Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 72, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND NONLINEAR STABILITY OF SOLITARY WAVE SOLUTIONS FOR COUPLED SCHRÖDINGER-KDV SYSTEMS PENGXUE CUI, SHUGUAN JI Abstract. In this article, we consider the existence and nonlinear stability of the solitary wave solutions to the coupled Schrödinger-KdV system. By using the undetermined coefficient method, we construct the exact solitary wave solutions. Furthermore, we prove the nonlinear stability of such solitary wave solutions with respect to small perturbations by applying the classical stability theory developed by Benjamin [8] and Bona [9], and the spectral analysis method. 1. Introduction The interaction models between long waves and short waves play a fundamental role in a variety of physical settings, such as plasmas physics [19], diatomic lattice system [24], quantum mechanics [6] and fluid mechanics [20]. To describe the resonant interaction between gravity long wave and interface short wave on shallow water surface, when the group velocity of the short wave is close to the phase velocity of the long wave, Kawahara et al. [20] derived the coupled Schrödinger-KdV system i(ut + c0ux) + δ1uxx = αuv, vt + c1vx + δ2vxxx + β(v2)x + η(|u|2)x = 0, (1.1) where c0, c1, δ1, δ2, α, β, η are real constants, u(x, t) is a complex value function describing interface short wave and v(x, t) is a real value function describing gravity long wave. It is obvious that, with the transformation u→ u · exp ( − c0 2δ1 i(x− c0 2 t) ) , system (1.1) can be reduced to iut + δ1uxx = αuv, vt + c1vx + δ2vxxx + β(v2)x + η(|u|2)x = 0. (1.2) During the past several decades, the coupled Schrödinger-KdV system has re- ceived extensive attention because of its important physical background. For the Cauchy problem of (1.2), please see [7, 12, 21, 23] and references therein. Tsutsumi [21] proved the global well-posedness in the space Hk+ 1 2 (R) ×Hk(R)(k ∈ Z+) by using the conservation laws. Bekiranov et al. [7] used the Fourier restriction norm method to weaken the regularity assumptions on the initial data and obtained the 2010 Mathematics Subject Classification. 35Q55, 35Q53, 35B35. Key words and phrases. Schrödinger-KdV system; nonlinear stability; solitary wave solution. c©2021. This work is licensed under a CC BY 4.0 license. Submitted March 2, 2021. Published September 10, 2021. 1 2 P. CUI, S. JI EJDE-2021/72 local well-posedness in Hs(R) ×Hs− 1 2 (R) for any s > 0. Corcho and Linaves [12] improved the previous results of [7, 21] and obtained the local well-posedness in L2(R) × H− 3 4 +(R) and a global result in H1(R) × H1(R). Wu [23] extended the result of [12] and obtained the global well-posedness in Hs(R)×Hs(R) when s > 1 2 whether the system is in the resonant case or in the non-resonant case by the I-method of Colliander et al. (see [13, 14] for examples). Another issues of great concern for this model are the existence and stability of the solitary wave