# New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

Authors: Jun Su; Genjiu Xu
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

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.


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## Body

## 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 [1], Hirota’s bilinear method [2], the Darboux transformation [3], the sine-cosine method [4], G′/G-expansion method [5, 6], and the transformed rational function method [7] 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 [10–16].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 [17–26]. 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=2ln⁡fx=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 [2]:(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 Section2, 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^=ϕ10ϕ11⋯ϕ1N-1ϕ20ϕ21⋯ϕ2N-1⋮⋮⋱⋮ϕN0ϕN1⋯ϕNN-1,where(7)Φ=ϕ1,…,ϕNT,ϕi0=ϕi,ϕij=∂j∂xjϕi,j≥1,1≤i≤N.Solutions determined by u=2(ln⁡f)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 functionsϕi=ϕi(x,y,z,t), 1≤i≤N, 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 [10] (14)N-1^∑i=1Nλiit∑i=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.Theorem1 tells us that if a group of functions ϕi=ϕi(x,y,z,t), 1≤i≤N, 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=2ln⁡fx=2fxf=2N-2^,NN-1^.Remark 1.
From the compatibility conditionsϕi,xxt=ϕi,txx, 1≤i≤N, of conditions (8)–(11), we have the equality (18)∑j=1Nλijtϕj=0,1≤i≤N,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 matrixΛ=(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=2∂xlnΦ0,Φ1,…,ΦN-1=2∂xln⁡PΦ0,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=Jλ101Jλ2⋱⋱01Jλmn×nhas the following two types of blocks:(I)(23)Jλi=λi01λi⋱⋱01λiki×ki(II)(24)Jλi=Λi0I2Λi⋱⋱0I2Λili×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

SupposeΛ has the first type of Jordan blocks. Without loss of generality, let (25)Jλ1=λ101λ1⋱⋱01λ1k1×k1.In this case, if the eigenvalueλ1=0, J(λ1) becomes of the following form: (26)0010⋱⋱010k1×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,i≥1.Such functions ϕi(i≥1) 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=2∂xln⁡Wϕ1,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 k≠0 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, i≥1, 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=2∂xln⁡Wϕ1=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=2∂xln⁡Wϕ1,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=2∂xln⁡Wϕ1,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 eigenvalueλi≠0, J(λi) becomes of the following form: (33)λi01λi⋱⋱01λiki×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=C1isin⁡λix+ky+2z-4tλi+C2icos⁡λix+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!∂λiϕiλi⋮1ki-1!∂λiki-1ϕiλixx=λi01λi⋱⋱01λiki×kiϕiλi11!∂λiϕiλi⋮1ki-1!∂λiki-1ϕiλi,1j!∂λijϕiλiy=k1j!∂λijϕiλix,1j!∂λijϕiλiz=21j!∂λijϕiλix,1j!∂λijϕiλit=41j!∂λijϕiλixxx,0≤j≤ki-1.Therefore, through this set of eigenfunctions, we obtain a Wronskian solution to (1):(37)u=2∂xln⁡Wϕiλi,11!∂λiϕiλi,…,1ki-1!∂λiki-1ϕiλi,which corresponds to the first type of Jordan blocks with a nonzero real eigenvalue.Whenλi>0, we get positon solutions [29], and when λi<0, we get negaton solutions [30]. 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=2∂xln⁡Wϕ1λ1,11!∂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=2∂xln⁡Wϕ1λ1,11!∂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=2∂xln⁡Wϕ1,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(1≤i≤n) are arbitrary real constants. For example, a 1-soliton to (1) is given by(42)u=2∂xln⁡Wϕ1=2∂xln⁡cosh-λ1x+ky+2z-4tλ1+γ1=2-λ1tanh⁡θ1,where θ1=-λ1(x+ky+2z-4tλ1)+γ1.Similarly, we have a 2-soliton to (1):(43)u=2∂xln⁡Wcosh⁡θ1,sinh⁡θ2=2λ1-λ2sinh⁡θ1-θ2-sinh⁡θ1+θ2-λ1--λ2cosh⁡θ1+θ2--λ1+-λ2cosh⁡θ1-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.Figure 1
The shape of the 1-soliton to (1) with λ1=-2, y=0, t=1, γ1=0.Figure 2
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=2∂xln⁡Wϕ,∂λϕ,…,∂λk1-1ϕ,ϕ=cos⁡λx+ky+2z-4tλ+γλ,(45)u=2∂xln⁡Wϕ,∂λϕ,…,∂λk1-1ϕ,ϕ=sin⁡λx+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.Whenλ1>0, a 1-positon of zero-order reads(46)u=2∂xln⁡Wϕ1=2∂xln⁡cosλ1x+ky+2z-4tλ1+γ1=-2λ1tan⁡θ3,where θ3=λ1(x+ky+2z-4tλ1)+γ3. And a 1-positon of first-order is(47)u=2∂xln⁡Wcosθ3,∂λ1cosθ3=4λ11+cos⁡2θ32λ1x+ky+2z-12tλ1+sin⁡2θ3.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.Figure 3
The shape of the 1-positon of zero-order to (1) with λ1=2, y=0, t=1, γ3=0.Figure 4
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=2∂xln⁡Wϕ,∂λϕ,…,∂λk1-1ϕ,ϕ=cosh⁡-λx+ky+2z-4tλ+γ-λ,(49)u=2∂xln⁡Wϕ,∂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.Whenλ1<0, a 1-negaton of first-order reads(50)u=2∂xln⁡Wcosh⁡θ1,∂λ1cosh⁡θ1=4-λ11+cosh⁡2θ12-λ1x+ky+2z-4tλ1+sinh⁡2θ1,where θ1=-λ1(x+ky+2z-4tλ1)+γ1. And the 1-negaton of second-order is given by(51)u=2∂xln⁡Wcosh⁡θ1,∂λ1cosh⁡θ1,∂λ12cosh⁡θ1=-6λ1δ1+48tλ12sinh⁡θ1-4λ1-λ1δ12cosh⁡θ1+6-λ1sinh2⁡θ1cosh⁡θ1-2λ1δ12sinh⁡θ1+24tλ1-λ1--λ1δ1cosh⁡θ1+sinh⁡θ1cosh2⁡θ1,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.Figure 5
The shape of the 1-negaton of first-order to (1) with λ1=-4, y=0, t=0, γ1=0.Figure 6
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=2∂xln⁡Wϕ1λ,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=2∂xln⁡Wϕrational,ϕsoliton=-2λ1x+ky+2zcosh⁡θ1-λ1x+ky+2zsinh⁡θ1-cosh⁡θ1,urp=2∂xln⁡Wϕrational,ϕpositon=2λ2x+ky+2zcos⁡θ2λ2x+ky+2zsin⁡θ2+cos⁡θ2,usp=2∂xln⁡Wϕsoliton,ϕpositon=2λ2-λ1cosh⁡θ1cos⁡θ2-λ1sinh⁡θ1cos⁡θ2+λ2cosh⁡θ1sin⁡θ2,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=2∂xln⁡Wϕrational,ϕsoliton,ϕpositon=2qp,where (57)p=x+ky+2zλ2-λ1sinh⁡θ1cos⁡θ2+λ1λ2cosh⁡θ1sin⁡θ2+λ1-λ2cosh⁡θ1cos⁡θ2,q=x+ky+2z-λ1λ2λ1-λ2sinh⁡θ1sin⁡θ2+λ1-λ1sinh⁡θ1cos⁡θ2+λ2λ2cosh⁡θ1sin⁡θ2,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.


## 3.1. Rational Solutions

SupposeΛ has the first type of Jordan blocks. Without loss of generality, let (25)Jλ1=λ101λ1⋱⋱01λ1k1×k1.In this case, if the eigenvalueλ1=0, J(λ1) becomes of the following form: (26)0010⋱⋱010k1×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,i≥1.Such functions ϕi(i≥1) 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=2∂xln⁡Wϕ1,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 k≠0 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, i≥1, 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=2∂xln⁡Wϕ1=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=2∂xln⁡Wϕ1,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=2∂xln⁡Wϕ1,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 eigenvalueλi≠0, J(λi) becomes of the following form: (33)λi01λi⋱⋱01λiki×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=C1isin⁡λix+ky+2z-4tλi+C2icos⁡λix+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!∂λiϕiλi⋮1ki-1!∂λiki-1ϕiλixx=λi01λi⋱⋱01λiki×kiϕiλi11!∂λiϕiλi⋮1ki-1!∂λiki-1ϕiλi,1j!∂λijϕiλiy=k1j!∂λijϕiλix,1j!∂λijϕiλiz=21j!∂λijϕiλix,1j!∂λijϕiλit=41j!∂λijϕiλixxx,0≤j≤ki-1.Therefore, through this set of eigenfunctions, we obtain a Wronskian solution to (1):(37)u=2∂xln⁡Wϕiλi,11!∂λiϕiλi,…,1ki-1!∂λiki-1ϕiλi,which corresponds to the first type of Jordan blocks with a nonzero real eigenvalue.Whenλi>0, we get positon solutions [29], and when λi<0, we get negaton solutions [30]. 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=2∂xln⁡Wϕ1λ1,11!∂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=2∂xln⁡Wϕ1λ1,11!∂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=2∂xln⁡Wϕ1,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(1≤i≤n) are arbitrary real constants. For example, a 1-soliton to (1) is given by(42)u=2∂xln⁡Wϕ1=2∂xln⁡cosh-λ1x+ky+2z-4tλ1+γ1=2-λ1tanh⁡θ1,where θ1=-λ1(x+ky+2z-4tλ1)+γ1.Similarly, we have a 2-soliton to (1):(43)u=2∂xln⁡Wcosh⁡θ1,sinh⁡θ2=2λ1-λ2sinh⁡θ1-θ2-sinh⁡θ1+θ2-λ1--λ2cosh⁡θ1+θ2--λ1+-λ2cosh⁡θ1-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.Figure 1
The shape of the 1-soliton to (1) with λ1=-2, y=0, t=1, γ1=0.Figure 2
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=2∂xln⁡Wϕ,∂λϕ,…,∂λk1-1ϕ,ϕ=cos⁡λx+ky+2z-4tλ+γλ,(45)u=2∂xln⁡Wϕ,∂λϕ,…,∂λk1-1ϕ,ϕ=sin⁡λx+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.Whenλ1>0, a 1-positon of zero-order reads(46)u=2∂xln⁡Wϕ1=2∂xln⁡cosλ1x+ky+2z-4tλ1+γ1=-2λ1tan⁡θ3,where θ3=λ1(x+ky+2z-4tλ1)+γ3. And a 1-positon of first-order is(47)u=2∂xln⁡Wcosθ3,∂λ1cosθ3=4λ11+cos⁡2θ32λ1x+ky+2z-12tλ1+sin⁡2θ3.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.Figure 3
The shape of the 1-positon of zero-order to (1) with λ1=2, y=0, t=1, γ3=0.Figure 4
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=2∂xln⁡Wϕ,∂λϕ,…,∂λk1-1ϕ,ϕ=cosh⁡-λx+ky+2z-4tλ+γ-λ,(49)u=2∂xln⁡Wϕ,∂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.Whenλ1<0, a 1-negaton of first-order reads(50)u=2∂xln⁡Wcosh⁡θ1,∂λ1cosh⁡θ1=4-λ11+cosh⁡2θ12-λ1x+ky+2z-4tλ1+sinh⁡2θ1,where θ1=-λ1(x+ky+2z-4tλ1)+γ1. And the 1-negaton of second-order is given by(51)u=2∂xln⁡Wcosh⁡θ1,∂λ1cosh⁡θ1,∂λ12cosh⁡θ1=-6λ1δ1+48tλ12sinh⁡θ1-4λ1-λ1δ12cosh⁡θ1+6-λ1sinh2⁡θ1cosh⁡θ1-2λ1δ12sinh⁡θ1+24tλ1-λ1--λ1δ1cosh⁡θ1+sinh⁡θ1cosh2⁡θ1,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.Figure 5
The shape of the 1-negaton of first-order to (1) with λ1=-4, y=0, t=0, γ1=0.Figure 6
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=2∂xln⁡Wϕ1λ,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=2∂xln⁡Wϕrational,ϕsoliton=-2λ1x+ky+2zcosh⁡θ1-λ1x+ky+2zsinh⁡θ1-cosh⁡θ1,urp=2∂xln⁡Wϕrational,ϕpositon=2λ2x+ky+2zcos⁡θ2λ2x+ky+2zsin⁡θ2+cos⁡θ2,usp=2∂xln⁡Wϕsoliton,ϕpositon=2λ2-λ1cosh⁡θ1cos⁡θ2-λ1sinh⁡θ1cos⁡θ2+λ2cosh⁡θ1sin⁡θ2,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=2∂xln⁡Wϕrational,ϕsoliton,ϕpositon=2qp,where (57)p=x+ky+2zλ2-λ1sinh⁡θ1cos⁡θ2+λ1λ2cosh⁡θ1sin⁡θ2+λ1-λ2cosh⁡θ1cos⁡θ2,q=x+ky+2z-λ1λ2λ1-λ2sinh⁡θ1sin⁡θ2+λ1-λ1sinh⁡θ1cos⁡θ2+λ2λ2cosh⁡θ1sin⁡θ2,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.



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