With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a bilinear form and construct a bilinear Bäcklund transformation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the bilinear BT.

1. Introduction

Dynamical systems, such as those for the shallow waters [1, 2], plasmas and optical fiber communications [3–6], can often be described by the nonlinear evolution equations (NLEEs) [7–9] and studied by the relevant methods including the inverse scattering [1], Bäcklund transformation (BT) [10–13], and Hirota method [14–16]. Among them, the Hirota method [17, 18] is a direct tool for dealing with certain NLEEs and relevant soliton problems [19, 20]. Based on the bilinear form of a given NLEE, one can obtain the multisoliton solutions [21], bilinear auto-BTs [18], nonlinear superposition formulas, Lax pair, Wronskian formulation [22], and so on [23].

Reflecting the complex nonlinear phenomena in our real world [24–26], higher-dimensional NLEEs with their analytic solutions and integrable properties [27–29] have been of great interest. In fact, some (2+1)-dimensional NLEEs have been investigated with different methods, for example, the (2+1)-dimensional breaking soliton equation, Kadomtsev-Petviashvili equation, and (2+1)-dimensional Kaup-Kupershmidt equation [30–32]. However, for some (3+1)-dimensional NLEEs, the conventional integrability test fails [27], and then a natural problem is whether or not there exists BT for a given (3+1)-dimensional NLEE. Moreover, for the higher-dimensional NLEEs, finding a bilinear BT via the exchange formula is often difficult, even if possible [18, 21].

In this paper, we will study the following (3+1)-dimensional Jimbo-Miwa (JM) equation [32]:(1)uxxxy+3uyuxx+3uxuxy+2uyt-3uxz=0,where u is a real scalar function with four independent variables x, y, z, and t and the subscripts denote the corresponding partial derivatives. Seen as one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems [32, 33], (1) describes certain (3+1)-dimensional waves [13, 32] but does not have the Painlevé property [34] as defined in [35]. The soliton [36, 37], periodic [15], rational, and dromion solutions [38, 39] for (1) have been obtained. BTs and analytic solitonic solutions have been given in [13] with the truncated Painlevé expansion at the constant level term.

However, existing literature has not studied the bilinear BT and Bell-polynomial-typed BT of (1) as yet. Therefore, in this paper, by means of the Bell polynomials and Hirota bilinear method, we will obtain two BTs for (1), which are different from those in [13]. In Section 2, we will introduce some concepts on the Bell polynomials and their connection with the bilinear forms. In Section 3, using the Bell-polynomial expressions, we will derive a bilinear form of (1). In Section 4, based on this bilinear form, we will obtain a bilinear BT with soliton solutions and a Bell-polynomial-typed BT. Finally, our conclusions will be given in Section 5.

2. Preliminaries

Suppose that φ is C∞-function with respect to x, and set φθx=∂xθφ(θ=0,1,2,…). Then the Bell exponential polynomials are given as [40–43](2)Ynxφ≡Ynφ1x,φ2x,…,φnx=e-φ∂xneφ,where n=1,2,….

For example,(3)Y1x=φ1x,Y2x=φ2x+φ1x2,Y3x=φ3x+3φ1xφ2x+φ1x3,….Two-dimensional Bell polynomials are expressed as [40–43](4)Ymx,ntφ≡Ym,nφ1x,0t,φ0x,1t,…,φrx,st,…,φmx,nt=e-φ∂xm∂tneφ,φrx,st=∂xr∂tsφx,t,m=1,2,…;r=0,1,…,m;s=0,1,…,nwith φ hereby being C∞-function of x and t.

Based on the Bell polynomials given above, the binary Bell polynomials, namely, Y-polynomials, can be defined as [41](5)Ymx,ntv,w≡Ymx,ntφv,w=Ym,nφ1x,0t,φ0x,1t,…,φrx,st,…,φmx,ntφrx,st=vrx,st,if r+s is odd,wrx,st,if r+s is even,where the vertical line means that the elements on the left-hand side are chosen according to the rule on the right-hand side, while v and w are the functions that replace φ in the corresponding positions of the Bell polynomials. For simplicity, we denote Ymx,nt(v,w) as Ymx(v,w) or Ynt(v,w) if n=0 or m=0, respectively.

