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Exact three-wave solutions for the (3 + l)-dimensional

Boiti-Leon-Manna-Pempinelli equation

Hongcai Ma1,2*, Yongbin Bai1 and Aiping Deng1

"Correspondence: hongcaima@hotmail.com 1 Department of Applied Mathematics, Donghua University, Shanghai, 201620, China 2Faculty of Science, Ningbo University, Ningbo, 315211, China

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.

Keywords: (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation; extended three-soliton method; periodic wave; MAPLE

ft Springer

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 Backlund 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

©2013 Ma et al.; licensee Springer. This is an Open Access article distributed under the terms ofthe Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

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 + l)-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 [3l]:

where u = u(x,y, t) and subscripts represent partial differentiation with respect to the given variable. Boiti etal. [31] also discussed the Painleve property, Lax pairs and some exact solutions of (2 + 1)-dimensional BLMP. Through the Backlund 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 Backlund transformation of a potential BLMP system by using the standard truncated Painleve 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 etal. [37] constructed a Backlund 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

uyt + uzt + uxxxy + uxxxz — 3 Ux (uxy + Uxz) — 3 Uxx (uy + uz) = 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 , ut, ux, uz, utt, utx, utz, uxx, uxz, uzz,.. • ) = 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

DmDXa(t,x)-b(t,x) = — dy^a(t + s,x + y)b(t _ s,x _y)|s=o,y=o. (4)

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

F (D, Dt, Dx, Dz, Dtt, Dtx, Dtz,...)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(f) + a_i exp(_9) +ai exp(9) + a2 sinh(n), (6)

where f = p1(x + nz + fay + «1t), n = P2(x + Y2z + fay + a2t), 9 = p3(x + Y3z + fty + a3t) and Pi, au fa, yi (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(n) cos(f), sinh(n) exp(9), sinh(n) exp(_9), cosh(n) sin(f), cosh(n) exp(9), cosh(n) exp(_9), sin(f) exp(9), sin(f) exp(_9), cos(f) exp(9), cos(f) exp(_9) to zero, we can derive a set of algebraic equations for a_1, a1, a2, pi, ai, fai, yi (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, a1, a2, pi, ai, fa, yi (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-Pempi-nelli equation Eq. (2):

uyt + uzt + uxxxy + uxxxz _ 3ux(uxy + uxz) _ 3uxx(uy + uz) = 0,

or, equivalently,

(uy + uz)t + (uy + uz)xxx _ 3ux(uy + uz)x _ 3uxx(uy + uz) = 0. (7)

Under the dependent variable transformation,

u = -2 (lnf )x, (8)

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

_fyft _fzft _fxxxfy _ 3fxxyfx + 3fxxfxy _fxxxfz _ 3fxxzfx + 3fxxfxz +fytf +fztf +fxxxyf +fxxxzf =

0. (9)

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

(DyDt + DzDt + DyDl + D„Dl)f f = 0. (10)

According to the methodology in Section 2, we can derive a set of algebraic equations for a_i, ai, «2, Pi, ai, Pi, Yi (i = 1,2,3)

sinh(rç) cos(f ) :

_3p2_p2Yi2 + P2«2^2 -P2aiYi + P2«2Y2 -pfciA - 3plplP2 - 3p2p2^ - 3p2p2Y2

+ Pi A + PlA + Pi Yi + P2 Y2 = 0, sinh(rç) exp(0), sinh(rç) exp(-0) :

3p!p2 Y2 + 3p2p3 Y3 + P^Y2 + P4A + 3p3p2ft + ptfo + 3p|p3^3 + P3 Y3 + P2AY3

+ p\a2Y2 + P^A + P2 Y3a3 = 0, cosh(rç) sin(f ) :

-3p2^i + 3p2A + p2Yi - 3p2Yi + 3p\Y2 -p2Y2 + A ai + Yia2 + Y2ai

+ Aa2 -P2A + P2^i = 0,

cosh(rç) exp(0), cosh(rç) exp(-0) :

+ 3p2^2 + Aa3 + 3p2^3 + Y3a2 + p"^Y3 + Y2a3 + p23Y2 + P2A + 3p\Y2 + p2A + 3p3y3 = 0, sin(f ) exp(0), sin(f ) exp(-0) :

-Pa + plfo + p2Y3 -P33Yi + 3p2A + 3p2Yi -p3A - Aai - 3p2Y3

- Yia3 - 3p23P3 - Y3ai = 0, cos(f ) exp(0), cos(f ) exp(-0) :

p2Aa3 -piWi - 3p2p2P3 + p2Y3a3 - 3p2piAi + PjAi - 3p2p3Y3 + P4A

- 3PaP2 Yi + Pi Yi -P2aiAi + P4Y3 = 0,

and constant term:

-«2p2a2 A - a^a2Y2 -P^iA -P2aiYi - ialpjfo - 4«2p4y2 + i6«_ip2«iA + i6a_ip2 aiY3 + 4a-iP2Aaia3 + la-ip^aa + 4p4A + IpjVi = 0.

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

a_i = a_i, «i = «i, a2 = a2, Pi = 0, P2 = 0, P3 = P3, ai = ai, a2 = a2, a3 = -p3,

(a) t=-8 (b) t=0 (c) t=8

Figure 1 The propagation of solution u1 with a-1 = -1, a1 = 1, p3 = 1, ~y3 = 1, z =1.

fag + na2 + Y2«i y

Pi =--, P2 = P2< P3 = -Y3>

Yi = Yi, Y2 = Y2, Y3 = Y3. In this case, we obtain the single soliton solution

ui = -2—a-ip3£-2 + aiP3e---, (ii)

1 + a-ie-P3[x+Y3(z-y)-p21] + aieP3[x+Y3(z-y)-p3t\

where a-i, ai, p3, y3 are free constants. The propagation of solution ui is described in Figure i. Case 2.

a-i = a-i, ai = ai, a2 = a2, pi = pi, P2 = 0, p3 = 0, ai = pp a2 = a2, «3 = «3,

a p3«2 + Y3«2 + Y2«3 „ „

Pi = -Yi, P2 =--, P3 = P3,

Yi = Yi, Y2 = Y2, Y3 = Y3.

Then we obtain new periodic solutions as follows:

u = 2Pi sin(pi [x + Yi(z - y) +p\i]) (2)

cos(pi[x + yi(z - y) +p21]) + a-i + ai'

where a-i, ai, pi, yi are free constants. The propagation of solution u2 is described in Figure 2. Case 3.

a-i = 0, ai = ai,

Pi= Pi,

ai = ai, «2 = -p2, P2 = -Y2,

a2 = a2, Pi = 0, p2= P2, P3= P3,

«3 = -pi

P3 = - Y3, Yi = Yi, Y2 = Y2, Y3 = Y3.

We then obtain

al^[«k^)-^] + a2p2 sinh(p2[x + y2(z -y)-p2t]) U3 = — 2-2-, (i3)

1 + a1eP3[x+y3(z—y)—P3t] + azcosh(pz[x + Yz(z-y) -p\t])

where a1, a2, p2, p3, Y3 are arbitrary constants. Case 4.

a-1 = -—, ai = ai, a2 = a2, pi = P3 i, P2 = 0, P3 = P3,

ai = -p2, a2 = -3p2, a3 = -p2,

Pi = -Yi - - Y3, P2 = P2, P3 = P2, Yi = Yi, Y2 = Y2, Y3 = Y3. We then obtain a complexiton solution

-p3isin(£) - p3e'p3(X+43Z+P3y-p2t) + amepi^Yi^y^

U4 = -2--—-, (i4)

cosfê ) + e^"*™2* + «1^+ Y2Z+A2У_p2t)

where f = ip3(x + Yiz + (- Yi - A3 - Y3)y _p3t) and ai,p3, a3, p3, y1, Y3 are free constants. Case 5.

a_i = a_i, «i = «i, «2 = «2, Pi = Pi, P2 = 0, P3 = P3, ai = p2, a2 = a2, a3 = a3, Ai = - Yi, P2 = - Y2, A3 = - Y3, Yi = Yi, Y2 = Y2, Y3 = Y3.

We then obtain new periodic cross-kink solutions

-pi sin(fi)-a_1p3e_P2(x+ + а1p3eP2(x+ Y3Z_ Y3y+a3t)

U5 cos(f1) + a_1eP2(x+ V2Z_Y2У+a2t) + aleP2(x+Y2Z_Y2У+a2t) , ( )

where = p1(x + Y1z - Y1y + P21) and a_1, a1, p1, p3, a3, y3 are free constants.

Case 6.

Pi Yi + PiPi + aifirs + fl^PsPs

«-i =--77—--, «i = al> «2 = a2,

4(P3 + Y3)p3«I

Pi= Pi, P2= P3, P3= P3, ai = -3p2+ pj2, «2 = 3p2 -p3, «3 = 3p2 -P2, Pi = Pi, P2=-Y2- Y3 - P3, P3 = P3, Yi = Yi, Y2 = Y2, Y3 = Y3.

