Skip to main content

Advertisement

Exact solutions and conservation laws of a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation

Article metrics

  • 2238 Accesses

  • 13 Citations

Abstract

In this paper we study a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation. This equation is an extension of the well-known Kadomtsev-Petviashvili equation, which describes weakly dispersive and small amplitude waves propagating in quasi-two-dimensional media. We first obtain exact solutions of the BKP equation using the multiple-exp function and simplest equation methods. Furthermore, the conservation laws for the BKP equation are constructed by using the multiplier method.

1 Introduction

It is well known that many phenomena in science and engineering, especially in fluid mechanics, solid state physics, plasma physics, plasma waves and biology, are described by the nonlinear partial differential equations (NLPDEs). Therefore the investigation of exact solutions of NLPDEs plays an important role in the study of NLPDEs. For this reason, during the last few decades, researchers have established several methods to find exact solutions to NLPDEs. Some of these methods include the inverse scattering transform method [1], the Bäcklund transformation [2], the Darboux transformation [3], the Hirota bilinear method [4], the ( G /G)-expansion method [5], the homogeneous balance method [6], the variable separation approach [7], the tri-function method [8, 9], the sine-cosine method [10], the Jacobi elliptic function expansion method [11, 12], the exp-function expansion method [13] and the Lie symmetry method [1416].

The purpose of this paper is to study one such NLPDE, namely the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation, that is given by [17]

u x x x y +α ( u x u y ) x + ( u x + u y + u z ) t ( u x x + u z z )=0,
(1.1)

where α is a real-valued constant. This is a nonlinear wave equation in three spatial (x,y,z) and one temporal coordinate (t).

It is well known that the Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in quasi-two-dimensional media [18]. The KP hierarchy of B-type possesses many integrable structures as the KP hierarchy. The (3+1)-dimensional nonlinear generalized BKP equation

u y t u x x x y 3 ( u x u y ) x +(3 u x x +3 u z z )=0,
(1.2)

was studied in [1820] by different approaches. In [17] a new form of the (3+1)-dimensional BKP equation given by (1.1) was investigated and it was shown, using the simplified form of the Hirota method, that one- and two-soliton solutions exist for (1.1). Also, specific constraints were developed that guarantee the existence of multiple soliton solutions for (1.1).

In this paper we employ the multiple exp-function method [21] and the simplest equation method [22, 23] to obtain some exact solutions of (1.1). In addition to this, conservation laws are constructed for (1.1) using the multiplier method [24].

2 Exact solutions of (1.1)

In this section we employ two methods of solution.

2.1 Exact solutions using the multiple exp-function method

In this subsection we employ the multiple exp-function method and obtain exact explicit one-wave and two-wave solutions of (1.1). For details of the method, the reader is referred to the paper [20], in which this method was introduced. So, following the method and using the notation of [20], for a one-wave solution, we have

p = A 0 + A 1 e k 1 x + l 1 y + m 1 z ω 1 t , q = B 0 + B 1 e k 1 x + l 1 y + m 1 z ω 1 t

and the resulting one-wave solution is

u(x,y,z,t)= p q ,

with

A 1 = ( 6 k 1 B 0 + α A 0 ) B 1 α B 0 , m 1 = θ k 1 , ω 1 = k 1 3 ,

where θ is any root of θ 2 + k 1 2 θ+ k 1 2 +1=0.

Likewise, for a two-wave solution, we have

p = 2 k 1 e k 1 x + l 1 y + m 1 z ω 1 t + 2 k 2 e k 2 x + l 2 y + m 2 z ω 2 t p = + 2 A 12 ( k 1 + k 2 ) e k 1 x + l 1 y + m 1 z ω 1 t e k 2 x + l 2 y + m 2 z ω 2 t , q = 1 + e k 1 x + l 1 y + m 1 z ω 1 t + e k 2 x + l 2 y + m 2 z ω 2 t + A 12 e k 1 x + l 1 y + m 1 z ω 1 t e k 2 x + l 2 y + m 2 z ω 2 t

and the resulting two-wave solution is

u(x,y,z,t)= p q ,

where

A 12 = 1 , k 1 = 1 , k 2 = 1 , l 1 = 1 , l 2 = 1 , α = 3 , m 1 = θ , ω 1 = 12 6 m 2 2 m 2 2 + 4 θ m 2 + θ m 2 2 ( 2 + m 2 ) 2 , ω 2 = m 2 2 2 + m 2

and θ is any root of 2 θ 2 ( m 2 6)θ+2 m 2 2 +6 m 2 +12=0.