As one special kind of Y-polynomials, P-polynomials only possess the even-order partial differential terms and, with q=w-v, are defined as [40, 41](6)Pmx,ntq≡Ymx,ntφ0,q=Ym,nφ1x,0t,φ0x,1t,…,φrx,st,…,φmx,ntφrx,st=0,if r+s is odd,qrx,st,if r+s is even,which vanish unless n+m is even.

According to the above, the lower-order P-polynomials can be given as(7)P0q=1,P2xq=q2x,Px,tq=qxt,P4xq=q4x+3q2x2,….

For a given pair of exponentials,(8)F=expfx,t,G=expgx,t,where f and g are C∞-functions of x and t, while the Hirota D-operators are defined as [17, 42, 43](9)DxmDtnF·G≡∂∂x-∂∂x′m∂∂t-∂∂t′nFx,t×Gx′,t′x′=x,t′=t,where x′, t′ are the formal variables.

It has been found that there exist some relations between the binary Bell polynomials and the Hirota D-operators [40, 41]. When v=lnF/G, w=lnFG, a binary Bell polynomial can be transformed into a bilinear term according to the identity [40, 41](10)FG-1DxnDtmF·G=Ynx,mtv=lnFG,w=lnFG.Likewise, when w=2lnG, P-polynomials can be associated with the Hirota D-operators according to the identity [40, 41](11)G-2DxnDtmG·G=Ynx,mtv=0,w=2lnG=Pmx,ntq.

3. Bilinear Form

We will next investigate (1), to be written in P-polynomial form with one independent variable. Based on the relation between the binary Bell polynomials and Hirota bilinear operators, namely, identities (10) and (11), (1) can be translated into the corresponding bilinear forms.

Consider the following scale transformations:(12)x⟶λkx′,y⟶λly′,z⟶λαz′,t⟶λβt′,u⟶λμu′,where λ, k, l, α, β, and μ are the real constants. Invariance of (1) under such transformations requires that μ=-k, α=l+2k, and β=3k.

Notice that if we require that μ=-k, we have to set u=cqx in (1) and obtain(13)q4x,y+3cqxyq3x+3cq2xq2x,y+2qxyt-3q2x,z=0,where c is an arbitrary constant.

In order to express (13) with P-polynomials, we choose c=1. Then(14)P3x,yq+2Pytq-3Px,zqx=0,whose corresponding bilinear form is (15)Dx3Dy+2DyDt-3DxDzG·G=0.

Therefore, we get the bilinear form of (1), which is (14) with P-polynomials or (15) with the bilinear operators. We note that (15) is the same as that in [15], but the method that we used is different from that in [15].

4. Bell-Polynomial-Typed BT and Bilinear BT with Soliton Solutions

To construct a BT, we express (1) with P-polynomials:(16)Eq=P3x,yq+2Pytq-3Px,zq.

Based on E(q) we will derive the Bell-polynomial-typed BT under the homogenous constraints between the primary and replica fields instead of using exchange formulae.

Using the Bell polynomials, we have(17)cv,w=Eq′-Eqq=2lnG,q′=2lnF=Ew+v-Ew-vw=lnFG,v=lnF/G=w+v3x,y+3w+vxyw+v2x+2w+vyt-3w+vxz-w-v3x,y+3w-vxyw-v2x+2w-vyt-3w-vxz=2v3x,y+6wxyvxx+6vxywxx+4vyt-6vxz.

Note that (18)Y2x,y=v2x,y+vyw2x+2vxwxy+vyvx2,Y3x=v3x+3vxw2x+vx3.Therefore, substituting (18) into (17), we have(19)cv,w2=v3x,y+3wxyvxx+3vxywxx+2vyt-3vxz=av3x,y+1-av3x,y+2vyt-3vxz+3wxyv2x+3vxyw2x=aY2x,yx-avyw2x+2vxwxy+vyvx2x+1-aY3xy-1-a3vxw2x+vx3y+2Yty-3Yzx+3wxyv2x+3vxyw2x=aY2x,y-3Yzx+1-aY3x+2Yty+A,where(20)A=3wxyv2x+3vxyw2x-avyw2x+2vxwxy+vyvx2x-1-a3vxw2x+vx3y,and a is an arbitrary constant.

Further computation shows that(21)A=2avxyw2x+a-3vxw2x,y+2a-3vx2vxy-2a-3v2xwxy-avyw3x-2avxvyv2x=2avxY2xy+-2a-32vxY2xy-avyY2xx+a-32vxw2x,y+-2a+3v2xwxy.Hereby, if we choose a=3/2 and set Y2x=σ, then(22)A=2avxY2xy=3τvxy,cv,w=2aY2x,y-3Yzx+1-aY3x+2Yty+6σvxy.Hence, (23)cv,w2=32Y2x,y-3Yz+3-bσYyx+-12Y3x+2Yt+bσYxy.Moreover, a decomposition of (23) leads to the following Bell-polynomial-typed BT:(24)Y2x=σ,32Y2x,y-3Yz+3-bσYy=τ,-12Y3x+2Yt+bσYx=δ,where b, σ, τ, and δ are the arbitrary constants.

Using the connection between the Bell polynomials and bilinear operators, we give a bilinear BT between G and G′ as (25)Dx2G′·G=σG′·G,32Dx2Dy-3Dz+3-bσDyG′·G=τG′·G,-12Dx3+2Dt+bσDxG′·G=δG′·G.

As an application, we derive the one-soliton solutions from a trivial solution, by virtue of the bilinear BT, that is, (25). Taking τ=0, δ=0, and G′=1 in (25), we get(26)Gxx=σG,(27)32Gxxy-3Gz+3-bσGy=0,(28)-12G3x+2Gt+bσGx=0.Substituting (26) into (28), we have(29)b-12G3x+2Gt=0,and hence we take (30)G=eξ+e-ξ,where ξ=kx+ly+mz+ωt+ξ0 and ξ0 is a nonzero constant. On the other hand, we can choose σ=k2 and substitute it into (26) and (29), which implies that(31)ω=14-b2k3.Similarly, substituting (26) into (27) leads to(32)92-bk2Gy=3Gz.Solving (32), we obtain (33)m=32-b3k2l.Finally, we can present the one-soliton solutions of (1) as (34)u=2lnGx=2ktanhξ,where ξ=kx+ly+(3/2-b/3)k2lz+(1/4-b/2)k3t+ξ0, while the parameters k, l, b, and ξ0 are all arbitrary constants.

5. Discussions and Conclusions

We have investigated the (3+1)-dimensional Jimbo-Miwa equation, that is, (1). With the aid of the Bell polynomials and Hirota bilinear operators, we have derived bilinear form (16) of (1) and then constructed a new BT, that is, (25), with the Bell polynomials and symbolic computation. The bilinear form and BT are important integrable property for the nonlinear evolution equations. Moreover, a BT often can be cast into the Lax pair for integrable equations. It may be possible to construct the bilinear BTs for the (3+1)-dimensional Jimbo-Miwa equation via the exchange formulae; however, the computation is tedious. Bell-polynomial-typed BTs (24) have been constructed hereby and then cast into bilinear BTs (25), which help us avoid the difficulties in using the exchange formulae. As an application, one-soliton solutions (34) have been obtained via BT (25). The existence of solution obtained via solving this BT indicates that (24) or (25) are genuine ones.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere thanks to Mr. Q. X. Qu and Mr. K. Sun for their helpful suggestions. This work has been supported by the National Natural Science Foundation of China under Grant no. 11272023, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant no. 2011BUPTYB02.

AblowitzM. J.ClarksonP. A.BarnettM. P.CapitaniJ. F.von zur GathenJ.GerhardJ.Symbolic calculation in chemistry: selected examplesLüX.ZhuH.-W.MengX.-H.YangZ.-C.TianB.Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communicationsLüX.Bright-soliton collisions with shape change by intensity redistribution for the coupled Sasa-Satsuma system in the optical fiber communicationsTianB.GaoY. T.ZhuH. W.Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computationTianB.GaoY. T.Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmasYanZ.-Y.ZhangH.-q.Symbolic computation and new families of exact soliton-like solutions to the integrable Broer-Kaup (BK) equations in (2+1)-dimensional spacesLüX.PengM.Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physicsLüX.PengM.Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas modelRogersC.ShadwickW. F.LüX.New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada–Kotera modelLüX.LiJ.Integrability with symbolic computation on the Bogoyavlensky-Konoplechenko model: BELl-polynomial manipulation, bilinear representation, and Wronskian solutionTianB.GaoY.-T.HongW.The solitonic features of a nonintegrable (3+1)-dimensional Jimbo-Miwa equationZhangJ. F.WuF. M.Bäcklund transformation and multiple soliton solutions for the (3+1)-dimensional Jimbo-Miwa equationZhaQ. L.LiZ. B.Multiple periodic-soliton solutions for (3+1)-dimensional Jimbo-Miwa equationLüX.Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformationHirotaR.Exact solution of the korteweg—de Vries equation for multiple collisions of solitonsHirotaR.A new form of Bäcklund transformations and its relation to the inverse scattering problemHirotaR.SatsumaJ.A variety of nonlinear network equations generated from the bäcklund transformation for the toda latticeHirotaR.HuX.-B.TangX.-Y.A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and Lax pairsHirotaR.MatsunoY.LiC.-X.MaW.-X.LiuX.-J.ZengY.-B.Wronskian solutions of the Boussinesq equation—solitons, negatons, positons and complexitonsAratynH.FerreiraL. A.ZimermanA. H.Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbersWazwazA.Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equationsGaoY. T.TianB.New family of overturning soliton solutions for a typical breaking soliton equationYuJ.LouZ.A (3+1)-dimensional Painlevé integrable model obtained by deformationChenJ. B.Finite-gap solutions of 2+1
dimensional integrable nonlinear evolution equations generated by the Neumann systemsWardR. S.Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable systemZhangJ. F.Multiple soliton-like solutions for (2+1)-dimensional dispersive Long-Wave equationsMikhailovA. V.YamilovR. I.Towards classification of (2+1)-dimensional integrable equationsJimboM.MiwaT.Solitons and infinite dimensional Lie algebrasTangX.-Y.LinJ.Conditional similarity reductions of Jimbo-Miwa equations via the classical Lie group approachDorrizziB.GrammaticosB.RamaniA.WinternitzP.Are all the equations of the KP hierarchy integrable?WeissJ.TaborM.CarnevaleG.The Painlevé property for partial differential equationsBaiC.-L.ZhaoH.Some special types of solitary wave solutions for (3+1)-dimensional Jimbo-Miwa equationWazwazA.-M.Multiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff, Jimbo-Miwa and {YTSF} equationsXuG. Q.The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in (3+1)-dimensionsMaS.-H.FangJ.-P.HongB.-H.ZhengC.-L.New exact solutions and interactions between two solitary waves for (3+1)-dimensional jimbo–miwa systemGilsonC.LambertF.NimmoJ. J.WilloxR.On the combinatorics of the Hirota D-operatorsLambertF.SpringaelJ.On a direct procedure for the disclosure of Lax pairs and Bäcklund transformationsLüX.TianB.SunK.WangP.Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one tau-functionLüX.LinF. H.QiF. H.Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions