# Dynamics Behavior of Lumps and Interaction Solutions of a (3+1)-Dimensional Partial Differential Equation

Authors: Bo Ren
Journal: Complexity
 (2019)

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## Abstract

In this work, we investigate a linear partial differential equation in (3+1)-dimensions. We construct lump solutions which localize in all directions in the(x,y,z)-space. By combining the trigonometric, hyperbolic and exponential functions with a quadratic function, diversity interaction solutions such as interacted lumps with periodic waves and interacted lumps with multisoliton are generated. The phenomena of interaction solutions between a lump and a multisoliton and between a lump and a multikink soliton are presented by figures. The results expand understanding dynamical behavior of the (3+1)-dimensional partial differential equations.


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## Body

## 1. Introduction

Nonlinear partial differential equations (NLPDEs) have attracted an increasingly attention from mathematics and physicists. Finding exact solutions to NLPDEs is one of the fundamental topics in this field [1–4]. Among these exact solutions, lump solutions, another kind of rational solutions, was first discovered by Manakov et al. [5]. Lump waves are found to be localized in all directions of the space. The subject of lump waves has a widespread application in the field of mathematical physics and engineering [6–9].Due to an important physical significance of lump waves, many high-dimensional nonlinear partial differential equations admit lump solutions, such as the Ishimori-I equation [10], the Kadomtsev-Petviashvili (KP) equation [11], the extended KP equation [12], the KP-Boussinesq equation [13, 14], the Hirota bilinear equation [15], the generalized Hirota-Satsuma-Ito equation [16], and the generalized KP equation [17]. Furthermore, interaction solutions among solitons and other kinds of complicated waves are studied by the localization procedure related to the nonlocal symmetry [18–20]. Similar to these interaction solutions, interactive lump-kink [21–24] and lump-soliton [25–27] of the nonlinear evolution equations have been studied by combining a positive quadratic function with an exponential function. A special rogue wave of the (2+1)-dimensional Korteweg-de Vries equation [28] and the KP equation [29] is obtained by combining the positive quadratic function and the hyperbolic cosine function. In the (3+1)-dimensional case, only lump-type solutions which are rationally localized in almost all but not all directions in space [25–27, 30–33] can be presented. In addition, the lump, interaction between a lump and periodic wave, and interaction between a lump and one soliton of a (3+1)-dimensional linear partial differential equation (PDE) are explored by symbolic computations [34]. It will be interesting to enlarge this category of the (3+1)-dimensional PDE that possesses lump solutions. In this paper, the main purpose of this work is to construct a (3+1)-dimensional linear PDE which possesses lump solutions. In the meanwhile, interaction solutions between lumps and multisoliton are obtained by introducing an ansätz function. This kinds of interaction solutions to the (3+1)-dimensional PDE have not been reported in other studies.This paper is organized as follows. In Section2, we study a class of (3+1)-dimensional linear PDE. The lumps, interaction between lumps and periodic waves, and interaction solutions between lumps and multisoliton are derived by using an ansätz function. Some concluding remarks will be given in the last section.


