Mathematical Problems in Engineering

Volume 2014, Article ID 156483, 5 pages

http://dx.doi.org/10.1155/2014/156483

## Joint Application of Bilinear Operator and F-Expansion Method for ()-Dimensional Kadomtsev-Petviashvili Equation

Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

Received 5 August 2014; Accepted 2 November 2014; Published 23 November 2014

Academic Editor: Marcelo M. Cavalcanti

Copyright © 2014 Shaolin Li et al. 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

The bilinear operator and F-expansion method are applied jointly to study ()-dimensional Kadomtsev-Petviashvili (KP) equation. An exact cusped solitary wave solution is obtained by using the extended single-soliton test function and its mechanical feature which blows up periodically in finite time for cusped solitary wave is investigated. By constructing the extended double-soliton test function, a new type of exact traveling wave solution describing the assimilation of solitary wave and periodic traveling wave is also presented. Our results validate the effectiveness for joint application of the bilinear operator and F-expansion method.

#### 1. Introduction

In the past few decades, much effort has been devoted to the investigation of dynamical behaviours of nonlinear evolution equation. Traveling wave, one of the spatial dynamics analyses, always plays a significant role and attracts more and more of the experts’ and scholars’ attention. There has been much literature on traveling wave of nonlinear evolution equation due to the abundant type of nonlinear traveling wave and some well-known concepts (e.g., solitary wave [1–3], periodic wave [4, 5], kink wave [6], cusped wave [7], etc.) have been used and generalized extensively. To understand the inherent essence and evolution mechanism of these nonlinear traveling waves, seeking the exact traveling wave solutions has been recognized. In recent years, much efforts have been spent on this task and many significant methods have been established such as variational iteration method [8], homotopy perturbation method [9, 10], Fan subequation method [11, 12], exp-function method [13], Hirota’s bilinear method [14, 15], -expansion method [16, 17], and F-expansion method [18–20]. In most of the existing literature, authors always study the improvement of the adopted method to obtain more forms of solutions. However, to the best of our knowledge, how to realize the joint applications of different methods is still challenging and open work. In this paper, we choose the classical nonlinear evolution equation, ()-dimensional Kadomtsev-Petviashvili (KP) equation, as an example to validate the effectiveness of the proposed method.

The ()-dimensional KP equation [14] is written as where and measure the positive and negative transverse dispersion effects. Equation (1) with and are called KP-I equation and KP-II equation, respectively. In recent years, kinds of research fields and solution types of KP equation have been studied extensively in various aspects [21–23]; exact multiple solitary wave solution, periodic solitary wave solution, quasi-periodic solutions, and so forth have been obtained. In the past works, the resonance interaction phenomenon between periodic solitary wave and line soliton was investigated and spatial-temporal bifurcation and deflexion of solitary wave were exhibited [14, 15].

The rest of the paper is organized as follows. In Section 2, we combine the bilinear operator with F-expansion method to solve KP equation. By single-soliton test approach, a new type of solitary wave solution which possesses cusped structure is obtained. In Section 3, an exact expression describing the interaction of solitary wave and periodic traveling wave is obtained by the extending of double-soliton test approach. Conclusions are drawn in Section 4.

#### 2. Cusped Solitary Wave Solution

Introduce an independent transformation where is an unknown real function. Substituting (2) into (1), we can obtain the bilinear form of KP equation: where Hirota’s bilinear operator “” is defined by

Consider the traveling wave transformation where , and are nonzero constants and is a phase constant. Equation (3) is converted to an ODE:

Generally, letting , we can obtain an exact single solitary wave solution of bilinear equation (3). In this case, we consider the extended single-soliton test function where satisfies the following auxiliary equation:

The coefficients are undetermined constants and and are undetermined positive integers. To determine the values of and , balancing the lowest order nonlinear term with the highest nonlinear terms in (6), we have a relation of and :

From (9), we conclude that and is an arbitrary positive integer. As a test, is taken into account; (7) and (8) are reduced into where and .

