Cite this Article
Li Xiang-Zheng, Zhang Jin-Liang, Wang Ming-Liang. Solitary Wave Solutions of KP equation, Cylindrical KP Equation and Spherical KP Equation. Communications in Theoretical Physics, 2017, 67(2): 207  
Permissions
Solitary Wave Solutions of KP equation, Cylindrical KP Equation and Spherical KP Equation
Li Xiang-Zheng1, †, Zhang Jin-Liang1, Wang Ming-Liang1, 2
School of Mathematics and Statistcs, Henan University of Science and Technology, Luoyang 471023, China
School of Mathematics and Statistcs, Lanzhou University, Lanzhou 730000, China

 

† Corresponding author. E-mail: 57546388@qq.com

Supported by the National Natural Science Foundation of China under Grant No. 11301153 and the Doctoral Foundation of Henan University of Science and Technology under Grant No. 09001562, and the Science and Technology Innovation Platform of Henan University of Science and Technology under Grant No. 2015XPT001

Abstract
Abstract

Three (2+1)-dimensional equations–KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same KdV equation by different transformation of variables respectively. Since the single solitary wave solution and 2-solitary wave solution of the KdV equation have been known already, substituting the solutions of the KdV equation into the corresponding transformation of variables respectively, the single and 2-solitary wave solutions of the three (2+1)-dimensional equations can be obtained successfully.

PACS: 05.45.Yv;04.20.Jb
Keyword:2-Solitary wave solution;KP equation;cylindrical KP equation;spherical KP equation;transformation of variables
1 Introduction

Many famous nonlinear evolution equations such as Korteweg-de Vries (KdV), Modified KdV (mKdV), Kadomstev–Perviashvili (KP), Coupled KP and Zakharov–Kuznetsov (ZK) have been obtained in nonlinear propagation of dust-acoustic wave, especially, the dust-acoustic solitary wave (DASW) in space and laboratory plasma.[1] It is known that the transverse perturbations always exist in the higher-dimensional system. Anisotropy is introduced into the system and the wave structure and stability are modified by the transverse perturbation. Recent theoretical studies for ion-acoustic/dust-acoustic waves show that the properties of solitary waves in bounded nonplanar cylindrical/spherical geometry differ from that in unbounded planar geometry. A dissipative cylindrical/spherical KdV is obtained by using the standard reductive perturbation method in Ref. [2]. The cylindrical KP equation (CKP) has been introduced by Johnson[3–4] to describe surface wave in a shallow incompressible fluid. A spherical KP (SKP) equation is obtained by using the standard reductive perturbation method.[5]

We consider the classical KP equation in the form

where s is a constant. The KP equation is also derived using reductive perturbation method in superthermal dusty plasma and the steady state solution has been given.[6]

The cylindrical KP equation (CKP)

where α is a constant, has also been investigated to obtain decay mode solutions by means of Hirota method and simplified homogeneous balance method respectively.[7–8]

We also consider the classical spherical KP equation (SKP) in the form

where v0 is a constant. An exact solitary wave solution of SKP, that demonstrates the amplitude and wave velocity of solitary wave are uniquely determined by the parameters of the system and only depending on the initial conditions, has been obtained in Ref. [5].

The KdV equation has been researched by many authors. The multi-soliton solutions of the KdV equation with general variable coefficients have been completely obtained by homogeneous balance principle.[9–10] Some solutions, which possess movable singular points while their energies are only redistributed without dissipation, of KdV equation have been obtained through the modified bilinear Bäcklund transformation.[11] An n-soliton solution with the bell shape has been obtained in Ref. [12], whose stationary height is an arbitrary constant c. The rational solutions, solitary wave solutions, triangular periodic solutions, Jacobi periodic wave solutions and implicit function solutions of KdV equation have been constructed by means of an extended algebraic method.[13] If a transformation of variables between the KdV equation and other equation can be constructed, the results of KdV equation above can be used directly.

In the present paper we aim to find solitary wave solutions of KP Eq. (1), CKP Eq. (2) and SKP Eq. (3). The paper is organized as follows: In Sec. 2, the KP Eq. (1), CKP Eq. (2) and SKP Eq. (3) are reduced to the same KdV equation by different transformation of variables respectively. In Sec. 3, the single solitary wave solutions and 2-solitary wave solutions of KP Eq. (1), CKP Eq. (2) and SKP Eq. (3) can be obtained in terms of the corresponding transformation of variables respectively, since the solutions of the KdV equation have been known already. In Sec. 4, some conclusions are made.

2 Reduction of KP, CKP and SKP
2.1 Reduction of KP

In Eq. (1) we assume that

where q1 = q1 (y, t) is a function to be determined later. Substituting Eq. (4) into Eq. (1), yields an equation as follows

Setting the coefficients of wξ and wξξ to zero, yields

The system (6) admits the following solution:

where λ is a nonzero arbitrary constant. Using Eq. (7) the expression (4) becomes

And after integrating (5) with respect to ξ once and taking the constant of integration to zero, Eq. (5) becomes the classical KdV equation for w = w(ξ,t)

From the discussion above, we come to the conclusion that the KP Eq. (1) for u = u(x, y, t) is reduced to the KdV Eq. (9) for w = w(ξ, t) by using the transformation of variables (8), if w(ξ, t) is a solution of KdV Eq. (9), substituting it into Eq. (8), then we have the exact solution of the KP Eq. (1).

2.2 Reduction of CKP

In Eq. (2) we assume that

where q2 = q2(y, t) is a function to be determined later. Substituting Eq. (10) into Eq. (2), yields an equation as follows

Setting the coefficients of wξ and wξξ to zero, yields

The system (12) admits a solution:

Using Eq. (13) the expression (10) becomes

And after integrating (11) with respect to ξ once and taking the constant of integration to zero, Eq. (11) becomes the classical KdV Eq. (9) for w(ξ, t).

From the discussion above, we come to the conclusion that the CKP Eq. (2) for u = u(x, y, t) is reduced to the KdV Eq. (9) for w = w(ξ, t) by using the transformation of variables (14), if w(ξ, t) is a solution of KdV Eq. (9), substituting it into (14), then we have the exact solution of CKP Eq. (2).

2.3 Reduction of SKP

In Eq. (3), we assume that

where q3 = q3(y, t) is a function to be determined later. Substituting Eq. (15) into Eq. (3), yields an equation as follows

Setting the coefficients of wξ and wξξ to zero, yields

The system (17) admits a solution:

where µ is a nonzero arbitrary constant. Using Eq . (18) the expression (15) becomes

and after integrating Eq. (16) once, taking the constant of integration to zero, Eq. (16) becomes the classical KdV Eq. (9).

From the discussion above, we come to the conclusion that the SKP Eq. (3) for u = u(x, y, t) is reduced to the KdV Eq. (9) for w = w(ξ, t) by using the transformation of variables (19), if w(ξ, t) is a solution of KdV Eq. (9), and substituting it into Eq. (19), we have the exact solution of the SKP Eq. (3).

3 Solitary Wave Solutions of KP, CKP and SKP

In previous section, the KP Eq. (1), CKP Eq. (2) and SKP Eq. (3) have been reduced into the same classical KdV Eq. (9) by the transformation (8), (14), and (19), respectively. The KdV Eq. (9) is of physically importance and its solutions have been known for many researchers, for instance, according to Ref. [9], the KdV Eq. (9) has single solitary wave

where k and x0 are arbitrary parameters. And KdV Eq. (9) also has 2-soliton solution

where ki and xi are arbitrary parameters, , (i = 1; 2), a12 = (k1 − k2)2/(k1 + k2)2.

3.1 Solitary Wave Solutions of KP

Substituting Eq. (20) into Eq. (8) we have the single solitary wave solution for KP Eq. (1), which is expressed by

where k and x0 are arbitrary parameters.

Substituting Eq. (21) into Eq. (8) we have the 2-solitary solution for KP Eq. (1), which is expressed by

where ki and xi are arbitrary parameters, , (i = 1, 2), a12 = (k1 − k2)2 / (k1 + k2)2.

3.2 Solitary Wave Solutions of CKP

Substituting Eq. (20) into Eq. (14) we have the single solitary wave solution for CKP Eq. (2), which is expressed by

where k and x0 are arbitrary parameters. The solution (24) is shown in Figs. 1–4 with k = 0.2, x0 = 3, α = 1.

Fig. 1 Plots of solution (24) with t = 0.
Fig. 2 Plots of solution (24) with t = 0.1.
Fig. 3 Plots of solution (24) with t = 0.5.
Fig. 4 Plots of solution (24) with t = 1.

Substituting Eq. (21) into Eq. (14) we have the 2-solitary solution for CKP Eq. (2), which is expressed by

where ki and xi are arbitrary parameters,

The solution (25) is shown in Figs. 5–8 with α = 1, x1 = −2, x2 = 2, k1 = 0.2, k2 = 0.3.

Fig. 5 Plots of solution (25) with t = 0.
Fig. 6 Plots of solution (25) with t = 0.5.
Fig. 7 Plots of solution (25) with t = 1.
Fig. 8 Plots of solution (25) with t = 2.
3.3 Solitary Wave Solutions of SKP

Substituting Eq. (20) into Eq. (19) we have the single solitary wave solution for SKP Eq. (3), which is expressed by

where k and x0 are arbitrary parameters.

Substituting Eq. (21) into Eq. (19) we have the 2-solitary solution for SKP Eq. (3), which is expressed by

where ki, xi are arbitrary parameters,

4 Conclusion

In this paper, by making corresponding transformation of variables, the KP equation, cylindrical KP equation and spherical KP equation are all reduced to the same classical KdV equation, which can be solved by using homogeneous balance method[9–10] to obtain single solitary wave solution and 2-soliton solution. Substitutiong the solitary solutions of the KdV equation into the corresponding transformation of variables, we have the solitary wave solutions of the KP equation, cylindrical KP equation and spherical KP equation, respectively. It is interesting to research CKP equation and SKP equation but avoid the singularity point analysis when t = 0. The analysis in the present paper may be extended to other works to make further progress.

Reference
[1] Wang H.Y. Zhang K.B. J. Sichuan Normal Univ. 36 2013 911 in Chinese
[2] Xue J.K. Phys. Lett. A 322 2004 225
[3] Johnson R.S. J. Fluid Mech. 97 1980 701
[4] Johnson R.S. A Modern Introduction to the Mathematical Theory of Water Waves Cambridge University Press Cambridge 1997
[5] Xue J.K. Phys. Lett. A 314 2003 479
[6] Saini N.S. Nimardeep Kaur Gill T.S. Advan. Space Res. 55 2015 2873
[7] Deng S.F. Appl. Math. Comput. 218 2012 5974
[8] Wang M.L. Zhang J.L. Li X.Z. Appl. Math. Lett. 62 2016 29
[9] Wang M.L. Wang Y.M. Phys. Lett. A 287 2001 211
[10] Zhong W.P. Xie R.H. Belié M. Petrovié N. Chen G. Yi L. Phys. Rev. A 78 2008 023821
[11] Chen Z.Y. Bi J.B. Chen D.Y. Commun. Theor. Phys. 41 2004 397
[12] Guo F.K. Zhang Y.F. Commun. Theor. Phys. 46 2006 577
[13] Yang X.L. Tang J.S. Commun. Theor. Phys. 48 2007 1