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.
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 with a proper parameter , 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 with 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 spaces and , where and is compactly and densely embedded in . The operator is linear and the spectrum of satisfies . In addition, the interpolated space is a space between and with . Let be linear operators including a simple parameter. Here map into and depend continuously on .
For the convenience of studying, the Swift-Hohenberg equation has been rewritten in the following form: with initial condition and Dirichlet boundary conditions on , where is defined as , which is a trilinear and continuous function from to , . Here, we define . The operator is closed self-adjoint linear operator with dense domain . In addition, is self-adjoint operator with an orthonormal basis of eigenfunctions in and corresponding eigenvalues .
Assume that , where is finite. Let the topological complement of be and the function maps to . Then we have the high frequency part and low frequency part , where and . Thus, for any .
3. Approximation of Parameter Manifolds
According to the approximation methods in [5], using backward-forward systems of Swift-Hohenberg equation and , we assume the following reduced system: where , , , and . Equation (4) is the linear part of (3) in space , which is assumed and seen as the first iteration level in of reduced system (4)-(5). From the above system, the existence and specific presentation forms of manifolds under parameter are given by the following theorem with the initial values of and .
Theorem 1. Consider the Swift-Hohenberg equation (3), where is a trilinear function. Assume also that for all . Let , . If, for all , when , , then the pullback limit for the solution of (5) exists and is given by where is the solution of (4) and Moreover, has the following analytic expression: where ,
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 where 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 subspace and take 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 where , , and Similar to the process in Section 2, there is a pullback limit of solution for the above system (10)–(11) written as .
Then projecting (3) into the subspace , we have where and , with and being the canonical projectors associated with and , respectively.
Considering and substituting the pullback limit of the first lever system (10)–(11) into the above equation, we obtain reduced equation, which provides an approximation solution of the Swift-Hohenberg equation projected onto the low-frequency parts.
From (8), we know that the coefficients of contained in the expansion of are decayed. Therefore, the analytic representation of can be given from (8); the nonlinear term is given by such that and Then we can obtain the result as follows: where , , and Similar to the above processing, the analytic formula of can be given, which has a complex formula in form. Then, there are some difficulties by using directly the analytic formula of to obtain the vector when varies in in spite of any case. So, replacing with when initial condition is at , where is obtained by the above backward-forward system (4)-(5), we consider the following substitutive reduced equation: where is the initial datum; is given by the following system: Now, we assume that Then, when and , taking inner product with and on both sides of (18), respectively, we obtain that with and , where and , , given via the following system: with , , and , That is to say,
From the above process, the numeration approximation system of solutions on finite dimensional space 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 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (no. 11771444); Guangxi Natural Science Foundation (no. 2017GXNSFAA198221); Promotion of the Basic Capacity of Middle and Young Teachers in Guangxi Universities (no. 2017KY0340); the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (no. 2018061); Key Laboratory Open Fund Project of the Simulation of Manufacturing Systems and Informational Enterprise (no. 2016ZD008); Yue Qi Young Scholar Project, China University of Mining and Technology (Beijing); and CSC.