Abstract and Applied Analysis

Volume 2013 (2013), Article ID 301645, 7 pages

http://dx.doi.org/10.1155/2013/301645

Research Article

New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation

XiaoHua Liu and CaiXia He

School of Science, Guizhou Minzu University, Guiyang, Guizhou 550025, China

Received 24 July 2013; Accepted 13 September 2013

Academic Editor: Ziemowit Popowicz

Copyright © 2013 XiaoHua Liu and CaiXia He. 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.

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.

1. Introduction

For the investigation of traveling wave solutions to nonlinear partial differential equations, which have been the subject of study in various branches of mathematical physical sciences such as physics, biology, and chemistry, many effective methods (see, e.g., [1–12]) have been presented, such as inverse scattering transform method [1], Hirota’s method [2], Backlund and Darboux transformation method [5], the Jacobi (Weierstrass) elliptic function method [7], undetermined coefficient method [9, 10],-expansion method and others.

Following the work of Hirota and Satsuma in [13], recently, considerable attention has been focused on the study of coupled nonlinear partial differential equation (see, e.g., [14–19]) that can be solved exactly. However, less work on the coupled version of the higher KdV equation seems to have been reported.

In [20], Guha-Roy has been presented a system of coupled nonlinear wave equations as follows

where the subscripts refer to partial differentiations with respect to the indicated variables, and ,,,,, are arbitrary parameters. Equation (1) is the coupled version of combined form of the higher (modified) KdV equation and KdV equation. It is interesting to point out that, as is outlined in Wadati [21], (1) shares properties with the KdV and the modified KdV equation, under certain conditions.

Guha-Roy [22] supposed that with ,, by transforming to that Weierstrauss elliptic function, some exact solitary wave solutions in special conditions to (1) are obtained. Lu et al. [23] obtain the kink-antikink solitary wave solutions of (1) by using a truncated expansion. However, there has not been found any literature on the general analysis of the existences of the solitary wave solution and the exact expressions of the solitary wave solution with nonzero asymptotic value for (1).

In this work, we investigate generally the existence of bell-shaped, kink-shaped solitary wave solutions and periodic wave solutions by using the theory of planar dynamical system under different parameter conditions. It follows that we obtain the new exact expressions of all three bell-shaped solitary wave solutions, and one kink-shaped solitary wave solution with nonzero asymptotic value for (1) by undetermined coefficient methods. These solutions obtained here cannot be deduced from [20, 23].

2. Existence of the Bounded Traveling Wave Solutions to (1)

By introducing an analogue of the stream function, Guha-Roy et al. [18] have shown that if one of the solutions of some coupled nonlinear equations is of the traveling wave type, then the other must also exhibit the same form. Keeping this in mind, we choose a new variable , whereis the wave speed, such that and , substituting them to (1) yields

where the prime denotes the derivative with respect to.

Integrating the second equation of (2), we have

where is the integration constant treated as an arbitrary parameters. In order to have a regular everywhere, we have to impose . It may be noted that satisfies the following boundary conditions:

as . Condition (4) is different from those considered by Guha-Roy [20], in Guha-Roy’s, was found to be vanished in the infinity.

Thus, (3) reduces to

This shows that is directly related to . By (6) and the first equation of (2), we get, after rearrangement,

where , , , and is an arbitrary integration constant.

After the translation , we rewrite (7) to

where and .

It follows and that (8) is equivalent to the two dimensional system

which has the first integral

System (9) is a four-parameter planar dynamical system depending on the parameter group . Because of the phase orbits defined by the vector fields of system (9) that determine all traveling wave solutions of (8), we will investigate the phase portraits of (9) in the phase plane as the parameters ,,, are changed.

We point out that here we are considering a physical model where only bounded traveling waves solutions are meaningful, so that we only pay attention to the bounded solutions of system (9).

To investigate the equilibrium points of system (9), we need to find all real zeros of the function . Suppose that and , clearly, has three real zeros at most, denoted by ,, and . Therefore, system (9) has three equilibrium points at , , at most.

Let

where , , is the coefficient matrix of the linearized system of (9) at equilibrium point , . At this equilibrium point, we obtain the determinant of matrix which is

By the theory of planar dynamical systems [24–26] for an equilibrium point of a planar dynamical (Hamiltonian) system, if , then the equilibrium pointis a saddle point; if , then the equilibrium pointis a center point; and if and the Poincare index of the equilibrium point is zero, then the equilibrium pointis a cusp point. So, we have

(1)for and , there exists three equilibrium points of system (9) at , , with . The points andare center points, is a saddle point. There is two homoclinic orbits to the saddle point , in which there exists a family of periodic orbits surrounding the center and . The phase portrait is shown in Figures 1(b), 1(c), and 1(e).(2)For and , there exists two equilibrium points of system (9) at. If , the point is a cusp point, is a center point; if , the point is a center point, is a cusp point. There is a homoclinic orbit to the cusp point , in which there exists a family of periodic orbits surrounding the center (or ). The phase portrait is shown in Figures 1(a) and 1(d).(3)For and , there exists three equilibrium points of system (9) at with . The points and are saddle points, is a center point. If , There is a homoclinic orbit to the saddle points and , respectively, if . There are two heteroclinic orbits connecting the saddle point and , in which there exists a family of periodic orbits surrounding the center . The phase portrait is shown in Figures 2(b), 2(c), and 2(e).(4)Forand, there exists two equilibrium points of system (9). If, the point is a cusp point,is a saddle point; if, the pointis a saddle point,is a cusp point. There does not exist bounded orbits. The phase portraits are shown in Figures 2(a) and 2(d).

