Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components

In this article, the authors consider the orbital stability of periodic traveling wave solutions for the coupled compound KdV and MKdV equations with two components \begin{document}$ \begin{equation*} \left\{ \begin{aligned} u'>Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period \begin{document}$ L $\end{document} for the coupled compound KdV and MKdV equations. Then, combining the orbital stability theory presented by Grillakis et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution with period \begin{document}$ L $\end{document} is orbitally stable. As the modulus of the Jacobian elliptic function \begin{document}$ k\rightarrow 1 $\end{document} , we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the coupled compound KdV and MKdV equations from our work. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition \begin{document}$ v = 0 $\end{document} , called the compound KdV and MKdV equation.

1. Introduction. As is well known, the coupled nonlinear equations in which a KdV structure is embedded occur naturally in shallow water wave problems. Guha-Roy et al. [7,8,9] have studied the coupled nonlinear partial differential equations that can be solved exactly. The following coupled version of compound KdV and MKdV equations with two components u t + αvv x + βu 2 u x + γu xxx + λuu x = 0, v t + α(uv) x + 2vv x = 0 (1) 12 XIAOXIAO ZHENG AND HUI WU model the physical problem of describing the strong interaction of two-dimensional long internal gravity waves propagating on neighboring pycnocline in a stratified fluid, where α, β, γ, λ are arbitrary constants. When the variable v = 0, Eqs. (1) can reduce to the compound KdV and mKdV equation or the Garder equation Eq.
(2) represents a model for wave propagation in a one-dimensional nonlinear lattice and has widespread applications in the field of solid-state physics, plasma physics, fluid physics, and quantum field theory [19,20,5]. Recently, Eq.(2) has attracted great attention of many mathematicians and physicists. By the elementary integral method, Dai et al. [6] got the approximate solutions for the solitary waves with zero asymptotic value for Eq.(2). By using many other different methods, a various of exact solitary solution for Eq. (2) have been obtained [17,13]. More recently, Zhang and Shi et al. [21] considered orbital stability of solitary waves with zero and nonzero asymptotic value for the compound KdV and MKdV equation (2).
Moreover, in 1981, Hirota and Satsuma [14] presented a coupled Korteweg-de Vries equation and indicated the equation exhibited a soliton solution and three basic conserved quantities. Eqs. (3) describe an interaction of two long waves with different dispersion relations [4]. As one of the variable v = 0, Eqs. (3) can be reduced to the well-known KdV equation. In the recent years, there have been many profound results on the orbital stability of solitary waves, cnoidal waves and dnoidal waves for the systems (3) and its generalization [10,1,2]. Guo and Chen [10] studied the orbital stability for solitary waves for Eqs.(3) by applying the abstract results of Grillakis et al. [11] and detailed spectral analysis. Angulo [1,2] obtained the existence of non-trivial smooth curve of cnoidal and dnoidal periodic traveling waves solutions respectively, and proved the nonlinear stability of these waves solutions.
When α = 1, λ = −1, the coupled compound KdV and MKdV equations (1) become As we known, even if the stability of solitary waves for the compound KdV and mKdV equation and some types of the coupled nonlinear partial differential equations have been studied, but the orbital stability of solitary wave and periodic wave of the coupled version of compound KdV and MKdV equations with two components have not been studied. In this paper, we will study the existence and orbital stability of periodic traveling wave solutions of dnoidal type for Eqs. (4). We focus on solutions for (4) of the form where c ∈ R, ξ = x − ct, φ c , ψ c : R → R are smooth functions with the same fundamental period L > 0. Because the stability in view here refers to perturbations of the periodic-wave profile itself, a study of the initial-value problem for Eqs.(4) is necessary. Similar to Theorem [12], we have the following general lemma regarding the existence of solutions to the initial value problem of Eqs.(4).
Next, based on the classical Grillakis, Shatah and Strauss theory [11], we study orbital stability of the periodic wave solutions (5). Firstly, we prove that there exist smooth periodic traveling wave solutions (5) for Eqs. (4), where φ c , ψ c are smooth function with given period L > 0.
The rest of this paper is organized as follows. In section 2, we devote to prove the existence of a smooth curve of dnoidal wave solutions for Eqs. (4). Section 3 studies the spectral analysis of one certain self-adjoint operator with a crucial role to obtain our stability result. In section 4, we show our stability result of the dnoidal waves solutions for system (4).
2. Existence of dnoidal wave solutions for the coupled compound KdV and MKdV equations. In this section, we devote to show the existence of a smooth curve of dnoidal wave solutions of the form (5) for the coupled compound KdV and MKdV equations (4).
Substituting the form of solutions in (5) into the system (4), we obtain that Integrating the system (9) with respect to ξ once, and assuming the integration constant of the second equation of (9) being zero, we obtain φ = φ c , ψ = ψ c have to satisfy the following ordinary differential system where E 0 is an arbitrary integration constant. By the second equation in (10), we have that for all ψ = 0. Then, substituting (11) into the first equation of (10) and assuming Next, we prove that there is an explicit periodic solution which will depend on Jacobian elliptic functions for Eq. (12). Multiplying (12) by φ and integrating once, we get that φ must satisfy where A φ is a needed nonzero integration constant. For convenience, we make We know that solutions of Eq.(13) depend on the roots of the polynomial F (φ). For − 3c 2 β < A φ < 0, we have Hence, F (t) has the real and symmetric roots ±η 1 and ±η 2 . Without loss of generality, we assume that 0 < η 2 < η 1 . Hence, we can write Since β > 0, the left side of (14) is not negative. Then, we obtain that η 2 ≤ φ ≤ η 1 and the η ,

STABILITY OF PERIODIC SOLUTIONS TO COUPLED COMPOUND KDV-MKDV EQS. 15
From the first equation of (15), we have c > 0. Define ρ = φ η1 and k 2 = And then, define a new variable χ through the relation ρ 2 = 1 − k 2 sin 2 χ, from (15) and (16), we get According to the definition of the Jacobi elliptic function snoidal, we get that has the solution sin(χ(ξ)) = sn( Hence, using the fact that k 2 sn 2 + dn 2 = 1, we obtain and ρ(0) = 1. Substituting the form of ρ(ξ) to the definition ρ = φ η1 , we get the dnoidal wave solution Substituting (19) into (11), we have Since dn has fundamental period 2K, namely, dn(u; k) = dn(u + 2K; k), where K = K(k) represents the complete elliptic integral of first kind, we obtain that φ and ψ have fundamental period Then, from (15), , and fundamental period T φ = T ψ can be seen as a function of variable η 2 only, that is Next, we will show that

XIAOXIAO ZHENG AND HUI WU
For L > 0 fixed, we choose c > 0 such that √ c > π L . From the analysis given above, there exists a unique η 2 ≡ η 2 (c) such that the fundamental period of the dnoidal wave φ = φ(·; η 1 (c); η 2 (c)) and ψ = ψ(·; η 1 (c); η 2 (c)) will be Then, on the basis of the limitation dn(x, 1) = sech(x), the formulae (19) and (20) lose its periodicity and we get a waveform with a single hump and with "infinity period" of the form which are the classical solitary wave solutions for the coupled compound KdV and MKdV equations.
In the following, by applying the implicit function theorem, we show that the Theorem 1.2 holds.
Next, we show that ∂ η Λ < 0 in Ω. Differentiating (24) with respect to η, we have Hence, the function k(η, c) decreases strictly with respect to η. Then, according to the relation
Since c 0 is chosen arbitrarily in the interval I = ( π 2 L 2 , +∞), from the uniqueness of the function Λ, it follows that we can extend Λ to ( π 2 L 2 , +∞). Using the smoothness of the function involved, we can immediately obtain part(2) of Theorem 1.2. Corollary 2.2. The map Π : I(c 0 ) → B(η 2,0 ) is a strictly decreasing function. Therefore, from (24), k(c) strictly increases with respect to c.
3. Spectral analysis. Before starting the spectral analysis of a linear operator, we firstly derive the operator L 1 . Differentiating (12) with respect to x, we have (−∂ 2 x + 2c − βφ 2 )φ x = 0, then, we define the operator L 1 = −∂ 2 x + 2c − βφ 2 , that is, L 1 φ x = 0. Our main purpose in this section is devoted to consider the spectral properties related to the linear operator which play a key role in the proof of the orbital stability of dnoidal wave solutions, where φ is the dnoidal wave solution (19) with the fundamental period L and c ∈ ( π 2 L 2 , +∞). We write L 1 = L 2 + M 2 , where L 2 = − d 2 dx 2 + 2c. Since M 2 is relatively compact with respect to L 2 , we have σ ess (L 1 ) = σ ess (L 2 ), following from Wely , s essential spectrum theorem. The spectrum properties for operators L 1 in the following theorem are obtained with the assistance of the periodic eigenvalue problem considered on [0, L] L 1 χ = λχ, and a semi-periodic eigenvalue problem considered on [0, L] L 1 χ = µχ, (see (4.2)-(4.8) in [22]). L 2 per ([0, L]) defined in (30) has first three simple eigenvalues λ 0 , λ 1 and λ 2 , where λ 1 = 0 is the second one with associated eigenfunction φ . Moreover, the remainder of the spectrum for operator L 1 consists of a discrete set of eigenvalues which are double.
To complete the proof of Theorem 3.1, we need to study the semi-periodic problem (32) and get the first two eigenvalues ϑ 0 , ϑ 1 related to problem (32). Similar to Eqs.(33), the semi-periodic problem (32) can also turn to the Lamé s equation in (33) with the conditions Ψ(0) = −Ψ(2K), Ψ (0) = −Ψ (2K), and the eigenvalues ϑ i associated to problem (32) are related to the µ i by the relation between ϑ i and µ i
From Theorem 3.1, we know that the operator L 1 has one simple negative eigenvalue, zero is the second one and simple with associated eigenfunction φ x , and the rest of its spectrum is positive away from zero.
We know that the dual space of X is . Define the pairing < ·, · > between X and X * by Then, there is a natural isomorphism I : X → X * defined by < If, g >= (f, g). By (36) and (37), we obtain that Define one-parameter groups of unitary operator T on X by T (s)U (·) = U (· + s), for U (·) ∈ X, s ∈ R.
In this section, we shall consider the orbital stability of solitary waves T (ct)Φ c (x) of (4). We will prove that Eqs.(4) are a Hamiltonian system, and satisfy the conditions of the general orbital stability theory proposed by Grillakis et al. [11]. Note that Eqs.(4) are invariant under T (·), we define the orbital stability as follows: is stable in X by the periodic flow generated by the coupled nonlinear wave equation (4), if for every > 0, there exists a δ( ) > 0 such that for any Otherwise, we say that T (s)Φ c is called orbitally unstable. Define By (38) and (43), we can verify that E(U ) is invariant under operator T , namely, and E(U ) is conserved, namely, for any t ∈ R, Note that the system (4) can be written as the following Hamiltonian system: where U = (u, v) T , J is a skew-symmetrically linear operator defined by is the Frechet derivative of E.
From (47), we get the Frechet derivative of E (U ) such that T (0) = JB. Then, we can define the following conserved functional Q(U ) Define an operator H c : X → X * by Since c is fixed, we write Φ for Φ c , and we have From the definition of inner product (37), we know that H c is a self-adjoint operator in the sense of H * c = H c . In effect, the operator I −1 H c is bounded self-adjoint on X. Then, the spectrum of H c is constituted by the real numbers λ such that H c −λI is not invertible.
For any y = (y 1 , y 2 ) ∈ X, by (52), we have H c y, y = (−∂ 2 x + 2c − βφ 2 )y 1 − ψ(y 1 + y 2 ), y 1 + −ψ(y 1 + y 2 ), y 2 where L 1 = −∂ 2 x + 2c − βφ 2 . In order to get the spectrum properties of operator H c , we make ψ < 0. Since k 2 sn 2 + dn 2 = 1, we β . In the condition of 0 < c < 6 β , according to Theorem 3.1, for any Ψ = (y 1 , y 2 ), we choose y 1 = Ψ 0 ( where λ 0 < 0 defined in Theorem 3.1 is the eigenvalue of L 1 . Then, operator H c has one simple negative eigenvalue with associated eigenfunction Ψ − . If β 1 denotes the second eigenvalue of operator H c , then by min-max characterization of eigenvalues, we have From (55), we have From Theorem 3.1, we know zero is the second one and simple with associated eigenfunction φ x . Combining the above inequality, we have that β 1 = 0 is the second eigenvalue of operator H c with associated eigenfunction T (0)Φ c , namely, Moreover, from Theorem 3.1, we know that the rest of spectrum for operator L 1 is positive away from zero. Let Again using min-max characterization of eigenvalues, we get that the third eigenvalue of H c is strictly positive, that is, for any ζ ∈ P defined by (58), there exist δ 3 > 0 such that where δ 3 is independent of ζ. Let From (57) and (60), we obtain Z is contained in the kernel of H c . Combining (54) and (61), we have H c U, U = H c k 2 Ψ − , k 2 Ψ − = −k 2 2 Ψ − , Ψ − < 0 for any U ∈ N . Hence, according to the above discussion, the space X in which we work can be divided into a direct sum X = N + Z + P , that is the assumption 3.3 in [11] holds.
Let v = 0 in our work, we can also obtain the orbital stability results of solitary wave solution for the compound KdV equation in the sense of limit.