Orbital stability of solitary waves to the coupled compound KdV and MKdV equations with two components

which clearly has Hamiltonian form. In this paper, we mainly consider the orbital stability and instability of solitary waves with zero or nonzero asymptotic value for this equations. Precisely, we first obtain two explicitly exact solitary waves with zero asymptotic value and four explicitly exact solitary waves with nonzero asymptotic value. Secondly, we conclude some results on the orbital stability of solitary waves with zero or nonzero asymptotic value. To this aim, in order to overcome the difficulty in studying orbital stability of solitary waves with nonzero asymptotic value, we use a translation transformation to transfer this problem into solitary waves with zero asymptotic value for a reduced nonlinear coupled equations. Then by applying the classical orbital stability theory presented by Grillakis et al. and Bona et al., we obtain the orbital stability and instability of solitary waves with zero asymptotic value for the new equations. We finally derive some results on orbital stability of solitary waves with zero or nonzero asymptotic value. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition v = 0, called the compound KdV and MKdV equation, which have been studied by Zhang et al.


Introduction
The coupled Korteweg-de Vries equations were presented firstly by Hirota and Satsuma [1] in 1981, which indicated Eq.(1.1) exhibited a soliton solution and three basic conserved quantities. System (1.1) is used to describe the interaction of two long waves with different dispersion relations [2]. If v = 0, then system (1.1) is reduced to the wellknown KdV equation. From then on, the coupled nonlinear wave equations draw much more attention from mathematicians. In recent years, system (1.1) has been extensively studied. Many profound results have already been obtained, on the orbital stability of solitary waves, cnoidal waves and dnoidal waves for the system (1.1) and its generalization, see Refereces [3][4][5] and the reference therein.
As we known, the following compound KdV and MKdV equation or the Gardner equation u t + βu 2 u x + γu xxx + λuu x = 0, (1.2) has also been well studied during these decades. This equation is presented as 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 [6][7][8]. In plasma physics, Eq.(1.2) describes the small amplitude propagation of ion acoustic waves without Landau damping. When β = 0, Eq.(1.2) becomes the famous KdV equation, which is a classical model describing the one-way wave propagating in fluid. When λ = 0, Eq.(1.2) becomes MKdV equation, which is applied to describe the sonic wave propagating in some non-harmonic lattices and Alfven wave in the plasma free cold collision. Recently, Eq.(1.2) has been widely studied in physics and mathematics, see for example [9][10][11][12][13][14][15]. Using Hirota's method, the modified method of full approximation, a special transformation based on the similarity variables, mapping approach and Fan's direct algebraic method, [9][10][11][12][13][14] studied the conservation laws, N-soliton, exact solitary solution etc. for Eq.(1.2). Zhang and Shi et al. [15] presented four explicit exact solitary waves with nonzero asymptotic value and two explicit exact solitary waves with zero asymptotic value for Eq.(1.2), and applied the orbital stability theory presented by Grillakis-Shatah-Strauss [16,17] to consider orbital stability of these solitary waves solutions. Alejo [18] presented local well-posedness results in the classical Sobolev space H s (R) with s > 1 4 for the Cauchy problem of the Gardner equation and proved that the soliton was orbitally stable in the energy space using the standard techniques given by Zhidkov [19]. Muñoz [20] studied the stability of multi-kink solutions of the Gardner equation. Andrade and Pastor [21] established sufficient conditions for the orbital stability of periodic traveling wave solutions for one-dimensional dispersive equations by combining Lyapunov stability theorem and GSS orbital stability theory, and gave several applications for well known dispersive equations, such as KdV equation, MKdV euqation and Gardner equation et al. Moreover, Alves et al. [22] dealt with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves and studied orbital stability of periodic waves for KdV eqaution, generalized KdV equation and Kawahara equation et al. The stability results revealed the behaviour of solution for KdV-type equation, and guided us to understand the evolution mechanism of physical quantity or state of these equations.
As is well known, the coupled nonlinear equations in which a KdV structure are embedded occur naturally in shallow water wave problems. Guha-Roy et al. [23][24][25] have studied the coupled nonlinear partial differential equations that can be solved exactly. Even if the stability of solitary waves with zero asymptotic value has been widely studied, few results are known on the orbital stability of solitary waves with nonzero asymptotic value. Moreover, the stability of solitary waves with nonzero asymptotic value cannot be easily obtained. As far as we know, 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 are concerned with the following coupled version of compound KdV and MKdV equations with two components where α, β, λ ∈ R are arbitrary constants.  [15]. In this paper, we will apply the general theory of orbital stability presented by Grillakis et al. [16,17] to study orbital stability and instability of Eqs. (1.3). To overcome the difficulty of studying orbital stability of solitary waves with nonzero asymptotic value for Eqs.(1.3), we use a translation transformation to transfer this problem into solitary waves with zero asymptotic value for a reduced nonlinear coupled equations. Applying the translation transformation u = ϕ + D and v = ψ + D to Eqs.(1.3), we have (1.5) By direct computation, we obtain that if u(ξ) is a solitary wave with D asymptotic value, and v(ξ) is a solitary wave with D asymptotic value of Eqs. Because the stability in view here refers to perturbations of the solitary wave profile itself, a study of the initial value problem (1.3) is necessary. Similar to Theorem 1-2 in [26] and Theorem in [27], we can obtain the existence of solutions to the initial value problem of Eqs.(1.4) and (1.5). Theorem 1.1.
We define the orbital stability as follows: Definition 1.1. The solitary waves T (ct)Φ(x) are orbitally stable if for any ε > 0, there exists δ > 0 with the following property: If U 0 − Φ(x) X < δ and U(t) is a solution of (1.4) and (1.5) in some interval [0, t 0 ) with U(0) = U 0 , then U(t) can be continued to a solution in 0 ≤ t < +∞, and