(a)   

(b)

(c)

(d)

(e)

Figure 1: The phase portraits of system (9) for.

(a)  

(b)

(c)

(d)

(e)

Figure 2: The phase portraits of system (9) for.

Because orbits cannot be changed by the transformation, it follows that (6), (7), (8), and the above discussion the following.

Theorem 1. Suppose that , wave speed , and integration constantsatisfy ; then

(1)when, (1) has two bell-shaped solitary wave solutions and uncountable infinite many periodic traveling wave solutions in the case of, , and, respectively, (see Figures 1(b), 1(c), and 1(e)).(2)When, (1) has one bell-shaped solitary wave solution and uncountably infinite many periodic traveling wave solutions in the case ofand, respectively, (see Figures 1(a), and 1(d)).

Theorem 2. Suppose that , wave speed , and integration constant satisfy ; then

(1)when, (1) has one bell-shaped solitary wave solution and uncountably infinite many periodic traveling wave solutions in the case of and , respectively, and two kink-shaped solitary wave solutions and uncountably infinite many periodic traveling wave solutions in the case of (see Figures 2(b), 2(c), and 2(e)).(2)When, (1) does not exist bounded traveling wave solutions (see Figures 2(a), and 2(d)).

3. Exact Explicit Representations of Bell-Shaped and Kink-Shaped Solitary Wave Solutions

According to the discussion in Section 2, we assume that (7) has solution with the following form:

where ,,, and are undetermined real parameters and is arbitrary constant.

Substituting (13) and , into (7), by using the linear independence of , , we obtain that the following algebraic equations with ,,, and :

Suppose thatsatisfy

By solving the above equations; we have the following.

Case 1. For , we have

Case 2. For, we have

By (6), (13), and the above conclusions, we can write the exact solutions of (1).

Theorem 3. For , , is an arbitrary constant; suppose that is real and satisfies ; then

(1) when , and , (1) has the following bell-shaped solitary wave solutions:

where .

(2) When and , (1) has the following bell-shaped solitary wave solutions

Remark 4.   , , denotes the solitary wave solutions taking “+” in expression of (16) and (17), , , is similar.

Substituting into solutions of expressions (18) yields

where and are denoted by (7). Solution (20) is the solitary wave solution (17) and (7) of Guha-Roy [20].

Substituting and into solutions of expressions (19) yields

where and are denoted by (7). Solution (21) is the solitary wave solution (20) and (7) of Guha-Roy [20].

Solutions (18) and (19) cannot be obtained by the method used in Guha-Roy [20].

For , , and , the roots ,, and of satisfy , , where ,, and are the roots of . After computation, we know that , , and , so, solution in expression (18) is less than and in expression (18) is over . Then, two homoclinic orbits at a saddle point can be denoted by solutions in (18), in here, denotes the left hand homoclinic orbit and denotes the right hand homoclinic orbit in Figure 1(b). Similarly, we can explain that solutions denote the two homoclinic orbits in Figure 1(e).

For , , and , the roots ,, and of satisfy , , where ,, and are the roots of . and are saddle points and is a center point. After computation, we know that , and , by yields , . It follows and , that , so is unbounded. Therefore, the homoclinic orbit at a saddle point in Figure 2(b) is denoted by in expression (18), is unbounded. Similarly, we can explain that solutions denote the homoclinic orbits in Figure 2(e).

For , and , point and are corresponding to the centers and of system (9), respectively, points is corresponding to a saddle point of system (9), where , ; then denotes the two homoclinic orbits in Figure 1(c).

If we suppose that (7) has solutions with the following form:

where ,, and are undetermined real parameters and is arbitrary constant.

Substituting (22) into (7) and using the linear independence of , yields

By solving the above equations, we obtain

It follows (22), (24) and (6) that

Theorem 5. Suppose thatis an arbitrary constant, and, (1) has the following kink-shaped solitary wave solutions:

Remark 6. In [20], expression (13) can be rewritten as follows:

where. Comparing expression (26) with in expression (25), we know that the solution (13) solved by Guha-Roy [20] is in accordance with solutionsof expression (25) in the case of. But the general solutions (25) cannot be solved by Guha-Roy [20].

For , , and , points and are two saddle points of system (9), and is a center point, where , . According to yields , ; thus,and of expression (25) are corresponding to the two heteroclinic orbits in Figure 2(c).

