# Exact Solutions for (3+1)-Dimensional Potential-YTSF Equation and Discrete Kadomtsev-Petviashvili Equation

Authors: Yan Wang; Zhenhui Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

By employing Hirota bilinear method, we mainly discuss the (3+1)-dimensional potential-YTSF equation and discrete KP equation. For the former, we use the linear superposition principle to get its N exponential wave solutions. In virtue of some Riemann theta function formulas, we also construct its quasiperiodic solutions and analyze the asymptotic properties of these solutions. For the latter, by using certain variable transformations and identities of the theta functions, we explicitly investigate its periodic waves solutions in terms of one-theta function and two-theta functions.


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## Body

## 1. Introduction

There has been considerable interest in seeking exact solutions to nonlinear differential or discrete equations. Since the exact solutions can help the physicists to well understand the mechanism of the complicated physical phenomena and dynamic processes modeled by these equations. In recent years, various approaches for constructing exact solutions are currently available such as inverse scattering transformation [1], Hirota direct method [2], the Bäcklund transformation [3], and algebra-geometric method [4–6].Among vast exact solutions, linear superposition principle [7] and periodic or quasiperiodic wave solution [8] play a significant role in explaining the physical applications of these systems. The former method can be applied to exponential traveling waves of Hirota bilinear equations and is helpful in generating N-wave solutions to soliton equations, particularly those in higher dimensions. The latter is also called algebra-geometric solutions or finite gap solution; it is often obtained based on the inverse spectral theory and algebra-geometric method. The algebra-geometric theory, however, needs Lax pairs and is also involved in complicated analysis procedures on the Riemann surfaces. It is rather difficult to directly determine the characteristic parameters of waves, such as frequencies and phase shifts, for a function with given wave numbers and amplitudes. Based on the Hirota forms, Nakamura proposed a convenient way to find a kind of explicit quasiperiodic solution of nonlinear equations [9, 10]; it does not need any Lax pair and Riemann surface for the given nonlinear equation and is also able to find the explicit construction of multiperiodic wave solutions. Recently, Fan et al. [11–13] extended this method and gave a uniform approach to establish the periodic solutions of nonlinear differential and difference equations. However, there are only a few works [14, 15] available for constructing multiperiodic wave solutions of discrete equations, since more constraint equations need to be satisfied, but the parameters in the bilinear form of these discrete equations are insufficient; thus, it is difficult to construct multiperiodic wave solutions, even for two-periodic wave solution.In this paper, we will mainly consider3+1-dimensional potential-YTSF equation and discrete KP equation. First, we use a linear superposition principle to generate N-wave solutions of the former then use the above discussing method to construct quasiperiodic or periodic wave solutions of these two equations. In an appropriate limiting procedure, the soliton solutions are also obtained from the quasiperiodic solutions.


## 2.(3+1)-Dimensional Potential-YTSF Equation

(
3
+
1
)-Dimensional potential-YTSF equation, first introduced by Yu et al. [16], may be written as
(1)-4xt+uxxxz+4uxuxz+2uxxuz+3uyy=0.
The exact solitary-wave and periodic solutions have been addressed by means of the auto-Bäcklund transformation [17], the generalized projective Riccati equation method [18], the extended homoclinic test technique [19], and so on. Using the dependent variable transformation
(2)ζ=x+dz,u=2(lnf)ζ,
we transform (1) to the bilinear form
(3)[-4DζDt+dDζ4+3Dy2+c]f·f=0,
where d is an arbitrary positive constant and Hirota bilinear operators Dζ, Dy, and Dt are defined by
(4)DζmDtm′f(ζ,t)·g(ζ,t)=(∂ζ-∂ζ′)m(∂t-∂t′)m′∣ζ′=ζ,t′=t,
which have a nice property when acting on exponential functions
(5)DζmDtm′eη1·eη2=(k1-k2)m(w1-w2)m′·eη1+η2,
with ηj=kjζ+ljy+wjt, j=1,2.


### 2.1. N Exponential Wave Solutions

Let us consider the special casec=0 in (3) and introduce N-wave testing function
(6)f=∑j=1N‍ϵjfj=∑j=1N‍ϵjekjζ+ljy+wjt,1≤j≤N,
where εj, kj, lj, and wj are arbitrary constants. Substituting (6) into (3) and using (5) give
(7)∑j=1N‍∑j=1N[-4(ki-kj)(wi-wj)+d(ki-kj)4‍hh+3(li-lj)2]fi·fj=0.
If the following condition
(8)[-4(ki-kj)(wi-wj)+d(ki-kj)4+3(li-lj)2]fi·fj=0
is satisfied, then any linear combination of the N exponential wave solutions eηj, 1≤j≤N solves (3). By inspection, a solution to (8) is
(9)wj=dkj3,lj=±dkj2,1≤j≤N.
Therefore, (1) has the following N-wave solution
(10)u=2(lnf)ζ,f=∑j=1N‍ϵjfj=∑j=1N‍ϵjekj(x+dz)±dkj2y+dkj3t,hhhhhhhhhhhhhhhhhh1≤j≤N,
where ϵj, kj are arbitrary constants. Especially assume that ϵ1=ϵ2=ϵ3=1, k1=-k2, and k3=0 in (10); then, we get
(11)f=1+(ek1(x+dz)+dk13t+e-k1(x+dz)-dk13t)e±dk12y,u=2(lnf)ζ=2k1sinh(k1(x+dz)+dk13t)e±dk12y(1/2)+cosh(k1(x+dz)+dk13t)e±dk12y.


### 2.2. Quasiperiodic Solutions

The Riemann theta functions of genus one [20] are defined by
(12)Θ1(ξ,τ)=-i∑m=-∞∞‍(-1)meπiξ(2m+1)+πiτ(m+(1/2))2,Θ2(ξ,τ)=∑m=-∞∞‍eπiξ(2m+1)+πiτ(m+(1/2))2,Θ3(ξ,τ)=∑m=-∞∞‍e2mπiξ+πiτm2,Θ4(ξ,τ)=∑m=-∞∞‍(-1)me2mπiξ+πiτm2,
respectively, where ξ=kζ+ly+wt. Letting Θk(x)=Θk(x,τ) (k=1,2,3,4) and using the identities
(13)Θ3(x+y)Θ3(x-y)Θ22(0)=Θ32(x)Θ22(y)+Θ42(x)Θ12(y),Θ4(x+y)Θ4(x-y)Θ22(0)=Θ42(x)Θ22(y)+Θ32(x)Θ12(y),
we can obtain the Hirota derivatives of Θ3(x), Θ4(x):
(14)Dx2Θ3(x)·Θ3(x)=∂y2Θ3(x+y)Θ3(x-y)∣y=0=b1Θ42(x)+b2Θ32(x),Dx4Θ3(x)·Θ3(x)=∂y4Θ3(x+y)Θ3(x-y)∣y=0=c1Θ42(x)+c2Θ32(x),Dx2Θ4(x)·Θ4(x)=∂y2Θ4(x+y)Θ4(x-y)∣y=0=b2Θ42(x)-b1Θ32(x),Dx4Θ4(x)·Θ4(x)=∂y4Θ3(x+y)Θ3(x-y)∣y=0=c2Θ42(x)-c1Θ32(x),
where
(15)b1=2Θ32(0)Θ42(0),b2=2Θ2′′(0)Θ2(0),b3=2(Θ3′′(0)Θ3(0)+Θ4′′(0)Θ4(0)),b4=2Θ2(4)(0)Θ2(0),c1=2b1(b2+b3),c2=32b22+b4.
Next, we will calculate analytically the quasiperiodic solutions using the Hirota derivatives in (14).Case 1.
In the bilinear equation (3), we suppose f(x,y,z,t) to be
(16)f(x,y,z,t)=Θ3(ξ),ξ=k1ζ+l1y+w1t,
where k1,  l1, and w1 are arbitrary constants. Substituting (16) into (3) and setting the coefficients of the terms Θ32(ξ), Θ42(ξ) to be zero, we obtain a system of algebraic equations
(17)(-4k1w1+3l12)b2+dk14c2+c=0,(4k1w1-3l12)b1-dk14c1=0.
Solving (17), we find
(18)w1=3l12b1+dk14c14k1b1,c=-(3l12b2+dk14c1)+b2b1(3l12b1+dk14c1),
where b1, b2 and c1, c2 are given by (15). We have thus constructed a kind of quasiperiodic solution for (1)
(19)u=2ln(Θ3(k1ζ+l1z+w1t))ζ.
In the following, we further analyze the asymptotic property of (19).Proposition 1.
If the vector(w1,c)T is given by (18) and supposing ξ′=2πiξ+πiτ, k′=2πik, and Imτ→∞, then one has
(20)f=Θ3(ξ)=1+eπiτ[e2πiξ+e-2πiξ]+e4πiτ[e4πiξ+e-4πiξ]+⋯=1+eξ′+e2πiτe-ξ′+e2πiτe2ξ′…≈1+eξ′,asImτ→∞.
From (2), one gets the solution of (1) as follows:
(21)u=2(lnf)ζ⟶4kπieξ′1+eξ′=k′secξ′2e(ξ′/2).Case 2.
Assumef(x,y,z,t) in (3) to be
(22)f(x,y,z,t)=Θ4(ξ),ξ=k2ζ+l2y+w2t,
where k1, l1, and w1 are arbitrary constants. Setting the coefficients of the terms Θ32(ξ), Θ42(ξ) to be zero, we have
(23)(4k2w2-3l22)b2-dk24c1=0,(-4k2w2+3l22)b2+dk24c2+c=0.
Solving (23) yields
(24)w2=3l22b1+dk24c14k2b1,c=-(3l22b2+dk24c1)+b2b1(3l22b1+dk24c1),
where b1, b2, and c1, c2 are given by (15). Thus, the second kind of quasiperiodic solution of (1) is constructed as follows:
(25)u=2ln(Θ4(k2ζ+l2z+w2t))ζ.Proposition 2.
If the vector(w2,c)T is given by (24) and supposing ξ′=2πiξ+πiτ, k′=2πik, and Imτ→∞, then one has
(26)f=Θ4(ξ)=1-eπiτ[e2πiξ+e-2πiξ]+e4πiτ[e4πiξ+e-4πiξ]+⋯=1-eξ′-e2πiτe-ξ′+e2πiτe2ξ′…≈1-eξ′,asImτ⟶∞.
Using the variable transformation (2), one gets the solution of (1) as follows:
(27)u=2(lnf)ζ⟶-4kπieξ′1-eξ′=-k′cscξ′2e(ξ^′/2).


