Journal of Applied Mathematics

Volume 2013 (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

Lei Shi1,2

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 [1–3] 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 [10–12] 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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