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Theory and Modern Applications

Exact three-wave solutions for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

Abstract

In this paper, we consider a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. We employ the Hirota bilinear method to obtain the bilinear form of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Based on the bilinear form, we derive exact three-wave solutions by using an extended three-soliton method. In addition, we also get the trajectory of some solution with the help of MAPLE.

1 Introduction

Integrable systems and nonlinear evolution equations [1–9] have attracted much attention of mathematicians and physicists. Especially, exact solutions of nonlinear evolution equations play a pivotal role in the study of mathematical physical phenomena. Not only can these exact solutions describe many important phenomena in physics and other fields, but they can also help physicists to understand the mechanisms of the complicated physical phenomena. A variety of powerful methods have been employed to study nonlinear phenomena, such as the inverse scattering transform [10], the tanh function method [11], the extended tanh-function method [12], the homogeneous balance method [13], the auxiliary function method [14], and the exp-function method [15], the Pfaffian technique [16], the dressing method [17], the Bäcklund transformation method [18], the Darboux transformation [19], the generalized symmetry method, the tri-function method [20] and the G ′ /G-expansion method [21], the modified CK direct method [22].

Very recently, Dai et al. proposed a new technique called the three-wave approach to seek periodic solitary wave solutions for integrable equations [23]. The method is to use Frobenius’ idea [24] to reduce the PDE into integrable ODEs. Frobenius’ idea was successfully used to establish the transformed rational function method [25] and to solve the KPP equation [26]. In fact, the Tanh function method and the G ′ /G expansion method are special cases of the reduction idea raised in [26], say, the general Frobenius idea. Furthermore, a three-wave solution in (3+1)-dimension was obtained by using the multiple exp-function method [27, 28]. With the rapid development of computer technology and the help of symbolic computation, this approach is of utmost simplicity. Hence, it can be applied to many kinds of nonlinear evolution equations and higher-dimensional soliton equations. Zitian Li obtained periodic cross-kink wave solutions, doubly periodic solitary wave solutions and breather type of two-solitary wave solutions for the (3+1)-dimensional Jimbo-Miwa equation by this method [29]. Wang applied the method to a higher dimensional KdV-type equation [30].

The BLMP equation was first derived in [31]:

u y t + u x x x y −3 u x x u y −3 u x u x y =0,
(1)

where u=u(x,y,t) and subscripts represent partial differentiation with respect to the given variable. Boiti et al. [31] also discussed the Painlevé property, Lax pairs and some exact solutions of (2+1)-dimensional BLMP. Through the Bäcklund transformation, Bai and Zhao got some new solutions of the BLMP equation. By means of the multilinear variable separation approach, a general variable separation solution of the BLMP equation was derived in [32]. Liu proposed a simple Bäcklund transformation of a potential BLMP system by using the standard truncated Painlevé expansion and symbolic computation, and a solution of the potential BLMP system with three arbitrary functions was given in [33]. The symmetry, similarity reductions and new solutions of the (2+1)-dimensional BLMP equation were obtained in [34]. These solutions include rational function solutions, double-twisty function solutions, Jacobi oval function solutions and triangular cycle solutions. In [35], based on the binary Bell polynomials, the bilinear form for the BLMP equation was obtained. The new exact solutions were derived with an arbitrary function in y, and soliton interaction properties were discussed by the graphical analysis. The author in [36] discussed the BLMP equation and generalized breaking soliton equations by using the exponential function and obtained some new exact solutions of the equations. By using the modified Clarkson-Kruskal (CK) direct method, Li et al. [37] constructed a Bäcklund transformation of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation. Laurent Delisle and Masoud Mosaddeghi proposed the study of the BLMP equation from two points of view: the classical and the super symmetric. They constructed new solutions of this equation from Wronskian formalism and the Hirota method in [38].

In this paper, we consider the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

u y t + u z t + u x x x y + u x x x z −3 u x ( u x y + u x z )−3 u x x ( u y + u z )=0,
(2)

which was introduced by Darvishi in [39]. We apply the extended three-soliton method to the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, obtaining more exact solutions including a complexiton solution, periodic cross-kink solutions about it.

2 Methodology

In this section, we briefly highlight the main features of the extended three-soliton method. Let us consider a PDE for u(x,z,t) in the form

P(u, u t , u x , u z , u t t , u t x , u t z , u x x , u x z , u z z ,…)=0,
(3)

where P is a polynomial in its arguments. The solution method will also work for systems of nonlinear equations and high-dimensional ones.

Step 1. Firstly, we introduce the D-operator which was proposed by Hirota [40] and defined as

D t m D x n a(t,x)⋅b(t,x)= ∂ m ∂ s m ∂ n ∂ y n a(t+s,x+y)b(t−s,x−y) | s = 0 , y = 0 .
(4)

By transformation u=alnf, u= G ( f ) f and D-operator definition, Eq. (3) can be turned into

F(D, D t , D x , D z , D t t , D t x , D t z ,…)f⋅f=0,
(5)

where F is a polynomial in its arguments.

Step 2. To seek the three-wave solution of Eq. (3), let us consider the solution of Eq. (5) in the following form:

f=cos(ξ)+ a − 1 exp(−θ)+ a 1 exp(θ)+ a 2 sinh(η),
(6)

where ξ= p 1 (x+ γ 1 z+ β 1 y+ α 1 t), η= p 2 (x+ γ 2 z+ β 2 y+ α 2 t), θ= p 3 (x+ γ 3 z+ β 3 y+ α 3 t) and p i , α i , β i , γ i (i=1,2,3) are free constants to be determined later.

Step 3. Substituting Eq. (6) into Eq. (5), and collecting the coefficient of sinh(η)cos(ξ), sinh(η)exp(θ), sinh(η)exp(−θ), cosh(η)sin(ξ), cosh(η)exp(θ), cosh(η)exp(−θ), sin(ξ)exp(θ), sin(ξ)exp(−θ), cos(ξ)exp(θ), cos(ξ)exp(−θ) to zero, we can derive a set of algebraic equations for a − 1 , a 1 , a 2 , p i , α i , β i , γ i (i=1,2,3).

Step 4. Solving the set of algebraic equations defined by Step 3 with the help of MAPLE, we can derive parameters a − 1 , a 1 , a 2 , p i , α i , β i , γ i (i=1,2,3). Therefore, we can obtain abundant exact multi-wave solutions of Eq. (3).

3 Exact three-wave solutions for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

In this section, we consider the following (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation Eq. (2):

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

or, equivalently,

( u y + u z ) t + ( u y + u z ) x x x −3 u x ( u y + u z ) x −3 u x x ( u y + u z )=0.
(7)

Under the dependent variable transformation,

u=−2 ( ln f ) x ,
(8)

where f(x,y,z,t) is an unknown real function, system (2) is turned into

− f y f t − f z f t − f x x x f y − 3 f x x y f x + 3 f x x f x y − f x x x f z − 3 f x x z f x + 3 f x x f x z + f y t f + f z t f + f x x x y f + f x x x z f = 0 .
(9)

Equivalently, Eq. (9) can be mapped into the Hirota bilinear equation

( D y D t + D z D t + D y D x 3 + D z D x 3 ) fâ‹…f=0.
(10)

According to the methodology in Section 2, we can derive a set of algebraic equations for a − 1 , a 1 , a 2 , p i , α i , β i , γ i (i=1,2,3)

