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Traveling wave solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation.

1. Introduction

The Benjamin-Bona-Mahony (BBM) equation [1]

[u.sub.t] + [u.sub.x] + [uu.sub.x] - [u.sub.xxt] = 0 (1)

was proposed as the model for propagation of long waves where nonlinear dispersion is incorporated. The KadomtsevPetviashvili (KP) equation [2]

[([u.sub.t] + [auu.sub.x] + [u.sub.xxx]).sub.x] + [u.sub.yy] = 0 (2)

was given as the generalization of the KdV equation. In addition, both BBM and KdV equations can be used to describe long wavelength in liquids, fluids, and so forth. Combining the two equations, the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation

([u.sub.t] + [u.sub.x] - a[([u.sup.2]).sub.x] - [bu.sub.xxt]).sub.x] + [ku.sub.yy] = 0 (3)

was presented in [3] for further study. Some methods are developed and applied to find exact solutions of nonlinear evolution equations because exact solutions play an important role in the comprehension of nonlinear phenomena. For instance, extended tanh method, extended mapping method with symbol computation, and bifurcation method of dynamical systems are employed to study (3) [4-6], and some solitary wave solutions and triangle periodic wave solutions were obtained.

However, there is no method which can be applied to all nonlinear evolution equations. The research on the solutions of the KP-BBM equation now appears insufficient. Further studies are necessary for the traveling wave solutions of the KP-BBM equation. The purpose of this paper is to apply the bifurcation method [7-10] of dynamical systems to continue to seek traveling waves of (3). Firstly, we obtain bell-shaped solitary wave solutions involving more free parameters, and some results in [6] are corrected and improved. Then, we get some new periodic wave solutions in parameter forms of Jacobian elliptic function, and numerical simulation verifies the validity of these periodic solutions. The periodic wave solutions obtained in this paper are different from those in [5]. Furthermore, we find an interesting relationship between the bell-shaped waves and periodic waves; that is, the bell-shaped waves are limits of the periodic waves in forms of Jacobian elliptic function as modulus approaches 1.

This paper is organized as follows. First, we draw the bifurcation phase portraits of planar system according to the KP-BBM equation in Section 2. Second, bell-shaped solitary wave solutions to the equation under consideration are presented in Section 3. Third, periodic solitary waves are given in the forms of Jacobian elliptic function and numerical simulation is done. Finally, the relationship between the bell-shaped solitary waves and periodic waves is proved in Section 4.

2. Bifurcation Phase Portraits of System (6)

Suppose (3) possesses traveling wave solutions in the form u(x, t) = [phi]([xi]), [xi] = x + ry - ct, where c is the wave speed and r is a real constant. Substituting u(x, t) = [phi]([xi]), [xi] = x + ry - ct into (3) admits to the following ODE:

(1 + [kr.sup.2] - c)[phi]" - a([[sigma].sup.2])" + [bc[phi].sup.(4)] = 0, (4)

where the derivative is for variable [xi]. Integrating (4) twice with respect to [xi] and letting the first integral constant take value zero, it follows that

(1 + [kr.sup.2] - c)[phi]" - [a[phi].sup.2] + bc[phi]" = g, (5)

where g is the second integral constant.

Equation (5) is equivalent to the following two-dimensional system:

[d[phi]]/[d[xi]] = y, [dy/[d[xi]]] = [a[[phi].sup.2] -(1 + [kr.sup.2] -c)[phi] + g]/bc. (6)

It is obvious that system (6) has the first integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where h is the constant of integration.

Define [DELTA] = [(1 + [kr.sup.2] - c).sup.2] - 4ag. When [DELTA] > 0, there are two equilibrium points ([[phi].sub.1], 0) and ([[phi].sub.2], 0) of (6) on [phi]- axis, where [[phi].sub.1] = ((1 + [kr.sup.2] - c)- [square root of [DELTA]])/2a, [[phi].sub.2] = ((1 + [kr.sup.2] - c) + [square root of [DELTA]])/2a. The Hamiltonian H of ([[phi].sub.1], 0) and ([[phi].sub.2], 0) is denoted by [h.sub.1] = H([[phi].sub.1], 0) and [h.sub.2] = H([[phi].sub.2], 0).