solutions. It is known that, due to the effect of nonlinearity and dispersion, the coupled Schrödinger-KdV system usually possesses such kind of solutions. Please see [1, 2, 4, 5, 11, 25] for the related results. Chen [11] considered a special model with δ1 = α = c1 = η = 1 in (1.2) and obtained the orbital stability of solitary wave solutions by using the abstract method of Grillakis et al. [16, 17]. Then, for system (1.2) with α = η = −δ1 = −1, c1 = 0, δ2 = 2 and a certain range of values of β, by using the concentration compactness method, Albert and Angulo [1] proved that the system has a nonempty set of ground state solutions which is stable. For system (1.2) with δ1 = 1 and β = − 3 2α, Angulo [2] also proved the existence and stability of a nonempty set of solitary wave solutions by using the stability theory developed by Cazenave and Lions in [10] and the concentration compactness method. In this article, we consider the general model (1.2) and use the classical method of Benjamin [8] and Bona [9] to establish the results on the existence and orbital stability of solitary wave solutions. The results obtained in this paper can be regarded as a supplementary extension of [1, 2, 11]. The crucial idea of our proof is to show that solitary wave solutions is the local minimizer of the conserved functional for (1.2) via the detailed spectral analysis. The remainder of his paper is organized as follows. In Section 2, we construct the exact solitary wave solutions of Schrödinger-KdV system (1.2). In Section 3, we give the spectral analysis which is needed to prove the stability of solitary wave solutions. In Section 4, we complete the proof of the orbital stability of the solitary wave solutions for (1.2). Notation. The set of all real numbers is denoted by R. The norm of f ∈ Lp(R) is defined by ‖f‖Lp(R) = ( ∫ R |f | pdx)1/p for 1 ≤ p < ∞, and ‖f‖L∞(R) denotes the norm of f ∈ L∞(R) which is defined as the essential supremum of f on R. The inner product of two functions f, g in L2(R) is defined by (f, g) = ∫ R f(x)g(x)dx. The Fourier transform of f is denoted by f̂ which is defined as follows f̂(τ) = ∫ R f(x)e−iτxdx. For s ≥ 0, Hs(R) denotes the Sobolev space with the norm ‖f‖Hs(R) = (∫ R (1 + |ξ|2)s|f̂ |2dξ )1/2 . It is obvious that ‖f‖2H1(R) = ‖f‖2L2(R) + ‖f ′‖2L2(R). EJDE-2021/72 COUPLED SCHRÖDINGER-KDV SYSTEMS 3 2. Existence of solitary wave solutions to system (1.2) In this section, we seek the exact solitary wave solutions of system (1.2) of the form u(x, t) = e−iωtφ̃(ξ) = e−iωteiq(x−ct)φ(x− ct), v(x, t) = ϕ(ξ) = ϕ(x− ct), (2.1) where c, q, ω ∈ R, ξ = x− ct, and φ(ξ), ϕ(ξ) are real functions satisfying φ(ξ) → 0 and ϕ(ξ)→ 0 as |ξ| → +∞. Substituting (2.1) into (1.2), we obtain that (φ(ξ), ϕ(ξ)) satisfies δ1φ ′′ + i(2δ1q − c)φ′ + (ω + qc− δ1q2 − αϕ)φ = 0, δ2ϕ ′′ + βϕ2 − (c− c1)ϕ+ ηφ2 = 0. Noting that, both φ(ξ) and ϕ(ξ) are real functions, so we need to require q = c 2δ1 , which further reduces the above system to δ1φ ′′ + (ω + c2 4δ1 − αϕ)φ = 0, δ2ϕ ′′ + βϕ2 − (c− c1)ϕ+ ηφ2 = 0. (2.2) Thus, the solitary wave solutions of system (1.2) can be constructed by solving system (2.2). Theorem 2.1. If ω, α, β, c, c1, δ1, δ2, η ∈ R satisfy δ1αη > 0, 4δ1ω + c2 < 0, c1 − 4δ2( ω δ1 + c2 4δ2 1 ) > c. Then there exists a solitary wave solution of (1.2) of the form (2.1). Proof. Assume φ = d1 sech(d2ξ), where d1 and d2 will be determined in what follows. Then φ′′ = (d2 2 − 2d2 2 sech2(d2ξ))d1 sech(d2ξ) = ( − ω δ1 − c2 4δ2 1 + α δ1 ϕ ) φ. (2.3) By (2.2) and (2.3), we obtain α δ1 ϕ = −2d2 2 sech2(d2ξ) + d2 2 + ω δ1 + c2 4δ2 1 = −2d2 2 sech2(d2ξ), (2.4) d2 2 = − ω δ1 − c2 4δ2 1 . (2.5) Substituting (2.3)–(2.5) into the second equation of (2.2), we have 2δ1(c− c1)d2 2 α sech2(d2ξ) + 4d4 2βδ 2 1 α2 sech4(d2ξ) + 4δ1δ2d 4 2 α (3 sech4(d2ξ)− 2 sech2(d2ξ)) + ηd2 1 sech2(d2ξ) = (2δ1(c− c1)d2 2 α − 8δ1δ2d 4 2 α + ηd2 1 ) sech2(d2ξ) + (12δ1δ2d 4 2 α + 4d4 2βδ 2 1 α2 ) sech4(d2ξ) = 0. (2.6) 4 P. CUI, S. JI EJDE-2021/72 Combining (2.5) and (2.6), we obtain q = c 2δ1 , δ2 = −δ1β 3α , d1 = √ 2δ1 αη (− ω δ1 − c2 4δ2 1 ) ( c1 − c− 4δ2( ω δ1 + c2 4δ2 1 ) ) , d2 = √ − ω δ1 − c2 4δ2 1 . Thus, we have φ(ξ) = √ 2δ1 αη (− ω δ1 − c2 4δ2 1 ) ( c1 − c− 4δ2( ω δ1 + c2 4δ2 1 ) ) sech (√−4ωδ1 − c2 2δ1 ξ ) , ϕ(ξ) = 4ωδ1 + c2 2αδ1 sech2 (√−4ωδ1 − c2 2δ1 ξ ) . The proof is complete. � 3. Spectral analysis By (2.2) and Theorem 2.1, we have( − d2 dξ2 − ( ω δ1 + c2 4δ2 1 ) + 3α δ1 ϕ ) φ′ = 0,( − d2 dξ2 − ( ω δ1 + c2 4δ2 1 ) + α δ1 ϕ ) φ = 0, δ2ϕ ′′ + βϕ2 − (c− c1)ϕ+ ηφ2 = ( δ2 d2 dξ2 + δ2(4δ1ω + c2) δ2 1 + βϕ ) ϕ = 0. (3.1) Now, we define L1 = − d2 dξ2 − ( ω δ1 + c2 4δ2 1 ) + 3α δ1 ϕ, L2 = − d2 dξ2 − ( ω δ1 + c2 4δ2 1 ) + α δ1 ϕ, L3 = δ2 d2 dξ2 + δ2(4δ1ω + c2) δ2 1 + βϕ; (3.2) therefore L1φ ′ = 0, L2φ = 0, L3ϕ = 0. To prove the orbital stability of the solitary in next section, we study the spectra of the self-adjoint operators L1, L2 and L3. Theorem 3.1. Let δ2 < 0, φ and ϕ be the solitaty wave solutions given by Theorem 2.1. Then (i) operator L1 in (3.2) defined in H2(R) whose domain is L2(R) has exactly one negative eigenvalue which is simple; zero is the second simple eigen- value with eigenfunction φ′. Moreover, the remainder of the spectrum is constituted by a discrete set of eigenvalues; (ii) operator L2 in (3.2) defined in H2(R) whose domain is L2(R) has only non- negative eigenvalues and zero is the first one which is simple with eigen- function φ. Moreover, the remainder of the spectrum is constituted by a discrete set of eigenvalues; EJDE-2021/72 COUPLED SCHRÖDINGER-KDV SYSTEMS 5 (iii) operator L3 in (3.2) defined in H2(R) whose domain is L2(R) has only non- negative eigenvalues and zero is the first one which is simple with eigen- function ϕ. Moreover, the remainder of the spectrum is constituted by a discrete set of eigenvalues. Proof. Since x = 0 is a unique zero point of φ′, by using the Sturm-Liouville Theorem [15], we obtain that zero is the second eigenvalue of L1. Hence, L1 has a negative eigenvalue −σ2 whose corresponding