also there exists an orthogonal matrix A I such that T ID := A −1 I T R A I is diagonal, more precisely, where In view of the constants r 1,2 appearing in (59), the spectrum of T ID may change according to the sign of b (differently from the operator T RD ).
Corollary 7. Assume b > 0 and let ψ be the function in (51). The operator T ID defined in L 2 with domain H 2 has no negative eigenvalues. Zero is a simple eigenvalue with associated eigenfunction (0, 0, ψ). Moreover, the rest of the spectrum is positive, bounded away from zero and the essential spectrum is the interval [γ + 1, +∞).
Remark 5. Corollary 7 is not true if b < 0. Assume, for instance that a > 0. A simple inspection reveals that r 1 > r 3 . Since ψ < 0 we then have r 1 ψ < r 3 ψ. The comparison theorem implies that T r1 has at least one negative eigenvalue.
Corollary 8. Assume b > 0 and let ψ be the function in (51). The operator T I defined in L 2 with domain H 2 has no negative eigenvalues. Zero is a simple eigenvalue with associated eigenfunction A I (0, 0, ψ) ≡ (ψ, 2φ, 3ϕ) and the rest of the spectrum is positive and bounded away from zero.

Spectral stability.
This subsection is intended to study the spectral stability of the standing wave (4) with ψ, φ, and ϕ as above. The calculations run in much the way as those in Section 3. So, we give only the main steps. To start, we define V = T p (−γt)U − Ψ, where Ψ is as in (52) and T p (s) is similar to the transformation (38) with the modification that now if (w, v, u) is a solution of (2) so is (e is w, e 2is v, e 3is u), for any s ∈ R. Upon linearization, we get Our main theorem in this section reads as follows.
Theorem 4.2. Let Ψ be as in (52) and assume b > 0. Then the standing wave solution (4), with γ = (1 − β)/3, is spectrally stable in the sense of Definition 3.3, that is, the operator JT γ has no eigenvalues with positive real part.
Proof. As in the proof of Theorem 3.4 we can define T R and T I . In addition, JT γ has exactly max{n( T R ), n( T −1 (61) positive eigenvalues. By reasoning as in the proof of Theorem 3.4, we will have proved the theorem if we show that

5.
Standing waves for quadratic nonlinearities. The goal of this section is to show the existence of decaying to zero solutions for systems of the form where α, α 1 , and α 2 are positive real numbers. Our main theorem reads as follows.
As we said, the argument to show Theorem 5.1 is based on the Mountain Pass Theorem and some ideas from the concentration-compactness method. Our approach is inspired by [34], where the authors proved a similar result for the twocomponent system Before proceeding, we point out that for arbitrary values of α 1 , α 2 > 0, the existence of nontrivial solutions for (67), which are homoclinic to the origin, was established in [34] (see also [21,32], and [33] for the existence and stability of soliton and multisoliton solutions). Uniqueness of positive symmetric solutions for (67) was proved in [22]. Also, existence and nonlinear stability of periodic pulses (with respect to periodic perturbations) were addressed in [1].
The rest of this section is then devoted to prove Theorem 5.1. We start with some preliminaries.

Preliminary results.
For U = (w, v, u) ∈ H 1 we define the functional It is easy to see that I ∈ C 1 (H 1 , R) and for any U = (w, v, u) ∈ H 1 , V = (v 1 , v 2 , v 3 ) ∈ H 1 the Fréchet derivative of I at U applied at V is given by At this point, it should be noted that critical points of I are weak solutions of system (10) that decay to zero as x → ±∞. Moreover, as is well known (see e.g. [28]), such weak solutions are indeed C 2 classical solutions. Thus, in order to prove Theorem 5.1 we only have to prove that I has at least one nontrivial critical point in H 1 . Associated with the functional I is an equivalent norm in H 1 defined as In particular there exist constants C 1 , C 2 > 0 satisfying for any U ∈ H 1 . The inner product associated with the norm · I is Lemma 5.2. Under the hypothesis (F), we have where C * = max{a ijk ; 0 ≤ i, j, k ≤ 3, i + j + k = 3}.
Proof. By Young's inequality, we promptly deduce The conclusion then follow in view of the Sobolev embedding U L ∞ ≤ 2 U H 1 (see Lemma 2.1 in [34]). Now we recall the Mountain Pass Theorem (see e.g. [30]).
Remark 6. We have used that the coefficients a ijk are nonnegative in order to show Lemma 5.4. This assumption can be weakened if we assume the existence of U 0 ∈ H 1 satisfying R F (U 0 )dx > 0. As a consequence of the above lemma and Theorem 5.3, we obtain a sequence The idea now is to prove that the above sequence converges (in some sense) to a nontrivial critical point of the functional I. Proof. The proof is similar to that of Lemma 2.4 in [34]. So we omit the details.
Lemma 5.6. Let I be the functional in (68) and assume that U n = (w n , v n , u n ) ∈ H 1 satisfies (72). Then, there exists a positive constant d such that for n sufficiently large, sup x∈R |w n (x)| + sup x∈R |v n (x)| + sup x∈R |u n (x)| ≥ d.
Proof. From Lemma 5.5 there exists a constat B > 0 such that U n H 1 ≤ B, for all n ∈ N. From assumption (F), Thus, taking the limit, as n → ∞, in (73), in view of (72), we get 6c = lim n→∞ R (F w (U n )w n + F v (U n )v n + F u (U n )u n )dx.
In particular, taking into account that c > 0, for n large enough, 3c ≤ R |F w (U n )w n + F v (U n )v n + F u (U n )u n |dx.
Note that, after applying Young's inequality, there exists a constant C > 0 (depending only on a ijk ) such that R |F w (U n )w n |dx ≤ sup With Lemma 5.6 in hand we are able to show that the sequence U n = (w n , v n , u n ) ∈ H 1 satisfying (72) obtained with the help of the Mountain Pass Theorem converges (up to a translation) to a nontrivial critical point of I. Indeed, from Lemma 5.6, we may assume, without loss of generality, that for all n ∈ N, U n L ∞ = sup x∈R |w n (x)| + sup x∈R |v n (x)| + sup x∈R |u n (x)| ≥ d.
Next, multiplying the first equation in (85) by a and subtracting from the second one, we infer that where we used that u = bw. By using that 1/2 + b = a 2 /b and b + 1 = 1/b, we can write the last equation as −z + α 2 z + a b wz = 0, Since a/b > 0, this implies, as before, that z ≡ 0. This all together, imply that u and v must be multiple of w, that is, u = bw and v = aw. Substituting this in (85) and using the relations between a and b, we see that it reduces to the single equation w − α 2 w + a(b + 1)w 2 = 0. Since this equation has a unique solution, the proof is completed.
Remark 9. The uniqueness of positive solutions for (80) is not ruled out. Actually, we believe that such solutions are unique. However, we are unable to prove this result. It seems that the techniques employed in [10,21,22,31] does not apply for three-component systems.