DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2016/5420156 5420156 Research Article New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation http://orcid.org/0000-0001-9822-6160 Su Jun 1 http://orcid.org/0000-0002-1472-3950 Xu Genjiu 2 Karachalios Nikos I. 1 School of Science Xi’an University of Science and Technology Xi’an 710054 China xust.edu.cn 2 Department of Applied Mathematics Northwestern Polytechnical University Xi’an 710072 China nwpu.edu.cn 2016 2872016 2016 20 04 2016 06 06 2016 2016 Copyright © 2016 Jun Su and Genjiu Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.

National Natural Science Foundation of China 11402194 11501442 71271171 Shaanxi Provincial Education Department 2013JK0584 Science and Technology Research and Development Program in Shaanxi Province of China 2014KW03-01
1. Introduction

In recent years, the problem of finding exact solutions of nonlinear evolution equations (NLEEs) is very popular for both mathematicians and physicists. Because seeking exact solutions of NLEEs is of great significance in nonlinear dynamics, many methods such as the inverse scattering transformation , Hirota’s bilinear method , the Darboux transformation , the sine-cosine method , G/G-expansion method [5, 6], and the transformed rational function method  have been proposed. The Wronskian method which is based on the bilinear form of the NLEEs was proposed by Freeman and Nimmo in [8, 9]. It is a fairly powerful tool to construct exact solutions of NLEEs in terms of the Wronskian determinant. By means of the method, the exact solutions of some NLEEs are obtained .

The study of the BKP equation has attracted a considerable size of research work. These equations were studied using the Hirota method, the multiple exp-function algorithm, the Pfaffian technique, Riemann theta functions, the extended homoclinic test approach, and Bäcklund transformation by many authors . In this paper, based on the Wronskian method, the new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions of the (3+1)-dimensional generalized BKP equations are investigated.

In this paper, we will consider the following (3+1)-dimensional generalized BKP equation:(1)uyt-uxxxy-3uxuyx+6uxx-3uzz=0.When z=x, this (3+1)-dimensional generalized BKP equation reduces to the BKP equation [27, 28]:(2)uyt-uxxxy-3uxuyx+3uxx=0.By the dependent variable transformation(3)u=2lnfx=2fxf,the (3+1)-dimensional generalized BKP equation (1) becomes a bilinear form(4)DyDt-Dx3Dy+6Dx2-3Dz2f·f=0,where Dt, Dx, Dy, and Dz are the Hirota operators :(5)ffyt-fxxxy+6fxx-3fzz-fyft+fxxxfy+3fxxyfx-3fxxfxy-6fx2+3fz2=0.

We will show this (3+1)-dimensional generalized BKP equation has a class of Wronskian solutions with all generating functions for matrix entries satisfying a linear system of partial differential equations involving a free parameter. Rational solutions, solitons, positons, negatons, and interaction solutions to (1) among Wronskian determinant solutions are constructed and a few plots of particular solutions are made.

The paper is organized as follows. In Section 2, we derive a Wronskian formulation for the (3+1)-dimensional generalized BKP equation. In Section 3, Wronskian solutions to the (3+1)-dimensional generalized BKP equation are obtained. Section 4 presents the conclusion.

2. A Wronskian Formulation

The Wronskian technique is a powerful tool to construct exact solutions to bilinear differential or difference equations. To use the Wronskian technique, we adopt the compact notation introduced by Freeman and Nimmo [8, 9]:(6)Wϕ1,ϕ2,,ϕN=N-1^;Φ=N-1^=ϕ10ϕ11ϕ1N-1ϕ20ϕ21ϕ2N-1ϕN0ϕN1ϕNN-1,where(7)Φ=ϕ1,,ϕNT,ϕi0=ϕi,ϕij=jxjϕi,j1,1iN.Solutions determined by u=2(lnf)x with f=|N-1^| to the (3+1)-dimensional generalized BKP equation (1) are called Wronskian solutions.

Theorem 1.

Assuming that a group of functions ϕi=ϕi(x,y,z,t), 1iN, satisfies the following linear conditions(8)-ϕi,xx=j=1Nλijtϕj,(9)ϕi,y=kϕi,x,(10)ϕi,z=2ϕi,x,(11)ϕi,t=4ϕi,xxx,where k is an arbitrary nonzero constant, then the Wronskian determinant f=|N-1^| defined by (6) solves the bilinear equation (5).

Proof.