Exact solitary waves of the coupled compound KdV and MKdV equations
The method of seeking the solitary wave solutions are various, such as inverse scattering transform, Painlevé analysis, Hirota bilinear transform method, Exp-function method, similarity transformation and so on., but we study the orbital stability for one specific form of solution in this paper. For simplicity, we will seek the solution of sech-type by a direct method.
Considering the traveling wave solutions of Eqs.
where E 1 is an integration constant. Then, we obtain from (2.4) It is to be noted here that u would be regular everywhere (in particular, as v(ξ) → 0) provided E 1 vanishes. As a result, Eq.
By combining (2.6), (2.11) with (2.13) and (2.14), respectively, we get the following theorem: where u is the form of (2.11), v is the form of (2.6), and A, B, k and D are given by (2.13), u + represents the solution taking " + " in B of (2.13), while u − represents another case.
where u + denotes the solution taking " + " in front of radical sign in (2.15), u − denotes another case. Also, v + represents the solution taking " − " in front of radical sign in (2.16), v − represents another case.
Meanwhile, the exact solutions (u(ξ), v(ξ)) are the solitary waves with nonzero asymptotic value of Eqs.(1.3) as D 0 in Theorem 2.1. Since the asymptotic value D is the solution of a 3 D 3 + 3 2 a 2 D 2 − 3a 1 D = 0, we know that Theorem 2.1 gives the solitary waves (u(ξ), v(ξ)) with the following three asymptotic values . By some simple computations, we get the corollary as follows. .

(2.21)
In the condition of α > c 0. If λ + α 3 > 0 and c < 0, the coupled compound KdV and MKdV equations (1.3) have the solitary waves (u(ξ), v(ξ)) with (D 1 , c − αD 1 ) asymptotic value , 0. If λ + α 3 < 0 and c < 0, the coupled compound KdV and MKdV equations (1.3) have the solitary waves (u(ξ), v(ξ)) with where We will prove that system (3.1)-(3.2) are a Hamiltonian system, and satisfies the conditions of the general orbital stability theory proposed by Grillakis et al. [16]. Let U = (ϕ, ψ). The function space on which we shall work is defined by X = H 1 (R) × H 1 (R). Let the inner product of X be 3) , there exists a nature isomorphism I : X → X * defined by where ·, · denotes the pairing between X and X * Let T be one-parameter groups of unitary operator on X defined by T (s)U(·) = U(· − s), f or U(·) ∈ X, s ∈ R. (3.7) Differentiating (3.7) with respect to s at s = 0, we have (3.8) It follows from Theorem 2.1, (3.1) and (3.2) that there exist solitary waves T (ct)Φ(x) of (3.1) and (3.2) with Φ(x) defined by In this and the following sections, we shall consider the orbital stability of solitary waves T (ct)Φ(x) of (3.1) and (3.2).

Conclusion
In this article, we are interested in studying the stability of the solitary waves with nonzero asymptotic value for a coupled version of compound KdV and MKdV equations with two components (1.3). In order to overcome the difficulty of studying orbital stability of solitary waves with nonzero asymptotic value for the coupled compound KdV and MKdV equations, we use a translation transformation to transfer this problem into solitary waves with zero asymptotic value for a reduced nonlinear coupled equations. By applying the orbital stability theory presented by Grillakis et al., the results obtained by Bona et al. and detailed spectral analysis, we obtain the orbital stability and instability of solitary waves with zero asymptotic value for a reduced new coupled nonlinear equation. From Conclusion 4.1 and Theorem 4.1, it is easy to see that the orbital stability and instability of solitary waves with zero and nonzero asymptotic value are related to wave speed c. The influence regions of stability and instability are given separately in Conclusion 4.1 and Theorem 4.1.
As v = 0, we can also obtain the stability and instability of solitary waves for the compound KdV equation. Our work not only extends GSS methods to study the orbital stability and instability of solitary wave solutions with nonzero asymptotic value, but also includes and improves the results of Zhang et al. [15]. Moreover, the orbital stability of periodic solitary wave is an open problem, we will go on to study this problem.