PERIODIC SOLUTIONS AND HOMOCLINIC SOLUTIONS FOR A SWIFT-HOHENBERG EQUATION WITH DISPERSION

. We investigate the 1D Swift-Hohenberg equation with dispersion where σ,α,β and γ are constants. Even if only the stationary solutions of this equation are considered, the dispersion term − σu ξξξ destroys the spatial reversibility which plays an important role for studying localized patterns. In this paper, we focus on its traveling wave solutions and directly apply the dynamical approach to provide the ﬁrst rigorous proof of existence of the periodic solutions and the homoclinic solutions bifurcating from the origin without the reversibility condition as the parameters are varied.

Especially, Burke, Houghton and Knobloch [5] stressed that the equation (1) for σ = 0 admits multiple stationary spatially localized states in the snaking or pinning region and these are organized in the snakes-and-ladders structure. This 1648 SHENGFU DENG structure appears because the equation (1) has some good properties like the variational structure and in particular the spatial reversibility, i.e., the invariance of the equation (1) under R : ξ → −ξ, u → u.
Using the numerical method, they pointed out that the equation (2) loses the reversibility which destroys the pitchfork bifurcations responsible for the rung states such that the snakes-and-ladders structure breaks up into a stack of isolas. After this, the meromorphic exact solutions of (2) are also given in [17] with the aid of the Laurent series. As we know, of particular interest in these studies is the existence of periodic solutions and homoclinic solutions. In this paper, we directly apply the dynamical approach and present the first rigorous proof of existence of the periodic solutions and homoclinic solutions of (2) near the origin if some constants are adjusted. Unlike most of papers listed above, we do not require that the equation has the reversibility and the variational structure. Since we here focus only on the solutions bifurcating from the origin, we mention that the dominant nonlinear term in (2) is βu 2 , which means that the nonlinear term −γu 3 is not crucial. The plan of this paper is the following. In Section 2, we consider the case that the linear operator of the equation (2) has an eigenvalue 0, a pair of purely imaginary eigenvalues and an eigenvalue with a nonreal part. Applying the center manifold reduction theorem and the averaging theory, we show that the zero Hopf bifurcation will occur such that an unstable periodic solution bifurcates from the origin. Section 3 yields the existence of the homoclinic solution and the periodic solution if the linear operator has a double eigenvalue 0 and two eigenvalues with nonreal parts by means of the well-known Melnikov function. The existence of the periodic solution is also given in Section 4 if the linear operator has a pair of purely eigenvalues and two eigenvalues with nonreal parts.
2. Zero Hopf bifurcation. Let u 1 = u x , u 2 = u xx and u 3 = u xxx . Then the equation (2) is equivalently changed into a system with dimension four Before we study the zero Hopf bifurcation of the system (3), we need the following result (For example, see [13,32]), which provides a first order approximation for the periodic solutions of a periodic differential system by the averaging theory. Consider the differential systeṁ for X ∈ D, where the dot denotes the derivative with respect to x, D is an open subset of R n , x ≥ 0, F 1 (x, X ) and F 2 (x, X , ) are T −periodic in x, and the constant T is independent of . The averaged differential equation iṡ where (a) If p is an equilibrium of (5) and Det then there exists a T -periodic solution φ(x, ) of (4) such that φ(0, ) → p as → 0. (b) If the eigenvalues of the equilibrium p all have negative real part, the corresponding periodic solution φ(x, ) is asymptotically stable for sufficiently small. If one of the eigenvalues has positive real part, then φ(x, ) is unstable.
Apply this lemma together with the center manifold reduction theorem, and we obtain the existence of an unstable periodic solution emanating from the zero Hopf bifurcation of the system (3). and where µ > 0 is a small parameter and α 1 , c 1 are fixed constants. Then for µ > 0 sufficiently small there exists an unstable periodic solution of (2) that shrinks to the origin.
Remark 1. The condition (7) guarantees that the linear operator of (3) for µ = 0 has an eigenvalue zero and a pair of purely imaginary eigenvalues. The condition (8) confirms that the averaged differential equation has a nontrivial equilibrium.
Proof of Theorem 2.2. In order to investigate the zero Hopf bifurcation of the system (3) near the origin, we adjust the constants and first assume that where µ > 0 is a small parameter, and α 0 , c 0 are constants. Take (u, u 1 , u 2 , u 3 ) = µ(w, w 1 , w 2 , w 3 ) such that the system (3) is equivalent to Clearly, since σ = 0, its linear operator for µ = 0 has four eigenvalues which corresponding eigenvectors are The solution of (10) can be decomposed in terms of the above eigenvectors. Note that the system (10) is real. If let and plug it into (10), we obtain its equivalent systeṁ where Since σ = 0, the center manifold reduction theorem (see [13,32]) shows that (11) has a smooth reduction function B = Φ(µ, A, v 1 , v 2 ) near the origin, which satisfies Before we analyze the zero Hopf bifurcation of (11), we need the leading terms of B in (12). Assume (12) with respect to x and using the system (11) together with (13), we have a i = 0 for i = 1, · · · , 6 and In order to eliminate the affect of the terms of order we further assume that where α 1 and c 1 are constants, which implies Thus, the conditions in (9) become which is given in (7) of Theorem 2.2. The reduced equations of A, v 1 and v 2 can be changed intoẊ f 10 , Now we write the system (14) in the form of (4) so that we can apply Lemma 2.1. Letṽ 1 = r cos(θ) andṽ 2 = r sin(θ), and we havė where g 10 , g 20 and g 30 are of order µ 2 O(|(µ, θ, r,Ã)|), and Note that the above system is only well defined for r > 0 since r appears in the denominator of g 3 . Using the fact that θ = √ 2 + O(µ 3/2 ) > 0 for small µ > 0, we rewrite the system (15) into where H 10 (µ, θ, r,Ã) and H 20 (µ, θ, r,Ã) are of order µ 2 O(|(µ, θ, r,Ã)|), and The system (17) is exactly the form of (4) and 2π-periodic in the variable θ. According to (6), we define the bifurcation function B(w) by since we need r 0 > 0. Moreover, the Jacobian matrix DwB(w) of B(w) at w 0 has two eigenvalues Lemma 2.1 implies that there exists a periodic solution W p (µ, θ) of the system (17) with period 2π for sufficiently small µ > 0, which approaches to w 0 as µ → 0. Sincẽ λ 2 > 0, the periodic solution W p (µ, θ) is unstable. Now going back through the center manifold reduction theorem, the changes of variables and the rescaling which keep the instability of this periodic solution, we know that the original equation (2) for µ > 0 sufficiently small has one unstable periodic solution. Hence the proof is completed.
3. Double eigenvalue 0. In this section, we assume that Thus, the linear operator of the system (3) for µ = 0 has a double eigenvalue 0, . For simplicity, we suppose Then the corresponding eigenvectors and the generalized eigenvectors for the eigenvalues 0, λ 1 and λ 2 are

The dominant system of the system (25) is
which is the KdV system. For simplicity, we first assume that Obviously, the system (26) has a center equilibrium (0, 0) and a saddle equilibrium (− α0 β , 0). Moreover, it has a homoclinic solution connecting the saddle equilibrium and infinite many periodic solutions around the center equilibrium. In what follows, we will use the bifurcation method and in particular the Melnikov function to rigorously prove the persistence of the homoclinic solution and the periodic solutions when the higher order terms are added, that is, the system (25) owns the homoclinic solution and the periodic solution.
3.1. Persistence of homoclinic solutions. In order to move the saddle equilibrium (− α0 β , 0) to the origin, we let The system (25) is changed intô where the smooth functionR(µ,Â,B) =R 2 (µ,Ã,B) = O(µ). Its dominant system isÂ which is a Hamiltonian system with a Hamiltonian function It is easy to check that the system (29) has a homoclinic solution which exponentially approaches to the origin where a = 3α0 2β and b = √ −α0 2 √ 2 . Theorem 3.1. Suppose that (27) holds and Then the system (28) has a homoclinic solution for sufficiently small µ > 0.
Going back to the origin equation (2), we obtain the following theorem.
Then, for sufficiently small µ > 0, the equation (2) has a homoclinic solution u h (x) given by for x ∈ R where the smooth functions R 0 and R 1 are of order µ 3/2 , R 0 is independent of x, and R 1 exponentially tends to 0 as x → ±∞.
Proof of Theorem 3.1. The expressions of the homoclinic solution for the dominant system (29) have been obtained in (30). Consider the system (28) and we use the Melnikov function to prove the persistence of this homoclinic solution under small perturbation, which is not time-dependent and is defined by (For example, see [13,32]) where the smooth function M 1 (µ, α 0 , c 0 ) = O(µ). The bifurcation situation, when the saddle connection is preserved, is given by M 0 (µ, α 0 , c 0 ) ≡ 0, or, for µ > 0 small by the implicit function theorem which is given in (31). Thus this implies that the system (28) has a homoclinic solution for small µ > 0.

3.2.
Persistence of periodic solutions. Now we consider the persistence of periodic solutions of the system (25) and have the following theorem.

Theorem 3.3. Suppose that (27) is valid and
Then the system (25) has a periodic solution for sufficiently small µ > 0.
This above theorem and its proof imply the following theorem for the equation (2). and Then, for sufficiently small µ > 0, there exists a constant k 0 ∈ (0, 1) such that the equation (2) has a periodic solution u p (x) given by where sn is the Jacobi elliptic function, the smooth function R 2 is periodic in x and of order µ 3/2 , and Proof of Theorem 3.3. The Hamiltonian function of the dominant system (26) is It is clear that all the periodic solutions (Ã,B) T (x) of (26) in the (Ã,B)-plane must lie on the level curve for e ∈ (e 0 , e 1 ). In the following, we look for some e ∈ (e 0 , e 1 ) such that the corresponding periodic solution persists if the higher order terms are included. In order to do this, we need the exact expression of the periodic solution. Suppose that the curve H(Ã,B) = e intersects theÃ-axis with three points C 1 (Ã 1 , 0), C 2 (Ã 2 , 0) and C 3 (Ã 3 , 0). Then they satisfy It is easy to check that with θ = 1 3 arccos(1 + 24eβ 2 /α 3 0 ). Thus, (40) yields Due to the symmetry of the system (26), we take only the upper branch for the periodic solutions, i.e., The relationshipÃ =B and the properties of the elliptic integrals (see [6]) give for β > 0x or for β < 0 From (42) and (44) we obtain the periodic solutions of (26) given by with a period where K(k) = F ( π 2 , k) is the complete elliptic integral of the first kind, sn, cn and dn are the Jacobi elliptic functions, and Clearly, when e is increasing from e 0 to e 1 , θ is strictly increasing from 0 to π 3 . By (41), (43) and (45), we have This shows that k is strictly increasing in θ, which implies that k is strictly increasing in e. In the following, we shall use the parameter k rather than e. Thus, the problem of the persistence of the periodic solution for some fixed e ∈ (e 0 , e 1 ) can be changed into one for some k ∈ (0, 1).
Lemma 3.5. The period T (k) is strictly increasing with respect to k ∈ [0, 1) and The proof can easily be obtained by using the fact that K(k) is strictly increasing in k ∈ [0, 1) and which is given in Lemma 23 of [21].
In order to obtain the persistence of the periodic solution (P A (x), P B (x)) T when the higher order terms are added in (25), it is sufficient by the Melnikov theory to verify that for small µ > 0 is zero for some k ∈ (0, 1). Thus, we have where E(k) is the complete elliptic integral of the second kind, and It is clear to see that by the properties of E(k) and K(k) in [6] If σ > 2 √ 2, α0σ 2 < c 0 < 5α0σ 14 (or σ < −2 √ 2 and 5α0σ 14 < c 0 < α0σ 2 ), which is given in (37), we have This gives the existence of the periodic solutions of the system (25). The proof is completed.
Using a similar argument as above, we have the following results. Then the equation (2) has a homoclinic solution for sufficiently small µ > 0.
(3) If 0 < |σ| ≤ 2 √ 2, using a similar way, we get the reduced system which is the same as (23) so that Theorem 3.2, Theorem 3.4, and the above two remarks still hold.

4.
A pair of purely imaginary eigenvalues. In this section, we consider the case: the system (3) has a pair of purely imaginary eigenvalues and two eigenvalues with nonzero real parts. We have the following theorem.
where σ 1 is an arbitrary constant. For any positive constant A 0 , there exists a smooth functionc 0 (µ, σ 1 , A 0 ) such that andc 0 (µ, σ 1 , A 0 ) = O(µ). Then, for µ > 0 sufficiently small, the equation (2) has a periodic solution u p (x) given by where the smooth function R 3 is periodic in x and of order µ 2 , and Proof. Under the condition (47), the linear operator of the system (3) for µ = 0 has a pair of purely imaginary eigenvalues ±iλ 1 and two real eigenvalues ±λ 2 , whose eigenvectors are