EXISTENCE AND STABILITY OF A TWO-PARAMETER FAMILY OF SOLITARY WAVES FOR A 2-COUPLED NONLINEAR SCHRÖDINGER SYSTEM

In this paper, the existence and stability results for a two-parameter family of vector solitary-wave solutions (i.e both components are nonzero) of the nonlinear Schrödinger system { iut + uxx + (a|u|2 + b|v|2)u = 0, ivt + vxx + (b|u|2 + c|v|2)v = 0, where u, v are complex-valued functions of (x, t) ∈ R2, and a, b, c ∈ R are established. The results extend our earlier ones as well as those of Ohta, Cipolatti and Zumpichiatti and de Figueiredo and Lopes. As opposed to other methods used before to establish existence and stability where the two constraints of the minimization problems are related to each other, our approach here characterizes solitary-wave solutions as minimizers of an energy functional subject to two independent constraints. The set of minimizers is shown to be stable; and depending on the interplay between the parameters a, b and c, further information about the structures of this set are given.


Introduction. The nonlinear Schrödinger (NLS) equation
iu t + u xx ± |u| 2 u = 0, (1.1) where u is a complex-valued function of (x, t) ∈ R 2 arises in several applications. It has been derived in such diverse fields as deep water waves [30], plasma physics [31], nonlinear optical fibers [15,16], magneto-static spin waves [32], to name a few. The coupled nonlinear Schrödinger (CNLS) system iu t + u xx + (a|u| 2 + b|v| 2 )u = 0, iv t + v xx + (b|u| 2 + c|v| 2 )v = 0, (1.2) where u, v are complex-valued functions of (x, t) ∈ R 2 , and a, b, c ∈ R, arises physically under conditions similar to those described by (1.1) when there are two wavetrains moving with nearly the same group velocities [26,29]. The CNLS system 1006 NGHIEM V. NGUYEN AND ZHI-QIANG WANG also models physical systems in which the field has more than one components; for example, in optical fibers and waveguides, the propagating electric field has two components that are transverse to the direction of propagation. The CNLS system also arises in the Hartree-Fock theory for a double condensate. Readers are referred to the works [7,15,16,30,31] for the derivation as well as applications of this system. The system (1.2) has the following conserved quantities and It is our aim in this manuscript to prove existence and stability results for solitarywave solutions of (1.2). Such solutions are of the form v(x, t) = e i(ω2−σ 2 )t+iσx ψ(x − 2σt), (1.6) where ω 1 , ω 2 , σ ∈ R, and φ, ψ : R → R are functions that vanish at infinity in the sense that φ, ψ ∈ H 1 C . (Here H 1 C is the usual Sobolev space consisting of complexvalued, measurable functions such that both f and f x are in L 2 .) When σ = 0, solutions to (1.6) are usually referred to as standing-wave solutions. Notice that if u(x, t), v(x, t) as defined in (1.6) is a solution of (1.2), then (φ, ψ) solves the system of ordinary differential equations −φ + ω 1 φ = a|φ| 2 φ + b|ψ| 2 φ, −ψ + ω 2 ψ = c|ψ| 2 ψ + b|φ| 2 ψ. (1.7) In the last several years there have been intensive works in studying the existence of standing waves for nonlinear Schrödinger systems of the form studied in this paper, for example, see [2,3,4,5,6,12,13,19,22,27] and references therein. Few papers address the issue of stability of solitary-wave solutions to the CNLS systems [11,23,24,25,28]. The stability results obtained from those papers are all for oneparameter family of solitary waves. For example, in [11], the authors considered the variational problem of minimizing E with one constraint being the sum of the L 2 -norms of the two components, while in [23,24,25] the constraints were not independently chosen. In this paper we establish the existence and stability of a two-parameter family of solitary waves of (1.2).
