GENERALIZED MULTI-HUMP WAVE SOLUTIONS OF KDV-KDV SYSTEM OF BOUSSINESQ EQUATIONS

. The KdV-KdV system of Boussinesq equations belongs to the class of Boussinesq equations modeling two-way propagation of small-amplitude long waves on the surface of an ideal ﬂuid. It has been numerically shown that this system possesses solutions with two humps which tend to a periodic solution with much smaller amplitude at inﬁnity (called generalized two-hump wave solutions). This paper presents the ﬁrst rigorous proof. The traveling form of this system can be formulated into a dynamical system with dimension 4. The classical dynamical system approach provides the existence of a solution with an exponentially decaying part and an oscillatory part (small-amplitude periodic solution) at positive inﬁnity, which has a single hump at the origin and is reversible near negative inﬁnity if some free constants, such as the amplitude and the phase shit of the periodic solution, are activated. This eventually yields a generalized two-hump wave solution. The method here can be applied to obtain generalized 2 k -hump wave solutions for any positive integer k .

1. Introduction. We consider in this paper the following coupled system of two nonlinear dispersive wave equations in one space dimension η t + u ξ + (ηu) ξ + 1 6 u ξξξ = 0, which is called the KdV-KdV system in [3,5] because the dispersive terms are thirdorder spatial derivatives. This system is originally derived in [3] and later [1,21] from Boussinesq systems η t + u ξ + (ηu) ξ + au ξξξ − bη ξξt = 0, u t + η ξ + uu ξ + cη ξξξ − du ξξt = 0 for appropriate constants a, b, c and d. The system (1) has been proved to be a valid approximation to the full two-dimensional Euler equations for small-amplitude long for x = ξ − ct where c is the speed. Note that if (η(x), u(x)) T is a solution of (1) for some c > 0, then (η(x), −u(x)) T is also a solution propagating with speed −c, i.e. to the left, as t increases. Thus, we assume c > 0. It is easy to check that c = 1 is a critical point near which the rear parts of some eigenvalues change from zero to nonzero (see (8)). Hence we consider c as a parameter c = 1 + µ for small µ. Let a 1 = η, a 2 =η, a 3 = u and a 4 =u where the dot stands for the derivative with respect to x. Substituting (2) into (1) one obtains, after integrating once and taking the integration constants equal to zero, the system of nonlinear ordinary differential equationsȧ 1 = a 2 , a 2 = −6a 1 + 6(1 + µ)a 3 − 3a 2 3 , a 3 = a 4 , a 4 = 6(1 + µ)a 1 − 6a 3 − 6a 1 a 3 . ( Now, the solitary wave solutions of (1) correspond to the homoclinic solutions of (3). It is clear that the origin of (3) is a saddle-center equilibrium for µ > 0, that is, the linear operator around the origin has a positive eigenvalue, a negative eigenvalue and a pair of purely imaginary eigenvalues (see (8) or Figure 3). Since the stable and unstable manifolds are both one dimensional, it is very hard to prove that the stable and unstable manifolds will intersect to form the classical homoclinic solutions (see (1) of Figure 1). [5] numerically and theoretically verified that the system (3) actually possesses a generalized single-hump homoclinic solution by the standard Galerkin-finite element method and the theory of Lombardi [17] respectively, i.e., homoclinic solution has a main hump which does not tend to zero but to a small-amplitude periodic solution at infinity (see (2) of Figure 1). Moreover, [5] numerically obtained a generalized two-hump homoclinic solution for µ = 0.2 (see Figure 6 in [5] or Figure 2 here). The heights of the two humps are same and are about 0.44, and the distance of two humps is around 30. This implies that two humps are far apart. Does the system (3) really have a generalized twohump homoclinic solution with same heights? It is still an interesting open problem.
Intuitively, if two generalized single-hump homoclinic solutions are far away, they will not interact with each other so that we can paste these two solutions together and obtain a new solution with two humps. In this paper we will give a first rigorous proof. The basic idea is based on [5,10,17]. Note that the system (3) has a generalized single-hump homoclinic solution [5], denoted by (η s ,η s , u s ,u s ) T (x). We just look at the η-component η s (x) (the similar idea can be applied to u s (x)), which is even in x and exponentially tends to a periodic solution with small amplitude I as x → ±∞. The hump of η s (x) is at the origin. In order to get two-hump solutions, we have to break the evenness of the solution η s (x). We choose different amplitudes for the periodic solution in η s (x), that is, the amplitudes are taken as I + for x > 0 and I − for x < 0 respectively. The difference of I + − I − is nonzero and regarded as a new small parameter. Since the amplitudes are different, the modified solution, say η * s (x), exists only on [−τ, +∞), not on (−∞, +∞) where τ > 0 is large and can be chosen as we need. We might find a point x = −x 0 for large x 0 ∈ (0, τ ] where the derivative (η * s ) (−x 0 ) = 0. The reversibility of the system implies that η * s (x) can be smoothly extended on (−∞, +∞) so that it has two humps at x = 0 and x = −2x 0 . Therefore, the two-hump solutions are obtained and then the existence of the generalized two-hump wave solution with same heights observed in [5] is proved rigorously. The similar idea may be used to construct 2 k -hump solutions for any positive integer k. We would like to mention that our constructive method is different from other methods such as in [9,18,25] since we can provide more details of the obtained solutions such as the heights and the distance of two humps, and the small amplitude of the periodic solution, which play important roles in the applications. The main result can be summarized as follows.