## 2. Lumps and Interaction Solutions of a (3+1)-Dimensional PDE

We consider a (3+1)-dimensional linear PDE:(1)α1uxy+α2uxz+α3uxt+α4uyz+α5uty+α6utz+α7uxx+α8uyy+α9uzz+α10utt+α11ux++α12uy+α13uz+α14ut=0,where αi(1≤i≤14) are arbitrary constants.A kind of exact solutions assumes(2)u=vξ1,ξ2,ξ3,ξ4,where v is an arbitrary real function and ξi(1≤i≤4) are four linear variables:(3)ξi=aix+biy+ciz+dit+ei,where ai, bi, ci, di, and ei are real constants to be determined. By substituting (2) into (1), the linear PDE (1) becomes(4)∑i4∑j4wijvξiξj+∑i4mivξi=0,where wij and mi(1≤i≤j≤4) are quadratic and one-time functions of the parameters ai, bi, ci, di, and ei. By setting wij=0 and mi=0 for all present combinations of i and j, we can get the following equations on the parameters:(5)α1aibi+α2aici+α3aidi+α4bici+α5bidi+α6aidi+α7aiai+α8bibi+α9cici+α10didi=0,(6)α11ai+α12bi+α13ci+α14di=0,(7)α1aibj+ajbi+α2aicj+ajci+α3aidj+ajdi+α4bicj+bjci+α5bidj+bjdi+α6aidj+ajdi+α7aiaj+ajai+α8bibj+bjbi+α9cicj+cjci+α10didj+djdi=0.By solving the above determined equations, we can obtain several solutions of the constraint constants. We just list one case of the constraint constants:(8)b1=b4c1c4,b2=b4c2c4,b3=b4a3a4,c3=c4a3a4,d1=d4a1a4,d2=d4a2a4,d3=d4a3a4,α1=-α2c4b4-α5d4a4-α6c4d4a4b4,α8=-c4α4b4+α9c4b42,α10=-a4α7a4+α3d4d42,α11=-α14d4a4,α12=0,α13=0,The explicit exact solutions can be expressed as(9)uI=ln⁡fxx,or(10)uII=ln⁡fx,with(11)f=ξ12n1+ξ22n2+ξ32n3+gξ4,where n1, n2, and n3 are arbitrary natural numbers, ξi(1≤i≤4) are defined by(12)ξ1=a1x+b4c1c4y+c1z+d4a1a4t+e1,ξ2=a2x+b4c2c4y+c2z+d4a2a4t+e2,ξ3=a3x+b4a3c4y+c4a3a4z+d4a3a4t+e3,ξ4=a4x+b4y+c4z+d4t+e4,and the function g is an arbitrary function of ξ4. We can take(13)gξ4=A1,B1+C1cos⁡ξ4,D1+E1exp⁡ξ4,orF1cosh⁡ξ4,where A1, B1, C1, D1, E1, and F1 are constants to guarantee the positivity of f. By selecting the function g, we can obtain lump waves, interaction between a lump and periodic waves, interaction solution between a lump and one soliton, interaction solution between a lump and a one-kink soliton, and interaction solution between a lump and a two-stripe soliton to the linear PDE (1). To describe the interaction solution between a lump and one soliton, we set the parameters as(14)n1=n2=n3=1,D1=E1=1,a2=3,a3=-1,a4=1,b4=12,c1=2,c2=-1,c4=-12,d4=1,e1=1,e2=1,e3=2,e4=1.The 3D plot and density plot for the interaction solution between a lump and one soliton of (9) are shown in Figure 1. The 3D plot and density plot for the interaction solution between a lump and a one-kink soliton of (10) are depicted in Figure 2.Figure 1
The interaction solution between a lump and one soliton of (9). (a) 3-dimensional plot with the time t=0; (b) the corresponding density plot.
(a) (b)Figure 2
The interaction solution between a lump and a one-kink soliton of (10). (a) 3-dimensional plot with the time t=0; (b) the corresponding density plot.
(a) (b)To get another kind of exact solutions, we assume(15)u=vξ1,ξ2,ξ3,ξ4,ξ5,where v is an arbitrary real function, and ξi(1≤i≤5) are five linear variables:(16)ξi=aix+biy+ciz+dit+ei.By substituting (15) into (1), the linear PDE (1) becomes(17)∑i5∑j5wijvξiξj+∑i5mivξi=0,where wij and mi(1≤i≤j≤5) are quadratic and one-time functions of the parameters ai, bi, ci, di, and ei. By the accurate calculation, we can get several constraining equations on the various parameters. We list one case of the constraint parameters:(18)bq=b5aqa5,1≤q≤4,c4=a4c5c5,dp=apd5a5-cpα13α14+apc5α13α14a5,1≤p≤3,d4=d5a4a5,α2=α3α13α14+2α10α132c5a5α142-α4b5+α6d5a5+α132α10d5+α5b5-α6c5α14a5,α7=-α10α132c5a52α142-α10α12d5b5+2α10α13c5d5+α5α13b5c5a52α14-b5α12α5b5+α6c52a52α14-α3α13c5a5α14-α3α12b52a5α14+b5α4c5-α5d52a52-α10d52a52-α3d5a5-α1b52a5,α8=-α1a5+α4c5+α5d52b5+α5α102α14+α10α12d5α14b5+α12α6c5+α3a52α14b5,α9=α6α13α14-α10α132α142,α11=-α12b5a5-α13c5a5-α14d5a5.The explicit exact solutions can be expressed as(19)uI=ln⁡fxx,or(20)uII=ln⁡fx,with(21)f=ξ12n1+ξ22n2+ξ32n3+g1ξ4+g2ξ5,where n1, n2, and n3 are arbitrary natural numbers, and the functions g1 and g2 are arbitrary functions of ξ4 and ξ5, respectively. The lumps, interaction between lumps and periodic waves, and interaction between lumps and solitons can be obtained by selecting arbitrary functions g1 and g2. We can take(22)gkξk+4=Ak,Bk+Ckcos⁡ξk+4,Dk+Ekexp⁡ξk+4,orFkcosh⁡ξk+4,k=1,2,where Ak, Bk, Ck, Dk, Ek, and Fk are constants to guarantee the positivity of f. To describe the interaction solutions between a lump and two solitons and between a lump and a two-kink soliton, we set the parameters as(23)n1=n2=n3=1,Dk=Ek=1,α13=α14=1,a1=1,a2=1,a3=3,a4=3,a5=-3,b5=1,c1=2,c2=1,c3=1,c5=1,d5=1,e1=2,e2=1,e3=2,e4=2,e5=1,The 3D plot for the interaction solution between a lump and two solitons of (19) is depicted in Figure 3(a). The interaction solution between a lump and a two-kink soliton of (20) is shown in Figure 3(b).Figure 3