2.2 The simplest equation method

In this subsection we use the simplest equation method and obtain exact solutions of (1.1). This method was introduced by Kudryashov [22] and modified by Vitanov [23]. The simplest equations we use in this paper are the Bernoulli and Riccati equations. Their solutions can be written in elementary functions. For details, see, for example, [25].

Making use of the wave variable

ν= k 1 x+ k 2 y+ k 3 z+ k 4 t+ k 5 ,

where k i , i=1,,5 are constants, the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili (1.1) transforms to a fourth-order nonlinear ordinary differential equation (ODE)

k 2 k 1 3 F ′′′′ ( ν ) k 1 2 F ( ν ) + k 4 k 1 F ( ν ) k 3 2 F ( ν ) + k 2 k 4 F ( ν ) + k 3 k 4 F ( ν ) + 2 α k 2 k 1 2 F ( ν ) F ( ν ) = 0 .
(2.1)

Let us consider the solutions of ODE (2.1) in the form

F(ν)= i = 0 M A i ( G ( ν ) ) i ,
(2.2)

where G(ν) satisfies the Bernoulli and Riccati equations, M is a positive integer that can be determined by balancing procedure as in [23] and A 0 ,, A M are parameters to be determined.

2.2.1 Solutions of (1.1) using the Bernoulli equation as the simplest equation

The balancing procedure yields M=1 so the solutions of (2.1) are of the form

F(ν)= A 0 + A 1 G.
(2.3)

Substituting (2.3) into ODE (2.1) and making use of the Bernoulli equation and then equating the coefficients of the functions G i to zero, we obtain an algebraic system of equations. Solving this system with the aid of Mathematica, we obtain

α = 6 k 1 b A 1 , k 2 = k 1 2 + k 3 2 k 1 k 4 k 3 k 4 k 1 3 a 2 + k 4 .

As a result, a solution of (1.1) is

u(x,y,z,t)= A 0 + A 1 a { cosh [ a ( ν + C ) ] + sinh [ a ( ν + C ) ] 1 b cosh [ a ( ν + C ) ] b sinh [ a ( ν + C ) ] } ,

where ν= k 1 x+ k 2 y+ k 3 z+ k 4 t+ k 5 and C is a constant of integration.

2.2.2 Solutions of (1.1) using the Riccati equation as the simplest equation

The balancing procedure yields M=1, so the solutions of (2.1) are of the form

F(ν)= A 0 + A 1 G.
(2.4)

Substituting (2.4) into ODE (2.1) and making use of the Riccati equation, we obtain an algebraic system of equations by equating all coefficients of the functions G i to zero. Solving the algebraic equations, one obtains

α = 6 k 1 a A 1 , c = k 2 k 1 3 b 2 + k 1 k 4 k 3 2 + k 3 k 4 + k 2 k 4 k 1 2 4 k 2 k 1 3 a .

Hence solutions of (1.1) are

u(x,y,z,t)= A 0 + A 1 { b 2 a θ 2 a tanh [ 1 2 θ ( ν + C ) ] }

and

u(x,y,z,t)= A 0 + A 1 { b 2 a θ 2 a tanh ( 1 2 θ ν ) + sech ( θ ν 2 ) C cosh ( θ ν 2 ) 2 a θ sinh ( θ ν 2 ) } ,

where ν= k 1 x+ k 2 y+ k 3 z+ k 4 t+ k 5 and C is a constant of integration.

3 Conservation laws

In this section we construct conservation laws for (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (1.1). The multiplier method will be used [15, 24, 26]. First we recall some results that will be used in the computation of conserved vectors.

