# Existence and Approximation of Manifolds for the Swift-Hohenberg Equation with a Parameter

Authors: Yanfeng Guo; Chunxiao Guo; Donglong Li
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The existence and approximation of manifolds for the Swift-Hohenberg equation with a proper parameter have mainly been studied. Using the backward-forward systems from Swift-Hohenberg equation, the existence and specific representation forms of manifolds for Swift-Hohenberg equation with a parameter have been obtained. Meanwhile, we make use of technique of deposition of lower and higher frequency spaces of solutions and assume the reduced system to obtain the main numeration approximation system of approximation solution for the original system Swift-Hohenberg equation with a proper parameter.


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## Body

## 1. Introduction

Recently, many mathematicians and physicians are interested in manifolds of some partial differential equations (PDEs). Recently, various approximations of manifolds of PDEs have been proposed, for example, amplitude equations approaches and so on [1–5].In this paper, we have studied the manifolds of the real Swift-Hohenberg equation(1)ut=λu-1+∂xx2u-u3with a proper parameter λ, by which transverse pattern formations in both degenerate optical parametric oscillation and degenerate four-wave mixing are shown in the limit of small signal detuning [6].It is well known that the Swift-Hohenberg model was proposed to describe pattern formation in convection by Swift and Hohenberg in studying the convective instability in the Rayleigh-Bénard convection [7]. However, this mode does not adequately explain this phenomenon. Then some generalizations of this model have been derived in various branches in [8–12], such as nonlinear optics for lasers, magnetoconvection, and biological, chemical, and liquid-crystal light-valve experiment. In addition, the complex Swift-Hohenberg equation has been derived in [13, 14]. And it is very important to consider properties of real Swift-Hohenberg equation in some branches, such as physics, hydrodynamics, and nonlinear optics; see [15, 16].Many dynamical behaviors for the local and nonlocal one-dimensional Swift-Hohenberg equation, such as attractors and invariant manifolds, have been investigated in [17–21]. We know that some manifolds for partial differential equations are important to study the dynamical behaviors for them, which include stable and unstable manifolds [1, 22] and center manifolds [23, 24]. Some results about stable and unstable manifolds for Swift-Hohenberg equation have been obtained in [1]. However, there have been a few results about approximations of manifolds for Swift-Hohenberg equation with a proper parameter until now. Particularly, the numeration approximation system and some numerical solutions for Swift-Hohenberg equation with a proper parameter have not been considered by few authors. Here, approximation of manifolds has been mainly investigated by technique of deposition of solution between lower and higher frequency spaces. And we have used the numeration approximation system to solve the approximation solutions. This idea has been considered a few times to approximate manifolds for some system, especially for Swift-Hohenberg equation. Here, we will give the main idea of this method in the following process.In this paper, the existence and approximation of manifolds for the real Swift-Hohenberg equation(2)ut=λu-1+∂xx2u-u3with a proper parameter have been investigated by the idea in [5]. On the basis of existence of parameter manifold, the numeration approximation systems of iterative levels have been given by backward-forward systems. Then, some new results have been obtained. The results in this manuscript have been used to obtain the approximation solution of some real partial differential equations. This work can be extended to some real systems in a larger field of math. Firstly, the existence and the representation form of manifolds for Swift-Hohenberg equation with a proper parameters are given. Secondly, the numeration approximation system is considered based on manifolds with a proper parameter using deposition of solutions between lower and higher frequency spaces. Finally, we have obtained the corresponding numeration approximation system for numerical solutions of Swift-Hohenberg equation with proper parameters under some conditions.


## 2. Preliminaries

The work spaces are mainly given by the Hilbert spacesH1=D(A) and H=L2(0,π), where A=(1+Δ)2 and H1 is compactly and densely embedded in H. The operator A:H1→H is linear and the spectrum of -A satisfies σ(-A)<0. In addition, the interpolated space Hα is a space between H1 and H with α∈[0,1). Let Lλ=-A+λ be linear operators including a simple parameter. Here Lλ map Hα into H and depend continuously on λ.For the convenience of studying, the Swift-Hohenberg equation has been rewritten in the following form:(3)du=λu-1+∂xx2u-u3dt=Lλu+Fudtwith initial condition u(0,x)=u0(x)∈Hα and Dirichlet boundary conditions u(t,0)=u(t,π)=0 on D=[0,π], where F(u) is defined as F(u)=-u3, which is a trilinear and continuous function from Hα×Hα×Hα to H, α∈[0,1). Here, we define F(u1,u2,u3)=-u1u2u3. The operator A=(1+Δ)2 is closed self-adjoint linear operator with dense domain D(A)=H1=H4(0,π)∩H02(0,π). In addition, Lλ is self-adjoint operator with an orthonormal basis of eigenfunctions {ek=2/πsin⁡kx}k∈N in H and corresponding eigenvalues {βk(λ)=λ-(1-k2)2}k∈N.Assume thatHc=span{e1,e2,…,em}, where m is finite. Let the topological complement of Hc be Hαs and the function hpm maps Hc to Hαs. Then we have the high frequency part us(t)=Psu(t) and low frequency part uc(t)=Pcu(t), where Pc:H→Hc and Ps=I-Pc. Thus, u=Pcu+Psu=uc+us for any u∈L2(0,π).