Remark 1.
(1) The deviation η(x; x 0 ) of the free surface obtained in Theorem 1.1 is even in x and has two humps with same height 2µ at ±x 0 respectively (see Figure 2), which has small oscillations of amplitude at infinity. The amplitude of the oscillations at x near zero is of order O(µI + ) = O(µ n+3/2 ). The distance between two humps is 2x 0 = O(µ −1/2 | ln µ|). This is exactly the one observed in [5].
(2) If I − = I + , it means that the difference I + −I − is zero and then x 0 is infinity so that this solution becomes a generalized solitary wave solution.
The methods here can be applied to other types of Boussinesq equations and other systems, such as Bona-Smith system which has a generalized two-hump wave solutions with same heights numerically given in [2]. This paper is organized as follows. Section 2 reformulates the system (3) into a new dynamical system with the eigenvectors of the linear operator in (3) as a basis for the space R 4 . After doing the appropriate scaling and the change of variables, the dominant system is presented and its homoclinic solution H(x) is obtained. In Section 3, the periodic solution X p (x) of the whole system is given with the Fourier series technique and the amplitude I > 0 is chosen as a parameter. In order for twohump solutions, we intentionally choose I + for x > 0 and I − for x < 0 respectively to break the reversibility. In particular, the difference I + − I − is considered as an additional free constant. Section 4 writes the whole solution as the summation of H(x), X p (x) and the unknown perturbation term Z(x), and transforms the problem of the existence of generalized two-hump homoclinic solutions into the one for an integral equation with respect to Z(x) such that the fixed point theorem can be applied. The appropriate Banach spaces are constructed in Section 5 and Z(x) is decomposed into two parts: the reversible part Z e (x) and the remainder part Z r (x). Section 6 yields the estimates and the existence proof of Z e (x) on [0, +∞) and Z r (x) on [−2, +∞) by applying the fixed point theorem, while the existence of Z r (x) on [−τ, −2] is given in Section 7 for some large τ > 0. In Section 8, the obtained solution Z(x) on [−τ, +∞) is smoothly extended on (−∞, +∞) if some free constants such as I + − I − are adjusted, which gives the generalized two-hump homoclinic solution of (3). This finishes the proof of Theorem 1.1. Appendices present the proofs of some lemmas left in the previous sections.
Throughout this paper, M denotes a generic positive constant and B = O(C) means that |B| ≤ M |C|.