In the case of 1 + [kr.sup.2] - c < 0 and 1 + [kr.sup.2] - c > 0, the bifurcation phase portraits of system (6) see Figures 1 and 2 in [6], respectively, in which there are some homoclinic and periodic orbits of system (6). For our purpose, we redraw the homolinic and periodic orbits in this paper(see Figures 1-3).

3. Exact Explicit Expressions of Solitary Wave Solutions

In this section, we discuss bell-shaped wave solutions under g = 0 and g [not equal to] 0, respectively.

3.1. The Case g = 0. System (6) can be rewritten as

[d[phi]]/[d[xi]] = y, [dy/[d[xi]]] = [a[[phi].sup.2] -(1 + [kr.sup.2] - c)[phi]]/bc. (8)

The first integral of (8) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

When (1 + [kr.sup.2] - c)/bc < 0, there are two homoclinic orbits [[GAMMA].sub.1] and [[GAMMA].sub.2] (see Figures 1(a) and 1(b)). In [phi]-y plane, [[GAMMA].sub.1] and [[GAMMA].sub.2] can be described by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [[phi].sup.*] = 3(1 + [kr.sup.2] - c)/2a. That is,

y = [+ or -][square root of [2a/3bc] [[phi].sup.3] - [[[1 + [kr.sup.2] - c]]/bc] [[phi].sup.2]]. (11)

Substituting (11) into d[phi]/d[xi] = y and integrating along homoclinc orbits [[GAMMA].sub.1] and[[GAMMA].sub.2], respectively, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Completing the above integration, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Remark 1. [u.sub.1] (x, y, t) is a bright soliton solution when bc/a < 0 and a dark soliton solution when bc/a > 0. If the real number r in (13) takes value 1, then solution (13) is the same to solution (1.2) in [6]. Solution (1.1) in [6] is not a real solution of the KP-BBM equation; it is obvious that solution (1.1) tends to infinite as [xi] [right arrow] 0, and it does not satisfy the KP-BBM equation (3).

When (1 + [kr.sup.2] + c)/bc > 0, there are two homoclinic orbits [[GAMMA].sub.3] and [[GAMMA].sub.4] (see Figures 2(a) and 2(b)). Similarly solitary wave solutions according to [[GAMMA].sub.3] and [[GAMMA].sub.4] are obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Remark 2. If the real number r in (14) takes value 1, then solution (14) is the same to solution (1.4) in [6]. Solution (1.5) in [6] is not a real solution of the KP-BBM equation; it is easy to verify that it does not satisfy the KP-BBM equation (3).

3.2. The Case g [not equal to] 0. There are two homoclinic orbits [[GAMMA].sub.5] and [[GAMMA].sub.6] when g [not equal to] 0 (see Figures 3(a) and 3(b)). [[GAMMA].sub.5] and [[GAMMA].sub.6] can be described by

[y.sup.2] = [2a/3bc] [[phi].sup.3] - [[1 + [kr.sup.2] - c]/bc] [[phi].sup.2] [2g/bc] [phi] + 2h. (15)

When bc/a < 0, the corresponding homoclinic orbit [[GAMMA].sub.5] has a double zero point [[phi].sub.1] and a zero point [[phi].sub.3] on [phi]- axis (see Figure 3(a)), so (15) can be rewritten as

[y.sup.2] = [2a/3bc] [([phi] - [[phi].sub.1]).sup.2] ([phi] - [[phi].sub.3]); (16)

that is,

y = [+ or -] [square root of [2a/3bc] [([phi] - [[phi].sub.1]).sup.2] ([phi] - [[phi].sub.3])]. (17)

Substituting (17) into d[phi]/d[xi] = y and integrating along homoclinic orbits [[GAMMA].sub.5], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where ([[phi].sub.1] = (1 + [kr.sup.2] - c - [square root of [DELTA]])/2a and [[phi].sub.3] = (1 + [kr.sup.2] - c + 2 [square root of [DELTA]])/2a if a > 0 and bc < 0; then, completing (18) we get the following solution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

In (18), [[phi].sub.1] = (1 + [kr.sup.2] - c + [square root of [DELTA]])/2a and [[phi].sub.3] = (1 + [kr.sup.2] - c - 2 [square root of [DELTA]])/2a if a < 0 and bc > 0; then, completing (18) we get the following solution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Remark 3. If the real number r in (19) and (20) takes value 1, then solutions (19) and (20) are the same to solutions (1.6) and (1.8) in [6]. Solutions (1.7) and (1.9) in [6] are not real solutions of the KP-BBM equation, and it is easy to verify that they do not satisfy the KP-BBM equation (3).

When bc/a > 0, the corresponding homoclinic orbit [[GAMMA].sub.6] has a double zero point [[phi].sub.2] and a zero point [[phi].sub.4] on [phi]- axis (see Figure 3(b)), so (15) can be rewritten as

[y.sup.2] = 2a/3bc [([phi] - [[phi].sub.2]).sup.2] - ([phi] - [[phi].sub.4]); (21)

that is,

y = [+ or -] [square root of [2a/3bc] [([phi] - [[phi].sub.2]).sup.2] ([phi] - [[phi].sub.4])]. (22)

Substituting (22) into d[phi]/d[xi] = y and integrating along homoclinic orbits [[GAMMA].sub.6], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

where [[phi].sub.2] = (1 + [kr.sup.2] - c - [square root of [DELTA]])/2a and [[phi].sub.4] = (1 + [kr.sup.2] - c + 2 [square root of [DELTA]])/2a if a < 0 and bc < 0; then, completing (23) we get the solution [u.sub.3](x, y, t). In (23), [[phi].sub.2] = (1 + [kr.sup.2] - c + [square root of [DETLA]])/2a and [[phi].sub.4] = (1 + [kr.sup.2] - c - 2 [square root of [DELTA]])/2a if a > 0 and be > 0; then, completing (23) we get the solution [u.sub.4] (x, y, t).

4. Periodic Wave Solutions

So as to explain our work conveniently, in this section the Jacobian elliptic function sn(l, m) with modulus m will be expressed by snl. We discuss the periodic wave solutions under conditions g = 0 and g [not member of] 0, respectively.

4.1. The Case g = 0. When (1 + [kr.sup.2] + c)/bc < 0, system (6) has periodic orbits [[GAMMA].sub.7] and [[GAMMA].sub.8] (see Figures 1(a) and 1(b)). Their expressions are (9) on [phi]-y plane, where [h.sub.1] < h < [h.sub.2] (or [h.sub.2] < h < [h.sub.1]). Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (24)

then, we have the following results.

Claim 1. In the case of g = 0, (1 + [kr.sup.2] + c)/bc < 0 and [h.sub.1] < h < [h.sub.2] (or [h.sub.2] < h < [h.sub.1]); then, the function [f.sub.1]([phi]) must have three different real zero points.

Proof. Since [f.sub.1] ([phi]) is a cubic polynomial about [phi], we can use the Shengjin Theorem [11] to distinguish its solutions. We only discuss the case bc/a < 0, and the case bc/a > 0 is the same. Under the above conditions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

So (1 + [kr.sup.2] - c)/(6[a.sup.2] bc) < h < 0. In [f.sub.1]([phi]) the coefficients 2a/3bc < 0, (1 + [kr.sup.2] - c)/bc < 0, By Shengjing Theorem [11], it follows that the function [f.sub.1] ([phi]) has three different real zero points. Let A = [((1 + [kr.sup.2] - c)/bc).sup.2], B = -9(2a/3bc) x 2h, and C = 3((1 + [kr.sup.2] - c)/bc) x 2h; then, [B.sup.2]-4AC = 144[(a/bc).sup.2]h[h- [((1 + [kr.sup.2] - c).sup.3]/6[a.sup.2]bc)] < 0.

Let [r.sub.1] < [r.sub.2] < [r.sub.3] be three different real zero points of [f.sub.1]([phi]). Then Claim 1 means that (9) has three intersection points ([r.sub.1], 0), ([r.sub.2], 0), and ([r.sub.3], 0) on [phi]-axis. Therefore, (9) can be rewritten as

[y.sup.2] = 2a/3bc ([phi] - [r.sub.1])([phi] - [r.sub.2])([phi] - [r.sub.3]), (26)

where [r.sub.1] < 0 < [r.sub.2] < [phi] < [r.sub.3] when bc/a < 0 and [r.sub.1] < [phi] < [r.sub.2] < 0 < [r.sub.3] when bc/a > 0.

When bc/a < 0, the orbit [[GAMMA].sub.7] is according to a periodic solution of (6) and its expression is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

Substituting (27) into d[phi]/d[xi] = y and integrating along orbit [[GAMMA].sub.7], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

By formula (236) in [12], we have

[g.sub.1][sn.sup.-1] (sin [[psi].sub.1], [m.sub.5]) = [square root of -2a/3bc] [absolute value of [xi]], (29)

where [g.sub.1] = 2/ [square root of [r.sub.3] - [r.sub.1]], sin [[psi].sub.1] = [square root of ([r.sub.3] - [phi])/([r.sub.3] - [r.sub.2])], and [m.sub.5] = [square root of ([r.sub.3] - [r.sub.2])/([r.sub.3] - [r.sub.1])]. Solving (29), we get

[phi] = [r.sub.3] - ([r.sub.3] - [r.sub.2]) [sn.sup.2] [square root of - [a([r.sub.3] - [r.sub.1])]/6bc] [xi]. (30)

That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

where the modulus of sn is [m.sub.5] = [square root of ([r.sub.3] - [r.sub.2])/([r.sub.3] - [r.sub.1])].

Similarly, when bc/a > 0, the expression [[GAMMA].sub.8] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

Substituting (32) into d[phi]/d[xi] = y and integrating along orbit [[GAMMA].sub.8], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

The according periodic solution of [[GAMMA].sub.8] can be obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where the modulus of sn is [m.sub.6] = [square root of ([r.sub.2] - [r.sub.1])/([r.sub.3] - [r.sub.1])].

When (1 + [kr.sup.2] + c)/bc > 0, system (6) has periodic orbits [[GAMMA].sub.9] and [[GAMMA].sub.10] (see Figure 2). Their expressions are (9), where [h.sub.1] < h < [h.sub.2] (or [h.sub.2] < h < [h.sub.1]). Similarly, we can get the according periodic solutions of [[GAMMA].sub.9] and [[GAMMA].sub.10] as [u.sub.5] and [u.sub.6].

To verify validity of the periodic wave solutions, we take a = b = k = r= 1, c = -1, and h = -9/4 to make the conditions in Claim 1 satisfied. By simple calculation, we get that [r.sub.1] = -1.09808, [r.sub.2] = 1.5, and [r.sub.3] = 4.09808. The specific periodic wave solution is

u (x, y, t) = 4.090808 - [2.590808sn.sup.2] [square root of 5.19616/6] (x + y + t), (35)

where the modulus of sn is [square root of 2]/2.

4.2. The Case g [not equal to] 0. System (6) has periodic orbits [[GAMMA].sub.11] and [[GAMMA].sub.12] (see Figure 3). Their expressions are (7) on the [phi]-y plane, where [h.sub.1] < h < [h.sub.2] (or [h.sub.1] < h < [h.sub.2]). Let

[f.sub.2]([phi]) = 2a/3bc [[phi].sup.3] - 1 + [kr.sup.2] - /bc [[phi].sup.2] + 2g/bc [phi] + 2h; (36)

then, we have the following results about [f.sub.2]([phi]).

Claim 2. If g [not equal to] 0 and [h.sub.1] < h < [h.sub.2] (or [h.sub.2] < h < [h.sub.1]), then the function [f.sub.2]([phi]) must have three different real zero points.

Proof. We only prove the case bc/a < 0, and the case bc/a > 0 is the same. Under the above conditions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

So [f.sub.2]([[phi].sub.1]) x [f.sub.2]([[phi].sub.2]) = 4(h - [h.sub.1])(h - [h.sub.2]) < 0. For [f.sub.2]([phi]), we have [f.sub.2](-[infinity]) > 0, [f.sub.2]((p1) < 0, [f.sub.2]([[phi].sub.2]) > 0, and [f.sub.2](+[infinity]) < 0. Again, [f'.sub.2]([phi]) = (2a/bc)([phi] - [[phi].sub.1])([phi] - [[phi].sub.2]), which is monotonous in the intervals (-[infinity], [[phi].sub.1]), ([[phi].sub.1], [[phi].sub.2]), and ([[phi].sub.2], By zero point theorem of continuous function, there must be one real zero point of [f.sub.2]([phi]) that lies in each of the three intervals, proving the claim.

Let [c.sub.1] < [c.sub.2] < [c.sub.3] be three different real zero points of [f.sub.2]([phi]). Then Claim 2 means that (7) has three intersection points on [phi]-axis denoted by ([c.sub.1], 0), ([c.sub.2],0), and ([c.sub.3], 0). Then (7) can be rewritten as

[y.sup.2] = 2a/3bc ([phi] - [c.sub.1])([phi] - [c.sub.2])([phi] - [c.sub.3]), (38)

where [c.sub.1] < [[phi].sub.1] < [c.sub.2] < [[phi].sub.2] c3.

When bc/a < 0, the expression of periodic orbit [[GAMMA].sub.11] is

y=[+ or -][square root of (2a/3bc ([phi] - [c.sub.1]) ([phi] - [c.sub.2])([phi] - [c.sub.3]))], ([c.sub.1] < [c.sub.2] [less than or equal to] [phi] [c.sub.3]). (39)

When bc/a > 0, the expression of periodic orbit [[GAMMA].sub.12] is

y = [+ or -][square root of (2a/3bc ([phi] - [c.sub.1]) ([phi] - [c.sub.2])([phi] - [c.sub.3]))], ([c.sub.1] [less than or equal to] [phi] [less than or equal to] [c.sub.2] < [c.sub.3]). (40)

Substituting (39) and (40) into d[phi]/d[xi], = y and integrating along orbits [[GAMMA].sub.11] and [[GAMMA].sub.12], respectively, it is the same to the proceeding for solving [u.sub.5] and [u.sub.6] and we can get the according periodic solutions of [[GAMMA].sub.11] and [[GAMMA].sub.12] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41)

where the moduli for sn are [m.sub.7] = [square root of (([c.sub.3] - [c.sub.2])/([c.sub.3] - [c.sub.1]))] and [m.sub.8] = [square root of (([c.sub.2] - [c.sub.1])/([c.sub.3] - [c.sub.1]))] in (41).

For example, we take a = b = k = r= 1,c = -1, g = 2, and h = 3/2 such that the conditions in Claim 2 are satisfied. By simple calculation, we get that [c.sub.1] = 0.663975, [c.sub.2] = 1.5, and [c.sub.3] = 2.36603. The according periodic wave solution is

u(x, y, t) = 2.36603 - 0.86603[sn.sup.2] [square root of 1.702055/6 (x + y + t), (42)

where the modulus of sn is [square root of 0.86603/1702055].

5. Relationship between Bell-Shaped Waves and Periodic Waves

In Sections 3 and 4, the bell-shaped solitary wave and periodic wave solutions are obtained. Via further study, we find that there exists an interesting relationship between these two kinds of solutions; that is, the bell-shaped solutions are limits of the periodic waves in some sense. The results are detailed as follows.