## 3. Approximation of Parameter Manifolds

According to the approximation methods in [5], using backward-forward systems of Swift-Hohenberg equation and u=Pcu+Psu=uc+us, we assume the following reduced system: (4)duc1=Lλcuc1dt,uc10,x=ξ,ξ∈Hc,t∈-T,0,(5)dus1=Lλsus1+PsFuc1t-Tdt,us10=0,t∈0,T,where Lλc≔PcLλ, Lλs≔PsLλ, u(1)=uc(1)+us(1), and ξ∈Hc. Equation (4) is the linear part of (3) in space Hc, which is assumed and seen as the first iteration level in Hc of reduced system (4)-(5). From the above system, the existence and specific presentation forms of manifolds under parameter λ are given by the following theorem with the initial values of uc(1) and us(1).Theorem 1.
Consider the Swift-Hohenberg equation (3), where F is a trilinear function. Assume also that βj(λ)<0 for all j>m. Let I={1,…,m}, m=dim⁡(Hc). If, for all (k1,k2,k3)∈I3, <F(ek1,ek2,ek3),ej>≠0 when βk1+βk2+βk3-βj>0, j>m, then the pullback limit for the solution of (5) exists and is given by (6)hλ1ξ=limT→+∞us1T,x=∫-∞0e-tLλsPsFuc1t,ξdt,∀ξ∈Hc,where uc(1)(t) is the solution of (4) and (7)uc1t=etLλcξ.Moreover, hλ(1)(ξ) has the following analytic expression: (8)hλ1ξ=-∑j=m+1∞∑k1,k2,k3=1m1βk1+βk2+βk3-βjξk1ξk2ξk3Psek1ek2ek3,ejej,forj>m,where ξk=<ξ,ek>, k=1,…,m.Proof.
It is easy to obtain the solution of (4) given by (7). By using the variation-of-constants formula, we can formally obtain the solution of (5), which is given by (9)us1T,x=∫-T0e-tLλsPsFuc1tdt,where uc(1)(t) is the solution of (4) and has the form of (7). According to the conditions, we can obtain that limit of (9) exists. Plugging (7) into (9), we give the representation as (8).