2.
Formulations. Symbolically, the system (3) can be written aṡ where U = (a 1 , a 2 , a 3 , a 4 ) T , The system (6) is reversible with a reverser S defined by . This implies that a 1 (x) and a 3 (x) are even functions, and a 2 (x) and a 4 (x) are odd functions. It is easily seen that the eigenvalues of L are where λ 0 = √ 12 + 6µ. See Figure 3. In what follows, we assume that µ > 0 (in this case, the origin is a saddle-center equilibrium). Then the corresponding eigenvectors Note that the solutions of the real system (6) can be expressed in terms of the above eigenvectors, that is, whereÃ,B,ṽ 1 andṽ 2 are real functions of x. Thus, the system (6) is equivalent to the following real systeṁ Furthermore, we letÃ =Â +B 2 andB =B −Â 2 . Then (11) is changed intȯ Do the scaling and suppose that

MULTI-HUMP SOLUTIONS OF KDV-KDV SYSTEM
3677 which gives from (12) where X = (A, B, w 1 , w 2 ) T , and Clearly, the existence problem of generalized two-hump wave solutions of (1) is transformed into one of generalized two-hump homoclinic solutions of (14), and η(x), u(x) have the following forms In what follows, we pay our attention on the system (13). It is obvious that from (7) and (9) the reverser S is now given by The dominant system of (13)Ẋ = F (µ, X) has a homoclinic solution which satisfies where In the rest of this paper, we will use this homoclinic solution H(x) to construct a generalized two-hump homoclinic solution.
3. Periodic solutions. Since the generalized two-hump homoclinic solution to be proved exponentially approaches a periodic solution at infinity, this section applies the Fourier series to look for the periodic solutions of the system (13) with period 2π/(λ 0 + r 1 ). Assumex where r 1 is a small real constant to be determined later. The system (13) can be transformed into where the prime stands for the derivative with respect tox. Denote H m p (0, 2π) by the space of periodic functions ofx with period 2π such that their derivatives up to order m are in L 2 (0, 2π), whose norm is denoted by · m . The reversible periodic solutions of (21) in (H m p (0, 2π)) 4 can be expressed as where I > 0 is a free real constant. Plugging (22) into (21), setting each Fourier coefficient equal zero, and applying the fixed point theorem with suitable estimates, one can solve for A(x), B(x), w 1 (x) and w 2 (x) in (H m p (0, 2π)) 4 and the real constant r 1 as smooth functions of (µ, I) for small (µ, I) (more details can be seen in [14]), which can be written as (A, B, w 1 , w 2 )(x) = (A, B, w 1 , w 2 )(x; µ, I), r 1 = r 1 (µ, I).
Moreover, using the fact that the right sides of (21) are polynomials of degree 2, for I ∈ (0, I 1 ] and µ ∈ (0, µ 1 ] where the positive constants I 1 and µ 1 are small. Due to (20), we denote this periodic solution by which satisfies for j = 1, 2 and k = 3, 4 and any positive integer m, where X p [j] denotes the j-th component of X p . The Sobolev embedding theorem implies that (23) are also valid under the C m B (R)-norm, where C m B (R) is the space of continuously differentiable functions up to order m with a supremum norm.
A direct calculation shows that the dominant terms of the periodic solution X p (x) for small (µ, I) can be expressed as where In the following sections, we will prove the existence of a generalized two-hump homoclinic solution exponentially approaching the periodic solution X + p (x − θ + ) with amplitude I + > 0 and phase shift θ + as x → +∞ and another periodic solution X − p (x−θ − ) with amplitude I − > 0 and phase shift θ − on the left side of the homoclinic solution H(x) in (17). The reason to choose different periodic solutions is to break the reversibility. Otherwise, we obtain only one-hump solution. The difference of I + − I − will be used as one new parameter (see (69) and Lemma 8.1). It is also essential to introduce the phase shifts θ ± such that a matching condition (see (57)) between the decaying part and the oscillatory part at infinity holds. The purpose to take the phase shift θ − = −θ + for x < 0 instead of θ + for x > 0 is just to simplify the calculations later and only the amplitude of the periodic solution is changed. These two periodic solutions can be denoted by , where the plus sign is taken for x > 0 and the minus sign is used for x < 0, respectively.
Then, according to (24) and (25), we have the following lemma, which will be used later.
4. Integral equations. This section transforms the existence problem of twohump solutions for (13) into the one for an integral equation so that the fixed point theorem can be applied. Define smooth even cutoff functions σ(x) andσ(x) as follows andσ Since the dominant term H(x) in (17) and the small-amplitude periodic solutions X ± p (x − θ ± ) are obtained, the solution of (13) near H(x) can be assumed to have a form where Z(x) = (A, B, w 1 , w 2 ) T (x), (Here (A, B, w 1 , w 2 ) T is now denoted for Z, not for X in the previous sections since no confusion appears), is a perturbation term to be determined and exponentially decays to 0 as x → +∞. Substituting (29) into (13) yields the equation of Z(x) where L(x) = dF [µ, H], d means taking the Fréchet derivative, and The linear system of (30)Ż has four linearly independent solutions which satisfy for x ∈ R The adjoint system of (32) also has four linearly independent solutions and where ·, · denotes the Euclidean inner product on R 4 . The solution of (30) can be formally written as Thus, the existence problem of Z(x) is transformed into the fixed point problem of the operator E in (37).