Notation. For 1 ≤ p ≤ ∞, we denote by L p = L p (R) the space of all measurable functions f on R for which the norm |f | p = ∞ −∞ |f | p dx 1/p is finite for 1 ≤ p < ∞ and |f | ∞ is the essential supremum of |f | on R. Whether we intend the functions in L p to be real-valued or complex-valued will be clear from the context. H 1 C (R), as mentioned above, is the usual Sobolev space consisting of complex-valued, measurable functions on R such that both f and f are in L 2 , furnished with the norm We define the space X to be the Cartesian product For fixed s > 0 and t > 0, define the real number I = I(s, t) as follows: The set of minimizers for I(f, g) is For the single equation (1.1), stability of solitary waves is a direct consequence of the minimization problem of the energy functional subject to the one constraint of the L 2 −norm being kept constant. One crucial point in preventing dichotomy of minimizing sequences is establishing the strict sub-additivity of I, as is wellknown from Lions' pioneer work [20,21]. The strict sub-additivity of I seems to be much more challenging for the two-parameter variational problem posed in (1.8). Following the same approached used by Albert et.al. [1] which in turn relied on an argument due to [8,14], we utilize the fact that the H 1 −norms of some functions are strictly decreasing when the mass of the functions are symmetrically rearranged. The set of minimizers S s,t is shown to be stable; and depending on the interplay between the parameters a, b and c, further information about the structures of this set such as the minimizer (φ, ψ), the Lagrange multipliers ω 1 , ω 2 are given. The set S s,t form a true two-parameter family in the sense that if (s 1 , t 1 ) = (s 2 , t 2 ), then the two sets S s1,t1 and S s2,t2 are disjoint. To the best of our knowledge, such existence and stability results are the first for the system (1.2). (See also the Remark following Theorem 2.1 below.) After this paper was submitted we learnt the interesting work [17] in which a new type of solutions called multi-speeds solitary waves were constructed with each component behaving as a solitary wave to a scalar equation and the two components travelling in relatively large different speeds. The stability of these solutions are still unknown.
Naturally, prior to a discussion of stability should be a theory for the initialvalued problem itself. It has been proved (see, for example, [9,10]) that for all (u(x, 0), v(x, 0)) ∈ X, exists unique (u(x, t), v(x, t)) of (1.2) in C(R; X) emanating from (u(x, 0), v(x, 0)), and (u(x, t), v(x, t)) satisfies This manuscript is organized as follows. In Section 2, the main contributions of this manuscript are presented and discussed. The proofs of Theorems 2.1 and 2.2 are accomplished through several Lemmas and Propositions in Section 3. Lemmas 3.5 and 3.6 are crucial in establishing the proof of the relative compactness of minimizing sequences for the variational problem which defines the solitary wave solutions of (1.2). An immediate consequence of this fact is that the set of minimizers S s,t is stable. And depending on the interplay between the parameters a, b and c, further information about the structures of this set such as the minimizer (φ, ψ), the Lagrange multipliers ω 1 , ω 2 are given.
2. Statement of results. Our existence and stability results are as follows.
Theorem 2.1. Suppose a, b, c > 0. Then the following statements are true for all s > 0 and all t > 0.
1. The set of minimizers S s,t form a true two-parameter family in the sense that if (s 1 , t 1 ) = (s 2 , t 2 ), then the two sets S s1,t1 and S s2,t2 are disjoint. 2. Statements 2) and 3) say that when a, b, c > 0 with b between a and c, (1.7) still has vector solutions, and the set of minimizers S s,t is stable and consists In fact, by the non-existence result in [5] for (1.7), we must have 0 Our existence result also supplements those of [13] in which existence results were given for a range of the coupling constant b > 0 depending on a, c > 0, in terms of fixed ω 1 and ω 2 . Moreover, our stability result established here for a, b, c > 0 with b between a and c is new as this case has never been considered before. 3. When the two constraints are not independent, such as when it was proved in [23] that the set of minimizers consists of, up to translations, vector solutions with each component being multiple of the hyperbolic function sech. (Notice that in the condition (A2), the numbers a, c are allowed to be negative as well.) Regarding item 3) in the above Remark, we will show next that this is exactly the case for our set of minimizers S s,t when s, t are so restricted. Hence, the results here include those in [23].
. Then the following statements are true regarding the variational problem: In particular, the set S s,t is non-empty. 2. Each function (φ, ψ) ∈ S s,t is a solution of (1.7) for ω 1 = ω 2 = ω > 0, and therefore when substituted into (1.6) yields a (standing-wave) solitary-wave solution of (1.2). Moreover, 3. Variational problem. In preparation for the proof of Theorem 2.1, several Lemmas and Propositions are established first. Recall that a, b, c > 0 in Theorem 2.1. Our first Lemma states that the infimum must be finite and negative and that minimizing sequences are bounded uniformly.
Lemma 3.1. Every minimizing sequence for I(s, t) is bounded in X and Proof. Let (f n , g n ) ∈ X be a minimizing sequence. Using Gagliardo-Nirenberg inequality, the following estimates are clear: where C denotes various constants whose precise values are not of importance. Rewrite where an application of (3.1) is used to estimate the integral. As the norm of the minimizing sequence (f n , g n ) is bounded by itself but with a smaller power, it follows that the minimizing sequence must be bounded uniformly in X. A finite lower bound is now immediate using again (3.1) and the fact that (f n , g n ) is bounded.
To see that I(s, t) < 0, let (f, g) ∈ X such that |f | 2 2 = s and |g| 2 2 = t. For each Then |f r | 2 2 = s and |g r | 2 Thus, E(f r , g r ) < 0 for sufficiently small r.
Lemma 3.2. Let (f n , g n ) ∈ X be a minimizing sequence for I(s, t). Then for all sufficiently large n, i) if s > 0 and t ≥ 0, then there exists δ 1 > 0 such that |f n | 2 2 ≥ δ 1 ; ii) if s ≥ 0 and t > 0, then there exists δ 2 > 0 such that |g n | 2 2 ≥ δ 2 ; iii) if s > 0 and t > 0, then there exists δ 3 > 0 such that Proof. Suppose to the contrary that i) is false, then by passing to a subsequence if necessary, we may assume there exists a minimizing sequence such that lim n→∞ |f n | 2 = 0. By Gagliardo-Nirenberg inequality, Thus, Now, pick any ψ ∈ H 1 C (R) such that |ψ| 2 2 = s and let ψ r (x) = √ rψ(rx). Hence, for all n ∈ N, On the other hand, if we define then η < 0 for sufficiently small r > 0. Consequently, for all n ∈ N, But then a contradiction to (3.2) and (3.3). The case ii) can be proved similarly. To see iii), suppose the statement is false. By passing to a subsequence if necessary, we may assume that there exists a minimizing sequence (f n , g n ) for which Hence, where the minimum is attained On the other hand, which contradicts (3.4). Similar argument can be used to prove the other case.
Following approach used in [1], we will show in the next two Lemmas that the value of E(f, g) decreases when f and g are replaced by |f | and |g|, and when |f | and |g| are symmetrically rearranged. It is straightforward to see the next Lemma, using the fact that We recall here (see also [1,18]) the definition of symmetric decreasing rearrangement of a function. Let w : R → [0, ∞) be a non-negative function. If {x : w(x) > y} has finite measure m(w, y) for all y > 0, then the symmetric decreasing rearrangement w * of w is defined by Notice that if (f, g) ∈ X, then |f |, |g| ∈ H 1 , and thus |f | * and |g| * are well-defined.

Lemma 3.4.
For all (f, g) ∈ X, it must be true that Proof. Using the following important facts (for the proofs of those, see [18]): and b > 0, the Lemma is clear.
The next Lemma is crucial in obtaining the strict sub-additivity of the function I(s, t) needed in ruling out dichotomy of minimizing sequences. We refer readers to [1] for the proof of this.
Proof. Following closely the argument used in [1], we claim that for i = 1, 2, one can choose minimizing sequences (f  n | 2 2 = t i . To see this, we can take, without loss of generality, i = 1 as the case i = 2 is exactly the same. Moreover, we can assume that s 1 > 0 and t 1 > 0, as otherwise just simply take f (1) n ) be any minimizing sequence for I(s 1 , t 1 ). Since functions with compact support are dense in H 1 , and E : X → R is continuous, we can approximate (w    Notice next that if ψ is any non-negative, even, C ∞ , decreasing function for x ≥ 0 with compact support, then the convolution of ψ with any function f satisfying properties (i)-(iv) also satisfies (i)-(iv). Using "approximation to the identity" with n appropriately small for n large, then (w (4) n , z n ) satisfies all (i)-(v). Set n | 2 (which is possible as |w    n (x + x n ) have disjoint support, and g (1) n (x) andg (2) n (x) :≡ g (2) n (x + x n ) have disjoint support. Define: n +f (2) n * ; g n = g (1) n +g (2) n * .
In this case, it is well-known that where the minimum is achieved at φ s1 (x) = √ 2a where the minimum is achieved at φ t2 (x) = √ 2c t 2 4 sech( ct 2 4 x).

Thus, we have
The Lemma is hence proved.
As M n (r) is a uniformly bounded sequence of nondecreasing functions in r, one can show that it has a subsequence, which is still denoted as M n , that converges point-wisely to a nondecreasing limit function M Then 0 ≤ γ ≤ s + t.
The next Lemma says that vanishing of minimizing sequences cannot occur.
Proof. Let ρ, σ ∈ C ∞ (R) such that ρ 2 + σ 2 = 1 and ρ :≡ 1 on [−1, 1] and has support in [−2, 2]. Set, for ω > 0, We claim now that for > 0 given, ∃ω > 0 and a sequence y n such that, after passing to a subsequence, the functions n (x) = σ ω (x − y n ) f n (x), g n (x) ; satisfy To see (3.12), notice that where e(f n )(f n ) x denotes the real part of (f n )(f n ) x , because of the following: since for each n ≥ N , we can find y n such that Similarly, we have Consequently, Hence (3.12) follows. Now, if s 1 , t 1 , s − s 1 and t − t 1 are all positive, then by re-scaling f (i) n and g (i) n for i = 1, 2 so that which gives As all the scaling factors tend to 1 as n → ∞, If s 1 = 0 and t 1 > 0, then Similar estimates hold if t 1 , s−s 1 or t−t 1 are zero. Thus, in all the cases we have the limit inferior as n → ∞ of the left hand side of (3.12) ≥ I(s 1 , t 1 ) + I(s − s 1 , t − t 1 ). Consequently, I(s 1 , t 1 ) + I(s − s 1 , t − t 1 ) ≤ I(s, t) + C , which implies that I(s t , t 1 ) + I(s − s 1 , t − t 1 ) ≤ I(s, t), as > 0 is arbitrary.
The following Lemma rules out the possibility of dichotomy of minimizing sequences.
But this gives a contradiction to Lemma 3.9.
With all the above calculations at hand, we now proceed to prove Theorem 2.1.
Proof. (of Theorem 2.1) As the vanishing and dichotomy of minimizing sequences both have been ruled out, the Concentration Compactness Lemma [20,21] asserts that minimizing sequences must be compact up-to translations. Hence statement 1) follows directly. Moreover, since the functionals E and Q are all invariant under translations, an immediate consequence of the above result is that the set of minimizers S s,t is stable. (See, for example, [23].) Thus, statement 4) is also clear.
To see the validity of statements 2) and 3), notice that the Lagrange multiplier principle guarantees that there are real numbers ω 1 , ω 2 such that where the prime denotes the Fréchet derivative. Thus, the following equations (3.13) hold at least in the sense of distributions. A straightforward bootstrapping argument reveals that indeed (3.13) holds true in the classical sense as well.
Proof. (of Theorem 2. 2) It is clear that in order to prove Theorem 2.2, we only need to justify the validity of Lemmas 3.1, 3.2 and 3.6 in the presence of the condition a, c < 0 and b > 0 such that b 2 > ac. Recall that, for any fixed ω > 0, we now take We make the following claims. since c < 0, which is a contradiction to Lemma 3.1. The case ii) can be proved similarly.
To see iii), suppose the statement is false. By passing to a subsequence if necessary, we may assume that there exists a minimizing sequence (f n , g n ) for which lim inf n→∞ ∞ −∞ |f n | 2 − a 2 |f n | 4 − b|f n | 2 |g n | 2 dx ≥ 0. Hence, which again contradicts Lemma 3.1. Similar argument can be used to prove the other case.
Claim 3. Let a, c < 0 and b > 0 such that b 2 > ac. Then Lemma 3.6 is still valid.
The proof of Theorem 2.2 hence is now complete.