Classification of Single Traveling Wave Solutions to the Generalized Kadomtsev-Petviashvili Equation without Dissipation Terms in p = 2

Abstract

By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in p=2 is obtained.

Share and Cite:

X. Du and H. Xin, "Classification of Single Traveling Wave Solutions to the Generalized Kadomtsev-Petviashvili Equation without Dissipation Terms in p = 2," Advances in Pure Mathematics, Vol. 3 No. 9A, 2013, pp. 1-8. doi: 10.4236/apm.2013.39A1001.

1. Introduction

In mathematics and physics, the Kadomtsev-Petviashvili (KP) equation is a partial differential equation to describe nonlinear wave motion. It can be used to model water waves of long wavelength with weakly nonlinear restoring forces and frequency dispersion [1]. A number of modified forms of the KP equation have been studied [2-6]. In [1,7], the generalized Kadomtsev-Petviashvili equation without dissipation terms was given by

(1)

where are constants, ,. Some of modified form of the KP equation can be written in the form of Equation (1).

Many reliable methods are used in the literature to examine the completely integrable nonlinear evolution equations. The Hirota bilinear method, the Bäcklund transformation method, the inverse scattering method, the Painlevé analysis, the simplified Hirotas method established by Hereman et al. [8], and others were effectively used in [1-13]. Liu proposed a complete discrimination system for polynomial method [10-13]. That is, by using of elementary integral method and complete discrimination system for polynomial, the single wave solutions can be classified for some nonlinear differential equations which can be directly reduced to integral forms.

In this paper, we consider the following generalized Kadomtsev-Petviashvili equation without dissipation terms in:

(2)

where are constants,. By using Liu’s complete discrimination system for polynomial method, the classification of single traveling wave solutions to Equation (2) is obtained.

2. Classification of Solutions to Equation (2)

Take wave transformation

and

into Equation (1), the following nonlinear ordinary difference equation is given:

(3)

Integrating Equation (3) once with respect to, and setting the integral constant to zero yields:

(4)

Integrating Equation (4) twice yields

(5)

where are arbitrary constants.

Case 2.1., we substitute the transformation

into Equation (5) yields

(6)

where

(7)

Let

and is the discriminant of the polynomial. According to the classification of the roots of, there are three cases to be discussed.

Case 2.1.1., whenfrom Equation (6), we have

(8)

Case 2.1.2., when

from Equation (6), we have

(9)

(10)

(11)

When

from Equation (6), we have

(12)

(13)

(14)

where

,.

Case 2.1.3.. From Equation (6), we have

(15)

where

.

Case 2.2.. Substituting the transformation

into Equation (5) yields

(16)

where

(17)

If

we take

; ifwe take. The complete discrimination system for the third order polynomial is given as follows:

(18)

According to the classification of the roots of, there are four cases to be discussed.

Case 2.2.1.. Then

where are real constants, , and If, when and, or when and, from Equation (16), we have

(19)

when, and, or, we have

(20)

when, we have

(21)

If,when and, or when and, from Equation (16), we have

(22)

when, and, or, and, we have

(23)

when, we have

(24)

Case 2.2.2. . Then

where is a real constant. If, when, and, or, and, we have

(25)

If, when, and, or, and, we have

(26)

Case 2.2.3.. Then

where are different real constants. If, when, or, we have

(27)

(28)

where

.

If, when, and, we have

(29)

(30)

where

.

Case 2.2.4.

where are all real constants, and, and. we have

(31)

where

,

.

Case 2.3.. The Equation (5) becomes

(32)

where, and

,.

The complete discrimination system for the fifth order polynomial is given as follows:

(33)

According to the classification of the roots of, there are seven cases to be discussed.

Case 2.3.1., then

and are real numbers, From Equation (32), we have

(34)

(35)

(36)

(37)

where.

Case 2.3.2.

,

and are real numbers, From Equation (32), we have

(38)

(39)

where

Case 2.3.3.

,

are real numbers, From Equation (32), we have

(40)

(41)

where

Case 2.3.4.

,

.

Respectively, from Equation (32), we have

(42)

where

(43)

where the signs of and must satisfy

Case 2.3.5.,

.

Respectively, from Equation (32), we have

(44)

where we renew to queue the orders of, and, denote.When

or

(other cases can be written similarly, they are omitted), the meaning of every parameter in Equation (44) are given as follows:

(45)

Case 2.3.6.

where we renew to queue the orders of and 0, and denote When

, or

(other cases can be written similarly, they are omitted), we have

(46)

(47)

The signs are the same as the ones in Equation (45), furthermore,

(48)

Case 2.3.7.

Now we renew to queue the orders of and 0, and denote, we have

(49)

(50)

where

(51)

where the positive sign and negative sign for must satisfy

other signs are the same with the former.

From the description above, using elementary integral method and complete discrimination system for polynomial, we have obtained the solutions of equations (6), (16) and (32) that can be expressed by elementary functions and elliptic functions. What’s more, some solutions are explicit, but some solutions are implicit functions. So we can write concretely the exact traveling wave solutions of Equation (5) in some special cases. They are omitted for simplicity.

3. Conclusion

Using the complete discrimination system for polynomial method, we have obtained the classification of single traveling wave solutions to the generalized KadomtsevPetviashvili equation without dissipation terms in. With the same method, some of other evolution equations can be dealt with.

4. Acknowledgements

The project is supported by Scientific Research Fund of Education Department of Heilongjiang Province of China under Grant No. 12521049.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] M. J. Ablowits, “Lectures on the Inverse Scattering Transform,” Studies in Applied Mathematics, Vol. 58, No. 11, 1978, pp. 17-94. [2] G. C. Das and J. Sarma, “Evolution of Solitary Waves in Multicomponent Plasmas,” Chaos, Solitons and Fractals, Vol. 9, No. 6, 1998, pp. 901-911. http://dx.doi.org/10.1016/S0960-0779(97)00170-7 [3] X. Zhao, W. Xu, H. Jia and H. Zhou, “Solitary Wave Solutions for the Modified Kadomtsev-Petvisahvili Equation,” Chaos, Solitons, and Fractals, Vol. 74, 2007, pp. 465-475. http://dx.doi.org/10.1016/j.chaos.2006.03.046 [4] W. X. Ma, A. Abdeljabbar and M. G. Assad, “Wronskian and Grammian Solutions to a (3 + 1)-Dimensional Generalized KP Equation,” Applied Mathematics and Computation, Vol. 217, No. 24, 2011, pp. 10016-10023. http://dx.doi.org/10.1016/j.amc.2011.04.077 [5] A. M. Wazwaz, “Multiple-Soliton Solutions for a (3 + 1)-Dimensional Generalized KP Equation,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 2, 2012, pp. 491-495. http://dx.doi.org/10.1016/j.cnsns.2011.05.025 [6] W. X. Ma and A. Abdeljabbar, “A Bilinear Backlund Transformation of a (3 + 1)-Dimensional Generalized KP Equation,” Applied Mathematics Letters, Vol. 25, No. 10, 2012, pp. 1500-1504. http://dx.doi.org/10.1016/j.aml.2012.01.003 [7] G. B. Whitham, “Linear and Nonlinaer Wave,” John Wiley, New York, 1974. [8] W. Hereman and A. Nuseir, “Symbolic Methods to Construct Exact Solutions of Nonlinear Partial Differential Equations,” Mathematics and Computers in Simulation, Vol. 43, No. 1, 1997, pp. 13-27. http://dx.doi.org/10.1016/S0378-4754(96)00053-5 [9] W. X. Ma and A. Pekcan, “Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations,” Zeitschrift für Naturforschung A, Vol. 66, 2011, pp. 377-382. [10] C. S. Liu, “Applications of complete Discrimination System for Polynomial for Classifications of Traveling Wave Solutions to Nolinear Differential Equations,” Computer Physics Communications, Vol. 181, No. 2, 2010, pp. 317-324. http://dx.doi.org/10.1016/j.cpc.2009.10.006 [11] C. S. Liu, “Classification of All Single Traveling Wave Solutions to Calogero-Focas Equation,” Communications in Theoretical Physics, Vol. 48, 2007, pp. 601-604. http://dx.doi.org/10.1088/0253-6102/48/4/004 [12] C. S. Liu, “Exact Traveling Wave Solutions for (1 + 1)-Dimentional Dispersive Long Wave Equation,” Chinese Physical Society, Vol. 14, No. 9, 2005, pp. 1710-1716. [13] C. S. Liu, “Direct Integral Method, Complete Discrimination System for Polynomial and Applications to Classifications of All Single Traveling Wave Solutions to Nonlinear Differential Equations: A Survey,” arXiv:nlin/ 0609058v1, 2006.

 +1 323-425-8868 customer@scirp.org +86 18163351462(WhatsApp) 1655362766 Paper Publishing WeChat