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Journal of Applied Mathematics
Volume 2013, Article ID 926512, 8 pages
http://dx.doi.org/10.1155/2013/926512
Research Article

Bifurcation Analysis of a Coupled Kuramoto-Sivashinsky- and Ginzburg-Landau-Type Model

1School of Sciences, Nanjing Agricultural University, Nanjing 210095, China
2School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210046, China

Received 24 April 2013; Accepted 2 July 2013

Academic Editor: Abdul Hamid Kara

Copyright © 2013 Lei Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the bifurcation and stability of trivial stationary solution of coupled Kuramoto-Sivashinsky- and Ginzburg-Landau-type equations (KS-GL) on a bounded domain with Neumann's boundary conditions. The asymptotic behavior of the trivial solution of the equations is considered. With the length of the domain regarded as bifurcation parameter, branches of nontrivial solutions are shown by using the perturbation method. Moreover, local behavior of these branches is studied, and the stability of the bifurcated solutions is analyzed as well.

1. Introduction

The mathematical theory of pattern formation [13] has a wide range of applications. In the field of fluid mechanics, Rayleigh-Bénard convection is the most widely studied example. Like the thermally driven Bénard convection, the surface tension-driven Marangoni convection is also an interesting pattern formation of nonlinear system. With mass transfer from liquid phase to gas phase, a typical setup for the Marangoni convection is a liquid layer resting on a rigid surface, with a free deformable upper interface contacting an infinite layer of gas. Nevertheless, at present, we still find it hard to analyze the governing equations for the Marangoni convection, that is, Navier-Stokes equations and mass (surfactant) transfer equation. What we can only use is numerical simulation so far. There is little work that has been done on the nonlinear Marangoni convection.

In this paper, we consider a simplified model, which is proposed to capture important nonlinear features yet more amendable to analyse: whereis the amplitude for the Marangoni convection mode andis the interface deformation (real function). Equation (1) has been derived by Golovin et al. [4]. The constant is positive, withdenoting the Marangoni number, and represents the gradient (derivative) of surface tension with respect to surfactant concentration.is the critical Marangoni number at which the trivial stationary state becomes linearly unstable, andis a constant related to other system parameters.

Equation (1) without the interaction termis the well-known Ginzburg-Landau equation [5, 6], while the second equation without the termis the linearized version of the so-called Kuramoto-Sivashinsky equation [7]. Both the G-L equation and K-S equation have been extensively investigated as model examples of infinite dimensional dynamical systems. In [8], Kazhdan et al. have done numerical simulation of this coupled system of Kuramoto-Sivashinsky- and Ginzburg-Landau-type equations (hereafter, KS-GL system). Duan et al. [9] have discussed the existence and uniqueness of global solutions of this coupled system, using the contraction mapping principle and energy estimates. Despite these publications on KS-GL equation, the static bifurcations of the equation have not been thoroughly studied. In this paper, we focus on studying bifurcations of the KS-GL system. In [10], Xiao and Gao analyzed the bifurcations of the 1D Swift-Hohenberg equation with quintic nonlinearity. Two types of structures in the bifurcation diagrams are presented when the bifurcation points are closer, and their stabilities are analyzed. Li and Chen have applied singularity theory and the perturbation method to study the bifurcations of the 1D and 2D K-S equations and get the asymptotic expressions of the steady-state solution branches that have bifurcated from the equilibrium in [11, 12]. In this paper, we will use the methods in [1012] to discuss the bifurcated solutions. In (1),is complex. Namely, we can write . The additional phasemakes analysis very complicated. Here, we restrict our attention to invariant subspace in whichis real. We hope to return to the general case in future study. In this paper, we discuss steady solutions of the parabolic partial differential equations on the cylindrical domainsubject to the boundary conditions The steady-state equation of (2) reads as We will discuss the bifurcation of the trivial steady state of the equations.

2. Analysis of the Trivial Steady State

In this section, we study some properties of the linear problem associated with problem (4). Let   . We linearize the problem at the trivial solution , and then we have the corresponding differential operator matrix and the eigenvalue problem is

The eigenvaluesare given by If , then the corresponding eigenvectors are If , then for , the corresponding eigenvector is and for , the corresponding eigenvector is It will be convenient to rescale the spatial variable so that the domainmaps onto the fixed domain . Thus, we introduce the variablesand , , , , then omit the tildes, and we find that problem (4) becomes The corresponding eigenvalue problem at is It is easy to find that.

3. Bifurcation and Stability Analysis in Different Cases

Letbe a bifurcation parameter;is a bifurcation point. In this section, we discuss how many nontrivial solution branches will be bifurcated from the trivial solution nearand their asymptotic expression. Moreover, we will discuss the stability of the solution branches.

When, the eigenvector at of (12) is We set whereis a small parameter,, .

Substituting (14), (15) into (11) leads to where meansand means.

