## Study of Integrability and Exact Solutions for Nonlinear Evolution Equations

View this Special IssueResearch Article | Open Access

# Traveling Wave Solutions and Infinite-Dimensional Linear Spaces of Multiwave Solutions to Jimbo-Miwa Equation

**Academic Editor:**Weiguo Rui

#### Abstract

The traveling wave solutions and multiwave solutions to (3 + 1)-dimensional Jimbo-Miwa equation are investigated in this paper. As a result, besides the
exact bounded solitary wave solutions, we obtain the existence of two families
of bounded periodic traveling wave solutions and their implicit formulas by
analysis of phase portrait of the corresponding traveling wave system. We
derive the exact 2-wave solutions and two families of arbitrary finite *N*-wave
solutions by studying the linear space of its Hirota bilinear equation, which
confirms that the (3 + 1)-dimensional Jimbo-Miwa equation admits multiwave solutions of any order and is completely integrable.

#### 1. Introduction

Various nonlinear partial differential equations (NLPDEs) have been proposed to model different kinds of phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid-state physics, optical fibers, acoustics, mechanics, biology, and mathematical finance. Obviously, it is of significant importance to study the solutions of such NLPDEs from both theoretical and practical points of view. However, the solution spaces of nonlinear equations are infinite-dimensional and contain diverse solution structures, so it is usually a difficult job to determine the solutions to nonlinear NLPDEs.

A great idea to generate exact solutions of NLPDEs is to reduce the NLPDEs into some algebraic equations by assuming the solutions to have some special forms or satisfy some solvable simpler equations. This can be seen in, for example, the exp-function method [1], the tanh function method [2], the homogeneous balance method [3, 4], the auxiliary function method [5, 6], the sech-function method [7], the sine-cosine method [8, 9], the tanh-coth method [10], the Jacobi elliptic function method [11], the -expansion method, and the extended -expansion method [12]. Normally, it is not an easy task to solve these nonlinear algebraic equations reduced from NLPDEs because they involve many parameters. However, the rapid development of symbolic computation makes it relatively easy to solve these algebraic equations [13]. At the same time one should be very careful when applying these methods as different methods might give the same solutions. For example, solutions obtained by using the sech-function method, the tanh-coth method, and the exp-function method are actually the same [14, 15], because , for any function .

Recently, the planar dynamical system theorem has been employed to study the traveling wave solutions of NLPDEs [16, 17]. The best advantage of this approach is that the boundedness, periodicity, the shapes of the solutions, and even the singular traveling wave solutions of NLPDEs can be recognized easily from the corresponding orbits of their phase portraits under various different parametric conditions. Also the exact solutions can be derived at the same time. It is worth pointing out that Hirota’s bilinear method [18] is also an amazing method to find exact solutions to some NLPDEs. By some independent variable transformation, various nonlinear equations of mathematical physics are transformed into Hirota’s bilinear equations [18, 19], which possess some specific properties and thus might be applied to study the solution sets of nonlinear differential equations. Recently, even some programs have been designed and some algorithms have been proposed on searching for integrable bilinear equations [20–26]. Based on Hirota’s bilinear form, soliton solutions were obtained by the Hirota perturbation technique [18], the multiple exp-function algorithm [27], and other methods [28–31]. Even some generalized bilinear form has been proposed recently by Ma [32].

The -dimensional Jimbo-Miwa equation was first proposed and studied by Jimbo and Miwa [33]. Recently, this equation has attracted a great deal of attention which mainly focuses on its solutions, integrability properties and symmetries. Ma and Lee [34] proposed a direct approach to solve (1) by using rational function transformations. Li and Dai [35] applied the generalized Riccati equation method to look for its exact solutions. However, in [36], Kudryashov and Sinelshchikov pointed out some mistakes and proved that some solutions in [35] could be written in a uniform form and thus they are not new at all.

The dimensional Jimbo-Miwa equation can be transformed into the Hirota bilinear equation through the dependent variable transformation .

In this paper, we firstly study the bounded traveling wave solutions of the Jimbo-Miwa equation (1) by investigating the bifurcation and phase portraits of a planar cubic polynomial ordinary differential equation by using the planar dynamical system theory [16, 37]. We then explore the 1-wave and 2-wave solutions and the infinite-dimensional linear spaces of multiwave solutions to Jimbo-Miwa equation by employing Hirota’s bilinear method, thus confirming that Jimbo-Miwa equation possesses multiwave solutions of arbitrary order.

#### 2. Traveling Wave Solutions to the Dimensional Jimbo-Miwa Equation

To investigate the traveling wave solutions to the dimensional Jimbo-Miwa equation, we make the traveling wave transformation , , under which (1) is reduced to the nonlinear ordinary differential equation Integrating (1) once with respect to gives where is an arbitrary constant. Let ; then (1) becomes which is a second-order nonlinear ordinary differential equation. We will study the solutions of (5) by planar dynamical system method and thus derive the traveling wave solutions to the dimensional Jimbo-Miwa equation.

##### 2.1. Bounded Solutions of (5)

First, we rewrite (5) in a simpler and more general form, namely, where , , and . We now study the bifurcation and exact solutions of (6). Let ; then (6) is equivalent to the dynamical system which has the Hamiltonian Clearly, the phase orbits defined by the vector fields of system (7) determine all solutions of (6). The bounded solutions of (6) correspond to the bounded phase orbits of system (7), which we now investigate. Along the orbit corresponding to , Consequently, the general formula of the solutions of (6) can be expressed as

However, it is not easy to know the properties and the shapes of (10) which actually is determined by the parameters , , , and . Clearly, the abscissas of equilibrium points of system (7) are the zeros of . Obviously, the system has no bounded orbits when . We suppose that in order to study the bounded orbits of system (7). Denoting , then and are two equilibrium points of system (7). By the theory of planar dynamical system, we know that is a saddle point and is a center. Denote , and, by careful computation, we get Obviously, . corresponds to homoclinic orbits, and corresponds to the center and , where corresponds to a family of closed orbits surrounding the center , which are surrounded by a homoclinic orbit. That is to say, (10) defines bounded solutions if and only if . To be exact, (10) defines a family of periodic solutions when .

When , (10) defines a bounded solution which approaches as goes to infinity. Actually, where , so (10) can be reduced to from which we can get the exact solution as

By further simplification, (14) becomes which is an exact bounded solution of (5).

Thus, we have the following lemma.

Lemma 1. *The general second-order ODE (6) has bounded solutions if and only if . The bounded solutions can be expressed as (10) in an implicit form. In fact, provided , (10) defines a family of bounded periodic solutions and defines a bounded solution which approaches as goes to infinity and can be expressed explicitly as (15), where and is defined by (11).*

##### 2.2. Bounded Traveling Wave Solutions to the Dimensional Jimbo-Miwa Equation

According to the analysis and results in Section 2.1, we know that (5) has only two kinds of bounded solutions, among which one is a family of periodic solutions and another is a family of solutions approaching a fixed number as goes to infinity. Note that what we aim to study is the bounded traveling wave solutions to the -dimensional Jimbo-Miwa equation which are determined by and satisfies (5). So we have to investigate how we can get the bounded solutions to (3) from the bounded solution of (5).

Clearly, , and can be expressed implicitly as (10). By the geometry meaning of the integral and the properties of the solutions of (5), we get the traveling wave solutions to the dimensional Jimbo-Miwa equation. To get the bounded solution, we choose the integral constant to be 0; that is, in (15), and so that is, which is a family of exact bounded kink traveling wave solutions to the dimensional Jimbo-Miwa equation, where is an arbitrary constant.

However, we may not get bounded solutions from the family of periodic solutions of (5). It is easy to see that if is a periodic solution of (5), then is bounded if and only if , where is the period of the function . Recall that the period of the function which is defined by (10) with continuously depends on the parameters , , , and . So continuously depends on the parameters , , , and too. Let ; then is a continuous function of , , , and . We now prove the existence of the root of to get the existence of the bounded periodic traveling wave solutions to the dimensional Jimbo-Miwa equation.

By the theory of planar dynamical system, when , where in the case of and in the case of . The periodic solution satisfies when and when . Clearly, when and . So and thus when and . However, and thus when , and . There must exist at least one zero of in the region of the parameter space since is a continuous function of , , , and . The same happens in the region of the parameter space. So, we know that there exist at least two families of bounded periodic traveling wave solutions to the dimensional Jimbo-Miwa equation. Thus we have the following theorem.

Theorem 2. *The dimensional Jimbo-Miwa equation has two types of bounded traveling wave solutions as given below.*(1)*The dimensional Jimbo-Miwa equation has a family of exact bounded kink traveling wave solutions
* *where and are two arbitrary constants.*(2)*The -dimensional Jimbo-Miwa equation has at least two families of bounded periodic traveling wave solutions which are determined implicitly by (10) and , where and is an arbitrary constant.*

#### 3. Wave Solutions Linear Subspace of the Dimensional Jimbo-Miwa Equation

In this section we study the wave solutions to the dimensional Jimbo-Miwa equation by the linear superposition principle to Hirota bilinear equations which was proposed firstly by Ma and Fan [30].

Let wave variables , , where , . According to the linear superposition principle to Hirota bilinear equations [30, 31], we can get the following sufficient and necessary conditions for being a subspace of the Hirota bilinear equation (2).

Theorem 3. *Let , , and . Then, for any constants , solves the Hirota bilinear equation (2) if and only if
**
for any .*

It follows from Theorem 3 that is an -wave solution to the Hirota bilinear equation (2) if (20) holds for any . However, the corresponding is not necessarily an -wave solution to the dimensional Jimbo-Miwa equation (1) even if is linear independent. In fact, where , and . Obviously, the function above is an -wave solution to the dimensional Jimbo-Miwa equation (1) if , is independent; that is, , is linear independent, solves (20), and satisfies the dispersion relations. Here, satisfies the dispersion relations meaning that holds.

Recall that the goal of this paper is to investigate the multiwave solutions to the dimensional Jimbo-Miwa equation. Combining the conclusion of Theorem 3 with the analysis above, we can now state the following theorem on the -wave solution to the dimensional Jimbo-Miwa equation (1).

Theorem 4. *Let , , . For any constants and , suppose that
**
Then is an -wave solution to the dimensional Jimbo-Miwa equation (1), if is a linear independent solution set of equation (20) satisfying the dispersion relations.*

To get other -wave solutions to the dimensional Jimbo-Miwa equation, we study the linear independent solution sets to (20) which are required to satisfy the dispersion relations. Actually, (20) is a system possessing coupled equations (plus the dispersion conditions). It is usually not so easy to get the solutions of (20). Fortunately, the number of the equations is 1 when and it is 3 when , which might make it easy to get the solutions in these two cases. Clearly, 1-wave solution is the traveling wave solution. Let us check what kinds of traveling wave solutions we can obtain by the Hirota bilinear method first.

For the case in (22), we only need to find the independent solutions of the dispersion relation

That is to say, we get the traveling wave solutions to the dimensional Jimbo-Miwa equation from (20). Suppose that , are any arbitrary constants, and ; then satisfies (23), and thus where and are any arbitrary constants, is a -wave solution, that is, traveling wave solution to the dimensional Jimbo-Miwa equation.

By further computation, (24) could be rewritten as For , and, for , where is an arbitrary constant. Consequently, (24) can be rewritten as By parametric transformation, we can find that the traveling wave solution (28) is exactly the same as solution (19) in Section 2. However, (29) is a family of unbounded solutions.

##### 3.1. 2-Wave Solutions to the Dimensional Jimbo-Miwa Equation

To obtain the 2-wave solutions to the dimensional Jimbo-Miwa equation, we now study the solutions to the following coupled algebraic equations:

From (30), we get Clearly, from (31)–(33), we have from which we get Now, substituting (35) into (32), we obtain Consequently, we get the 2-wave solutions to the dimensional Jimbo-Miwa equation and we can now state the following theorem.

Theorem 5. *Let , , . For any constants and ,
**
is a -wave solution to the -dimensional Jimbo-Miwa equation (1), where , , , , and are arbitrary constants and the constants , , and are determined by (31), (35), and (36), respectively.*

##### 3.2. -Wave Solutions to the Dimensional Jimbo-Miwa Equation

To get the -wave solutions to the dimensional Jimbo-Miwa equation, we need to investigate the independent solution set of (20), which satisfies the dispersion relations. Generally, it is very difficult to get the solution if is greater than 3 in which case the number of (20) is . However, it might be possible to solve these equations by assuming some special relations between these parameters [30, 31].

Following the idea in [30, 31], suppose that , , where and , are parameters to be determined later. To get the possible solutions to (20), we choose to be or .

For the case when , substituting , , into (20) gives Obviously, (38) holds for arbitrary values of and if and only if , and satisfy the two equations Solving (39), we get , , where and are free parameters. It is easy to see that , , satisfy the dispersion condition. Consequently, we know that the dimensional Jimbo-Miwa equation (1) possesses -wave solution for any arbitrary positive integer . Thus, we have the following theorem.

Theorem 6. *For any arbitrary positive integer and for any different , ,
**
is a family of -wave solutions to the dimensional Jimbo-Miwa equation (1), where , , and and are arbitrary constants.*

For the case when , substituting ,, , into (20) gives Clearly, (41) holds for arbitrary values of and if and only if , and satisfy Solving (42), we obtain , where and are free parameters. Unfortunately, unlike the case when above, it is easy to see that , , do not satisfy the dispersion condition any more. However, if we let , then, for any , satisfies the dispersion relations because is a solution set of (20).

Consequently, besides the family of -wave solution (40), the dimensional Jimbo-Miwa equation (1) has another group of -wave solution for any arbitrary positive integer and so we have the following theorem.

Theorem 7. *For any arbitrary positive integer and for any different , ,
**
is a family of -wave solutions to the dimensional Jimbo-Miwa equation (1), where , , and and are arbitrary constants.*

#### 4. Concluding Remarks

The dynamical system theory was employed to study the traveling wave solutions of the dimensional Jimbo-Miwa equation (1). The multiwave solutions were investigated by studying the linear space of the corresponding Hirota bilinear equation. The exact formulas of two families of multiwave solutions of any order were obtained as well. Wazwaz [38] employed Hirota’s bilinear method and derived 1-wave and 2-wave solutions to this equation and stated that it is completely integrable and it admits multiple-soliton solutions of any order. In this paper we explicitly obtained two different families of -wave solutions to the dimensional Jimbo-Miwa equation (1) given by (40) and (44) and this confirmed the statement given in [38] that the Jimbo-Miwa equation (1) admits multiple-wave solutions of any order .

#### Conflict of Interests

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

#### Acknowledgments

This work is supported by the Nature Science Foundation of China (no. 11101371). This paper is motivated by nice discussion with Professor Wen-Xiu Ma during the authors’ visit at University of South Florida in March 2013. The authors would like to express their sincere appreciations to Professor Ma and the anonymous reviewers.