eigenfunction is χ, satisfying L1χ = −σ2χ, 〈χ, χ〉 = 1. Similarly, φ and ϕ have no zero point in R, then zero is the first eigenvalue of L2 and L3 by the Sturm-Liouville Theorem. Furthermore, noting (3.2), we have 3α δ1 ϕ→ 0, as |x| → +∞, α δ1 ϕ→ 0, as |x| → +∞, βϕ→ 0, as |x| → +∞. Then by Weyl’s essential spectral Theorem [18], we have σess(L1) = [−( ω δ1 + c2 4δ2 1 ),+∞), σess(L2) = [−( ω δ1 + c2 4δ2 1 ),+∞), σess(L3) = [ δ2(4δ1ω + c2) δ2 1 ,+∞), where ω δ1 + c2 4δ21 < 0 and δ2 < 0. The theorem is proved. � Now let us do further study on the properties of operators L1, L2 and L3, which will be used later in the proof of stability. To do so, we need the following lemma. Lemma 3.2 ([22]). Let L be a self-adjoint operator having exactly one negative eigenvalue λ0 with corresponding ground state eigenfunction f0 ≥ 0. Define −∞ < α ≡ min f (Lf, f), where ‖f‖L2(R) = 1 and (f,R) = 0. We assume (R, f0) 6= 0 and R ∈ N⊥(L). Then α ≥ 0 if (L−1R,R) ≤ 0. Theorem 3.3. Under the conditions of Theorems 2.1 and 3.1, we have inf{(L2ψ,ψ) : ψ ∈ H1(R), ‖ψ‖L2(R) = 1, (ψ, φϕ) = 0} := ι1 > 0. (3.3) Proof. By Theorem 3.1, we know that L2 is a nonnegative operator, so it is obvious that ι1 ≥ 0. In what follows, we suppose that ι1 = 0. Firstly, we prove that the infimum of (3.3) can be attained. Let {ψi} be a sequence of H1(R)-functions with ‖ψi‖L2(R) = 1, (ψi, φϕ) = 0 and (L2ψi, ψi)→ ι1 as i→∞. It follows that ‖ψi‖H1(R) is bounded for any i ≥ 0. Then there is a subsequence of {ψi} which is still denoted by itself such that ψi ⇀ Φ weakly in H1(R). Now, since the classical embedding 6 P. CUI, S. JI EJDE-2021/72 H1(R) ↪→ L2(R) is compact, we obtain that Φ satisfies ‖Φ‖L2(R) = 1 and (Φ, φϕ) = 0. Furthermore, since weak convergence is lower semi-continuous, it follows that ι1 ≤ (L2Φ,Φ) < lim inf i→∞ (L2ψi, ψi) = ι1. Therefore, the infimum ι1 of (3.3) is attained at some admissible function Φ 6= 0. Thus, there exists a function Φ with ‖Φ‖L2(R) = 1, (Φ, φϕ) = 0 and (L2Φ,Φ) = 0. Next, from the theory of Lagrange multipliers, there are real constants k1, k2 such that L2Φ = k1Φ + k2φϕ. Because (L2Φ,Φ) = 0 and (Φ, φϕ) = 0, we obtain k1 = 0. And since L2φ = 0, we have k2 ∫ R φ2ϕdξ = (L2Φ, φ) = 0, which implies k2 = 0. Then L2Φ = 0. There is a real constant k3 6= 0 such that Φ = k3φ. But 0 = (Φ, φϕ) = k3 ∫ R φ2ϕdξ 6= 0, which is a contradiction. Therefore the minimum ι1 > 0. The proof is complete. � Remark 3.4. From Theorem 3.3 and the specific form of L2, we have that if f ∈ H1(R) satisfies (f, φϕ) = 0, then (L2f, f) ≥ δ2‖f‖2H1(R). Theorem 3.5. Under the conditions of Theorems 2.1 and 3.1, if c1 − 8δ2( ω δ1 + c2 4δ2 1 ) > c, (3.4) then: (i) inf { (L1ψ,ψ) : ψ ∈ H1(R), ‖ψ‖L2(R) = 1, (ψ, φ) = 0 } := ι2 = 0; and (ii) inf { (L1ψ,ψ) : ψ ∈ H1(R), ‖ψ‖L2(R) = 1, (ψ, φ) = 0, (ψ, (φϕ)′) = 0} := ι3 > 0. Proof. The solitary wave solution φ given by Theorem 2.1 is a bounded function which implies that ι2 is finite. And since (φ′, φ) = 0, L1φ ′ = 0, we have ι2 ≤ 0. Furthermore, we can obtain ι2 = 0 by proving ι2 ≥ 0 in virtue of Lemma 3.2. According to Theorem 3.1, we obtain that the operator L1 satisfies the condition of Lemma 3.2. So, we only need to find a function χ satisfying L1χ = φ and (χ, φ) ≤ 0. In fact, we define the mapping µ → φµ ∈ H1(R), where µ = −( ωδ1 + c2 4δ21 ). By differentiating (3.1) with respect to µ, it yields − ∂2 ∂x2 dφ dµ + φ− ( ω δ1 + c2 4δ2 1 ) dφ dµ + 3α δ1 ϕ dφ dµ = 0. Thus χ = −dφdµ satisfies L1χ = φ . Namely, χ = L−1 1 φ. Furthermore, we have (χ, φ) = (−dφ dµ , φ) = −1 2 d dµ ∫ R φ2dξ EJDE-2021/72 COUPLED SCHRÖDINGER-KDV SYSTEMS 7 = −1 2 d dµ ∫ R 2δ1 αη µ(c1 − c+ 4δ2µ) sech2(ξ)dξ = − δ1 αη (c1 − c+ 8δ2µ) ∫ R sech2(ξ)dξ. By (3.4) and the conditions of Theorem 2.1, we know (χ, φ) < 0. Then, according to Lemma 3.2, we obtain ι2 ≥ 0. Therefore ι2 = 0. The proof of (i) is complete. By (i), we have ι3 ≥ 0. In what follows, we suppose that ι3 = 0. By using the similar proof of Theorem 3.3, we can obtain an admissible function Φ satisfying ‖Φ‖L2(R) = 1, (Φ, φ) = 0, (Φ, (φϕ)′) = 0 and (L1Φ,Φ) = 0. Next, from the theory of Lagrange multipliers, there are real constants k4, k5, k6 such that L1Φ = k4Φ + k5φ+ k6(φϕ)′. From (L1Φ,Φ) = 0, (Φ, φ) = 0 and (Φ, (φϕ)′) = 0, we obtain k4 = 0. Since L1φ ′ = 0, (φ, φ′) = 0, we have k6 ∫ R φ′(φϕ)′dξ = −3k6η (c1 − c+ 4δ2(− ω δ1 − c2 4δ21 )) ∫ R (φ′)2φ2dξ = 0. By (3.4), we obtain k6 = 0. Thus L1Φ = k5φ. Since L1χ = φ with χ = −dφdµ , we have L1(Φ − k5χ) = 0. Therefore there exists a real constant k7 6= 0 such that Φ − k5χ = k7φ ′. Since (χ, φ) 6= 0, (φ′, φ) = 0 and (Φ, φ) = 0, we obtain k5 = 0. That is, Φ = k7φ ′. But 0 = (Φ, (φϕ)′) = k7(φ′, (φϕ)′) = −3k7η (c1 − c+ 4δ2(− ω δ1 − c2 4δ21 )) ∫ R (φ′)2φ2dξ 6= 0, which is a contradiction. Therefore ι3 > 0. The proof is complete. � Remark 3.6. From (ii) in Theorem 3.5 and the specific form of L1, we have that if f ∈ H1(R) satisfies (f, φ) = 0 and (f, (φϕ)′) = 0, then (L1f, f) ≥ δ1‖f‖2H1(R). 4. Orbital stability To obtain the stability of the solitary wave solutions, we rewrite (1.2) in the Hamiltonian form dU dt = JE′(U), U = (u, v) ∈ X, where X = H1 complex(R)× L2 real(R), J is a skew-symmetrical matrix operator by J = ( − i 2 0 0 − η α ∂ ∂x ) , E(U) = ∫ R ( δ1|ux|2 + αv|u|2 + αc1 2η v2 + αβ 3η v3 − αδ2 2η v2 x ) dx, (4.1) E′(U) = ( −2δ1uxx + 2αuv αδ2 η vxx + αc1 η v + αβ η v 2 + αu2 ) . And the inner product in X is (~u,~v) = Re ∫ R ( u1v̄1 + u1xv̄1x + u2v2 ) dx, ~u = (u1, u2), ~v = (v1, v2) ∈ X. (4.2) 8 P. CUI, S. JI EJDE-2021/72 The dual space of X is X∗ = H−1 complex(R) × L−2 real(R). There exists a natural isomorphism I : X → X∗, defined by 〈I~u,~v〉 = (~u,~v), (4.3) where 〈~u,~v〉 = Re ∫ R (u1v̄1 + u2v2) dx, ~u = (u1, u2), ~v = (v1, v2) ∈ X. (4.4) From (4.2)–(4.4), we obtain I = ( 1− ∂2 ∂x2 0 0 1 ) . In the remainder of this paper, we will use the method of Benjamin [8] and Bona [9] to prove the orbital stability of the solitary wave solution Ψ = (φ̃(ξ), ϕ(ξ)) with φ̃(ξ) = ei c 2δ1 ξφ(ξ) given by Theorem 2.1. First of all, let us give the