3.1 Preliminaries

Consider a k th-order system of PDEs given by

E α (x,u, u ( 1 ) ,, u ( k ) )=0,α=1,,m,
(3.1)

with n independent variables x=( x 1 , x 2 ,, x n ) and m dependent variables u=( u 1 , u 2 ,, u m ). Here u ( 1 ) , u ( 2 ) ,, u ( k ) denote the collections of all first, second, … , k th-order partial derivatives. That is, u i α = D i ( u α ), u i j α = D j D i ( u α ), , respectively, where the total derivative operator with respect to x i is given by

D i = x i + u i α u α + u i j α u j α +,i=1,,n.
(3.2)

The n-tuple T=( T 1 , T 2 ,, T n ), T j A, j=1,,n, where is the space of differential functions, is a conserved vector of (3.1) if T i satisfies

D i T i | ( 3.1 ) =0
(3.3)

and equation (3.3) defines a local conservation law of system (3.1).

The Euler-Lagrange operator, for each α, is defined as

δ δ u α = u α + s 1 ( 1 ) s D i 1 D i s u i 1 i 2 i s α ,α=1,,m.
(3.4)

A multiplier Λ α (x,u, u ( 1 ) ,) has the property that

Λ α E α = D i T i
(3.5)

hold identically. The right-hand side of (3.5) is a divergence expression. The determining equation for the multiplier Λ α is given by

δ ( Λ α E α ) δ u α =0.
(3.6)

After obtaining the multipliers, we can calculate the conserved vectors by using a homotopy formula [24].

3.2 Construction of conservation laws for (1.1)

We now construct conservation laws for (3+1)-dimensional nonlinear BKP equation (1.1). We obtain a multiplier of the form

Λ=C u x +f(t,y,z),

where C is an arbitrary constant and f is any solution of f z z f t z f t y =0. Corresponding to the above multiplier, we obtain the following conserved vectors:

T 1 t = 1 2 ( u x z u u x y u + u x 2 ) , T 1 x = 1 2 ( u z z u + u t z u + u t y u + α u x 2 u y + 2 u x u x x y u x 2 ) , T 1 y = 1 6 ( 3 u t u x + α u x 3 3 u x x 2 ) , T 1 z = 1 2 ( u x z u + u t u x + u x ( u z ) )

and

T 2 t = 1 2 ( f z ( u ) f y u + u x f + u z f + u y f ) , T 2 x = 1 4 ( α f y u x u + 3 α u x u y f α u x y f u 2 f t u 4 u x f + 3 u x x y f + 2 u t f f y u x x ) , T 2 y = 1 4 ( α u x 2 f + α u x x f u 2 f t u + u x x x f + 2 u t f ) , T 2 z = 1 2 ( 2 f z u f t u 2 u z f + u t f ) .

Remark 1 Due to the presence of the arbitrary function f in the multiplier, one can obtain infinitely many conservation laws.

4 Concluding remarks

In this paper we studied (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (1.1). Exact solutions of the BKP equation were found using two distinct methods, namely the multiple-exp function method and the simplest equation method. Also, the conservation laws for the BKP equation were derived by using the multiplier method.

References

  1. 1.

    Ablowitz MJ, Clarkson PA: Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge; 1991.

  2. 2.

    Gu CH: Soliton Theory and Its Application. Zhejiang Science and Technology Press, Zhejiang; 1990.

  3. 3.

    Matveev VB, Salle MA: Darboux Transformation and Soliton. Springer, Berlin; 1991.

  4. 4.

    Hirota R: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge; 2004.

  5. 5.

    Wang M, Xiangzheng LX, Jinliang ZJ:The ( G /G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008, 372: 417-423. 10.1016/j.physleta.2007.07.051

  6. 6.

    Wang M, Zhou Y, Li Z: Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A 1996, 216: 67-75. 10.1016/0375-9601(96)00283-6

  7. 7.

    Lou SY, Lu JZ: Special solutions from variable separation approach: Davey-Stewartson equation. J. Phys. A, Math. Gen. 1996, 29: 4209-4215. 10.1088/0305-4470/29/14/038

  8. 8.