#### References

- J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons & Fractals*, vol. 30, no. 3, pp. 700–708, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. J. Parkes and B. R. Duffy, “An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations,”
*Computer Physics Communications*, vol. 98, no. 3, pp. 288–300, 1996. View at: Publisher Site | Google Scholar - M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,”
*Physics Letters A*, vol. 216, no. 1–5, pp. 67–75, 1996. View at: Google Scholar - Z. Feng, “Comment on ‘on the extended applications of homogeneous balance method’,”
*Applied Mathematics and Computation*, vol. 158, no. 2, pp. 593–596, 2004. View at: Publisher Site | Google Scholar | MathSciNet - S. Zhang and T. Xia, “A generalized new auxiliary equation method and its applications to nonlinear partial differential equations,”
*Physics Letters A*, vol. 363, no. 5-6, pp. 356–360, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Zhang, “New exact travelling wave solutions of nonlinear evolution equation using a sub-equation,”
*Chaos, Solitons & Fractals*, vol. 39, no. 2, pp. 873–881, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. X. Ma, “Travelling wave solutions to a seventh order generalized KdV equation,”
*Physics Letters A*, vol. 180, no. 3, pp. 221–224, 1993. View at: Publisher Site | Google Scholar | MathSciNet - C. Yan, “A simple transformation for nonlinear waves,”
*Physics Letters A*, vol. 224, no. 1-2, pp. 77–84, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A.-M. Wazwaz, “A sine-cosine method for handling nonlinear wave equations,”
*Mathematical and Computer Modelling*, vol. 40, no. 5-6, pp. 499–508, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. B. Lan and K. L. Wang, “Exact solutions for two nonlinear equations. I,”
*Journal of Physics A. Mathematical and General*, vol. 23, no. 17, pp. 3923–3928, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. J. Parkes, B. R. Duffy, and P. C. Abbott, “The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations,”
*Physics Letters A*, vol. 295, no. 5-6, pp. 280–286, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. L. Cai, Q. C. Wang, and J. J. Huang, “A modified F-expansion method for solving breaking soliton equation,”
*International Journal of Nonlinear Science*, vol. 2, pp. 122–128, 2006. View at: Google Scholar - W. Hereman and W. Zhuang, “Symbolic computation of solitons with Macsyma,” in
*Computational and Applied Mathematics, II (Dublin, 1991)*, pp. 287–296, North-Holland, Amsterdam, The Netherlands, 1992. View at: Google Scholar | Zentralblatt MATH | MathSciNet - N. A. Kudryashov, “Seven common errors in finding exact solutions of nonlinear differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 9-10, pp. 3507–3529, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Zhang and X. Huo, “On the exp-function method for constructing travelling wave solutions of nonlinear equations,” in
*Nonlinear and Modern Mathematical Physics: Proceedings of the First International Workshop*, vol. 1212, pp. 280–285, American Institute of Physics, Melville, NY, USA, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. B. Li,
*Singular Traveling Wave Equations: Bifurcations and Exact Solutions*, Science Press, Beijing, China, 2013. - L. Zhang, L.-Q. Chen, and X. Huo, “The effects of horizontal singular straight line in a generalized nonlinear Klein-Gordon model equation,”
*Nonlinear Dynamics*, vol. 72, no. 4, pp. 789–801, 2013. View at: Publisher Site | Google Scholar | MathSciNet - R. Hirota,
*The Direct Method in Soliton Theory*, vol. 155 of*Cambridge Tracts in Mathematics*, Cambridge University Press, Cambridge, Mass, USA, 2004. View at: Publisher Site | MathSciNet - J. Hietarinta, “Hirota's bilinear method and soliton solutions,”
*Physics AUC*, vol. 15, part 1, pp. 31–37, 2005. View at: Google Scholar - Y. Ye, L. Wang, Z. Chang, and J. He, “An efficient algorithm of logarithmic transformation to Hirota bilinear form of KdV-type bilinear equation,”
*Applied Mathematics and Computation*, vol. 218, no. 5, pp. 2200–2209, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhou, J. Fu, and Z. Li, “An implementation for the algorithm of Hirota bilinear form of PDE in the Maple system,”
*Applied Mathematics and Computation*, vol. 183, no. 2, pp. 872–877, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhou, J. Fu, and Z. Li, “Maple packages for computing Hirota's bilinear equation and multisoliton solutions of nonlinear evolution equations,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 92–104, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X.-D. Yang and H.-Y. Ruan, “A Maple package on symbolic computation of Hirota bilinear form for nonlinear equations,”
*Communications in Theoretical Physics*, vol. 52, no. 5, pp. 801–807, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Hietarinta, “A search for bilinear equations passing Hirota's three-soliton condition. II. mKdV-type bilinear equations,”
*Journal of Mathematical Physics*, vol. 28, no. 9, pp. 2094–2101, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Hietarinta, “A search for bilinear equations passing Hirota's three-soliton condition. III. Sine-Gordon-type bilinear equations,”
*Journal of Mathematical Physics*, vol. 28, no. 11, pp. 2586–2592, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Hietarinta, “A search for bilinear equations passing Hirota's three-soliton condition. IV. Complex bilinear equations,”
*Journal of Mathematical Physics*, vol. 29, no. 3, pp. 628–635, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W.-X. Ma, T. Huang, and Y. Zhang, “A multiple exp-function method for nonlinear differential equations and its application,”
*Physica Scripta*, vol. 82, no. 6, Article ID 065003, 2010. View at: Publisher Site | Google Scholar - W.-X. Ma and W. Strampp, “Bilinear forms and Bäcklund transformations of the perturbation systems,”
*Physics Letters A*, vol. 341, no. 5-6, pp. 441–449, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W.-X. Ma, R. Zhou, and L. Gao, “Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in $(2+1)$ dimensions,”
*Modern Physics Letters A*, vol. 24, no. 21, pp. 1677–1688, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W.-X. Ma and E. Fan, “Linear superposition principle applying to Hirota bilinear equations,”
*Computers & Mathematics with Applications*, vol. 61, no. 4, pp. 950–959, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W.-X. Ma, Y. Zhang, Y. Tang, and J. Tu, “Hirota bilinear equations with linear subspaces of solutions,”
*Applied Mathematics and Computation*, vol. 218, no. 13, pp. 7174–7183, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. X. Ma, “Generalized bilinear differential equations,”
*Studies in Nonlinear Sciences*, vol. 2, no. 4, pp. 140–144, 2011. View at: Google Scholar - M. Jimbo and T. Miwa, “Solitons and infinite-dimensional Lie algebras,”
*Publications of the Research Institute for Mathematical Sciences*, vol. 19, no. 3, pp. 943–1001, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W.-X. Ma and J.-H. Lee, “A transformed rational function method and exact solutions to the $3+1$ dimensional Jimbo-Miwa equation,”
*Chaos, Solitons & Fractals*, vol. 42, no. 3, pp. 1356–1363, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Li and Z. Dai, “Abundant new exact solutions for the $(3+1)$-dimensional Jimbo-Miwa equation,”
*Journal of Mathematical Analysis and Applications*, vol. 361, no. 2, pp. 587–590, 2010. View at: Publisher Site | Google Scholar | MathSciNet - N. A. Kudryashov and D. I. Sinelshchikov, “A note on “Abundant new exact solutions for the $(3+1)$-dimensional Jimbo-Miwa equation,”
*Journal of Mathematical Analysis and Applications*, vol. 371, no. 1, pp. 393–396, 2010. View at: Publisher Site | Google Scholar | MathSciNet - S. N. Chow and J. K. Hale,
*Methods of Bifurcation Theory*, Springer, New York, NY, USA, 1981. - A.-M. Wazwaz, “Multiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff, Jimbo-Miwa and YTSF equations,”
*Applied Mathematics and Computation*, vol. 203, no. 2, pp. 592–597, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Lijun Zhang and C. M. Khalique. 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.