Function spaces.
In order to use the fixed point theorem for the operator E in (37), we define three function spaces as follows where τ > 0 is a large constant, n is a positive integer, and α is a constant. Here the constants τ and α will be fixed later. From the definition of H r , we see that f ∈ H r may be exponentially large near negative infinity for large τ . In order to control this bad property, we will adjust the difference I + − I − (see (94)). Due to the reversibility of the system (13), we decompose the solution Z(x) accordingly, and let H 1 = H e ⊕ H r and H 2 = H d ⊕ H r . For f ∈ H 1 , it means that f ∈ C 2n ([−τ, +∞)) and can be decomposed as where with a norm · 0 given by and with a norm · 1 given by where which norms are defined by for Z r only well defined with |x| ≤ 2 (in this case H is denoted by H 0 ) and for Z r only well defined with x ∈ [−τ, −2] (in this case, H 1 stands for H). If no confusion arises, we sometimes use (A r , B r , w 1r , w 2r ) for (A er , B dr , w 1er , w 2dr ).
From the definitions f er (x) and f dr (x) in Section 5, one obtains that A direct calculation shows that from (31) and (37) Notice that F 3 (t) + F 3 (−t) and F 4 (t) − F 4 (−t) will have a factor I + − I − , and we will use I + − I − to control the term e 2 √ 6µτ (see (94) and (95)). Hence, we assume which implies Before we can proceed further, we solve the equation (57) for θ + . The plus sign in F 3 and F 4 is taken since the integral is from zero to positive infinity. For the sake of simplicity, we in advance assume that Lemma 6.1. Suppose that (60) is valid and (57) is equivalent to where Φ(θ + , µ, I + ) is differentiable with respect to its arguments. Moreover, it and its derivatives with respect to θ + are uniformly bounded for small (θ + , µ, I + ).
Assume that for x ≥ −2 r (x), and where I 0 andl are positive constants, and the constant a 0 will be fixed later. Clearly, (38) and (60) are satisfied. Now consider H 0 as a product space of (Z e ,Ẑ r ) and choose a closed ballB 2 (0) in H 0 such that forẐ = (Z e ,Ẑ r ) inB 2 (0) According to Lemma 6.2, we obtain that forẐ,Ẑ ∈B 2 (0) which implies thatÊ is a contraction mapping onB 2 (0) for small µ > 0. Therefore, E has a unique fixed pointẐ ∈ H 0 for x ∈ [−2, ∞). This shows that (37) has a unique solution Z ∈ H 0 for x ∈ [−2, ∞) satisfying The more exact estimates for A e (x) and w 1e (x) will be needed later.
Lemma 6.3. Under the assumption (69), we have for small µ > 0 and then A e e + w 1e e ≤ M µ.
Proof. A direct calculation shows that by (30), (69) and (71) Notice that w 1e (x) is even. It is straightforward to see that Since w 1e (x) exponentially tends to 0 as x → ∞, we have with integration by parts Substituting (75) into (74) with integration by parts twice again yields which implies that w 1e e ≤ M µ. Using this result with a similar method, we obtain which further implies that The proof is completed.
From (89) and (92), it is easy to obtain the following lemma.
where X 1 andX 3 are the first and third components of Z 1 respectively.
Take a closed setB 3 (0) in H 4 2 such that for Z ∈B 3 (0) Lemma 7.1 shows that Ξ is a contraction mapping onB 3 (0) for small µ > 0. Therefore, (93) has a unique solution Z ∈B 3 (0), which is smooth with respect to its arguments and satisfies (96) together with the more accurate estimates From the relationship between Z and Z r (see (76), (77), (79) and (81)), it is ob- . Hence, the existence and smoothness of Z r (x) for x ∈ [−τ, −2] are obtained. Moreover, Z r (x) satisfies where the norm · 1 is defined in (46). Thus, (37) has a unique smooth solution In order to smoothly extend the obtained solution Z(x) on (−∞, +∞), we need more details about A(|x|) − A(−|x|) and B(|x|) + B(−|x|). Using the above results, we have the following lemma. and The proof is given in Appendix 3.