We then obtain new periodic cross-kink solutions

_ „ -Pi "hfe) + ^^Yt^3 e &2 + aiP3ee2 + g^p3 cosh(m)

U5 — 2 222222 , (i^)

cosfe) - PlYi+P4(;3++agP23a+ia2P3p3 e-fl2 + aie®2 + «2 sinh(m)

where & = Pi(x + Yiz + Piy + (-3p2 + P2)t), 02 = P3(x + Y3Z + P3y + (3P2 -P23)t), n2 = P3(x + Y2Z +(-Y2 - Y3-P3)y +(3P2 -P2)t) and «i, «2, Pi, P3, Pi, P3, Yi, Y2, Y3 are free constants. Case 7.

fl_i = 0, ai = «i, «2 = «2, Pi = ¿P2, P2 = P2,

„ 2 „ 2 3P3P2 + p3 - 3p|p2 + 3p3 P3= P3, ai = -4p2, a2 = -4p2, «3 = - 1 12 -—-—,

2 2 P 3

a P3p3 + P3Y3 + P2Yi „ P3p3 + P2Y2 + P3Y3 a a

Pi =--, P2 =--, P3 = P3,

Yi = Yi, Y2 = Y2, Y3 = Y3. We then obtain a new complexiton solution

, „ 3P3P2+Pi-3P3P2+3P2

„ -iP2 sinfe) + alPзePз(x+Ysz+P3y- -3 t) + «2P2 cosh(n3) _ u7 = -2-:—2 3,2—71-, (i7)

cosfe) + a1eP3(x+V3Z+^y--"-'-r-3-1 f) + « sinh(n

where & = p1(x + Yiz - P3p3+PPY3+P2Y1 y - 4-2i), n3 = Pi(x + m - y - 4-2i) and

«i> a2, p2, p3, p3, y1) y2, y3 are free constants. Case 8.

fl-1 = fl-i, «1 = «1, «2 = «2, Pi = Pi, P2 = P2,

-3 = 0, ai = Pp a2 = -p2, «3 = a3,

Pi = -Yi, P2 = -2, P3 = P3,

Yi = Yi, Y2 = Y2, Y3 = Y3. We then obtain new periodic cross-kink solutions

-Pi sin (Pi (x + Yi z - Yiy + P2t)) + «2P2 cosh(P2(x + Y2Z - Y2 y - P2^)) cos(Pi(x + Yiz - Yiy + P2t)) + «-i + «i + «2 sinh(P2(x + Y2z - Y2y -P\t)) '

where «_i, «i, «2, Pi, P2, a3, p3, y1, y2 and y3 are free constants.

(a) t=0 (b) t=3 (c) t=5

Figure 3 The propagation of solution u3 with o1 = 1, o2 = 1, p2 = 1, P3 = 1, Y2 = 1, Y3 = 1, z = 1-

Case 9.

(alfa + fai + yi + alft)^l

«—i =-77;;-r-n-«1 = «1, «2 = «2, Pi = IP2,

4(P3 + Y3)p3«i

P2= P2, P3= Pз, = -3P2 -p2, a2 = —3p2 — P2, a3 = —3P2 — P2,

Pi = A, A = —Pi — Y2 — Yi, P3 = P3, Yi = Yi, Y2 = Y2, Y3 = Y3.

We then obtain a complexiton solution

—P sin(^4) — ^^YY+t 6—04 + aiP3e6i + «2P2 00Sh(^4)

Hg = —2-2-2-, (!")

cosfe) + a2fi+Pi+Yi+2a2Yi e—04 + «ie04 + ««2 cosh(n4)

4(P3+Y3)p2«I

where = ip2(x + Yiz + Piy + (—3P2 — p2)t), 04 = P3(x + Y3^ + P3y + (—3p2 — P23)t), n4 = P2(x + Y2Z + (—Pi — Y2 — Yi)y + (—3p2 — P2)t) and «i, a 2, P2, P3, Pi, P3, Yi, Y2, Y3 are free constants. Figures 3, 4, 5, 6 described the solution of H3, H4, H6 and h8 respectively.

Remark 1 Noting if we set fa = — in Case i 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.

(a) t=-20 (b) t=0 (c) t=20

Figure 5 The propagation of solution u6 with a1 = 1, a2 = 1, p1 = 1, p3 = 1, = 1, ^3 = 1, y = 1, y2=-1, y3 = 1, z =1.

(a) t=-10 (b) t=0 (c) t=10

Figure 6 The propagation of solution u8 with a-1 = 1, a1 =1, a2 = 1, p1 =1, p2 = 1, y1 = 1, y2 = -1, z =1.

Remark 2 Noting sinh(ix) = i sin(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 + l)-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 + l)-dimensional Boiti-Leon-Manna-Pempinelli equation and also that the (3 + l)-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.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Allauthors contributed equally and significantly in writing this paper. Allauthors read and approved the finalmanuscript.

Acknowledgements

The authors are in debt to thank the anonymous referees for helpfulsuggestions. The work is supported by the National

NaturalScience Foundation of China (project No. 11371086), the Fund of Science and Technology Commission of

Shanghai Municipality (project No. ZX201307000014) and the FundamentalResearch Funds for the Central Universities.

Received: 11 July 2013 Accepted: 26 September 2013 Published: 18 Nov 2013

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10.1186/1687-1847-2013-321

Cite this article as: Ma et al.: Exact three-wave solutions for the (3 + 1 )-dimensional Boiti-Leon-Manna-Pempinelli equation. Advances in Difference Equations 2013, 2013:321

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