## 4. Approximation of Solution Based Reduced System

Assume that the subspaceHc=span{e1,e2} and take 9<λ≤64. Now, the numeration system of approximation of solution is investigated using manifold under proper parameter by parts of solution in lower frequency spaces according to the reduced system of (3). In order to investigate the approximation of solution, the derivation process of an approximation system is given as follows.In order to replace the nonlinear term with the pullback limit of solution in high-frequency spaces, we consider the system(10)duc1=Lλcuc1dt,uc1t,x=ξt,ξt∈Hc,t∈t-T,t,(11)dus1=Lλsus1+PsFuc1t-Tdt,us1t,x=0,t∈t,t+T,where Lλc≔PcLλ, Lλs≔PsLλ, and (12)ξt=ξt,e1e1+ξt,e2e2=ξ1te1+ξ2te2∈Hc.Similar to the process in Section 2, there is a pullback limit of solution for the above system (10)–(11) written as hλ(1)(ξ(t)).Then projecting (3) into the subspace Hc, we have (13)duc=Lλcuc+PcFuc+usdt,where uc=Pcu and us=Psu, with Pc and Ps being the canonical projectors associated with Hc and Hs, respectively.Consideringus(t)=Psu(t) and substituting the pullback limit hλ(1)(ξ(t)) of the first lever system (10)–(11) into the above equation, we obtain reduced equation, (14)dξt=Lλcξ+PcFξt+hλ1ξtdt,which provides an approximation solution of the Swift-Hohenberg equation projected onto the low-frequency parts.From (8), we know that the coefficients of {ek(x)} contained in the expansion of hλ(1) are decayed. Therefore, the analytic representation of hλ(1) can be given from (8); the nonlinear term Fk1,k2,k3j=<F(ek1,ek2,ek3),ej> is given by -<ek1ek2ek3,ej> such that (15)Fk1k2k3j=12πfork1+k2+k3=j,Fk1k2k3j=-12πfork1+k2-k3=jork1-k2+k3=jor-k1+k2+k3=j,and (16)Fk1k2k3j=0forj≥7.Then we can obtain the result as follows: (17)hλ1ξ=-∑j=3∞∑k1,k2,k3=121βk1+βk2+βk3-βjξk1ξk2ξk3Psek1ek2ek3,ejej=-12πD3111ξ13e3-32πD3122ξ1ξ22e3+32πD4112ξ12ξ2e4+32πD4122ξ1ξ22e5+12πD6222ξ23e6where ξ1=<ξ,e1>, ξ2=<ξ,e2>, and Dj(k1k2k3)=1/βk1+βk2+βk3-βj. Similar to the above processing, the analytic formula of hλ(1)(ξ(t)) can be given, which has a complex formula in form. Then, there are some difficulties by using directly the analytic formula of hλ(1)(ξ(t)) to obtain the vector PcF(ξ(t)+hλ(1)(ξ(t))) when ξ(t) varies in Hc in spite of any case. So, replacing us(1)[ξ(t)](t+T) with hλ(1)(ξ(t)) when initial condition is ξ(t) at t, where us(1) is obtained by the above backward-forward system (4)-(5), we consider the following substitutive reduced equation: (18)dξt=Lλcξt+PcFξt+us1ξtt+Tdt,ξ0=ϕ,t>0,where ϕ is the initial datum; us(1)[ξ(t)] is given by the following system: (19)duc1=Lλcuc1ds,uc1ss=t=ξt,s∈t-T,t,(20)dus1=Lλsus1+PsFuc1s-Tds,us1ss=t=0,s∈t,t+T.Now, we assume that (21)ξt,x=ξ1te1x+ξ2te2x=y11te1x+y21te2x=uc1t,x.Then, when N≥7 and t∈[t,t+T], taking inner product with e1 and e2 on both sides of (18), respectively, we obtain that (22)dξ1t=β1λξ1t-32πξ13+2ξ1ξ22-ξ12-ξ22y31-2ξ1ξ2y41-ξ22y51+3πξ1∑i=3Nyi1yi+21-yi12+3πξ2∑i=3Nyi1yi+31-yi1yi+11+12π∑i=3N∑j=i+4Nyi1yj1yj-i-11-∑i=3N∑j=i+2Nyi1yj1yj-i+11+1π∑i=3N∑j=3Nyi1yj1yi+j+11-yi1yj1yi+j-11dt,t>0,(23)dξ2t=β2λξ1t-32πξ23+2ξ12ξ2-ξ12y41-2ξ1ξ2y51-y31-ξ22y61+3πξ1∑i=3Nyi1yi+31-yi1yi+11+3πξ2∑i=3Nyi1yi+41-yi12+12π-∑i=3N∑j=i+1Nyi1yj1yj-i+21+∑i=3N∑j=i+5Nyi1yj1yj-i-21+12π∑i=3N∑j=3Nyi1yj1yi+j+21-yi1yj1yi+j-21dt,t>0,with ξ1(0)=<ϕ,e1> and ξ2(0)=<ϕ,e2>, where ξt=ξ1(t)e1+ξ2(t)e2 and yj(1), j=3,…, given via the following system: (24)dy11s=β1λy11sds,s∈t-T,tdy21s=β2λy21sds,s∈t-T,t;dy31s=β3λy31s+12πy11s-T3-3y11s-Ty21s-T2ds,s∈t,t+T;dy41s=β4λy41s+32πy11s-T2y21s-Tds,s∈t,t+T;dy51s=β5λy51s+32πy11s-Ty21s-T2ds,s∈t,t+T;dy61s=β6λy61s+12πy21s-T3ds,s∈t,t+T;dyj1s=βjλyj1sds,s∈t,t+T,j=7,…,N;dyj1=0,s∈t,t+T,j=N+1,…with y1(1)(s)|s=t=ξ1(t), y2(1)(s)|s=t=ξ2(t), and yj(1)(s)|s=t=0, j=3,…. That is to say, (25)y11s=eβ1s-tξ1t,y21s=eβ2s-tξ2t,y31s=ξ132π3β1-β2e3β1s-T-t-e-3β1T-3ξ1ξ222πβ1+2β2-β3eβ1+2β2s-T-t-e-β1+2β2T,y41s=3ξ12ξ22π3β1+β2-β4e2β1+β2s-T-t-e-2β1+β2T,y51s=3ξ1ξ222π2β2+β1-β5e2β2+β1s-T-t-e-2β2+β1T,y61s=ξ232πe3β3s-T-t-e-3β3T,yj1=0,j=7,….From the above process, the numeration approximation system of solutions on finite dimensional spaceHc is given according to manifold under the proper parameter for Swift-Hohenberg equation, which can be done by computer to numerate it based on this reduced system according to above idea. It is important and beneficial in some points of view to obtain partial dynamics in approximation sense on the Hc modes in modeling of partial differential equations in practice. In fact, it is possible to achieve good modeling performances of solution from these results.


## 5. Conclusion

Therefore, existence of manifolds under proper parameters can be given for solution of some nonlinear partial differential equations. Furthermore, we have obtained the numeration approximation systems of iterative levels for solution on the basis of existence of parameter manifolds. However, the mathematical proofs in this manuscript maybe hold for the complex Swift-Hohenberg equation, which need us to further consider how to give some corresponding concepts in complex spaces. Although it is important to use the numerical solutions for some nonlinear real partial differential equations to model some practical problems according to the approximation solution in this paper, this idea can be extensively applied to solve other real partial differential equations.



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