EXISTENCE OF GENERALIZED HOMOCLINIC SOLUTIONS FOR A MODIFIED SWIFT-HOHENBERG EQUATION

. In this paper, we investigate the modiﬁed steady Swift-Hohenberg equation where k > 0, α and ε are constants. We obtain a homoclinic solution about the dominant system which will be proved to deform a reversible homoclinic solution approaching to a periodic solution of the whole equation with the aid of the Fourier series expansion method, the ﬁxed point theorem, the reversibility and adjusting the phase shift. And the homoclinic solution approaching to a periodic solution of the equation are called generalized homoclinic solution.


1.
Introduction. Since the Rayleigh-Bénard convection model was given, the study for the effects of thermal fluctuations on a fluid near the Rayleigh-Bénard instability has been attracted wide attention. One of the most important results is the proposition of Swift−Hohenberg equation [29], which is given by where b > 0 is a constant. Actually, this model can also arise in the study of plasma confinement in torial device [18], viscous film flow and bifurcating solutions of the Navier-Stokes equations [25]. And this model can describe the dynamics for spiral waves and many pattern formations such as spatially periodic rolls, hexgonal cell structures and so on, all of which have been observed in different physical chemical and biological context [1,2,28,19,33,22]. It has also been studied a great deal both analytically and numerically [3,23,14,15,20,6,4].
In 2003, Doelman et al. [10] first studied the modified Swift−Hohenberg equation below for a pattern formation system with two unbounded spatial directions that are near the onset of instability, where k > 0, µ and α are constants. Clearly when α = 0, the equation (2) becomes the usual Swift−Hohenberg equation. The additional term α|∇u| 2 , reminiscent of the Kuramoto-Sivashinsky equation, which arises in the study of various pattern formation phenomena involving some kind of phase turbulence or phase transition [17,26], breaks the symmetry u → −u. There are some wonderful works about the modified Swift-Hohenberg including the bifurcation analysis [31] and the proof of the existence of attractors. For example, [27] and [32] prove the existence of the global attractor, [23] shows the existence of the pullback attractor, [21] presents the existence of the uniform attractors, and [34] gives the numerical solution for this equation by Fourier spectral method. In 2013, Deng [3,6] investigated the steady Swift-Hohenberg equation for its homoclinic solutions.
In 2016, Deng [4] investigated the 1D Swift-Hohenberg equation with dispersion and obtained the existence of the periodic solutions and the homoclinic solutions bifurcating from the origin. In this paper, we consider the 1D modified steady Swift-Hohenberg and prove the existence of the generalized homoclinic solution which is defined by a solitary wave solution exponentially approaching to a periodic solution at infinity. To the best of our knowledge, the study of the generalized homoclinic solutions for the modified Swift-Hohenberg equation has not been done. Setting ε = µ − k, we change equation (2) into u t + ku xxxx + 2ku xx + αu 2 x − εu + u 3 = 0, where the space dimension is 1, whose steady equation is Equation (3) can be written as a system of ODEs by letting u 1 = u x , u 2 = u 1x , u 3 = u 2x . The linear operator of system (4) at (0, 0, 0, 0) has a double eigenvalue 0 and a pair of purely imaginary eigenvalues ± √ 2i for ε = 0, two pairs of purely imaginary eigenvalues for ε < 0, and a positive eigenvalue, a negative eigenvalue and a pair of purely imaginary eigenvalues for ε > 0. When ε > 0, the origin is a saddle-center equilibrium.
Moreover system (4) is reversible under the reverser S defined by which means that if (u, u 1 , u 2 , u 3 )(x) is a solution of (4), then S(u, u 1 , is also a solution. We call the solution (u, u 1 , u 2 , u 3 ) reversible if which means that u and u 2 are even, while u 1 and u 3 are odd.
Through the analysis above and inspired by [5,6,9,11,12,13,24], we guess that system (4) or the equation (3) has a generalized homoclinic solution. In this paper, we study the expression of the generalized homoclinic solution and rigorously prove its existence with the aid of the reversibility and the fixed point theorem.
Our paper is organized as follows. In Section 2, system (4) is changed into an equivalent system with dimension 4 and a homoclinic solution of the dominant system is given. In Section 3, by the fixed point theorem, Fourier series expansion technique method and the reversibility, we prove that there exists a periodic solution for the whole system. In Section 4, we give our main result about the existence of a generalized homoclinic solution for the equation (3) by adjusting the phase shift.
2. Homoclinic solution for the dominant system. In this section, we change the system (4) into a real system with dimension 4 and get a homoclinic solution of its dominant system for the real equivalent system.
In the case ε = 0, we consider the linear operator of (4) at (0, 0, 0, 0) and the corresponding eigenvectors and generalized eigenvectors are Then by letting (u, where Clearly, system (5) has the reversibility with S(Y, W, U, V ) = (Y, −W, U, −V ). For convenience, we write the system (5) as Considering the dominant system of system (5) we get a homoclinic solution for ε > 0, which satisfies and for x ∈ (−∞, +∞). In Section 4, we will prove that this homoclinic solution deforms into a generalized homoclinic solution.
3. Existence of the periodic solution for system (5). This section proves the existence of the periodic solution of the system (5) with Fourier series by constructing a contraction mapping. The general theory about a reversible system can be found in the book [16]. More details can also be seen in [7,8]. Take where r 1 is a small real constant to be determined later and change system (5) into where

Now we express the reversible solution with the Fourier series
and calculate each mode in the Fourier series expression for n = 1, To get accurate estimations, we denotē Now we first solve for Y n , W n , U n (n = 1), V n (n ≥ 2) with a fixed U 1 in system (12), and then solve (13) for r 1 . 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 m (0, 2π), in which the norm is denoted by · m , and define spaces Then we define a mapping Θ(A, B, E, F ;w) from Here for simplification, we take U 1 = γ > 0. Assuming thatBr(0) is a closed ball with a radiusr in the space H 1 , we have the following lemma. (0) and for any small boundedw andr, Θ is smooth in its arguments and satisfies Assume thatr Then Θ is a contraction mapping onBr(0) for smallw, wherer 1 is a fixed constant. Thus Θ has a unique fixed point which is a smooth function ofw and satisfies Using the same discussion we can show that (17) is in H m (0, 2π) and satisfies (18) with H m (0, 2π)-norm for any integer m > 0. For convenience we use In the following, we solve (13) for r 1 . Substituting (19) into (13), we get that is Clearly, (0, 0, 0, 0) is a trivial periodic solution of (11) while γ = 0, which means that [P 2 (ε, α, r 1 , Y p , W p , U p , V p )] 1 γ=0 = 0 and then has a factor γ. Then g(ε, α, r 1 , γ) is smooth in its arguments. Furthermore, we can prove that g is a contraction mapping satisfying |g| ≤ M ε. Thus g has a unique fixed point as a smooth function for small (ε, α, γ), which satisfies Therefore, system (5) has a periodic solution in H m (0, 2π). By the relation τ = √ 2(1 + r 1 )x, we write the periodic solution Then X ε,α,γ (x) is a reversible periodic solution of (5) which satisfies for any integer m > 0. The Sobolev embedding theorem gives that (22) holds also in C m B (R)-norm, which is a space of continuously differentiable functions up to order m with a supreme norm.

Main result.
In the section, we firstly give the expression of the generalized homoclinic solution of system (3) in the following theorem aided by a cut-off function, then prove its existence by using the fixed point theorem and the reversibility, and adjusting the phase shift.
then the equation (3) has a reversible generalized homclinic solution for x ∈ (−∞, +∞), where the phase shift θ is a constant with θ = O(ε), ς(x) is a smooth even cut-off function with ς(x) = 0 for |x| ≤ 1 and ς(x) = 1 for |x| ≥ 2. Here T (x; ε, α) and D(x; ε, α) are smooth functions in their arguments, and T (x; ε, α) is a periodic solution with a period 2π/ √ 2(1 + r 1 ). T (x; ε, α) and D(x; ε, α) satisfy 2k ), and M is a generic constant. It is clear that (16) holds while β = 1 8 . To obtain the existence of the solution approaching to the periodic solution X ε,α,γ (x) obtained in Section 3, we establish a generalized homoclinic solution (see in (24)) for the system (5) depending on the homoclinic and period solutions obtained respectively in Section 2 and Section 3, and prove its existence. In Section 4.1, we firstly prove that there exists a generalized homoclinic solution of (5) for only x ∈ [0, +∞). Then in Section 4.2, by the reversibility of the systems we prove the existence of a generalized homoclinic solution of (5) for x ∈ (−∞, +∞). Finally, by the equivalent relation in (6), we obtain Theorem 4.1.
We firstly discuss the solution for the following linear equation about Z(x), which has four linearly independent solutions Then we have for x ∈ [0, ∞) and The adjoint equation of (29) has four linearly independent solutions given by Then we have for x ∈ [0, ∞) and where ·, · denotes the Euclidean inner product on R 4 . It is not difficult to verify that Z(x) satisfies the following expression We will prove that such 2k ) and consider (35) as a fixed point problem in a Banach space For Z, Z 1 , Z 2 ∈ E ν , we have the following estimation.
for j = 1, 2 and k = 2, 3, where f [j] means the j−th component of f .
If letBr(0) ∈ E ν be a small ball with radiusr = O(ε (1/2+β) ), then from Lemma 25 we can show that F is a contraction onB r ∈ E ν for small ε > 0. This yields that (35) has a unique solution Z(x; ε, α, γ) satisfying Using the same argument as that for (41) and an extension of a contraction mapping principle [30], we can show that Z is smooth in its arguments. Thus, we have showed thatŪ (x; ε, α, γ) defined in (24) exists for x ≥x 0 with any fixed x 0 ∈ [0, ∞), which will be used to obtain a reversible homoclinic solution of (2.7) for x ∈ (−∞, ∞) in the following section.

4.2.
Reversible generalized homoclinic solution for x ∈ (−∞, ∞). In this section, we extend the range for x from [0, ∞) to (−∞, ∞) so that the existence of the generalized homoclinic solution of (5) is obtained. This problem is equivalent to solve the following equation By (8) and the definition of ς(x) in (25), it is easy to check that (42) is equivalent toũ Using (32) and (35), we know that (43) holds automatically. Thus, we only ensure that the equation (44) holds for some θ.
where ϕ is differentiable with respect to its arguments, ϕ and its derivative with respect to θ are uniformly bounded for small bounded ε.
The proof of Lemma 4.4 will be shown in Section 5.
Proof of Lemma 4.4. At first we estimate U p and V p , and then by (27) and (35) we get the proof of Lemma 4.4. Let where r 1 , U p and V p are given in (10) and (21) respectively, which yields Thus, Y p , W p , C p ,C p is a periodic solution of the following system We can express C p (τ ) as For the coefficient of e iτ in C p (τ ) is C 1 = γ. Thus γ = 1 2π 1 + K(r 1 ) γ − 1 2π 2π 0 e −is w(s)ds + w √ 2(1 + r 1 )x .