EXISTENCE OF GENERALIZED HOMOCLINIC SOLUTIONS OF A COUPLED KDV-TYPE BOUSSINESQ SYSTEM UNDER A SMALL PERTURBATION∗

This paper considers the coupled KdV-type Boussinesq system with a small perturbation uxx = 6cv − 6u − 6uv + εf(ε, u, ux, v, vx), vxx = 6cu − 6v − 3u + εg(ε, u, ux, v, vx), where c = 1 + μ, μ > 0 and ε are small parameters. The linear operator has a pair of real eigenvalues and a pair of purely imaginary eigenvalues. We first change this system into an equivalent system with dimension 4, and then show that its dominant system has a homoclinic solution and the whole system has a periodic solution if the perturbation functions g and h satisfy some conditions. By using the contraction mapping theorem, the perturbation theorem, and the reversibility, we theoretically prove that this homoclinic solution, when higher order terms are added, will persist and exponentially approach to the obtained periodic solution (called generalized homoclinic solution) for small ε and μ > 0.


Introduction
Bona, Chen and Saut [5,6] derived a four-parameter family of Boussinesq systems v t + u ξ + (uv) ξ + au ξξξ − bv tξξ = 0, u t + v ξ + uu ξ + cv ξξξ − du tξξ = 0 (1.1) to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal or near the shore with a flat bottom, where v represents the elevation from the equilibrium position and u represents the horizontal velocity in the flow at height βh (h is the undisturbed depth of the liquid and β ∈ [0, 1]). Here the parameters a, b, c and d are required to satisfy the following consistency conditions which implies a + b + c + d = 1 3 . This system has also be obtained in two space dimensions over a variable bottom for the generation and propagation of tsunami waves [23]. (1.1) has been studied mathematically and numerically in many literatures and numerous interesting results have been obtained (for example, see [1-4, 6, 9-12, 16, 18, 22, 25, 26]).
Bona and Chen [4] pointed out that if b = d = 1 6 , a = c = 0, the system (1.1) is a good candidate to model the two-dimensional surface water waves. When b = d = 0, the system (1.1) become a KdV-type Boussinesq system and [1] investigated its boundary stabilization posed on a bounded domain with the condition a = c > 0. If a = c = 1 6 and b = d = 0, a purely coupled KdV-type Boussinesq system is given in [5] as v t + u ξ + (uv) ξ + 1 6 u ξξξ = 0, u t + v ξ + uu ξ + 1 6 v ξξξ = 0, (1.2) which is free of the presumption of unidirectionality, while the classical coupled KdV equation assumes that the waves travel only in one direction. There are a lot of research papers devoted to the study of various properties of the system (1.2). For example, [24] discussed the exponential decay of the total energy; [6] considered its Hamiltonian structure and its initial-value problem; [7,8] stressed that (1.2) does not possess classical solitary waves decaying to zero at infinity, and they investigated numerically its generalized solitary wave solutions-solitary wave solutions exponentially approaching to a periodic solution with a small amplitude at infinity. In this paper, we study the system (1.2) and theoretically prove the existence of generalized solitary wave solutions. Let x = ξ−ct where the constant c is the speed. Integrating the system (1.2) and taking the constant of integration equal to zero, we obtain the ordinary differential system Since the real world always has some small noises or disturbances, here we are specially interested in the solutions of the system (1. 3) under small perturbations, that is, we will focus on the following system where ε is a small parameter and the conditions for the smooth real functions f and g will be given as we need (see (1.5) or Section 2). The linearized operator of (1.4) at the origin has two pairs of purely imaginary eigenvalues for c < 1; a double eigenvalue 0 and a pair of purely imaginary eigenvalues for c = 1; and a positive eigenvalue, a negative eigenvalue and a pair of purely imaginary eigenvalues for c > 1. As we know, a bifurcation will appear if an eigenvalue transversely cross the purely imaginary axis. In this paper, we are interested in the travelling speed c close to 1. Using a dynamical system approach, we will rigorously prove the existence of a generalized homoclinic solution-homoclinic solutions exponentially tending to a periodic solution for c > 1 but close to 1, which corresponds to a generalized solitary wave solution of (1.2) under a small perturbation. We would like to mention that Lombardi [19][20][21] has obtained some good results about the generalized homoclinic solutions for the general system with the aid of the norm form theory and the complex analysis. Our method is basically similar to ones in [19][20][21]. The main theorem can be stated as follows.
Theorem 1.1. Write c = 1 + µ (µ > 0) and assume that f (ε, 0, 0, 0, 0) = 0, g(ε, 0, 0, 0, 0) = 0, (1.5) There exist constants µ 0 and γ > 0 such that for each µ ∈ (0, µ 0 ), if we suppose that with a positive constant α 2 , then the system (1.4) has an even generalized homoclinic solution where the phase shift θ is of order O(µ 1/24 ), ς(x) is a smooth even cut-off function with ς(x) = 0 for |x| ≤ 1 and ς(x) = 1 for |x| ≥ 2. Here D 1 , D 2 , T 1 and T 2 are smooth functions in their arguments, T 1 and T 2 are periodic with a period π/( √ 3(1+ r 1 )) for some constant r 1 = O(µ), and D 1 , D 2 , T 1 and T 2 satisfy uniformly with respect to the parameter µ that The paper is organized as follows. Section 2 changes the system (1.4) into a system with a dimension 4. A homoclinic solution of the dominant system is given. In Section 3, some lemmas are presented. The contraction mapping theorem and the perturbation method are applied to prove that the obtained homoclinic solution deforms to a generalized homoclinic solution exponentially approaching to a periodic solution for x ∈ [0, ∞). In Section 4, using the reversibility and adjusting the phase shift, we extend this generalized homoclinic solution to x ∈ (−∞, ∞), which yields the proof of Theorem 1.1. Section 5 yields the proofs of two lemmas left in previous sections.
Throughout this paper, M denotes a positive constant and B = O(C) means that |B| ≤ M |C|.

Homoclinic solutions
where c = 1 + µ is used and µ is a small parameter. Under the assumption (1.5), we know that the origin is an equilibrium and the system (2.1) is reversible with a reverser S defined by , we call the solution (u, u 1 , v, v 1 ) reversible. This means that u(x) and v(x) are even functions, and u 1 (x) and v 1 (x) are odd functions. The reversibility will play an important role in the existence of the generalized homoclinic solution.
For µ = 0 and ε = 0, the linear operator of the system (2.1) has a double eigenvalue 0 and a pair of purely imaginary eigenvalues ±2 √ 3i. The corresponding eigenvectors and the generalized eigenvectors are given by and they satisfy Note that the solution of (2.1) can be expressed in terms of the above vectors. Since the system (2.1) is real and η 3 , η 4 are complex, we let U = (u, u 1 , v, v 1 ) T and Then (2.1) is equivalent to the real system for (A, B, V 1 , V 2 ), which is (2.7) By (2.2), (2.4) and (2.5), the reverser S is now given by The assumption (1.5) implies Thus, the system (2.6) is reversible with the reverser S. Symbolically, the system (2.6) can be written as (2.11) The dominant system of (2.6) is which has a homoclinic solution (2.14) Moreover H(x) satisfies the following inequality In Section 3 and Section 4, we will prove that this homoclinic solution deforms into a generalized homoclinic solution for the whole system (2.10).

Generalized homoclinic solution for x ∈ [0, ∞)
In this section, by using the contraction mapping theorem, we will demonstrate that for x ∈ [0, ∞), (2.10) has a generalized homoclinic solution exponentially approaching to a periodic solution X µ,ε,γ (x) as x → ∞.
First we look for the periodic solution X µ,ε,γ (x). Let τ = 2 √ 3(1 + r 1 )x for a constant r 1 to be determined. By the reversibility defined in (2.8), we can assume that the reversible periodic solution of (2.10) with a period 2π has the form Choose V 1,1 as a small positive real parameter (We use γ to denote V 1,1 in the following). Plugging the above expressions into (2.10) and making the coefficient of each term in the Fourier series equal, together with the contraction mapping theorem, we can solve for r 1 , A n , B n , V 1,n (n = 1) and V 2,n as functions of (x, µ, ε, γ). The result is given by the following lemma.
with any positive constants α 1 and α 2 , then the system (2.10) has a reversible smooth periodic solution and r 1 is a smooth function of (µ, ε, γ) such that Here f [j] denotes the j-th component of f , and the norm · denotes C m B (R)norm, which is a space of continuously differentiable functions up to order m with a supremum norm and any positive integer m.
The proof is given in Section 5. Assume that the solution of the system (2.10) has the following form where H(x) and X µ,ε,γ are defined in (2.15) and (3.2) respectively, the phase shift where Z(x) is a perturbation term to be determined, which exponentially tends to 0 as x → ∞, so thatÛ (x) is a solution of (2.10) that approaches to the periodic solution X µ,ε,γ (x + θ) as x → ∞. Plugging (3.5) into (2.10) yields where
It is easy to obtain that the solution Z(x) of (3.7) exists if x is in a finite interval and an initial condition is given. In order to prove the existence of Z(x) for x ≥ 0 with decay to zero at infinity, we change (3.7) to an integral equation and then use the contraction mapping theorem to prove the existence of a fixed point of the integral equation.
First, consider the linear equation which has four linearly independent solutions Moreover,  The adjoint equation of (3.10) has four linearly independent solutions given by (3.14) Moreover, For any x ∈ [0, ∞), we have where ·, · denotes the Euclidean inner product on R 4 .
The solution of (3.7) that decays to zero at infinity can be expressed as √ 6µ) and consider (3.18) as a fixed point problem in a Banach space Then we have Lemma 3.3. Under the assumption (3.1), for Z, Z 1 , Z 2 ∈ E v , the function F satisfies for j = 1, 2, and k = 3, 4 where f [j] means the j−th component of f .

Proof. By (3.12), (3.15), (3.18) and Lemma 3.2, it is obtained that for
for j = 3, 4. Note that (3.22) corresponds to the estimate of ṽ ν , which yields the second inequality of (3.19). Replace ṽ ν in (3.20) and (3.21) with (3.22), and we obtain the first inequality of (3.19). The remaining estimates can be similarly obtained. The proof is completed. Now we are ready to use the contraction mapping theorem to look for a fixed point of the mapping F. For simplicity, we take LetB r (0) ⊂ E ν be a closed ball in E ν with radius r = O(µ 1+α1/2 ). It is easy to see that F is a contraction mapping onB r (0) ∈ E v for small µ > 0 under the assumption (3.1). This yields that (3.18) has a unique solution Z(x; µ, ε, γ) satisfying With a more subtle estimate, we can get for x ∈ [0, ∞) Using the same argument as that for (3.25) and an extension of a contraction mapping principle [27], we can show that Z(x; µ, ε, γ) is smooth in its arguments. Note that the solution Z of (3.18) exists if x belongs to any finite interval and the initial condition is given so thatÛ (x; θ, µ, ε, γ) defined in (3.5) exists for x ≥x 0 with any fixedx 0 ∈ (−∞, ∞).
The proof of Lemma 4.1 is given in Section 5. Again using the contraction mapping theorem and Lemma 4.1, we can solve (4.4) for θ as a smooth function of µ, ε and γ, which gives that the equation (4.3) is valid.

Appendix
In this section, we will give the proofs of Lemma 3.1 and Lemma 4.1.

Proof of Lemma 3.1
The general theory for the existence of periodic solutions of a reversible system can be found in the book [17]. Here we need the exact estimates for some terms which will play an important role in the construction of the generalized homoclinic solution. We just sketch the proof. More details can be seen in [13,14]. The main method for this purpose is to use the Fourier series expansion technique and the contraction mapping theorem. Let where r 1 is a small real constant to be determined later. Then system (2.10) is changed into where Since we look for a reversible periodic solution, we assume that B n sin nτ, V 2,n sin nτ.
Let H m (0, 2π) be a space of periodic functions of τ with a period 2π such that their derivatives up to order m are in L 2 (0, 2π), and the norm is denoted by · m . Fix γ and define spaces f n sin nτ ∈ H 1 (0, 2π) f 1 = 0}.
Using a similar argument as above, we may show thatP is differentiable with respect to its arguments and its derivative with respect to θ is also uniformly bounded for bounded θ and small µ > 0, which gives the equation (4.4). The lemma is proved.