definition of orbital stability. Definition 4.1. We say that the orbit generated by Ψ = (φ̃, ϕ), ΩΨ := {(eiθφ̃(·+ y), ϕ(·+ y)) : (y, θ) ∈ R× [0, 2π)} (4.5) is stable in X = H1 complex(R) × L2 real(R) by the flow of (1.2), if for every ε > 0, there is δ(ε) > 0 such that, for any (u0(x, t), v0(x, t)) ∈ X satisfying ‖u0 − φ̃‖H1(R) < δ, ‖v0 − ϕ‖L2(R) < δ, the solution of the Schrödinger-KdV equations (1.2) with initial data u(0) = u0, v(0) = v0 exists globally and satisfies inf y∈R,θ∈[0,2π) ‖eiθu(·+ y, t)− φ̃‖H1(R) < ε, inf y∈R ‖v(·+ y, t)− φ‖L2(R) < ε, for any t ∈ R. Otherwise, we say that Ψ = (φ̃, ϕ) is unstable in X. For the proof of orbital stability, we need to introduce two energy functions. Let T1 and T2 be the one-parameter group of unitary operator on X defined by T1(s1)U(·) = U(· − s1),∀s1 ∈ R, U(·) = (u(·), v(·)) ∈ X, T2(s2)U(·) = (e−is2u(·), v(·)),∀s2 ∈ R, U(·) = (u(·), v(·)) ∈ X. (4.6) From (4.6), we obtain T ′1(0) = ( − ∂ ∂x 0 0 − ∂ ∂x ) , T ′2(0) = ( −i 0 0 0 ) . By requiring T ′1(0) = JB1 and T ′2(0) = JB2, we can obtain B1 = ( −2i ∂∂x 0 0 −αη ) , B2 = ( 2 0 0 0 ) . Then, we define Q1(U) = 1 2 〈B1U,U〉 = ∫ R Im(uxū)dx+ α 2η ∫ R v2dx, (4.7) Q2(U) = 1 2 〈B2U,U〉 = ∫ R |u|2dx, (4.8) EJDE-2021/72 COUPLED SCHRÖDINGER-KDV SYSTEMS 9 where U(·) = (u(·), v(·)) ∈ X. It is easy to verify that E(U), Q1(U) and Q2(U) are invariant under the transformation of T1 and T2 (see [16, 17] for details), that is, E(T1(s1)T2(s2)U) = E(U), Q1(T1(s1)T2(s2)U) = Q1(U), Q2(T1(s1)T2(s2)U) = Q2(U), (4.9) for any s1, s2 ∈ R, where U(t) = (u(t), v(t)) is a flow of (1.2) with E(u(t), v(t)) = E(u(0), v(0)) = E(u0, v0), Q1(u(t), v(t)) = Q1(u(0), v(0)) = Q1(u0, v0), Q2(u(t), v(t)) = Q2(u(0), v(0)) = Q2(u0, v0). (4.10) To investigate the orbital stability, we need to use some related results on the local and global well-posedness of the initial value problem of (1.2) which is actually studied extensively in [7, 12, 21, 23]. So we omit the details here and enter into the study of orbital stability directly. Theorem 4.2. Under the conditions of Theorem 2.1, if δ2 < 0, δ1 > 0, β > 0, c1 + 10δ2(− ω δ1 − c2 4δ2 1 ) > c, (4.11) then the orbit ΩΨ given by (4.5) is orbitally stable in X = H1(R) × L2(R) with respect to the flow of the nonlinear Schrödinger-KdV system (1.2). Proof. The main idea of our proof is based on the method of Benjamin [8], Bona [9], and Weinstein [22]. Let us start with the declaration, for any initial data (u0, v0) ∈ H1(R) × H1(R), (u(t), v(t)) is the global solution of Schrödinger-KdV system (1.2) with initial value (u0, v0). If we define Ωt(y, θ) = ‖eiθ(T3u)′(·+ y, t)− φ′‖2L2(R) + µ‖eiθ(T3u)(·+ y, t)− φ‖2L2(R), where µ = −( ωδ1 + c2 4δ21 ) and T3u = e−i c 2δ1 (x−ct)u(x, t), then the error of the solution (u(t), v(t)) from ΩΨ is measured by ρ((u(t), v(t)),ΩΨ) = √ inf (y,θ)∈R×[0,2π) Ωt(y, θ). So, by using the standard arguments in [8, 9], there is an interval I = [0, T ] such that the infimum of Ωt(y, θ) is reached in (y, θ) = (y(t), θ(t)) for any t ∈ I. Then we have ( ρ((u(t), v(t)),ΩΨ) )2 = Ωt(y(t), θ(t)). (4.12) Now, let us consider the perturbation of the solitary wave solutions Ψ = (φ̃, ϕ) which can be written as eiθu(x+ y, t) = φ̃+ γ̃1(x, t), v(x+ y, t) = ϕ+ γ2(x, t), (4.13) with φ̃ = ei c 2δ1 (x−ct)φ, y = y(t) and θ = θ(t) are determined by (4.12). For ease of calculation, we denote γ̃1(x, t) = ei c 2δ1 (x−ct)γ1(x, t) = ei c 2δ1 (x−ct)(p(x, t) + iq(x, t)) with real functions p(x, t), q(x, t). 