If we suppose that (7) has solutions with the following form:

where ,,, andare undetermined real parameters andis arbitrary constant.

By using (27) and its derivations, it follows (7) that

Further, we obtain the following results.

Theorem 7. Suppose that , and is an arbitrary constant, and satisfies integration constant ; then (1) has the following bell-shaped solitary wave solutions:

Remark 8. By the hypothesisand , we know that , and that , . If we takein solution (29), solutionof expression (29) is corresponding to the homoclinic orbit in Figure 1(d). If we take in solution (29), solutionof expression (29) is corresponding to the homoclinic orbit in Figure 1(a).

4. Discussion and Conclusion

In this paper, we obtain all the three bell-shaped and one kink-shaped solitary wave solutions of (1) by using three different undetermined coefficient methods. The conclusions have not been deduced from the method reported by Guha-Roy. The method is simple and can be applied to solve many couple nonlinear equations such as Ito equation, Ito-type equation, and coupled KdV equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project is supported by Fund of Guizhou Science and Technology Department (2138) and technical innovation talents support plan of Guizhou Education Department (KY092).

References

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991.

R. Hirota, “Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.

J.-H. He, “Variational iteration method—some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007.

M. L. Wang, Y. B. Zhou, and Z. B. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, pp. 67–75, 1996.

G. L. Lamb,, “Bäcklund transformations for certain nonlinear evolution equations,” Journal of Mathematical Physics, vol. 15, no. 12, pp. 2157–2165, 1974.

A. M. Wazawaz, “New traveling wave solutions of differential physical structures to generalized BBM equation,” Physics Letters A, vol. 355, no. 4-5, pp. 358–362, 2006.

E. V. Krishnan, “On the Itô-type coupled nonlinear wave equation,” Journal of the Physical Society of Japan, vol. 55, no. 11, pp. 3753–3755, 1986.

S. Zhang, “A generalized new auxiliary equation method and its application to the -dimensional breaking soliton equations,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 510–516, 2007.

E. Yomba, “A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations,” Physics Letters A, vol. 372, no. 7, pp. 1048–1060, 2008.

F. Kangalgil and F. Ayaz, “New exact travelling wave solutions for the Ostrovsky equation,” Physics Letters A, vol. 372, no. 11, pp. 1831–1835, 2008.

M. Wang, X. Li, and J. Zhang, “The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.

X. Liu, W. Zhang, and Z. Li, “Application of improved -expansion method to traveling wave solutions of two nonlinear evolution equations,” Advances in Applied Mathematics and Mechanics, vol. 4, no. 1, pp. 122–130, 2012.

R. Hirota and J. Satsuma, “Soliton solutions of a coupled Korteweg-de Vries equation,” Physics Letters A, vol. 85, no. 8-9, pp. 407–408, 1981.

B. A. Kupershmidt, “A coupled Korteweg-de Vries equation with dispersion,” Journal of Physics A, vol. 18, no. 10, pp. L571–L573, 1985.

M. Ito, “Symmetries and conservation laws of a coupled nonlinear wave equation,” Physics Letters A, vol. 91, no. 7, pp. 335–338, 1982.

S. Kawamoto, “Cusp soliton solutions of the Itô-type coupled nonlinear wave equation,” Journal of the Physical Society of Japan, vol. 53, no. 4, pp. 1203–1205, 1984.

D. C. Lu and G. J. Yang, “Compacton solutions and peakon solutions for a coupled nonlinear wave equation,” International Journal of Nonlinear Science, vol. 4, no. 1, pp. 31–36, 2007.

C. Guha-Roy, B. Bagchi, and D. K. Sinha, “Traveling-wave solutions and the coupled Korteweg-de Vries equation,” Journal of Mathematical Physics, vol. 27, no. 10, pp. 2558–2560, 1986.

A. Cavaglia, A. Fring, and B. Bagchi, “-symmetry breaking in complex nonlinear wave equations and their deformations,” Journal of Physics A, vol. 44, no. 32, Article ID 325201, 2011.

C. Guha-Roy, “Solitary wave solutions of a system of coupled nonlinear equations,” Journal of Mathematical Physics, vol. 28, no. 9, pp. 2087–2088, 1987.

M. Wadati, “Wave propagation in nonlinear lattice I, II,” Journal of the Physical Society of Japan, vol. 38, pp. 673–686, 1975.

C. Guha-Roy, “Exact solutions to a coupled nonlinear equation,” International Journal of Theoretical Physics, vol. 27, no. 4, pp. 447–450, 1988.

B. Q. Lu, Z. L. Pan, B. Z. Qu, and X. F. Jiang, “Solitary wave solutions for some systems of coupled nonlinear equations,” Physics Letters A, vol. 180, no. 1-2, pp. 61–64, 1993.

S. N. Chow and J. K. Hale, Method of Bifurcation Theory, Springer, New York, NY, USA, 1981.

J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, NY, USA, 1983.

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, NY, USA, 1991.