sinh ( η ) cos ( ξ ) : − 3 p 2 2 p 1 2 γ 1 2 + p 2 2 α 2 2 β 2 − p 1 2 α 1 γ 1 + p 2 2 α 2 γ 2 − p 1 2 α 1 β 1 − 3 p 1 2 p 2 2 β 2 − 3 p 2 2 p 1 2 β 1 − 3 p 1 2 p 2 2 γ 2 + p 1 4 β 1 + p 2 4 β 2 + p 1 4 γ 1 + p 2 4 γ 2 = 0 , sinh ( η ) exp ( θ ) , sinh ( η ) exp ( − θ ) : 3 p 3 2 p 2 2 γ 2 + 3 p 2 2 p 3 2 γ 3 + p 2 4 γ 2 + p 3 4 β 3 + 3 p 3 2 p 2 2 β 2 + p 2 4 β 2 + 3 p 2 2 p 3 2 β 3 + p 3 4 γ 3 + p 3 2 β 3 γ 3 + p 2 2 α 2 γ 2 + p 2 2 α 2 β 2 + p 3 2 γ 3 α 3 = 0 , cosh ( η ) sin ( ξ ) : − 3 p 1 2 β 1 + 3 p 2 2 β 2 + p 2 2 γ 1 − 3 p 1 2 γ 1 + 3 p 2 2 γ 2 − p 1 2 γ 2 + β 2 α 1 + γ 1 α 2 + γ 2 α 1 + β 1 α 2 − p 1 2 β 2 + p 2 2 β 1 = 0 , cosh ( η ) exp ( θ ) , cosh ( η ) exp ( − θ ) : β 3 α 2 + 3 p 2 2 β 2 + β 2 α 3 + 3 p 3 2 β 3 + γ 3 α 2 + p 2 2 γ 3 + γ 2 α 3 + p 3 2 γ 2 + p 3 2 β 2 + 3 p 2 2 γ 2 + p 2 2 β 3 + 3 p 3 2 γ 3 = 0 , sin ( ξ ) exp ( θ ) , sin ( ξ ) exp ( − θ ) : − β 1 α 3 + p 1 2 β 3 + p 1 2 γ 3 − p 3 2 γ 1 + 3 p 1 2 β 1 + 3 p 1 2 γ 1 − p 3 2 β 1 − β 3 α 1 − 3 p 3 2 γ 3 − γ 1 α 3 − 3 p 3 2 β 3 − γ 3 α 1 = 0 , cos ( ξ ) exp ( θ ) , cos ( ξ ) exp ( − θ ) : p 3 2 β 3 α 3 − p 1 2 α 1 γ 1 − 3 p 1 2 p 3 2 β 3 + p 3 2 γ 3 α 3 − 3 p 3 2 p 1 2 β 1 + p 1 4 β 1 − 3 p 1 2 p 3 2 γ 3 + p 3 4 β 3 − 3 p 3 2 p 1 2 γ 1 + p 1 4 γ 1 − p 1 2 α 1 β 1 + p 3 4 γ 3 = 0 ,

and constant term:

− a 2 2 p 2 2 α 2 β 2 − a 2 2 p 2 2 α 2 γ 2 − p 1 2 α 1 β 1 − p 1 2 α 1 γ 1 − 4 a 2 2 p 2 4 β 2 − 4 a 2 2 p 2 4 γ 2 + 16 a − 1 p 3 4 a 1 β 3 + 16 a − 1 p 3 4 a 1 γ 3 + 4 a − 1 p 3 2 β 3 a 1 α 3 + 4 a − 1 p 3 2 γ 3 a 1 α 3 + 4 p 1 4 β 1 + 4 p 1 4 γ 1 = 0 .

Solving the above algebraic equations with the help of MAPLE gives the following solutions.

Case 1.

a − 1 = a − 1 , a 1 = a 1 , a 2 = a 2 , p 1 = 0 , p 2 = 0 , p 3 = p 3 , α 1 = α 1 , α 2 = α 2 , α 3 = − p 3 2 , β 1 = − β 2 α 1 + γ 1 α 2 + γ 2 α 1 α 2 , β 2 = β 2 , β 3 = − γ 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

In this case, we obtain the single soliton solution

u 1 =−2 − a − 1 p 3 e − p 3 [ x + γ 3 ( z − y ) − p 3 2 t ] + a 1 p 3 e p 3 [ x + γ 3 ( z − y ) − p 3 2 t ] 1 + a − 1 e − p 3 [ x + γ 3 ( z − y ) − p 3 2 t ] + a 1 e p 3 [ x + γ 3 ( z − y ) − p 3 2 t ] ,
(11)

where a − 1 , a 1 , p 3 , γ 3 are free constants. The propagation of solution u 1 is described in Figure 1.

Figure 1
figure 1

The propagation of solution u 1 with a − 1 =−1 , a 1 =1 , p 3 =1 , γ 3 =1 , z=1 .