Letting the coefficient ofvanish in (16), (17) gives where.

Taking the inner product of (16) with, we obtain For the first term of (20), integrate it by part and from the boundary condition, and we have Substituting (21) to (20), we get

From the aforementioned orthogonality condition, we know that Taking the inner product of (19) with, we obtain Because of the boundary condition, orthogonality condition (24), we get from (25).

Comparing (22) and (26), for, we deduce the following important relation:

Substituting (27) to (18), we get Forsatisfies the boundary condition, we calculate from (28) that

Next, we want to calculate and substitute (27) to (19); we get the following ODE: By lengthy computations and the boundary conditions, we deduce that Taking the inner product withof (31), from (24), we know that which gives us Substituting (29), (35) to (23), we have from which we get

From the previous discussion, we obtain with the normal condition

When, then we have at this time,

Letting the coefficient ofof (16), (17) vanish, we get where , and .

Taking the inner product of (42) with, we obtain from which we know that

Next, we discuss the expression and the stability of the bifurcated solutions in different cases.

Case 1. ,.
Substituting (41), (45) to (42), we have Because of the orthogonality condition and boundary condition we get Substituting (41) to (43), we have Taking the inner product of (50) with, we obtain forsatisfies the orthogonality condition and the boundary condition we know which forces From (45) and (55), we get or Substituting (55) to (50), we get which gives us As has been discussed previously, we have the following.
Fornear, there are nontrivial steady-state solution branches of (11) bifurcated from the trivial solution: ,,satisfy (56), (57). ., are arbitrary constants.
Next, we wish to consider the stability of the nontrivial solutions given in (60).
Considering the eigenvalue problem since , , , we assume Substituting (62) to (61), we get On the branches given in (60), equating the coefficient ofin (63) to 0, we obtain From (64) and the boundary conditions we get Using the second-order term ofin (63), we obtain and considering the boundary condition we deduce that (67) has solution if and only if; in this case, we have (). Taking the sign ofinto account in (15), we consideronly. Therefore, we have So in Case 1, on the branches (60), the eigenvalue ofis The corresponding eigenfunction is From (70), we know that the eigenvalue of the linearized operatorat the nontrivial solutions given in (60) is negative if, and positive if. Finally, for small , if, then the corresponding solution branches are stable; if, then the corresponding solution branches are unstable; if, we need higher-order items ofto determine the result.
Thus, we have proved the following theorems.

Theorem 1. Fornear, there are nontrivial steady-state solution branches of (4) bifurcated from the trivial solution: where,,,.

Remark 2. Since , when , is a supercritical bifurcation point; when ,is a supercritical bifurcation point.

Theorem 3. The eigenvalue of the linearized operator at the nontrivial solutions given in Theorem 1 is negative if , positive if . Finally, for small , if , then the corresponding solution branches are stable; if , then the corresponding solution branches are unstable.

Case 2. ,.
If,, then; taking the sign ofinto account in (15), we consideronly. In this case, from (40), (41), (45), (49), (56), and (59), we have Similarly discussed as in Case 1 previously, we get

Remark 4. Since, there are no solution branches of (4) bifurcated from the trivial solution in Case 2.

Case 3. ,.
If,, then,; taking the sign ofinto account in (15), we consideronly. In this case, from (27), (31), and (37), we have Letting the coefficient ofof (16) vanish, we get Taking the inner product of (76) with, we obtain from which we get Letting the coefficient ofof (17) vanish, we get from which we get Substituting (78), (80) to the orthogonality condition we get Thus, we have that fornear, there are nontrivial steady-state solution branches of (11) bifurcated from the trivial solution:
Next, we wish to consider the stability of the nontrivial solutions given in (83).
Considering the eigenvalue problem since , , , we assume that Substituting (85) to (84), we get On the branches given in (83), equating the coefficient ofin (86) to 0, we obtain From (87) and the boundary conditions we get We find thatis a contradiction to, so there are no solution branches of (4) bifurcated from the trivial solution in Case 3.

Case 4. ,.
If,, then; taking the sign ofinto account in (15), we consideronly. In this case, from (27), (31), and (37), we have Similarly discussed as in Case 1 previously, we get

Remark 5. Since, there were no solution branches of (4) bifurcated from the trivial solution in Case 4.

Remark 6. In this paper, trivial stationary solution and bifurcations at it are considered for the KS-GL equations with Neumann’s boundary conditions on a finite domain. The results we get in this paper are new and original. It would be very interesting and much more complicated to try and extend our study to stationary patterns and the corresponding bifurcations. Another interesting but challenging direction would be the case in several space dimensions.

Acknowledgments

The work is supported by the Youth Science and Technology Innovation Fund of Nanjing Agriculture University (no. KJ2010026), the College Graduate Student Scientific Research Innovation Plan of Jiangsu Province (CXLX12_0390) and the Fundamental Research Funds for the Central Universities (no. Y0201100265).

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