Substituting (10) into (6), setting the coefficients of all powers of to zero, we get a nonlinear algebraic system of coefficients , and and , and . Solving it, we obtain where , and are arbitrary nonzero reals and . Under the condition of to solve (10), we get where .

Substituting (11) and (12) into (10), by (2), we obtain an exact traveling wave solution of KP equation as follows:

A solitary wave which possesses a cusped structure is shown by (13), whose amplitude oscillates with the evolution of time. To guarantee the regularity of solitary wave, we should avoid the denominator of (13) equalling zero. So, an inequality is taken into account:

For simplicity, if , we can conclude that when ; when . Choosing a set of parameters we exhibit a waveform of regular cusped solitary wave expressed by (13). From Figure 1, it is observed that the amplitude of cusped solitary wave periodically oscillates along the -axis.

However, if , it is inevitable that the denominator of (13) equals zero for some values of , and . In other words, the equation is satisfied; the irregularity of cusped solitary wave appears. From Figures 2, 3, and 4, these singular phenomena exhibit that irregular solitary wave blows up in finite time, where we only change to in (15).

#### 3. Interaction of Solitary Wave and Periodic Wave

Let where and . Theoretically, we can obtain the double solitary wave solution of KP equation (1).

In this case, we introduce an extended double-soliton test function: where and . The unknown real function satisfies the following auxiliary equation:

The parameters , and are nonzero constants to be determined, and , and are phase constants. is a real number that stands for the resonant factor of traveling wave.

Substituting (18) and (19) into (3) and collecting the coefficients of , , and , one yields a nonlinear algebraic system of parameters , and .

In particular, , , and are taken to solve the nonlinear algebraic system whose and are undetermined constants; we obtain

For (19), when , , and , it allows the fundamental solution as follows:

According to the expressions of , and in (20) and the significance of coefficient for KP equation, it is necessary that . Substituting (20) and (21) into (18), by (2), we obtain an exact traveling wave solution for KP-I equation: where and , , and are arbitrary nonzero real constants, and , and satisfy

The solution expressed by (22) is a new type of traveling wave solution for KP-I equation. Choosing a set of parameters the assimilation of solitary wave and periodic traveling wave is exhibited by Figures 5 and 6.

#### 4. Conclusions

In this paper, we consider the joint application of bilinear operator and F-expansion method. Choosing the KP equation as an example, we obtain a new type of solitary wave solution which possesses cusped structure by the single-soliton test approach. The regular and irregular parametric relationships of cusped solitary wave solution are discussed; an interesting phenomenon is found where irregular cusped solitary wave periodically blows up in finite time. Furthermore, an extended double-soliton test method is applied to obtain a new type of exact traveling wave solution of KP-I equation. By numerical simulation of waveform, a nonlinear phenomenon describing the dynamical behavior of assimilation for solitary wave and periodic traveling wave is found. To our knowledge, it has not yet been found until now. The above results obtained in this paper validate the effectiveness of joint application of bilinear operator and F-expansion method.

#### Conflict of Interests

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

#### Acknowledgments

The work described in this paper was fully supported by NSFC (11161020) and Yunnan Educational Science Foundation (08Y0336).

#### References

- M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,”
*Physics Letters A*, vol. 199, no. 3-4, pp. 169–172, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Ebaid, “Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method,”
*Physics Letters A*, vol. 365, no. 3, pp. 213–219, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Song, X. Hou, and J. Cao, “Solitary wave solutions and kink wave solutions for a generalized KDV-mKDV equation,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5942–5948, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Zhou, M. Wang, and T. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,”
*Physics Letters A*, vol. 323, no. 1-2, pp. 77–88, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. A. Abdou, “Exact periodic wave solutions to some nonlinear evolution equations,”
*International Journal of Nonlinear Science*, vol. 6, no. 2, pp. 145–153, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Dai, J. Liu, X. Zeng, and Z. Liu, “Periodic kink-wave and kinky periodic-wave solutions for the Jimbo-Miwa equation,”
*Physics Letters. A*, vol. 372, no. 38, pp. 5984–5986, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Li and Y. Zhang, “Exact loop solutions, cusp solutions, solitary wave solutions and periodic wave solutions for the special CH-DP equation,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 4, pp. 2502–2507, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. A. Abdou and A. A. Soliman, “Variational iteration method for solving BURger's and coupled BURger's equations,”
*Journal of Computational and Applied Mathematics*, vol. 181, no. 2, pp. 245–251, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. M. Zayed, T. A. Nofal, and K. A. Gepreel, “On using the homotopy perturbation method for finding the travelling wave solutions of generalized nonlinear Hirota-Satsuma coupled KdV equations,”
*International Journal of Nonlinear Science*, vol. 7, no. 2, pp. 159–166, 2009. View at Google Scholar · View at MathSciNet - M. Madani, M. Fathizadeh, Y. Khan, and A. Yildirim, “On the coupling of the homotopy perturbation method and Laplace transformation,”
*Mathematical and Computer Modelling*, vol. 53, no. 9-10, pp. 1937–1945, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D. Feng and K. Li, “Exact traveling wave solutions for a generalized Hirota-Satsuma coupled KdV equation by Fan sub-equation method,”
*Physics Letters A*, vol. 375, no. 23, pp. 2201–2210, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Xie, S. Tang, and D. Feng, “New exact solutions to the generalized KdV equation with generalized evolution,”
*Pramana*, vol. 78, no. 4, pp. 499–511, 2012. View at Publisher · View at Google Scholar · View at Scopus - A. Bekir and E. Aksoy, “Exact solutions of nonlinear evolution equations with variable coefficients using exp-function method,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 430–436, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z. Dai, S. Li, Q. Dai, and J. Huang, “Singular periodic soliton solutions and resonance for the Kadomtsev-Petviashvili equation,”
*Chaos, Solitons & Fractals*, vol. 34, no. 4, pp. 1148–1153, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z.-D. Dai, S.-L. Li, D.-L. Li, and A.-J. Zhu, “Periodic bifurcation and soliton deflexion for kadomtsev-petviashvili equation,”
*Chinese Physics Letters*, vol. 24, no. 6, pp. 1429–1432, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. Wang, X. Li, and J. Zhang, “The ${G}^{\prime}/G$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,”
*Physics Letters A*, vol. 372, no. 4, pp. 417–423, 2008. View at Google Scholar - R. Abazari, “Application of (
*G*′/*G*)-expansion method to travelling wave solutions of three nonlinear evolution equation,”*Computers and Fluids*, vol. 39, no. 10, pp. 1957–1963, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. A. Abdou, “The extended F-expansion method and its application for a class of nonlinear evolution equations,”
*Chaos, Solitons & Fractals*, vol. 31, no. 1, pp. 95–104, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Ebaid and E. H. Aly, “Exact solutions for the transformed reduced Ostrovsky equation via the $F$-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions,”
*Wave Motion*, vol. 49, no. 2, pp. 296–308, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Kumar and F. Chand, “Applications of extended F-expansion and projective Ricatti equation methods to (2+1)-dimensional soliton equations,”
*AIP Advances*, vol. 3, no. 3, Article ID 032128, 2013. View at Publisher · View at Google Scholar · View at Scopus - D.-Q. Xian and H.-L. Chen, “Symmetry reductions and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 70–79, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. V. Manakov and P. M. Santini, “Wave breaking in solutions of the dispersionless Kadomtsev-Petviashvili equation at at finite time,”
*Theoretical and Mathematical Physics*, vol. 172, no. 2, pp. 1118–1126, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Klein and J.-C. Saut, “Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations,”
*Journal of Nonlinear Science*, vol. 22, no. 5, pp. 763–811, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus