Orbital stability of periodic waves for theKlein-Gordon-Schrödinger system

This article deals with the existence and orbital stability of a 
two--parameter family of periodic traveling-wave solutions for the 
Klein-Gordon-Schrodinger system with Yukawa interaction. The 
existence of such a family of periodic waves is deduced from the 
Implicit Function Theorem, and the orbital stability is obtained 
from arguments due to Benjamin, Bona, and Weinstein.


1.
Introduction. In this work we establish the existence and orbital stability of periodic traveling-wave solutions associated with the following Klein-Gordon-Schrödinger system (KG-NLS henceforth): where u = u(x, t) is a complex-valued function, v = v(x, t) is a real-valued function, m is a real constant, and ∆ stands for the Laplace operator. System (1) is often referred to by physicists as the Klein-Gordon-Schrödinger system with Yukawa interaction. In the physical context, u represents a complex scalar nucleon field interacting with a real scalar meson field represented by v, and m depends on the meson mass. Furthermore, systems similar to (1) have been used to describe the dynamics of coupled electrostatic upper-hybrid and ion-cyclotron waves in a uniform magnetoplasma (see [31], [35] and references therein).
The Cauchy problem associated with (1) has been studied in recent years. In the 3-dimensional case, Baillon and Chadam [7], using Strichartz and energy estimates, deduced the existence of global smooth solutions. Fukuda and Tsutsumi [13] discussed the initial-boundary value problem posed on a regular domain and, by appealing to Galerkin's method, obtained the existence of global strong smooth solutions. Recently, Colliander, Holmer, and Tzirakis [12] have put forward a new 222 FÁBIO NATALI AND ADEMIR PASTOR method in order to establish global existence for a general class of dispersive equations. As an application of the method they proved the existence of global solutions in low regularity L 2 -based Sobolev spaces (see also Pecher [28] and Tzirakis [32]). In the 1-dimensional case, by using Kato's theory, Rabsztyn [29] established local well-posedness in the energy space H 1 (R) × H 1 (R) × L 2 (R). Global well-posedness is then deduced thanks to energy conservation. Most recently, by applying the the I-method combined with Strichartz-type estimates, Tzirakis [32] showed global well-posedness below the energy space.
The existence and orbital stability (in the energy space) of standing-wave solutions have also been considered in the literature. In the 3-dimensional case, Ohta [27] obtained the existence and stability of stationary states by using the variational approach introduced by Cazenave and Lions [11]. Later, Kikuchi and Ohta [21], [22] established the existence of standing waves, still applying a variational approach. They proved orbital stability if the frequency is sufficiently large [22] and orbital instability if the frequency is sufficiently small [21]. In the 1-dimensional case, Ohta [26] proved the existence and orbital stability of solitary-wave solutions of the form u(x, t) = e iµt ϕ(x − λt), v(x, t) = ψ(x − λt), where λ 2 = 1 and µ > 1/4. Recently, in the periodic case, the authors [24] considered the existence and orbital stability/instability of standing-wave solutions of the form where φ c (x) = β 2 + (β 3 − β 2 )cn 2 β 3 − β 1 12 x; k , The function cn(·; k) represents the Jacobian elliptic function of cnoidal type, k is the elliptic modulus, and β 1 , β 2 , and β 3 are smooth functions depending on the parameter c. The approaches to obtain the stability/instability results were those ones developed by Grillakis et al. [16], [17] and Grillakis [18], [19]. In recent years, efforts have been made on the stability theory of periodic travelingwaves solutions and, apparently, it has gained high visibility in the work by Angulo, Bona, and Scialom [2], who, adapting the Grillakis et al. theory [18], established a complete stability theory of cnoidal-wave solutions for the Korteweg-de Vries (KdV) equation In addition, by exploring Benjamin, Bona, and Weinstein's ideas (see [8], [9], [33]), Angulo [1] has studied the existence and orbital stability of periodic standing-and traveling-wave solutions for the cubic NLS (see also [14], [15]) and modified KdV equations. Recently, by using the Grillakis et al. [17] approach, Natali and Pastor [25] established the orbital stability/instability of periodic standing waves for the Klein-Gordon equation Next, for dispersive evolution equations in a general form Angulo and Natali in [4] have established a theory for the study of the nonlinear orbital stability of positive and periodic traveling-wave solutions of the form u(x, t) = ϕ(x − ct). In (6), p ≥ 1 is an integer and M is a Fourier multiplier operator defined via the Fourier transform as where the symbol β is measurable, locally bounded, and even. The approach in [4] determined, for instance, the first proof of the nonlinear stability of a family of periodic traveling-wave solutions for the Benjamin-Ono equation where H represents the periodic Hilbert transform defined by For complementary references see [3], [5], and [6].