Obviously, we have (12)fx=N-2^,N,fxx=N-3^,N-1,N+N-2^,N+1,fxxx=N-4^,N-2,N-1,N+2N-3^,N-1,N+1+N-2^,N+2.Using conditions (9), (10), and (11), we get that (13)fy=kN-2^,N,fxy=kN-3^,N-1,N+N-2^,N+1,fxxy=kN-4^,N-2,N-1,N+2N-3^,N-1,N+1+N-2^,N+2,fxxxy=kN-5^,N-3,N-2,N-1,N+3N-4^,N-2,N-1,N+1+2N-3^,N,N+1+3N-3^,N-1,N+2+N-2^,N+3,fz=2N-2^,N,fzz=2N-3^,N-1,N+N-2^,N+1,ft=4N-4^,N-2,N-1,N-N-3^,N-1,N+1+N-2^,N+2,fyt=4kN-5^,N-3,N-2,N-1,N-N-3^,N,N+1+N-2^,N+3.

Under (8), it is not difficult to obtain  (14)N-1^i=1Nλiiti=1NλiitN-1^=i=1NλiitN-1^2=N-2^,N+1-N-3^,N-1,N2=N-1^N-5^,N-3,N-2,N-1,N-N-4^,N-2,N-1,N+1+2N-3^,N,N+1-N-3^,N-1,N+2+N-2^,N+3.Therefore, (15)ffyt-fxxxy+6fxx-3fzz=3kN-1^N-5^,N-3,N-2,N-1,N-N-4^,N-2,N-1,N+1-2N-3^,N,N+1-N-3^,N-1,N+2+N-2^,N+3=3kN-2^,N+1-N-3^,N-1,N2-12kN-3^,N,N+1N-1^-fyft+fxxxfy+3fxxyfx-6fx2+3fz2=12kN-3^,N-1,N+1N-2^,N-3fxxfxy=-3kN-2^,N+1-N-3^,N-1,N2-12kN-3^,N-1,NN-2^,N+1.Substitution of the above results into (4) finally leads to the following Plücker relation:(16)DyDt-Dx3Dy+6Dx2-3Dz2f·f=-12kN-3^,N,N+1N-1^+12kN-3^,N-1,N+1N-2^,N-12kN-3^,N-1,NN-2^,N+1=0.

Theorem 1 tells us that if a group of functions ϕi=ϕi(x,y,z,t), 1iN, satisfies the linear conditions in (8)–(11), then we can get a solution f=|N-1^| to the bilinear BKP equation (4). The corresponding solution of (1) is(17)u=2lnfx=2fxf=2N-2^,NN-1^.

Remark 1.

From the compatibility conditions ϕi,xxt=ϕi,txx, 1iN, of conditions (8)–(11), we have the equality (18)j=1Nλijtϕj=0,1iN,and thus it is easy to see that the Wronskian determinant W(ϕ1,ϕ2,,ϕN) becomes zero if there is at least one entry λij satisfying λij(t)0.

Remark 2.

If the coefficient matrix Λ=(λij) is similar to another matrix M=(μij) under an invertible constant matrix P, let us say Λ=P-1MP, then Φ~=PΦ solves (19)-Φ~xx=MΦ~,Φ~y=kΦ~x,Φ~z=2Φ~x,Φ~t=4Φ~xxx,and the resulting Wronskian solutions to (1) are the same: (20)uΛ=2xlnΦ0,Φ1,,ΦN-1=2xlnPΦ0,PΦ1,,PΦN-1=uM.Based on Remark 1, we only need to consider case of (8)–(11) under dΛ/dt=0, that is, the following conditions:(21)-ϕi,xx=j=1Nλijtϕj,ϕi,y=kϕi,x,ϕi,z=2ϕi,x,ϕi,t=4ϕi,xxx,where Λ=(λij) is an arbitrary real constant matrix. Moreover, Remark 2 tells us that an invertible constant linear transformation on Φ in the Wronskian determinant does not change the corresponding Wronskian solution, and thus, we only have to solve (21) under the Jordan form of Λ.

3. Wronskian Solutions

In principle, we can construct general Wronskian solutions of (1) associated with two types of Jordan blocks of the coefficient matrix Λ. But it is not easy. In this section we will present a few special Wronskian solutions to the generalized BKP equation, together with examples of exact solutions.

It is well known that the corresponding Jordan form of a real matrix (22)Λ=Jλ101Jλ201Jλmn×nhas the following two types of blocks:

(I) (23)Jλi=λi01λi01λiki×ki

(II) (24)Jλi=Λi0I2Λi0I2Λili×li,Λi=αi-βiβiαi,I2=1001,

where λi, αi, βi are all real constants. The first type of blocks has the real eigenvalue λi with algebraic multiplicity ki(Σi=1mki=N), and the second type of blocks has the complex eigenvalue λi±=αi±βi-1 with algebraic multiplicity li.

3.1. Rational Solutions

Suppose Λ has the first type of Jordan blocks. Without loss of generality, let (25)Jλ1=λ101λ101λ1k1×k1.

In this case, if the eigenvalue λ1=0, J(λ1) becomes of the following form: (26)0010010k1×k1.From condition (21), we get(27)ϕ1,xx=0,-ϕi+1,xx=ϕi,ϕi,y=kϕi,x,ϕi,z=2ϕi,x,ϕi,t=4ϕi,xxx,i1.Such functions ϕi(i1) are all polynomials in x, y, z, and t, and a general Wronskian solution to the (3+1)-dimensional generalized BKP equation (1) (28)u=2xlnWϕ1,ϕ2,,ϕk1is rational and is called a rational Wronskian solution of order k1-1.

From (27), we solve ϕ1,xx=0, ϕ1,y=kϕ1,x, ϕ1,z=2ϕ1,x, ϕ1,t=4ϕ1,xxx and have (29)ϕ1=c1+c2x+ky+2z,where c1, c2, and k0 are all real constants. Similarly, by solving -ϕi+1,xx=ϕ1, ϕi+1,y=kϕi+1,x, ϕi+1,z=2ϕi+1,x, ϕi+1,t=4ϕi+1,xxx, i1, then two special rational solutions of lower-order are obtained after setting some integral constants to be zero.

( 1) Zero-Order. When c1=0, c2=1, ϕ1=x+ky+2z, we have the corresponding Wronskian determinant f=W(ϕ1)=x+ky+2z and the associated rational Wronskian solution of zero-order:(30)u=2xlnWϕ1=2x+ky+2z.

( 2) First-Order. Taking c1=0, c2=-1, ϕ1=-(x+ky+2z), we have ϕ2=(1/6)(x+ky+2z)3+4t. In this case, the corresponding Wronskian determinant is f=W(ϕ1,ϕ2)=-(1/3)(x+ky+2z)3+4t, and the rational Wronskian solution of first-order reads(31)u=2xlnWϕ1,ϕ2=2x+ky+2z21/3x+ky+2z3-4t.

( 3) Second-Order. Taking ϕ1=x+ky+2z, ϕ2=-(1/6)(x+ky+2z)3-4t, we have ϕ3=(1/120)(x+ky+2z)5+2(x+ky+2z)2t. Then the Wronskian determinant is f=W(ϕ1,ϕ2,ϕ3)=-(1/45)(x+ky+2z)6+(4/3)(x+ky+2z)3t+16t2, and the rational Wronskian solution of second-order is given by(32)u=2xlnWϕ1,ϕ2,ϕ3=-2/15x+ky+2z5+4x+ky+2z2t-1/45x+ky+2z6+4/3x+ky+2z3t+16t2.

3.2. Solitons, Positons, and Negatons

If the eigenvalue λi0, J(λi) becomes of the following form: (33)λi01λi01λiki×ki.We start from the eigenfunction ϕi(λi) determined by(34)-ϕiλixx=λiϕiλi,ϕiλiy=kϕiλix,ϕiλiz=2ϕiλix,ϕiλit=4ϕiλixxx.General solutions to this system in two cases of λi>0 and λi<0 read as(35)ϕiλi=C1isinλix+ky+2z-4tλi+C2icosλix+ky+2z-4tλi,ϕiλi=C3isinh-λix+ky+2z-4tλi+C4icosh-λix+ky+2z-4tλi,respectively, where C1i, C2i, C3i, and C4i are arbitrary real constants. By an inspection, we find that (36)-ϕiλi11!λiϕiλi1ki-1!λiki-1ϕiλixx=λi01λi01λiki×kiϕiλi11!λiϕiλi1ki-1!λiki-1ϕiλi,1j!λijϕiλiy=k1j!λijϕiλix,1j!λijϕiλiz=21j!λijϕiλix,1j!λijϕiλit=41j!λijϕiλixxx,0jki-1.Therefore, through this set of eigenfunctions, we obtain a Wronskian solution to (1):(37)u=2xlnWϕiλi,11!λiϕiλi,,1ki-1!λiki-1ϕiλi,which corresponds to the first type of Jordan blocks with a nonzero real eigenvalue.

When λi>0, we get positon solutions , and when λi<0, we get negaton solutions . If we suppose Λ have n different nonzero real eigenvalues, in which there are l positive real eigenvalues and n-l negative real eigenvalues, then a more general positon can be obtained by combining l sets of eigenfunctions associated with different λi>0: (38)u=2xlnWϕ1λ1,11!λ1ϕ1λ1,,1k1-1!λ1k1-1ϕ1λ1;;ϕlλl,11!λlϕlλl,,1kl-1!λlkl-1ϕlλl.Similarly, a more general negaton can be obtained by combining n-l sets of eigenfunctions associated with different λi<0: (39)u=2xlnWϕ1λ1,11!λ1ϕ1λ1,,1k1-1!λ1k1-1ϕ1λ1;;ϕn-lλn-l,11!λn-lϕn-lλn-l,,1kn-l-1!λn-lkn-l-1ϕn-lλn-l.This solution is called an l-positon of order (k1-1,k2-1,,kl-1) or n-l-negaton of order (k1-1,k2-1,,kn-l-1). If l=n or l=0, we simply say that it is an n-positon of order n or an n-negaton of order n.

( 1) Solitons. An n-soliton solution is a special n-negaton:(40)u=2xlnWϕ1,ϕ2,,ϕn,with ϕi being given by(41)ϕi=cosh-λix+ky+2z-4tλi+γi,iodd,ϕi=sinh-λix+ky+2z-4tλi+γi,ieven,where λ1<λ2<<λn<0 and γi(1in) are arbitrary real constants. For example, a 1-soliton to (1) is given by(42)u=2xlnWϕ1=2xlncosh-λ1x+ky+2z-4tλ1+γ1=2-λ1tanhθ1,where θ1=-λ1(x+ky+2z-4tλ1)+γ1.

Similarly, we have a 2-soliton to (1):(43)u=2xlnWcoshθ1,sinhθ2=2λ1-λ2sinhθ1-θ2-sinhθ1+θ2-λ1--λ2coshθ1+θ2--λ1+-λ2coshθ1-θ2,where θi=-λi(x+ky+2z-4tλi)+γi, i=1,2. Figures 1 and 2 of three-dimensional plots show the n-soliton to (1) defined by (40) on the indicated specific regions, with specific values being chosen for the parameters.

The shape of the 1-soliton to (1) with λ1=-2, y=0, t=1, γ1=0.

The shape of the 2-soliton to (1) with λ1=-4, λ2=-1, y=0, t=0, γ1=γ2=0.

( 2) Positons. Two kinds of special positons of order k1-1 are(44)u=2xlnWϕ,λϕ,,λk1-1ϕ,ϕ=cosλx+ky+2z-4tλ+γλ,(45)u=2xlnWϕ,λϕ,,λk1-1ϕ,ϕ=sinλx+ky+2z-4tλ+γλ,where λ>0 and γ is an arbitrary function of λ. But these two kinds of positons are equivalent to each other, due to the existence of the arbitrary function γ.

When λ1>0, a 1-positon of zero-order reads(46)u=2xlnWϕ1=2xlncosλ1x+ky+2z-4tλ1+γ1=-2λ1tanθ3,where θ3=λ1(x+ky+2z-4tλ1)+γ3. And a 1-positon of first-order is(47)u=2xlnWcosθ3,λ1cosθ3=4λ11+cos2θ32λ1x+ky+2z-12tλ1+sin2θ3.Figures 3 and 4 of three-dimensional plots show the special positons to (1) defined by (44) on the indicated specific regions, with specific values being chosen for the parameters.

The shape of the 1-positon of zero-order to (1) with λ1=2, y=0, t=1, γ3=0.

The shape of the 1-positon of first-order to (1) with λ1=1.5, y=0, t=1, γ3=0.

( 3) Negatons. Two kinds of special negatons of order k1-1 are(48)u=2xlnWϕ,λϕ,,λk1-1ϕ,ϕ=cosh-λx+ky+2z-4tλ+γ-λ,(49)u=2xlnWϕ,λϕ,,λk1-1ϕ,ϕ=sinh-λx+ky+2z-4tλ+γ-λ,where λ<0 and γ is an arbitrary function of -λ. Similarly, these two kinds of negatons are equivalent to each other.

When λ1<0, a 1-negaton of first-order reads(50)u=2xlnWcoshθ1,λ1coshθ1=4-λ11+cosh2θ12-λ1x+ky+2z-4tλ1+sinh2θ1,where θ1=-λ1(x+ky+2z-4tλ1)+γ1. And the 1-negaton of second-order is given by(51)u=2xlnWcoshθ1,λ1coshθ1,λ12coshθ1=-6λ1δ1+48tλ12sinhθ1-4λ1-λ1δ12coshθ1+6-λ1sinh2θ1coshθ1-2λ1δ12sinhθ1+24tλ1-λ1--λ1δ1coshθ1+sinhθ1cosh2θ1,where δ1=x+ky+2z-12tλ1. Figures 5 and 6 of three-dimensional plots show the special negatons to (1) defined by (48) on the indicated specific regions, with specific values being chosen for the parameters.

The shape of the 1-negaton of first-order to (1) with λ1=-4, y=0, t=0, γ1=0.

The shape of the 1-negaton of second-order to (1) with λ1=-1, y=0, t=0, γ1=0.

3.3. Interaction Solutions

We are now presenting examples of Wronskian interaction solutions among different kinds of Wronskian solutions to the (3+1)-dimensional generalized BKP equation (1).

Let us assume that there are two sets of eigenfunctions(52)ϕ1λ,ϕ2λ,,ϕlλ;ψ1μ,ψ2μ,,ψmμassociated with two different eigenvalues λ and μ, respectively. A Wronskian solution(53)u=2xlnWϕ1λ,ϕ2λ,,ϕlλ;ψ1μ,ψ2μ,,ψmμis said to be a Wronskian interaction solution between two solutions determined by the two sets of eigenfunctions in (52).

In what follows, we would like to show a few special Wronskian interaction solutions. Let us first choose different sets of eigenfunctions: (54)ϕrational=x+ky+2z,ϕsoliton=cosh-λ1x+ky+2z-4tλ1+γ1,ϕpositon=cosλ2x+ky+2z-4tλ2+γ2,where λ1<0, λ2>0, and γi(i=1,2) are arbitrary real constants.

Through three Wronskian interaction solutions between any two of a rational solution, a single soliton and a single positon read as(55)urs=2xlnWϕrational,ϕsoliton=-2λ1x+ky+2zcoshθ1-λ1x+ky+2zsinhθ1-coshθ1,urp=2xlnWϕrational,ϕpositon=2λ2x+ky+2zcosθ2λ2x+ky+2zsinθ2+cosθ2,usp=2xlnWϕsoliton,ϕpositon=2λ2-λ1coshθ1cosθ2-λ1sinhθ1cosθ2+λ2coshθ1sinθ2,where θ1=-λ1(x+ky+2z-4tλ1)+γ1 and θ2=λ2(x+ky+2z-4tλ2)+γ2.

One Wronskian interaction solution involving the three eigenfunctions is given by(56)ursp=2xlnWϕrational,ϕsoliton,ϕpositon=2qp,where (57)p=x+ky+2zλ2-λ1sinhθ1cosθ2+λ1λ2coshθ1sinθ2+λ1-λ2coshθ1cosθ2,q=x+ky+2z-λ1λ2λ1-λ2sinhθ1sinθ2+λ1-λ1sinhθ1cosθ2+λ2λ2coshθ1sinθ2,θ1=-λ1x+ky+2z-4tλ1+γ1,θ2=λ2x+ky+2z-4tλ2+γ2.Of course, we have more general Wronskian interaction solutions among three or more kinds of solutions such as rational solutions, positons, solitons, breathers, and negatons. Roughly speaking, it increases the complexities of rational solutions, positons, solitons, and negatons, respectively, to add zero, positive, negative eigenvalues to the spectrum of the coefficient matrix.

4. Conclusion

In summary we have extended the Wronskian method to a (3+1)-dimensional generalized BKP equation by its bilinear form. Moreover, we obtained some rational solutions, solitons, positons, negatons, and their interaction solutions to this equation by solving the systems of linear partial differential equations. All these show the richness of the solution space of the (3+1)-dimensional generalized BKP equation and the resulting solutions are expected to help understand wave dynamics in weakly nonlinear and dispersive media.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant nos. 11402194, 11501442, and 71271171), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 2013JK0584), and the Science and Technology Research and Development Program in Shaanxi Province of China (Grant no. 2014KW03-01).

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