8. Existence of two-hump solutions. In previous sections, we have proved the existence of the solution X(x; µ, θ ± , I ± ) of (13) for x ∈ [−τ, +∞), which has a hump at the origin. See (17) and (29). The reversibility and the translational invariance indicate that X(x − x 0 ; µ, θ ± , I ± ) for x − x 0 ≥ −τ and SX(−x − x 0 ; µ, θ ± , I ± ) for −x − x 0 ≥ −τ are also solutions of (13) with any x 0 ∈ R. To have a reversible solution with two humps for (13), we just glue these two solutions together at x = −x 0 for some large x 0 ∈ (0, τ ]. This is equivalent to find x 0 so that the following equation holds where I is an identity mapping. If this x 0 can be found, we define Then X 1 (x; µ, θ ± , I ± ) is a reversible smooth solution of (13) with two humps. The definition of the reverser S given in (16) implies that (101) is equivalent to 2p (−x 0 + θ + ) = 0, where the cutoff function σ(−x 0 ) = 1 is used. Since −x 0 is near negative, the decompositions of B(x) and w 2 (x) yield Hence, two constants are needed to solve the above two equations. We will choose these two free constants x 0 and a 0 defined in (69). Let and Then where [x] denotes the largest integer less than or equal to x, which shows that n 0 is a large positive integer and Thus, we have c 1 → 1 as µ → 0 + , which implies that x 0 ∈ (0, τ ) by (95). It is straightforward to check that Finally, we are ready to solve (103) and (104) for β and a 0 , and have the following lemma.
Hence, (103) and (104) hold by choosing (x 0 , a 0 ) obtained. The uniqueness of the solution for an initial value problem implies that X 1 (x; µ, θ ± , I ± ) defined in (102) is a smooth solution of the system (13). It is obvious that SX 1 (−x; µ, θ ± , I ± ) = X 1 (x; µ, θ ± , I ± ), that is, the solution X 1 (x; µ, θ ± , I ± ) is a reversible solution of (13). Clearly, X 1 (x; µ, θ ± , I ± ) exponentially approaches periodic solutions with small amplitudes at infinity and has two humps at x = ±x 0 respectively. The distance between two humps is which becomes very large if µ is small. According to the relationships among the original system (1), the system (3) and the system (13), it is straightforward to see that which, together with (15) and (102), yields the proof of Theorem 1.1.

Remark 2.
With a similar idea, we may prove the existence of a generalized 2 mhump wave solution for any positive integer m.

Appendix 2.
Proof of Lemma 6.2. Since we consider the case x ≥ −2, the space H 0 with the norm · 0 in (49) for Z r (x) will be used. By (24), (25) and (31), using a similar method for (116), we have for x ≥ −2 and any positive integer k, whereσ(x) is defined in (28). Now we divide the proof into four steps.
Before we can use the the fixed point theorem, we have to replace the term w 1e e in the above inequality with the estimate of E 3e .
Substituting (61) and (62) into (138) and (139) by replacing A e e and w 1e e with E 1e e , E 3e e respectively yields the inequality (66). The rest of the inequalities can be similarly obtained. The proof is completed.

Appendix 3.
Proof of Lemma 7.2. Since the existence of Z(x) for x ∈ [−τ, ∞) has been proved, we will use the following results This yields (100). The proof is completed.