Case 2.

a − 1 = a − 1 , a 1 = a 1 , a 2 = a 2 , p 1 = p 1 , p 2 = 0 , p 3 = 0 , α 1 = p 1 2 , α 2 = α 2 , α 3 = α 3 , β 1 = − γ 1 , β 2 = − β 3 α 2 + γ 3 α 2 + γ 2 α 3 α 3 , β 3 = β 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

Then we obtain new periodic solutions as follows:

u 2 = 2 p 1 sin ( p 1 [ x + γ 1 ( z − y ) + p 1 2 t ] ) cos ( p 1 [ x + γ 1 ( z − y ) + p 1 2 t ] ) + a − 1 + a 1 ,
(12)

where a − 1 , a 1 , p 1 , γ 1 are free constants. The propagation of solution u 2 is described in Figure 2.

Figure 2
figure 2

The propagation of solution u 2 with a − 1 =1 , a 1 =1 , p 1 =−1 , γ 1 =1 , z=5 .

Case 3.

a − 1 = 0 , a 1 = a 1 , a 2 = a 2 , p 1 = 0 , p 2 = p 2 , p 3 = p 3 , α 1 = α 1 , α 2 = − p 2 2 , α 3 = − p 3 2 , β 1 = β 1 , β 2 = − γ 2 , β 3 = − γ 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

We then obtain

u 3 =−2 a 1 p 3 e p 3 [ x + γ 3 ( z − y ) − p 3 2 t ] + a 2 p 2 sinh ( p 2 [ x + γ 2 ( z − y ) − p 2 2 t ] ) 1 + a 1 e p 3 [ x + γ 3 ( z − y ) − p 3 2 t ] + a 2 cosh ( p 2 [ x + γ 2 ( z − y ) − p 2 2 t ] ) ,
(13)

where a 1 , a 2 , p 2 , p 3 , γ 3 are arbitrary constants.

Case 4.

a − 1 = 1 4 a 1 , a 1 = a 1 , a 2 = a 2 , p 1 = p 3 i , p 2 = 0 , p 3 = p 3 , α 1 = − p 3 2 , α 2 = − 3 p 3 2 , α 3 = − p 3 2 , β 1 = − γ 1 − β 3 − γ 3 , β 2 = β 2 , β 3 = β 2 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

We then obtain a complexiton solution

u 4 =−2 − p 3 i sin ( ξ ) − p 3 e − p 3 ( x + γ 3 z + β 3 y − p 3 2 t ) 4 a 1 + a 1 p 3 e p 3 ( x + γ 3 z + β 3 y − p 3 2 t ) cos ( ξ ) + e − p 3 ( x + γ 3 z + β 3 y − p 3 2 t ) 4 a 1 + a 1 e p 3 ( x + γ 3 z + β 3 y − p 3 2 t ) ,
(14)

where ξ=i p 3 (x+ γ 1 z+(− γ 1 − β 3 − γ 3 )y− p 3 2 t) and a 1 , p 3 , α 3 , β 3 , γ 1 , γ 3 are free constants.

Case 5.

a − 1 = a − 1 , a 1 = a 1 , a 2 = a 2 , p 1 = p 1 , p 2 = 0 , p 3 = p 3 , α 1 = p 1 2 , α 2 = α 2 , α 3 = α 3 , β 1 = − γ 1 , β 2 = − γ 2 , β 3 = − γ 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

We then obtain new periodic cross-kink solutions

u 5 =−2 − p 1 sin ( ξ 1 ) − a − 1 p 3 e − p 3 ( x + γ 3 z + β 3 y − p 3 2 t ) + a 1 p 3 e p 3 ( x + γ 3 z − γ 3 y + α 3 t ) cos ( ξ 1 ) + a − 1 e p 3 ( x + γ 3 z − γ 3 y + α 3 t ) + a 1 e p 3 ( x + γ 3 z − γ 3 y + α 3 t ) ,
(15)

where ξ 1 = p 1 (x+ γ 1 z− γ 1 y+ p 1 2 t) and a − 1 , a 1 , p 1 , p 3 , α 3 , γ 3 are free constants.

Case 6.

a − 1 = − p 1 2 γ 1 + p 1 2 β 1 + a 2 2 p 3 2 γ 3 + a 2 2 p 3 2 β 3 4 ( β 3 + γ 3 ) p 3 2 a 1 , a 1 = a 1 , a 2 = a 2 , p 1 = p 1 , p 2 = p 3 , p 3 = p 3 , α 1 = − 3 p 3 2 + p 1 2 , α 2 = 3 p 1 2 − p 3 2 , α 3 = 3 p 1 2 − p 3 2 , β 1 = β 1 , β 2 = − γ 2 − γ 3 − β 3 , β 3 = β 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

We then obtain new periodic cross-kink solutions

u 6 =−2 − p 1 sin ( ξ 2 ) + p 1 2 γ 1 + p 1 2 β 1 + a 2 2 p 3 2 γ 3 + a 2 2 p 3 2 β 3 4 ( β 3 + γ 3 ) p 3 a 1 e − θ 2 + a 1 p 3 e θ 2 + a 2 p 3 cosh ( η 2 ) cos ( ξ 2 ) − p 1 2 γ 1 + p 1 2 β 1 + a 2 2 p 3 2 γ 3 + a 2 2 p 3 2 β 3 4 ( β 3 + γ 3 ) p 3 2 a 1 e − θ 2 + a 1 e θ 2 + a 2 sinh ( η 2 ) ,
(16)

where ξ 2 = p 1 (x+ γ 1 z+ β 1 y+(−3 p 3 2 + p 1 2 )t), θ 2 = p 3 (x+ γ 3 z+ β 3 y+(3 p 1 2 − p 3 2 )t), η 2 = p 3 (x+ γ 2 z+(− γ 2 − γ 3 − β 3 )y+(3 p 1 2 − p 3 2 )t) and a 1 , a 2 , p 1 , p 3 , β 1 , β 3 , γ 1 , γ 2 , γ 3 are free constants.

Case 7.

a − 1 = 0 , a 1 = a 1 , a 2 = a 2 , p 1 = i p 2 , p 2 = p 2 , p 3 = p 3 , α 1 = − 4 p 2 2 , α 2 = − 4 p 2 2 , α 3 = − 3 p 3 p 2 2 + p 3 3 − 3 p 3 2 p 2 + 3 p 2 3 p 3 , β 1 = − p 3 β 3 + p 3 γ 3 + p 2 γ 1 p 2 , β 2 = − p 3 β 3 + p 2 γ 2 + p 3 γ 3 p 2 , β 3 = β 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

We then obtain a new complexiton solution

u 7 =−2 − i p 2 sin ( ξ 3 ) + a 1 p 3 e p 3 ( x + γ 3 z + β 3 y − 3 p 3 p 2 2 + p 3 3 − 3 p 3 2 p 2 + 3 p 2 3 p 3 t ) + a 2 p 2 cosh ( η 3 ) cos ( ξ 3 ) + a 1 e p 3 ( x + γ 3 z + β 3 y − 3 p 3 p 2 2 + p 3 3 − 3 p 3 2 p 2 + 3 p 2 3 p 3 t ) + a 2 sinh ( η 3 ) ,
(17)

where ξ 3 = p 1 (x+ γ 1 z− p 3 β 3 + p 3 γ 3 + p 2 γ 1 p 2 y−4 p 2 2 t), η 3 = p 2 (x+ γ 2 z− p 3 β 3 + p 2 γ 2 + p 3 γ 3 p 2 y−4 p 2 2 t) and a 1 , a 2 , p 2 , p 3 , β 3 , γ 1 , γ 2 , γ 3 are free constants.

Case 8.

a − 1 = a − 1 , a 1 = a 1 , a 2 = a 2 , p 1 = p 1 , p 2 = p 2 , p 3 = 0 , α 1 = p 1 2 , α 2 = − p 2 2 , α 3 = α 3 , β 1 = − γ 1 , β 2 = − γ 2 , β 3 = β 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

We then obtain new periodic cross-kink solutions

u 8 =−2 − p 1 sin ( p 1 ( x + γ 1 z − γ 1 y + p 1 2 t ) ) + a 2 p 2 cosh ( p 2 ( x + γ 2 z − γ 2 y − p 2 2 t ) ) cos ( p 1 ( x + γ 1 z − γ 1 y + p 1 2 t ) ) + a − 1 + a 1 + a 2 sinh ( p 2 ( x + γ 2 z − γ 2 y − p 2 2 t ) ) ,
(18)

where a − 1 , a 1 , a 2 , p 1 , p 2 , α 3 , β 3 , γ 1 , γ 2 and γ 3 are free constants.