    Yan ZY: The new tri-function method to multiple exact solutions of nonlinear wave equations. Phys. Scr. 2008., 78: Article ID 035001

  9. 9.

    Yan ZY: Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov-Kuznetsov equation in dense quantum plasmas. Phys. Lett. A 2009, 373: 2432-2437. 10.1016/j.physleta.2009.04.018

  10. 10.

    Wazwaz M: The tanh and sine-cosine method for compact and noncompact solutions of nonlinear Klein-Gordon equation. Appl. Math. Comput. 2005, 167: 1179-1195. 10.1016/j.amc.2004.08.006

  11. 11.

    Lu DC: Jacobi elliptic functions solutions for two variant Boussinesq equations. Chaos Solitons Fractals 2005, 24: 1373-1385. 10.1016/j.chaos.2004.09.085

  12. 12.

    Yan ZY:Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey-Stewartson-type equation via a new method. Chaos Solitons Fractals 2003, 18: 299-309. 10.1016/S0960-0779(02)00653-7

  13. 13.

    He JH, Wu XH: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 2006, 30: 700-708. 10.1016/j.chaos.2006.03.020

  14. 14.

    Bluman GW, Kumei S Applied Mathematical Sciences 81. In Symmetries and Differential Equations. Springer, New York; 1989.

  15. 15.

    Olver PJ Graduate Texts in Mathematics 107. In Applications of Lie Groups to Differential Equations. 2nd edition. Springer, Berlin; 1993.

  16. 16.

    Adem KR, Khalique CM:Exact solutions and conservation laws of a (2+1)-dimensional nonlinear KP-BBM equation. Abstr. Appl. Anal. 2013., 2013: Article ID 791863 10.1155/2013/791863

  17. 17.

    Wazwaz AM:Two forms of (3+1)-dimensional B-type Kadomtsev-Petviashvili equation: multiple-soliton solutions. Phys. Scr. 2012., 86: Article ID 035007

  18. 18.

    Wazwaz AM:Distinct kinds of multiple-soliton solutions for a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. Phys. Scr. 2011., 84: Article ID 055006

  19. 19.

    Shen HF, Tu MH: On the constrained B-type Kadomtsev-Petviashvili equation: Hirota bilinear equations and Virasoro symmetry. J. Math. Phys. 2011., 52: Article ID 032704

  20. 20.

    Ma WX, Abdeljabbar A, Asaad MG:Wronskian and Grammian solutions to a (3+1)-dimensional generalized KP equation. Appl. Math. Comput. 2011, 217: 10016-10023. 10.1016/j.amc.2011.04.077

  21. 21.

    Ma WX, Huang T, Zhang Y: A multiple exp-function method for nonlinear differential equations and its applications. Phys. Scr. 2010., 82: Article ID 065003

  22. 22.

    Kudryashov NA: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals 2005, 24: 1217-1231. 10.1016/j.chaos.2004.09.109

  23. 23.

    Vitanov NK: Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 2050-2060. 10.1016/j.cnsns.2009.08.011

  24. 24.

    Anco SC, Bluman GW: Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications. Eur. J. Appl. Math. 2002, 13: 545-566.

  25. 25.

    Adem AR, Khalique CM: Symmetry reductions, exact solutions and conservation laws of a new coupled KdV system. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 3465-3475. 10.1016/j.cnsns.2012.01.010

  26. 26.

    Anthonyrajah M, Mason DP: Conservation laws and invariant solutions in the Fanno model for turbulent compressible flow. Math. Comput. Appl. 2010, 15: 529-542.

Download references

Acknowledgements

MA and CMK would like to thank the organizing committee of the International Conference on the Theory, Methods and Application of Nonlinear Equations, held at Texas A&M University-Kingsville, USA, for their kind hospitality during the conference.

Author information

Correspondence to Chaudry Masood Khalique.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MA and CMK worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Keywords

  • (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation
  • multiple-exp function method
  • simplest equation method
  • conservation laws