Proposition 4. Let [u.sub.i] (i = 1,2, ..., 8) be solutions of (3), let k, r, a, b, c, and g be parameters in (5), and let [m.sub.i] (i = 5,6, 7, 8) be modulus of the Jacobian elliptic function sn; then, one has the following.

Case 1. When g = 0 and (1 + [kr.sup.2] - c)/bc < 0, for modulus [m.sub.i] [right arrow] 1 (i = 5, 6), the periodic waves [u.sub.5] and [u.sub.6] degenerate bell-shaped wave [u.sub.1].

Case 2. When g = 0 and (1 + [kr.sup.2] - c)/bc > 0, for modulus [m.sub.i] [right arrow] 1 (i = 5, 6), the periodic waves [u.sub.5] and [u.sub.6] degenerate bell-shaped wave [u.sub.2].

Case 3. When g [not equal to] 0 and bc/a < 0, for modulus [m.sub.7] [right arrow] 1, the periodic wave [u.sub.7] degenerates bell-shaped wave [u.sub.3].

Case 4. When g [not equal to] 0 and bc/a > 0, for modulus [m.sub.8] [right arrow] 1, the periodic wave [u.sub.8] degenerates bell-shaped wave [u.sub.4].

Here, we only prove Cases 1 and 3 for simplicity. The remaining cases are the same. In the following proofs, we use the property of elliptic function that sn [right arrow] tanh when the modulus m [right arrow] 1 [5,13].

Proof of Case 1. When [m.sub.5] = [square root of (([r.sub.3] - [r.sub.2])/([r.sub.3] - [r.sub.1]))] [right arrow] 1, it means [r.sub.1] = [r.sub.2] and sn = tanh; then, we calculate

[r.sub.1] = [r.sub.2] = 0, [r.sub.3] 3(1 + [kr.sup.2] -c)/2a. (43)

Substituting [r.sub.i] (i = 1,2,3) into u5 admits to u1 as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

When [m.sub.6] = [square root of (([r.sub.2] - [r.sub.1])/([r.sub.3] - [r.sub.1]))] [right arrow] 1, it means [r.sub.2] = [r.sub.3]; then, we calculate [r.sub.2] = [r.sub.3] = 0 and [r.sub.1] = 3(1 + [kr.sup.2] - c)/2a, and substituting [r.sub.i] (i = 1,2, 3) into [u.sub.6] we get [u.sub.6] = [u.sub.1].

Proof of Case 3. When [m.sub.7] = [square root of (([c.sub.3] - [c.sub.2])/([c.sub.3] - [c.sub.1]))] [right arrow] 1, it means [c.sub.1] = [c.sub.2] and sn = tanh; then, we calculate [c.sub.1] = [c.sub.2] = (1 + [kr.sup.2] - c - [square root of [nabla])]/2a and [c.sub.3] = (1 + [kr.sup.2] - c + [square root of [nabla])]/2a, and substituting [c.sub.i] (i = 1,2,3) into [u.sub.7] admits to [u.sub.3] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

The results provide a manner that we can get bell-shaped waves from periodic waves for some nonlinear development equations.

http://dx.doi.org/10.1155/2014/943167

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11201070) and Guangdong Province (no. 2013KJCX0189 and no. Yq2013161). The author would like to thank the editors for their hard working and the anonymous reviewers for helpful comments and suggestions.

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Zhengyong Ouyang

Department of Mathematics, Foshan University, Foshan, Guangdong 528000, China

Correspondence should be addressed to Zhengyong Ouyang; zyouyang_math@163.com

Received 26 March 2014; Accepted 17 April 2014; Published 18 June 2014

Academic Editor: Junling Ma
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Title Annotation:Research Article
Author:Ouyang, Zhengyong
Publication:Abstract and Applied Analysis
Article Type:Report
Date:Jan 1, 2014
Words:4187
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