Case 9.

a − 1 = ( a 2 2 β 1 + β 1 + γ 1 + a 2 2 γ 1 ) p 2 2 4 ( β 3 + γ 3 ) p 3 2 a 1 a 1 = a 1 , a 2 = a 2 , p 1 = i p 2 , p 2 = p 2 , p 3 = p 3 , α 1 = − 3 p 3 2 − p 2 2 , α 2 = − 3 p 3 2 − p 2 2 , α 3 = − 3 p 2 2 − p 3 2 , β 1 = β 1 , β 2 = − β 1 − γ 2 − γ 1 , β 3 = β 3 , γ 1 = γ 1 , γ 2 = γ 2 , γ 3 = γ 3 .

We then obtain a complexiton solution

u 9 =−2 − i p 2 sin ( ξ 4 ) − a 2 2 β 1 + β 1 + γ 1 + a 2 2 γ 1 4 ( β 3 + γ 3 ) p 3 a 1 e − θ 4 + a 1 p 3 e θ 4 + a 2 p 2 cosh ( η 4 ) cos ( ξ 4 ) + a 2 2 β 1 + β 1 + γ 1 + a 2 2 γ 1 4 ( β 3 + γ 3 ) p 3 2 a 1 e − θ 4 + a 1 e θ 4 + a 2 cosh ( η 4 ) ,
(19)

where ξ 4 =i p 2 (x+ γ 1 z+ β 1 y+(−3 p 3 2 − p 2 2 )t), θ 4 = p 3 (x+ γ 3 z+ β 3 y+(−3 p 2 2 − p 3 2 )t), η 4 = p 2 (x+ γ 2 z+(− β 1 − γ 2 − γ 1 )y+(−3 p 3 2 − p 2 2 )t) and a 1 , a 2 , p 2 , p 3 , β 1 , β 3 , γ 1 , γ 2 , γ 3 are free constants. Figures 3, 4, 5, 6 described the solution of u 3 , u 4 , u 6 and u 8 respectively.

Figure 3
figure 3

The propagation of solution u 3 with a 1 =1 , a 2 =1 , p 2 =1 , p 3 =1 , γ 2 =1 , γ 3 =1 , z=1 .

Figure 4
figure 4

The propagation of solution u 5 with a − 1 =1 , a 1 =1 , p 1 =1 , p 3 =1 , α 1 , γ 1 =1 , γ 3 =1 , z=1 .

Figure 5
figure 5

The propagation of solution u 6 with a 1 =1 , a 2 =1 , p 1 =1 , p 3 =1 , β 1 =1 , β 3 =1 , γ 1 =1 , γ 2 =−1 , γ 3 =1 , z=1 .

Figure 6
figure 6

The propagation of solution u 8 with a − 1 =1 , a 1 =1 , a 2 =1 , p 1 =1 , p 2 =1 , γ 1 =1 , γ 2 =−1 , z=1 .

Remark 1 Noting if we set β i =− γ i in Case 1 to Case 5 of the solutions above are special solutions of the equation, we can see that for an arbitrary function, u(x,y−z,t) is also a solution. However, the other cases are different.

Remark 2 Noting sinh(ix)=isin(x) and cos(ix)=cosh(x), the solutions presented in this paper can be obtained by using the multiple exp-function. Furthermore, we can get an N-soliton solution just by modifying the ansatz and using the exp expanding method [27].

4 Conclusion

In this paper, we obtained three-wave solutions to the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with the extended three-soliton method. All the presented solutions show remarkable richness of the solution space of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation and also that the (3+1)-dimensional integrable system may have very rich dynamical behavior. The considered solutions are of complexiton type [41]. There is also a generalized theory of the Bell polynomials method which describes the generalized bilinear differential equations [42, 43]. To our knowledge, our solutions are novel. They cannot be obtained just through the simple generalization of the (2+1)-dimensional BLMP equation. In fact, the extended three-soliton method is entirely algorithmic and involves a large amount of tedious calculations. However, the method is direct, concise and effective. Therefore, we can apply the method to the variety of dynamics of a higher-dimensional nonlinear system and many other types of a nonlinear evolution equation in further work.