(a) 3-dimensional plot of the interaction solution between a lump and two solitons (19) with the time t=0; (b) 3-dimensional plot of the interaction solution between a lump and a two-kink soliton (20) with the time t=0.
(a) (b)We can assume another kind of solutions as(24)u=vξ1,ξ2,ξ3,ξ4,ξ5,ξ6,where v is an arbitrary real function, and ξi(1≤i≤6) are six linear variables:(25)ξi=aix+biy+ciz+dit+ei.Based on the same process, we can get the following set of the constraining parameters:(26)b5=0,dp=apd5a5-α12bp+α13cpα14+α13apc5α14a5,p=1,2,3,4,6,αq=2α10α11+qα13c5α142a5+2α10α11+qd5-α4+qα13c5α14a5-α4+qd5a5+α3α11+qα14,q=1,2,α4=-2α10α12α13α142+α5α13+α6α12α14,αk=-α10αk+42α142+αk-3αk+4α14,k=8,9,α7=-α10α132c52a52α142-2α10α13c5d5a52α14-α3α13c5α14a5-α10d52a52-α3d5a5.The explicit exact solutions can be expressed as(27)uI=ln⁡fxx,or(28)uII=lnfx,with(29)f=ξ12n1+ξ22n2+ξ32n3+g1ξ4+g2ξ5+g3ξ6,where n1, n2, and n3 are arbitrary natural numbers, and functions g1, g2, and g3 are arbitrary functions of ξ4, ξ5, and ξ6, respectively. We can take(30)gkξk+4=Ak,Bk+Ckcos⁡ξk+4,Dk+Ekexp⁡ξk+4,orFkcosh⁡ξk+4,k=1,2,3,where Ak, Bk, Ck, Dk, Ek, and Fk are constants to guarantee the positivity of f. The lumps, interaction between lumps and periodic waves, and interaction between lumps and solitons can be derived by selecting arbitrary functions g1, g2, and g3. To describe interaction solution between a lump and three solitons, we set the parameters as(31)n1=n2=n3=1,Dk=Ek=1,α12=α13=α14=1,a1=1,a2=1,a3=2,a4=1,a5=12,a6=4,b1=2,b2=1,b3=3,c1=2,c2=1,c3=1,c4=2,c5=1,c6=1,d5=1,e1=2,e2=1,e3=2,e4=2,e5=1,e6=1.The graphical behavior of the interaction solution between a lump and three solitons of (27) is shown in Figure 4(a). The graphical behavior of the interaction solution between a lump and a three-kink soliton of (28) is analyzed in Figure 4(b). To the best of our knowledge, these interaction solutions of the (3+1)-dimensional PDE have not been reported in other studies.Figure 4
(a) 3-dimensional plot of the interaction solution between a lump and three solitons of (27) with the time t=0; (b) 3-dimensional plot of the interaction solution between a lump and a three-kink soliton of (28) with the time t=0.
(a) (b)Remark.From the constraining parameters of (8), (18), and (26), a determinant equation satisfies(32)a1b1c1d1a2b2c2d2a3b3c3d3a4b4c4d4=0.It is shown that the rational solutions of (9), (10), (19), (20), (27), and (28) fail to be localized in all directions in the whole (x,y,z,t)-space. The rational solutions of (9), (10), (19), (20), (27), or (28) are all lump waves, rationally localized in all directions in the (x,y,z)-space [34].


## 3. Conclusion

In summary, some novel interaction solutions of a (3 + 1)-dimensional PDE are considered by using the ansätz functions (2), (15), and (24). By combining a quadratic function with exponential, hyperbolic, or trigonometric functions, lumps, interaction between lumps and periodic waves and interaction solutions between lumps and multisoliton are found. The phenomena of interaction solution between a lump and a multisoliton for the (3+1)-dimensional PDE are generated as illustrative examples.In this paper, the behavior of interaction between a lump and three solitons has been displayed for the choices of specific parameters. We can get interaction solutions between lumps and multisoliton by using the following ansätz function:(33)u=vξ1,ξ2,ξ3,ξ4,ξ5,ξ6,…,ξk,k≥6,where v is an arbitrary real function, and ξi(1≤i≤k) are multilinear variables. The results obtained in this paper provide further evidence on the effectiveness of lump waves in (3+1)-dimensions. Besides, some exact solutions to the (3+1)-dimensional nonlinear evolution equation can be constructed by use of the test function method [35]. The study of those problems may be helpful in soliton theory.



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