Attention is now turned to describing our results. Our purpose is to consider the 1-dimensional case, and to study the existence and orbital (nonlinear) stability of periodic traveling-wave solutions for (1) of the form where ϕ ω,c , φ ω,c : R → R are smooth periodic functions with the same fixed period L > 0 and ω, c are real parameters to be determined later. It should be noted that the function ψ ω,c (ξ) = e icξ ϕ ω,c (ξ) (11) is not necessarily a periodic function (with period L). Thus, the relation 2qπ c = L, for some q ∈ N, is assumed in the whole paper. This assures that ψ ω,c is periodic with period L. Substituting the waves (10) into (1), it follows that ϕ ω,c and φ ω,c must satisfy the following nonlinear system of ordinary differential equations: where, for simplicity, it is assumed that m 2 = 1. In view of the nonlinearity of (12), it is a hard task to get explicit representation of solutions. However, it is substantially simpler in the case c 2 = 1. Indeed, by assuming c 2 = 1 and defining −σ := ω + 1/2 and ϕ ω,1 := ϕ σ , (12) reads as which reduces to the ordinary differential equation, For σ > π 2 /L 2 , an L−periodic solution of (14) is given in terms of the Jacobian elliptic function of dnoidal type, namely, where dn(·; k) denotes the dnoidal function, and β is a parameter depending smoothly on σ.
We next consider the case c 2 = 1, with c 2 ≈ 1. Particular calculations are given to the case c ≈ 1, since that for c ≈ −1 can be similarly dealt with. The main idea is quite simple: fix σ 0 > π 2 /L 2 ; once the solution ϕ σ0 = ϕ ω0,1 is obtained, the Implicit Function Theorem is then applied to extend the range of parameters (ω, c) to a small ball in R 2 centered at (ω 0 , 1). This requires a detailed spectral description of the operator arising in the linearization of (1) around the traveling wave (10) at (ω, c) = (ω 0 , 1). Furthermore, since σ 0 > π 2 /L 2 is arbitrarily fixed, the parameters can be extended to an even large set, say, O. As a consequence, a smooth curve of solutions for (12)  Before proceeding, it should be pointed out that the quantities and where it is written w = v t , are conserved quantities by the flow of (1). With the periodic waves given in (16) in hand, their nonlinear stability is at issue. We just consider periodic perturbations with the same period as the underlying wave.
As mentioned above, the abstract Grillakis et al. theory can be applied to obtain, in a successfully way, the nonlinear stability for a wide class equations. Meanwhile, it is not clear how to apply it in the present case. Indeed, it must be observed that solutions in (16) are critical points of the functional R := E − ωF − cG, that is, where δR stands for the Fréchet derivative of R and ψ ω,c is given in (11). Moreover, as is well known, the conclusion in the Stability/Instability Theorem of [17] is obtained from the exact number of negative eigenvalues of the linearized operator δ 2 R and the exact number of positive eigenvalues of the Hessian d (ω, c), where d is the real-valued function given by However, it seems difficult to get the number of positive eigenvalues of d (ω, c) because the parameter c is abduced from the Implicit Function Theorem. In order to overcome these difficulties, at least for c near ±1 with |c| < 1, we employ the pioneering ideas of Benjamin, Bona, and Weinstein, whose analysis do not require this type of information (see [8], [9], [33]).
Next, we describe the main step in [8], [9], [33] applied to our case. We consider perturbations of the periodic wave (ϕ ω,c , φ ω,c , −cφ ω,c ) by defining where where E, F, and G are defined in (17), (18), and (19), respectively. The key point is to show that In order to obtain inequality (22) it is necessary to analyze the non-positive spectrum (see Lemma 3.3) of the operators The paper is organized as follows. Section 2 is concerned with the existence of periodic traveling waves for (1) of the form (10). The nonlinear stability of the aforementioned waves is presented in Section 3.

Notation and Well-Posedness Results
where f is the Fourier transform of f . To simplify, if no confusion is caused, we denote · 0 by · . The notation f ⊥ g means that f is orthogonal to g with respect to the L 2 per -inner product. The symbols sn(·; k), dn(·; k), and cn(·; k) represent the Jacobian elliptic functions of snoidal, dnoidal, and cnoidal type, respectively. The quantities Re(z) and Im(z) are the real and imaginary parts of the complex number z. We use c ≈ 1 to mean that c is sufficiently close to 1. Moreover, c ≈ 1 − (c ≈ −1 + ) means c ≈ 1 and c < 1 (c ≈ −1 and −1 < c).
The question about local well-posedness in the space per ([0, L]) associated with system (1) can be established by a direct application of classical semigroup theory and density arguments, which are similar to the proof of the continuous case (see [29]). Moreover, one can obtain global well-posedness 226 FÁBIO NATALI AND ADEMIR PASTOR just by taking the advantage that (1) conserves the quantities F and E. In fact, we consider the Cauchy problem It is clear that (24) can be written in the abstract form is clearly self-adjoint. Then, −iA is skew-symmetric. Thus, from Stone's Theorem, −iA is the generator of a strongly continuous one-parameter group of unitary operators on X. Since G is a locally Lipschitz perturbation on X such that G maps D(A) into itself and is compact for r > s ≥ 0, by using standard density results combined with the arguments stated above, we establish the following result. (24) is locally well-posed for data (u 0 , v 0 , w 0 ) ∈ X in the mild sense. More precisely, there are T > 0 and a unique pair solution Moreover, for every 0 < T 1 < T the mapping We discuss about the global well-posedness associated with problem (24). Indeed, let U 0 = (u 0 , v 0 , w 0 ). Since F and E are conserved quantities, from the Gagliardo-Nirenberg inequality, we obtain where M 1 = M + C||u 0 || 6 . Thus, from (26), we obtain the uniform bound Next, by using (27) and since H 1 , we can proceed in a similar way, as made in inequality (26), in order to deduce, where M 2 = M 2 (||u 0 || 1 , ||v 0 || 1 , ||w 0 ||). These a priori estimates allow us to establish the following theorem.
, v(t)) be the pair solution in Theorem 1.1. Then, it can be extended to any interval of time.
2. Existence of periodic-wave solutions. The aim in this section is to show the existence of a two-parameter smooth branch of periodic-wave solutions for (12) of the form (16). Our idea is first to consider c 2 = 1 and then to apply the Implicit Function Theorem to extend the parameter c such that c 2 ≈ 1. Without loss of generality, we consider m 2 = 1.
Next result gives us a smooth curve (depending on ω) of dnoidal-wave solutions for (12) with c = 1.
Proof. The first part follows immediately from Proposition 1. Then, it remains to show (31). In fact, since β = 2K(k)/L and where E denotes the complete elliptic integral of the second kind (see e.g., [10]), we have where in the last inequality we used Proposition 1 and the fact that the function k ∈ (0, 1) → K(k)E(k) is a strictly increasing function. This argument completes the proof.

2.2.
Existence of periodic waves for (12). In this subsection, we show the existence of a smooth branch of periodic wave of (12) for c ≈ 1 (if c ≈ −1 the procedure is similar) such that these solutions bifurcate the dnoidal solutions found in the last subsection.
In fact, we start by analyzing the periodic eigenvalue problem, where for σ > π 2 /L 2 , ϕ σ is the dnoidal wave in Proposition 1. The proof of the next theorem can be found in [1] or [6] (see also [24]). ) has exactly one negative eigenvalues, which is simple; zero is a simple eigenvalue (with eigenfunction ϕ σ ). Moreover, the remainder of the spectrum is constituted by a discrete set of eigenvalues bounded away from zero. Remark 1. In order to show Theorem 2.1, one observes that the periodic eigenvalue problem (32) is equivalent (under a suitable transformation) to the periodic eigenvalue problem associated with the Lamé operator posed on the interval [0, 2K].
Our main result concerning the existence of periodic waves is the following.

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Let us show that the kernel of D (as an operator on L 2 per × L 2 per ) is generated by ( ϕ ω0 , ( ϕ 2 ω0 ) ). Indeed, let (f, g) = 0 such that D(f, g) = (0, 0), that is, Substituting the second equation in (34) in the first one, we immediately obtain L σ0 f = 0, where σ 0 = −ω 0 − 1/2 and L σ0 is defined in (32). Hence, from Theorem 2.1, we have f = θϕ ω0 , for some constant θ = 0. Moreover, from (34), g = θ(ϕ 2 ω0 ) . This proves our assertion. Thus, since ϕ ω0 is an even function, it follows that ( ϕ ω0 , ( ϕ 2 ω0 ) ) does not belong to Y e and so D is injective (as an operator on L 2 per,e × L 2 per,e ). Now, let us prove that, with domain H 2 per,e × H 2 per,e , D is also surjective on L 2 per,e × L 2 per,e . Indeed, D is clearly a self-adjoint operator. Thus spec(D) = spec disc (D)∪spec ess (D), where spec(D) denotes the spectrum of D, and spec disc (D) and spec ess (D) denote, respectively, the discrete and essential spectra of D. Since  H 2 per,e ([0, L]) is compactly embedded in L 2 per,e ([0, L]), the operator D has compact resolvent. Consequently, spec ess (D) = ∅ and spec(D) = spec disc (D) consists of isolated eigenvalues with finite algebraic multiplicities (see e.g., [20, Section III.6]). Therefore, since D is one-to-one, it follows that 0 is not an eigenvalue of D, and so it does not belong to spec(D). This means that 0 ∈ ρ(D), where ρ(D) is used to denote the resolvent set of D, and so, by definition, D is surjective.
Our main result concerning the orbital stability is as follows, Then for any c ≈ 1 − and ω ∈ R such that (ω, c) ∈ O, the orbit generated by . Note that Theorem 3.2 gives us a stability result just for c ≈ 1 − . We need this condition because we want to make the norm K 1/2 c (·) (see (41) below) equivalent to that of H 1 per . Before proving Theorem 3.2, we remind the reader that, as we have already pointed out in the introduction, we do not know how to use the Grillakis et al. theory. Thus, we use the ideas of Benjamin, Bona, and Weinstein [8], [9], [33] (see also [3]).

FÁBIO NATALI AND ADEMIR PASTOR
Then, by using the property that the infimum of Ω t is attained at (x 0 , θ 0 ) = (x 0 (t), θ 0 (t)), we obtain from (38) that P (x, t) = Re(ζ(x, t)) and Q(x, t) = Im(ζ(x, t)) must satisfy the following compatibility relations: Next, we define the continuous functional R defined on H 1 where E, F, and G are defined in (17), (18), and (19), respectively. Thus, from (38) and (12), we have where the operator K −1 c is defined as a Fourier multiplier by Now, we recall the papers due to Benjamin [8] and Bona [9], to establish "good" bounds for R. In order to estimate L ω,c P, P and L + ω,c Q, Q , it is necessary some technical results concerning the spectra of the linear operators L ω,c and L + ω,c . (i) L ω,c has a unique negative eigenvalue λ ω,c , which is simple with associated eigenfunction χ ω,c . (ii) L ω,c has a simple eigenvalue at zero with eigenfunction d dx ϕ ω,c . (iii) L + ω,c is a non-negative operator. Zero is a simple eigenvalue with eigenfunction ϕ ω,c . Moreover, the remainder of the spectra of L ω,c and L + ω,c are constituted by a discrete set of eigenvalues bounded away from zero.
Proof. First of all we observe that from Weyl's essential spectrum theorem, all operators we study here have only point spectrum.
(i) Note that for c ≈ 1 − , and therefore L ω,c possesses a negative eigenvalue, say λ ω,c . On the other hand, for where σ = −ω − 1/2 and L σ is defined in (32). Since ϕ ω,c → ϕ σ , as c → 1 − , uniformly in ξ ∈ [0, L], and K −1 c is a bounded operator, it is clear that the last two terms in (43) converge to 0, as c → 1 − , for any f such that f = 1. Hence, for any c ≈ 1 − and ε > 0 small enough for all f such that f = 1. Let χ σ be the eigenfunction associated with the unique negative eigenvalue of L σ (see Theorem 2.1). Then, if f ⊥χ σ , it follows from Theorem 2.1 that L σ f, f ≥ 0. Therefore, if λ 1 denotes the second eigenvalue of L ω,c , from the min-max principle (see e.g., [30, Theorem XIII.1]), we obtain Since ε > 0 is arbitrary, this proves (i).
(ii) From (12) and the definition of K −1 c we have that L ω,c d dx ϕ ω,c = 0. If we repeat the same steps as in (i), using the min-max principle we get, for c ≈ 1 − , that zero is a simple eigenvalue of L ω,c .
(iii) Since L + ω,c ϕ ω,c = 0 and ϕ ω,c > 0, it follows from Floquet theory (see e.g., [23]) that zero is the first eigenvalue of L + ω,c , and it is simple. This completes the proof of the lemma.
Remark 3. From the proof of Theorem 3.2, we see that one can establish the stability of the periodic waves given in Remark 2 for c ≈ −1 + .