References

  1. Korteweg D, de Vries G: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 1895, 39: 422-443.

    Article  Google Scholar 

  2. Toda K, Yu S:The investigation into new equations in (2+1)-dimensions. J. Nonlinear Math. Phys. 2001, 8: 272-277. 10.2991/jnmp.2001.8.s.47

    Article  MathSciNet  Google Scholar 

  3. Yu S, Toda K:Lax pairs, Painlevè properties and exact solutions of the alogero Korteweg-de Vries equation and a new (2+1)-dimensional equation. J. Nonlinear Math. Phys. 2000, 7: 1-13. 10.2991/jnmp.2000.7.1.1

    Article  MathSciNet  Google Scholar 

  4. Li C-X, Ma WX, Liu X-J, Zeng Y-B: Wronskian solutions of the Boussinesq equation-solitons, negatons, positons and complexitons. Inverse Probl. 2007, 23: 279-296. 10.1088/0266-5611/23/1/015

    Article  MathSciNet  Google Scholar 

  5. Ma WX, Maruno K: Complexiton solutions of the Toda lattice equation. Physica A 2004, 343: 219-237.

    Article  MathSciNet  Google Scholar 

  6. Ma WX, You Y: Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 2005, 357: 1753-1778. 10.1090/S0002-9947-04-03726-2

    Article  MathSciNet  Google Scholar 

  7. Wazwaz A-M: New higher-dimensional fifth-order nonlinear equations with multiple soliton solutions. Phys. Scr. 2011., 84: Article ID 025007

    Google Scholar 

  8. Wazwaz A-M:A new (2+1)-dimensional Korteweg-de Vries equation and its extension to a new (3+1)-dimensional Kadomtsev-Petviashvili equation. Phys. Scr. 2011., 84: Article ID 035010

    Google Scholar 

  9. Asaad MG, Ma WX:Extended Gram-type determinant, wave and rational solutions to two (3+1)-dimensional nonlinear evolution equations. Appl. Math. Comput. 2012, 219: 213-225. 10.1016/j.amc.2012.06.007

    Article  MathSciNet  Google Scholar 

  10. Ablowitz MJ, Clarkson PA: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York; 1991.

    Book  Google Scholar 

  11. Parkes EJ, Duffy BR: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun. 1996, 98: 288-300. 10.1016/0010-4655(96)00104-X

    Article  Google Scholar 

  12. Fan E: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 2000, 277: 212-218. 10.1016/S0375-9601(00)00725-8

    Article  MathSciNet  Google Scholar 

  13. 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

    Article  Google Scholar 

  14. Zhang S, Xia T: A generalized new auxiliary equation method and its applications to nonlinear partial differential equations. Phys. Lett. A 2007, 363: 356-360. 10.1016/j.physleta.2006.11.035

    Article  MathSciNet  Google Scholar 

  15. He J-H: An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int. J. Mod. Phys. B 2008, 22: 3487-3578. 10.1142/S0217979208048668

    Article  Google Scholar 

  16. Hirota R: Soliton solutions to the BKP equations. I. The Pfaffian technique. J. Phys. Soc. Jpn. 1989, 58: 2285-2296. 10.1143/JPSJ.58.2285

    Article  MathSciNet  Google Scholar 

  17. Zakharov V, Shabat A: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl. 1979, 13: 166-174.

    Article  MathSciNet  Google Scholar 

  18. Satsuma J, Kaup DJ: A Bäcklund transformation for a higher order Korteweg-de Vries equation. J. Phys. Soc. Jpn. 1977, 43: 692-697. 10.1143/JPSJ.43.692

    Article  MathSciNet  Google Scholar 

  19. Gu C, Hu H, Zhou Z: Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Kluwer Academic, London; 2005.

    Book  Google Scholar 

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

    Google Scholar 

  21. Wang M, Li X, Zhang J: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

    Article  MathSciNet  Google Scholar 

  22. Lou S, Ma H-C:Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems obtained from a simple direct method. J. Phys. A, Math. Gen. 2005, 38: L129-L137. 10.1088/0305-4470/38/7/L04

    Article  MathSciNet  Google Scholar 

  23. Dai Z, Lin S, Fu H, Zeng X: Exact three-wave solutions for the KP equation. Appl. Comput. Math. 2010, 216: 1599-1604. 10.1016/j.amc.2010.03.013

    Article  MathSciNet  Google Scholar 

  24. Ma WX, Wu H, He J-S: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A 2007, 364: 29-32. 10.1016/j.physleta.2006.11.048

    Article  MathSciNet  Google Scholar 

  25. Ma WX, Lee J-H:A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation. Chaos Solitons Fractals 2009, 42: 1356-1363. 10.1016/j.chaos.2009.03.043

    Article  MathSciNet  Google Scholar 

  26. Ma WX, Fuchssteiner B: Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation. Int. J. Non-Linear Mech. 1996, 31: 329-338. 10.1016/0020-7462(95)00064-X

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  28. Ma WX, Zhu Z:Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Comput. Math. 2012, 218: 11871-11879. 10.1016/j.amc.2012.05.049

    Article  MathSciNet  Google Scholar 

  29. Li Z, Dai Z, Liu J:Exact three-wave solutions for the (3+1)-dimensional Jimbo-Miwa equation. Comput. Math. Appl. 2011, 61: 2062-2066. 10.1016/j.camwa.2010.08.070

    Article  MathSciNet  Google Scholar 

  30. Wang C, Dai Z, Liang L: Exact three-wave solution for higher dimensional KdV-type equation. Appl. Comput. Math. 2010, 216: 501-505. 10.1016/j.amc.2010.01.057

    Article  MathSciNet  Google Scholar 

  31. Boiti M, Leon JJ-P, Pempinelli F: On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Probl. 1986, 2: 271-279. 10.1088/0266-5611/2/3/005

    Article  MathSciNet  Google Scholar 

  32. Bai C-J, Zhao H:New solitary wave and Jacobi periodic wave excitations in (2+1)-dimensional Boiti-Leon-Manna-Pempinelli system. Int. J. Mod. Phys. B 2008, 22: 2407-2420. 10.1142/S021797920803954X

    Article  MathSciNet  Google Scholar 

  33. Liu GT: Bäcklund transformation and new coherent structures of the potential BLMP system. J. Inn. Mong. Norm. Univ. 2008, 37: 145-148.

    Google Scholar 

  34. Liu N, Liu X:Symmetries, new exact solutions and conservation laws of (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Chin. J. Quantum Electron. 2008, 25: 546-552.

    Google Scholar 

  35. Luo L: New exact solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli equation. Phys. Lett. A 2011, 375: 1059-1063. 10.1016/j.physleta.2011.01.009

    Article  MathSciNet  Google Scholar 

  36. Zhang LL: Exact solutions of breaking soliton equations and BLMP equation. J. Liaocheng Univ. Nat. Sci 2008, 21: 35-38.

    Google Scholar 

  37. Li Y, Li D:New exact solutions for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Appl. Math. Sci. 2012, 6: 579-587.

    MathSciNet  Google Scholar 

  38. Delisle L, Mosaddeghi M: Classical and SUSY solutions of the Boiti-Leon-Manna-Pempinelli equation. J. Phys. A, Math. Theor. 2013., 46: Article ID 115203

    Google Scholar 

  39. Darvishi M, Najafi M, Kavitha L, Venkatesh M:Stair and step soliton solutions of the integrable (2+1) and (3+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equations. Commun. Theor. Phys. 2012, 58: 785-794. 10.1088/0253-6102/58/6/01

    Article  Google Scholar 

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

    Book  Google Scholar 

  41. Ma WX: Complexiton solutions to the Korteweg-de Vries equation. Phys. Lett. A 2002, 301: 35-44. 10.1016/S0375-9601(02)00971-4

    Article  MathSciNet  Google Scholar 

  42. Ma WX: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys. Conf. Ser. 2013., 411: Article ID 012021

    Google Scholar 

  43. Ma WX: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2011, 2: 140-144.

    Google Scholar 

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Acknowledgements

The authors are in debt to thank the anonymous referees for helpful suggestions. The work is supported by the National Natural Science Foundation of China (project No. 11371086), the Fund of Science and Technology Commission of Shanghai Municipality (project No. ZX201307000014) and the Fundamental Research Funds for the Central Universities.

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Ma, H., Bai, Y. & Deng, A. Exact three-wave solutions for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Adv Differ Equ 2013, 321 (2013). https://doi.org/10.1186/1687-1847-2013-321

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