# Theta Function Solutions of the 3 + 1-Dimensional Jimbo-Miwa Equation

Authors: Ting Su; Hui-Hui Dai
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The 3 + 1-dimensional Jimbo-Miwa equation can be written into a Hirota bilinear form by the dependent variable transformation. We give its one-periodic wave solution and two-periodic wave solution by utilizing multidimensional ellipticΘ-function. With the help of the solution curves, the asymptotic properties of the periodic waves are analyzed in detail.


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## Body

## 1. Introduction

Nonlinear phenomena arise in many physical problems in a variety of fields. Solutions of the governing nonlinear equations can shed light on these phenomena. There are various systematical methods for constructing solutions, for example, nonlinearization method of Lax pairs [1–3], extended F-expansion method [4–6] and homogeneous balance method [7–10], and dressing method and generalized dressing method [11–16]. It is well known that the Hirota method with the aid of Riemann-theta function is a good method to obtain periodic and quasiperiodic solutions. Nakamura [17, 18] used this method to study some famous equations such as KdV, KP, Boussinesq, and Toda. By extending the approach adopted by Nakamura, Dai et al. obtained the graphic quasiperiodic solutions for the KP equation for the first time [19] and later they also gave the quasiperiodic solutions for Toda lattice [20]. Recently, a lot of researchers have used this method to study various soliton equations [21–24].In the present paper, we consider the 3 + 1-dimensional Jimbo-Miwa equation(1)uxxxy+3uxyux+3uyuxx+2uyt-3uxz=0,which is the second member of a KP hierarchy [25]. It has important physics to describe 3 + 1-dimensional waves. In the last decade or so, many researchers have studied this equation. Multiple-soliton solutions of (1) and its extended version were given in Wazwaz [26]. Tang in [27] obtained its Pfaffian solution and extended Pfaffian solutions with the aid of the Hirota bilinear form. Su et al. [28] constructed its Wronskian and Grammian solutions. Multiple-front solutions for (1) were obtained by employing the Hirota bilinear method in [29]. Ozixs and Aslan in [30] derived exact solutions of (1) via Exp-function method. In [31], Ma and Lee have obtained rational solutions of (1) including travelling wave solutions, variable separated solutions, and polynomial solutions by using rational function transformation and Bäcklund transformation. Li et al. have utilized generalized three-wave method to derive explicit three-wave solutions, such as doubly periodic solitary wave solutions and breather type of two-solitary wave solutions for (1) in [32]. Zhang et al. have obtained generalized Wronskian solution in [33]. Dai et al. obtained new periodic kink-wave and kinky periodic wave solutions for (1) in [34]. Ma in [35] has derived exact solutions by using Lie point symmetry groups of (1). Tang and Liang in [36] have obtained two types of variable separation solutions and abundant nonlinear coherent structure by using multilinear variable separation approach. Ma et al. obtained new exact solutions for (1) by utilizing improved mapping approach [37]. Liu and Jiang by applying the extended homogeneous balance method have obtained new solutions of (1) in [38]. Authors in [39] obtained special solutions by using extended homogeneous balance method. Hu et al. [40] discussed 3-soliton and 4-soliton solutions with the aid of bilinear form of (1). In [41], some complexion type solutions of (1) are presented by using two kinds of transcendental functions. Authors presented rational solutions of (1) with the aid of the generalized Riccati equation mapping method [42]. By utilizing two-soliton method and bilinear method, cross kink-wave and periodic solitary solutions of (1) are given in [43]. With the help of the bilinear form, here we construct one-periodic and two-periodic solutions of this equation (1) by utilizing the method of Dai et al. [19].The paper is organized as follows. In Section2, we obtain the formula of one-periodic wave solutions and discuss its asymptotic behavior. Further, in order to analyze the solution, some solution curves are plotted. In the third section, two-periodic wave solutions, their asymptotic behaviors and the solution curves are given.


## 2. One-Periodic Wave Solution of the 3 + 1-Dimensional Jimbo-Miwa Equation and Its Asymptotic Form

We consider the 3 + 1-dimensional Jimbo-Miwa equation(2)uxxxy+3uxyux+3uyuxx+2uyt-3uxz=0.Substituting the transformation(3)u=u0+2ln⁡fxinto (2) and integrating once again, we derive the following bilinear form:(4)Dx3Dyf·ff2+2DtDyf·ff2-3DxDzf·ff2+c=0;namely,(5)GDx,Dt,Dy,Dzf·f=Dx3Dy+2DtDy-3DxDz+cf·f=0,where c is an integration constant.The Hirota bilinear differential operator is defined by [44](6)DxmDtnDym1Dzn1f·g=∂x-∂x′m∂t-∂t′n∂y-∂y′m1∂z-∂z′n1f·gx=x′,t=t′,y=y′,z=z′.The D-operator possesses the good property when acting on exponential functions: (7)DxmDtnDym1Dzn1eξ1·eξ2=p1-p2mρ1-ρ2nl1-l2m1μ1-μ2n1eξ1+ξ2,where ξj=pjx+ljy+μjz+ρjt+ξ0j,  j=1,2. More generally, we have(8)GDx,Dt,Dy,Dzeξ1·eξ2=Gp1-p2,l1-l2,ρ1-ρ2,μ1-μ2eξ1+ξ2.


### 2.1. One-Periodic Wave Solution

In what follows, we consider Riemann-theta function solution of (5):(9)f=∑n∈ZNeπiτn,n+2πiξ,n,where n=n1,n2,…,nNT, ξ=ξ1,ξ2,…,ξNT, τ is a symmetric matrix, and Imτ>0,  ξj=pjx+ljy+μjz+ρjt+ξoj, j=1,2,…,N.First, we consider one-periodic solution of (5) when N=1; then (9) is(10)f=∑n=-∞∞e2πinξ+πin2τ.Substituting (10) into (5) yields that(11)GDx,Dy,Dz,Dtf·f=GDx,Dy,Dz,Dt∑n=-∞∞e2πinξ+πin2τ·∑m=-∞∞e2πimξ+πim2τ=∑n=-∞∞∑m=-∞∞GDx,Dy,Dz,Dte2πinξ+πin2τ·e2πimξ+πim2τ=∑n=-∞∞∑m=-∞∞G2πin-mp,2πin-ml,2πin-mμ,2πin-mρe2πin+mξ+πin2+m2τ=∑m′=-∞∞∑n=-∞∞G2πi2n-m′p,2πi2n-m′l,2πi2n-m′μ,2πi2n-m′ρeπin2+n-m′2τe2πim′ξ=∑m′=-∞∞G¯m′e2πim′ξ=0,where m′=n+m.(12)G¯m′=∑n=-∞∞G2πi2n-m′p,2πi2n-m′l,2πi2n-m′μ,2πi2n-m′ρeπin2+n-m′2τ=∑n′=-∞∞G2πi2n′-m′-2p,2πi2n′-m′-2l,2πi2n′-m′-2μ,2πi2n′-m′-2ρeπin′2+n′-m′-22τ·e2πim′-1τ=G¯m′-2e2πim′-1τ=⋯=G¯0eπim′m′-1τ,m′is even,G¯1eπim′+1m′-2τ,m′is odd,where n′=n-1, which implies that if G¯0=G¯1=0, then G¯m′=0, m′∈N.First, we consider(13)G¯0=∑n=-∞∞256π4n4p3l-32π2n2ρl+48π2n2pμ+ce2πin2τ=0,G¯1=∑n=-∞∞16π42n-14p3l-8π22n-12ρl+12π22n-12pμ+ceπin2+n-12τ=0;let(14)δ1n=e2πin2τ,δ2n=eπin2+n-12τ,b11=∑n=-∞∞256π4n3p3-32π2n2ρδ1n,b12=∑n=-∞∞48π2n2pδ1n,Δ1=∑n=-∞∞δ1n,b22=∑n=-∞∞12π22n-12pδ2n,Δ2=∑n=-∞∞δ2n,b21=∑n=-∞∞16π42n-14p3-8π22n-12ρδ2n;then (13) can be written as (15)lb11+b12μ+cΔ1=0,lb21+b22μ+cΔ2=0.Solving this system, we get(16)l=μb22Δ1-b12Δ2b11Δ2-b21Δ1,c=μb12b21-b22b11b11Δ2-b21Δ1.Then we get one-periodic wave solution(17)u=u0+2ln⁡fx=u0+2fxf,where f and l, p, μ, and ρ are given by (10) and (16).We see that the well known soliton solution of the 3 + 1-dimensional Jimbo-Miwa equation can be obtained as limit of the periodic solution (17).Theorem 1.
Under the condition(Imτ→∞), the solution (10) of (5) tends to the following exact solution of (2) via (3):(18)u=u0+4πipe2πipx+ly+μz+ρt+ξ0~1+e2πipx+ly+μz+ρt+ξ0~,with l→-3pμ/(4π2p3-2ρ), as α=eπiτ→0, where ξ0~ is an arbitrary constant.Proof.
We writef as(19)f=1+αe2πiξ+e-2πiξ+α4e4πiξ+e-4πiξ+⋯,with ξ=px+ly+μz+ρt+ξ0.
If we make an arbitrary phase constant slightly asξ0=ξ0~-(1/2)τ and have small amplitude limit of α=eπiτ→0, then we derive proper limit (20)f⟶1+e2πipx+ly+μz+ρt+ξ0~.It is easy to obtain the (18). In fact, we have(21)b11=∑n=-∞∞256π4n4p3-32π2n2ρe2πin2τ=512π4p3-64π2ρα2+oα2,Δ1=∑n=-∞∞e2πin2τ=1+2α2+oα2,b12=∑n=-∞∞48π2n2pe2πin2τ=96π2pα2+oα2,Δ2=∑n=-∞∞eπi2n2-2n+1τ=2α+2α5+oα5,b22=∑n=-∞∞12π22n-12peπin2+n-12τ=24π2pα+oα,b21=∑n=-∞∞16π42n-14p3-8π22n-12ρeπi2n2-2n+1τ=32π4p3-16π2ρα+oα5.By utilizing (16), it is easy to deduce that(22)l⟶-3pμ4π2p3-2ρ,c⟶0.The one-periodic solution curves are presented in Figures 1 and 2 for u0=0, respectively, in two-dimensional and three-dimensional space. It is obvious that the solution is periodic and cuspon from the above solution graphs. It is different to the Pfaffian solutions and extended Pfaffian solutions derived by Tang in [27]. The results are different to one-soliton solution and two-soliton solutions represented by researchers in [28, 30–33]. There are some difference between new types of exact periodic solitary wave and kinky periodic wave solutions in [34] by Dai et al. and the solutions in the paper.Figure 1
z
=
0.6, t=2, y=2, p=0.1, μ=2, ρ=4, ζ~=0.8, and x∈-3,3.Figure 2
z
=
0.6, t=2, p=0.1, μ=2, ρ=4, ζ~=0.8, x∈-6,6, and y∈5,20.


## 2.1. One-Periodic Wave Solution

In what follows, we consider Riemann-theta function solution of (5):(9)f=∑n∈ZNeπiτn,n+2πiξ,n,where n=n1,n2,…,nNT, ξ=ξ1,ξ2,…,ξNT, τ is a symmetric matrix, and Imτ>0,  ξj=pjx+ljy+μjz+ρjt+ξoj, j=1,2,…,N.First, we consider one-periodic solution of (5) when N=1; then (9) is(10)f=∑n=-∞∞e2πinξ+πin2τ.Substituting (10) into (5) yields that(11)GDx,Dy,Dz,Dtf·f=GDx,Dy,Dz,Dt∑n=-∞∞e2πinξ+πin2τ·∑m=-∞∞e2πimξ+πim2τ=∑n=-∞∞∑m=-∞∞GDx,Dy,Dz,Dte2πinξ+πin2τ·e2πimξ+πim2τ=∑n=-∞∞∑m=-∞∞G2πin-mp,2πin-ml,2πin-mμ,2πin-mρe2πin+mξ+πin2+m2τ=∑m′=-∞∞∑n=-∞∞G2πi2n-m′p,2πi2n-m′l,2πi2n-m′μ,2πi2n-m′ρeπin2+n-m′2τe2πim′ξ=∑m′=-∞∞G¯m′e2πim′ξ=0,where m′=n+m.(12)G¯m′=∑n=-∞∞G2πi2n-m′p,2πi2n-m′l,2πi2n-m′μ,2πi2n-m′ρeπin2+n-m′2τ=∑n′=-∞∞G2πi2n′-m′-2p,2πi2n′-m′-2l,2πi2n′-m′-2μ,2πi2n′-m′-2ρeπin′2+n′-m′-22τ·e2πim′-1τ=G¯m′-2e2πim′-1τ=⋯=G¯0eπim′m′-1τ,m′is even,G¯1eπim′+1m′-2τ,m′is odd,where n′=n-1, which implies that if G¯0=G¯1=0, then G¯m′=0, m′∈N.First, we consider(13)G¯0=∑n=-∞∞256π4n4p3l-32π2n2ρl+48π2n2pμ+ce2πin2τ=0,G¯1=∑n=-∞∞16π42n-14p3l-8π22n-12ρl+12π22n-12pμ+ceπin2+n-12τ=0;let(14)δ1n=e2πin2τ,δ2n=eπin2+n-12τ,b11=∑n=-∞∞256π4n3p3-32π2n2ρδ1n,b12=∑n=-∞∞48π2n2pδ1n,Δ1=∑n=-∞∞δ1n,b22=∑n=-∞∞12π22n-12pδ2n,Δ2=∑n=-∞∞δ2n,b21=∑n=-∞∞16π42n-14p3-8π22n-12ρδ2n;then (13) can be written as (15)lb11+b12μ+cΔ1=0,lb21+b22μ+cΔ2=0.Solving this system, we get(16)l=μb22Δ1-b12Δ2b11Δ2-b21Δ1,c=μb12b21-b22b11b11Δ2-b21Δ1.Then we get one-periodic wave solution(17)u=u0+2ln⁡fx=u0+2fxf,where f and l, p, μ, and ρ are given by (10) and (16).We see that the well known soliton solution of the 3 + 1-dimensional Jimbo-Miwa equation can be obtained as limit of the periodic solution (17).Theorem 1.
Under the condition(Imτ→∞), the solution (10) of (5) tends to the following exact solution of (2) via (3):(18)u=u0+4πipe2πipx+ly+μz+ρt+ξ0~1+e2πipx+ly+μz+ρt+ξ0~,with l→-3pμ/(4π2p3-2ρ), as α=eπiτ→0, where ξ0~ is an arbitrary constant.Proof.
We writef as(19)f=1+αe2πiξ+e-2πiξ+α4e4πiξ+e-4πiξ+⋯,with ξ=px+ly+μz+ρt+ξ0.
If we make an arbitrary phase constant slightly asξ0=ξ0~-(1/2)τ and have small amplitude limit of α=eπiτ→0, then we derive proper limit (20)f⟶1+e2πipx+ly+μz+ρt+ξ0~.It is easy to obtain the (18). In fact, we have(21)b11=∑n=-∞∞256π4n4p3-32π2n2ρe2πin2τ=512π4p3-64π2ρα2+oα2,Δ1=∑n=-∞∞e2πin2τ=1+2α2+oα2,b12=∑n=-∞∞48π2n2pe2πin2τ=96π2pα2+oα2,Δ2=∑n=-∞∞eπi2n2-2n+1τ=2α+2α5+oα5,b22=∑n=-∞∞12π22n-12peπin2+n-12τ=24π2pα+oα,b21=∑n=-∞∞16π42n-14p3-8π22n-12ρeπi2n2-2n+1τ=32π4p3-16π2ρα+oα5.By utilizing (16), it is easy to deduce that(22)l⟶-3pμ4π2p3-2ρ,c⟶0.The one-periodic solution curves are presented in Figures 1 and 2 for u0=0, respectively, in two-dimensional and three-dimensional space. It is obvious that the solution is periodic and cuspon from the above solution graphs. It is different to the Pfaffian solutions and extended Pfaffian solutions derived by Tang in [27]. The results are different to one-soliton solution and two-soliton solutions represented by researchers in [28, 30–33]. There are some difference between new types of exact periodic solitary wave and kinky periodic wave solutions in [34] by Dai et al. and the solutions in the paper.Figure 1
z
=
0.6, t=2, y=2, p=0.1, μ=2, ρ=4, ζ~=0.8, and x∈-3,3.Figure 2
z
=
0.6, t=2, p=0.1, μ=2, ρ=4, ζ~=0.8, x∈-6,6, and y∈5,20.


## 3. Two-Periodic Wave Solution of the 3 + 1-Dimensional Jimbo-Miwa Equation and Its Asymptotic Behavior

We consider the two-periodic wave solution of the 3 + 1-dimensional Jimbo-Miwa equation (2). Substituting (9) (N=2) into (5), we obtain(23)Gf·f=∑m,n∈Z2GDx,Dt,Dy,Dze2πiζ,n+πiτn,n·e2πiζ,m+πiτm,m=∑m,n∈Z2G2πin-m,p,2πin-m,l,2πin-m,μ,2πin-m,ρe2πiζ,n+m+πiτm,m+τn,n=∑m′∈Z2∑n1,n2=-∞∞G2πi2n-m′,p,2πi2n-m′,l,2πi2n-m′,μ,2πi2n-m′,ρexp⁡πiτn-m′,n-m′+τn,n·exp⁡2πiξ,m′≡∑m′∈Z2G¯⁡m1′,m2′exp⁡2πiξ,m′=0,where m′=n+m. It is easy to calculate that(24)G¯m1′,m2′=∑n1,n2=-∞∞G2πi2n-m′,p,2πi2n-m′,l,2πi2n-m′,μ,2πi2n-m′,ρeπiτn-m′,n-m′+τn,n=∑n1,n2=-∞∞G2πi∑j=122nj′-mj′-2δjlpj,2πi∑j=122nj′-mj′-2δjllj,2πi∑j=122nj′-mj′-2δjlμj,2πi∑j=122nj′-mj′-2δjlρj·exp⁡πi∑j,k=12nj′+δjlτjknk′+δkl+mj′-2δjl-nj′+δjlτjkmk′-2δkl-nk′+δkl=G¯m1′-2,m2′e2πim1′-1τ11+2πim2′τ12,l=1,G¯m1′,m2′-2e2πim2′-1τ22+2πim1′τ21,l=2,which implies that if G¯0,0=G¯0,1=G¯1,0=G¯1,1=0, then G¯m1′,m2′=0 and f is an exact solution of (5).(25)G¯0,0=∑n1,n2=-∞∞2n1l1+2n2l216π42n-m0,p3-8π22n-m0,ρ+12π22n1μ1+2n2μ22n-m0,p+cδ0n,G-0,1=∑n1,n2=-∞∞2n1l1+2n2-1l216π42n-m1,p3-8π22n-m1,ρ+12π22n1μ1+2n2-1μ22n-m1,p+cδ1n,G-1,0=∑n1,n2=-∞∞2n1-1l1+2n2l216π42n-m2,p3-8π22n-m2,ρ+12π22n1-1μ1+2n2μ22n-m2,p+cδ2n,G-1,1=∑n1,n2=-∞∞2n1-1l1+2n2-1l216π42n-m3,p3-8π22n-m3,ρ+12π22n1-1μ1+2n2-1μ22n-m3,p+cδ3n.Letting(26)δjn=eπiτn-mj,n-mj+πiτn,n,m0=0,0T,m1=0,1T,m2=1,0T,m3=1,1T,we have (27)Al1l2μ1μ2=b→,where (28)aj1=∑n1,n2=-∞∞2n1-m1j16π42n-mj,p3-8π22n-mj,pδjn,aj2=∑n1,n2=-∞∞2n2-m2j16π42n-mj,p3-8π22n-mj,pδjn,aj3=∑n1,n2=-∞∞2n1-m1j12π22n-mj,pδjn,aj4=∑n1,n2=-∞∞2n2-m2j12π22n-mj,pδjn,bj=-c∑n1,n2=-∞∞δjn,j=0,1,2,3.From this, we have (29)l1=Δ1Δ,l2=Δ2Δ,μ1=Δ3Δ,μ2=Δ4Δ,where Δ=A and Δ1, Δ2, Δ3, and Δ4 are from Δ by replacing 1st, 2nd, 3rd, and 4th column with b→, respectively.Theorem 2.
Under the conditionα1→0, α2→0, the solution (9) (N=2) of (5) tends to the two-periodic solution of (2) via (3).(30)u=u0+4πip1eξ~1+p2eξ~2+p1+p2eξ1~+ξ2~+A1+eξ~1+eξ~2+eξ1~+ξ2~+A,where(31)e2πiτ12=eA=-l1-l24π4p1-p23-2π2ρ1-ρ2+3π2μ1-μ2p1-p2l1+l24π4p1+p23-2π2ρ1+ρ2+3π2μ1+μ2p1+p2,l24π2p23-2ρ2+3μ2p2=0,l14π2p13-2ρ1+3μ1p1=0,ξ~1=2πip1x+l1y+μ1z+ρ1t+ξ~01,ξ~2=2πip2x+l2y+μ2z+ρ2t+ξ~02,and ξ~01 and ξ~02 are arbitrary constants.Proof.
We writef as (32)f=1+e2πiξ1+e-2πiξ1eπiτ11+e2πiξ2+e-2πiξ2eπiτ22+e2πiξ1+ξ2+e-2πiξ1+ξ2eπiτ11+2τ12+τ22+⋯.Set (33)ξ1=p1x+l1y+μ1z+ρ1t+ξ01,ξ2=p2x+l2y+μ2z+ρ2t+ξ02,ξ01=ξ~01-τ112,ξ02=ξ~02-τ222,α1=eπiτ11,α2=eπiτ22.Thus (34)f=1+eξ1~+eξ2~+eξ1~+ξ2~+2πiτ12+oα1s1α2s2.As α1→0, α2→0, and s1+s2≥3, then f→1+eξ1~+eξ2~+eξ1~+ξ2~+2πiτ12.
In view of transformation (3), we have derived the solution (30) of (2). In what follows, we will certify (31). (35)G¯0,0=∑n1,n2=-∞∞2n1l1+2n2l216π42n-m0,p3-8π22n-m0,ρ+12π22n1μ1+2n2μ22n-m0,p+cδ0n=c+2l1+l2128π4p1+p23-16π2ρ1+ρ2+ce2πiτ11+2τ12+τ22+2l1+l2128π4p1+p23-16π2ρ1+ρ2+ce2πiτ11+2τ12+τ22+32l2π28π2p23-ρ2+48π2μ2p2+ce2πiτ22+32l1π28π2p13-ρ1+48π2μ1p1+ce2πiτ11+oα1s1α1s2=c+22l1+l2128π4p1+p23-16π2ρ1+ρ2+cα12α22e4πiτ12+32l2π28π2p23-ρ2+48π2μ2p2+cα22+32l1π28π2p13-ρ1+48π2μ1p1+cα12+oα1s1α1s2.As α1→0 and α2→0, then G¯0,0→0, with s1+s2>3. Thus, we derive c→0.(36)G¯0,1=∑n1,n2=-∞∞2n1l1+2n2-1l216π42n-m1,p3-8π22n-m1,ρ+12π22n1μ1+2n2-1μ22n-m1,p+cδ1n=2l216π4p23-8π2ρ2+12π2μ2p2+ceπiτ22+2l1-l216π42p1-p23-8π22ρ1-ρ2+12π22μ1-μ22p1-p2eπi2τ11+τ22-2τ12+oα1s1α2s2s1+s2≥3.From G¯0,1→0, we have(37)l24π2p23-2ρ2+3μ2p2=0.Similarly, (38)G¯1,0=∑n1,n2=-∞∞2n2l2+2n1-1l116π42n-m2,p3-8π22n-m2,ρ+12π22n2μ2+2n1-1μ12n-m2,p+cδ2n=2l116π4p13-8π2ρ1+12π2μ1p1+ceπiτ11+2l2-l116π42p2-p13-8π22ρ2-ρ1+12π22μ2-μ12p2-p1eπi2τ22+τ11-2τ12+oα1s1α2s2s1+s2≥3.From G¯1,0→0, we have (39)l14π2p13-2ρ1+3μ1p1=0.Then(40)G¯1,1=∑n1,n2=-∞∞2n1-1l1+2n2-1l216π42n-m3,p3-8π22n-m3,ρ+12π22n1-1μ1+2n2-1μ22n-m3,p+cδ3n=2l1+l216π4p1+p23-8π2ρ1+ρ2+12π2μ1+μ2p1+p2+ceπiτ11+τ22+2τ12+2l1-l216π4p1-p23-8π2ρ1-ρ2+12π2μ1+μ2p1+p2+ceπiτ11+τ22+3l2-l116π43p2-p13-8π23ρ2-ρ1+12π23μ2-μ13p2-p1+ceπiτ11-τ12+3τ22+oα1s1α2s2=2l1+l216π4p1+p23-8π2ρ1+ρ2+12π2μ1+μ2p1+p2+ceπi2τ12+2l1-l216π4p1-p23-8π2ρ1-ρ2+12π2μ1+μ2p1+p2+ceπiτ11+τ22+3l2-l116π43p2-p13-8π23ρ2-ρ1+12π23μ2-μ13p2-p1+ceπiτ11-τ12+3τ22+oα1s1α2s2,s1+s2≥3.Using G¯1,1→0, we obtain (41)e2πiτ12=eA=-l1-l24π4p1-p23-2π2ρ1-ρ2+3π2μ1-μ2p1-p2l1+l24π4p1+p23-2π2ρ1+ρ2+3π2μ1+μ2p1+p2.In order to show the solution character, we drop the solution curves of realu and imaginary u. Figures 3 and 4 plot the real and imaginary of u, respectively, in three-dimensional space. From the solution graphs, we can see that the solutions are periodic and cuspon. The derived two-periodic solutions in the paper are different to the two-solitary solutions in [32] and rational solutions presented by Ma and Lee in [31].Figure 3
z
=
0.1, t=0.5, p1=0.2, μ1=0.35, ρ1=0.9, ζ~01=1, p2=0.2, μ2=0.3, ρ2=0.6, ζ~02=0.9, x∈-10,10, and y∈-10,10.Figure 4
z
=
0.1, t=0.5, p1=0.2, μ1=0.35, ρ1=0.9, ζ~01=1, p2=0.2, μ2=0.3, ρ2=0.6, ζ~02=0.9, x∈-10,10, and y∈-10,10.



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*Source: 2924947-2017-06-22.xml*