10 P. CUI, S. JI EJDE-2021/72 Since the minimum of Ωt(y, θ) can be reached in (y, θ) = (y(t), θ(t)), we can obtain that ∂Ωt ∂θ |θ=θ(t) = 0 and ∂Ωt ∂y |y=y(t) = 0. Hence, ∂Ωt ∂θ |θ=θ(t) = −2 ∫ R (φ′′ − µφ)qdx = −2 ∫ R (φϕ)qdx = 0, ∂Ωt ∂y |y=y(t) = −2 ∫ R (φ′′′ − µφ′)pdx = −2 ∫ R (φϕ)′pdx = 0. From the above equations, we obtain the following compatibility relation between p(x, t) and q(x, t)∫ R (φ(x)ϕ(x))q(x)dx = 0, ∫ R (φ(x)ϕ(x))′p(x)dx = 0. (4.14) We define the continuous functional in X = H1(R)×H1(R): H(u, v) = E(u, v)− cQ1(u, v)− ωQ2(u, v), where E, Q1 and Q2 are the conserved functional given in (4.1), (4.7) and (4.8). According to (4.9) and (4.10), the values of E,Q1 and Q2 are invariant under translation and rotation. By (2.2), (4.13) and the classical embedding H1(R) ↪→ Lp(R), for any p ≥ 2, we have ∆H(u, v) = H(u, v)−H(φ̃, ϕ) = δ1〈L1p, p〉+ δ1〈L2q, q〉+ 2δ1〈L2φ, p〉+ α η 〈L3ϕ, γ2〉+ α 2η 〈L3γ2, γ2〉 + ∫ R αβ 2η ϕγ2 2 + αγ2(p2 + 2pφ+ q2) + α 2η ( c1 − c− δ2(4δ1ω + c2) δ2 1 ) γ2 2dx + ∫ R c2 4δ1 φ2 − 2αϕp2 + αβ 3η γ3 2dx = δ1〈L1p, p〉+ δ1〈L2q, q〉+ α 2η 〈L3γ2, γ2〉+ ∫ R c2 4δ1 φ2 − 2αϕp2 + αβ 3η γ3 2dx + ∫ R m1γ 2 2 + 2γ2 α(p2 + 2pφ+ q2) 2 + α2(p2 + 2pφ+ q2)2 4m1 dx + ∫ R m1γ 2 2 − α2(p2 + 2pφ+ q2)2 4m1 dx = δ1〈L1p, p〉+ δ1〈L2q, q〉+ α 2η 〈L3γ2, γ2〉 + ∫ R ( γ2 √ m1 + α(p2 + 2pφ+ q2)√ 4m1 )2 dx + ∫ R m1γ 2 2 − α2(p2 + 2pφ+ q2)2 4m1 + c2 4δ1 φ2 − 2αϕp2 + αβ 3η γ3 2dx, (4.15) where m1 := α 4η ( βϕ+ c1 − c− δ2(4δ1ω + c2) δ2 1 ) . Since ϕ < 0, by (4.11), we have∫ R −2αϕp2dx > 0, m1 > 0. EJDE-2021/72 COUPLED SCHRÖDINGER-KDV SYSTEMS 11 Thus, (4.15) can be reduced to ∆H(u, v) ≥ δ1〈L1p, p〉+ δ1〈L2q, q〉+ α 2η 〈L3γ2, γ2〉 + ∫ R ( γ2 √ m1 + α(p2 + 2pφ+ q2)√ 4m1 )2 dx − C0‖γ1‖4H1(R) + C1‖γ2‖2L2(R) − C2‖γ2‖3L2(R), (4.16) where C1 and C2 are positive constants. Now let us estimate the terms 〈L1p, p〉, 〈L2q, q〉 and 〈L3γ2, γ2〉, where p(x, t), q(x, t) satisfy the compatibility relation (4.14). We first estimate 〈L1p, p〉. Since Q2(U) is invariant, we consider the normaliza- tion ‖u0‖L2(R) = ‖φ‖L2(R) for every t ∈ [0, T ]. According to (4.13), we have∫ R φ2dx = ‖u(t)‖2L2(R) = ‖γ1(t) + φ(t)‖2L2(R) = ∫ R (p+ φ)2 + q2dx. Thus, we obtain ∫ R (p2 + q2)dx = −2 ∫ R pφdx. That is ‖γ1‖2L2(R) = −2(p, φ), for any t ≥ 0. Without loss of generality, we suppose that ‖φ‖2L2(R) = 1. To estimate 〈L1p, p〉, we define the following two variables p‖ = (p, φ)φ = −1 2 [‖p‖2L2(R) + ‖q‖2L2(R)]φ, p⊥ = p− p‖. By (4.14), it is easy to see that (p⊥, (φϕ)′) = ∫ R p(φϕ)′ − 1 2 (‖p‖2L2(R) + ‖q‖2L2(R))φ(φϕ)′dx = 3‖γ1‖2L(R) 1− c− 4 3β(−ω − c2 4 ) ∫ R φ3(x)φ′(x)dx = 0, (4.17) and (p⊥, φ) = ∫ R pφ+ 1 2 (‖p‖2L2(R) + ‖q‖2L2(R))φ 2dx = 0. (4.18) Combining (4.17), (4.18) with Theorem 3.5, we have (L1p⊥, p⊥) ≥ C3‖p⊥‖2H1(R) ≥ C3‖p‖2H1(R) − C4‖γ1‖4H1(R). (4.19) Then, noting that (L1φ, φ) < 0, we can obtain (L1p‖, p‖) ≥ −C5‖γ1‖4H1(R). (4.20) Furthermore, by the Cauchy-Schwarz inequality and the definition of L1, we have (L1p⊥, p‖) = (p⊥, L1p‖) = 1 2 ‖γ1‖2L(R)(p⊥, L1φ) ≥ −C6‖γ1‖3H1(R) − C7‖γ1‖4H1(R). (4.21) Hence, by (4.19)–(4.21), we obtain (L1p, p) ≥ D1‖p‖2H1(R) −D2‖γ1‖3H1(R) −D3‖γ1‖4H1(R), (4.22) where Di > 0 for i = 1, 2, 3. 12 P. CUI, S. JI EJDE-2021/72 Next, according to Theorem 3.3, (4.14) and the specific form of L2, there is a D4 > 0 such that (L2q, q) ≥ D4‖q‖2H1(R). (4.23) Finally, by Theorem 3.1, we have 〈L3γ2, γ2〉 ≥ 0. (4.24) Thus substituting (4.22)–(4.24) into (4.16), we have ∆H(u, v) ≥ C̃1‖γ1‖2H1(R) − C̃2‖γ1‖3H1(R) − C̃3‖γ1‖4H1(R) + C1‖γ2‖2L2(R) − C2‖γ2‖3L2(R) ≥ b1‖γ1‖21,µ − b2‖γ1‖31,µ − b3‖γ1‖41,µ + b4‖γ2‖2L2(R) − b5‖γ2‖3L2(R) = b1‖γ̃1‖21,µ − b2‖γ̃1‖31,µ − b3‖γ̃1‖41,µ + b4‖γ2‖2L2(R) − b5‖γ2‖3L2(R) := g(‖γ̃1‖1,µ, ‖γ2‖L2(R)), (4.25) where g(s, z) = b1s 2 − b2s3 − b3s4 + b4z 2 − b5z3 with bi > 0 for i = 1, 2, 3, 4, 5 and ‖γ̃1‖21,µ = ‖γ̃1 ′‖2L2(R) + µ‖γ̃1‖2L2(R). Obviously, g(0, 0) = 0 and g(s, z) > 0 for (s, z) 6= (0, 0) belonging to some sufficiently small neighborhood of (0, 0). From (4.25), we can immediately get the result of stability of Theorem 4.2. In fact, let ε > 0, from the continuity of H(u, v) on S = {u0 ∈ H1(R), v0 ∈ L2(R) : ‖u0‖L2(R) = ‖φ‖L2(R)} and the continuity of the mapping ρ((u(t), v(t)),ΩΨ) in time, there is a δ(ε) > 0 such that if (u0, v0) ∈ S and ‖u0 − φ̃‖H1(R) < δ(ε), ‖v0 − ϕ‖L2(R) < δ(ε), then g(‖γ̃1‖1,µ, ‖γ2‖L2(R)) ≤ ∆H(u(t), v(t)) = ∆H(u0, v0) ≤ g(ε, ε), (4.26) for all t ∈ [0, T ]. By (4.26) and the continuity of inf(y,θ)∈R×[0,2π) Ωt(y, θ) as a function of t, we have ‖γ̃1‖1,µ < ε, ‖γ2‖L2(R) < ε. (4.27) Similar to the proof of [3, Theorem 6.1], we obtain that (4.27) still holds for all t > 0. Thus we know that the orbit ΩΨ is stable in X for the perturbations which are small in H1 and L2-norm, respectively. The proof is complete. � Acknowledgements. We are grateful to the anonymous referees for their valuable comments and suggestions. 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Peng; Orbital stability of periodic traveling wave solutions to the generalized Zakharov equations, Acta Math. Sci. Ser. B, 37 (2017), 998–1018. Pengxue Cui School of Mathematics and Statistics and Center for Mathematics and Interdisci- plinary Sciences, Northeast Normal University, Changchun 130024, China Email address: cuipx0205@vip.163.com Shuguan Ji (corresponding author) School of Mathematics and Statistics and Center for Mathematics and Interdisci- plinary Sciences, Northeast Normal University, Changchun 130024, China Email address: jisg100@nenu.edu.cn 1. Introduction Notation 2. Existence of solitary wave solutions to system (??) 3. Spectral analysis 4. Orbital stability Acknowledgements References