## 2.1. N Exponential Wave Solutions

Let us consider the special casec=0 in (3) and introduce N-wave testing function
(6)f=∑j=1N‍ϵjfj=∑j=1N‍ϵjekjζ+ljy+wjt,1≤j≤N,
where εj, kj, lj, and wj are arbitrary constants. Substituting (6) into (3) and using (5) give
(7)∑j=1N‍∑j=1N[-4(ki-kj)(wi-wj)+d(ki-kj)4‍hh+3(li-lj)2]fi·fj=0.
If the following condition
(8)[-4(ki-kj)(wi-wj)+d(ki-kj)4+3(li-lj)2]fi·fj=0
is satisfied, then any linear combination of the N exponential wave solutions eηj, 1≤j≤N solves (3). By inspection, a solution to (8) is
(9)wj=dkj3,lj=±dkj2,1≤j≤N.
Therefore, (1) has the following N-wave solution
(10)u=2(lnf)ζ,f=∑j=1N‍ϵjfj=∑j=1N‍ϵjekj(x+dz)±dkj2y+dkj3t,hhhhhhhhhhhhhhhhhh1≤j≤N,
where ϵj, kj are arbitrary constants. Especially assume that ϵ1=ϵ2=ϵ3=1, k1=-k2, and k3=0 in (10); then, we get
(11)f=1+(ek1(x+dz)+dk13t+e-k1(x+dz)-dk13t)e±dk12y,u=2(lnf)ζ=2k1sinh(k1(x+dz)+dk13t)e±dk12y(1/2)+cosh(k1(x+dz)+dk13t)e±dk12y.


## 2.2. Quasiperiodic Solutions

The Riemann theta functions of genus one [20] are defined by
(12)Θ1(ξ,τ)=-i∑m=-∞∞‍(-1)meπiξ(2m+1)+πiτ(m+(1/2))2,Θ2(ξ,τ)=∑m=-∞∞‍eπiξ(2m+1)+πiτ(m+(1/2))2,Θ3(ξ,τ)=∑m=-∞∞‍e2mπiξ+πiτm2,Θ4(ξ,τ)=∑m=-∞∞‍(-1)me2mπiξ+πiτm2,
respectively, where ξ=kζ+ly+wt. Letting Θk(x)=Θk(x,τ) (k=1,2,3,4) and using the identities
(13)Θ3(x+y)Θ3(x-y)Θ22(0)=Θ32(x)Θ22(y)+Θ42(x)Θ12(y),Θ4(x+y)Θ4(x-y)Θ22(0)=Θ42(x)Θ22(y)+Θ32(x)Θ12(y),
we can obtain the Hirota derivatives of Θ3(x), Θ4(x):
(14)Dx2Θ3(x)·Θ3(x)=∂y2Θ3(x+y)Θ3(x-y)∣y=0=b1Θ42(x)+b2Θ32(x),Dx4Θ3(x)·Θ3(x)=∂y4Θ3(x+y)Θ3(x-y)∣y=0=c1Θ42(x)+c2Θ32(x),Dx2Θ4(x)·Θ4(x)=∂y2Θ4(x+y)Θ4(x-y)∣y=0=b2Θ42(x)-b1Θ32(x),Dx4Θ4(x)·Θ4(x)=∂y4Θ3(x+y)Θ3(x-y)∣y=0=c2Θ42(x)-c1Θ32(x),
where
(15)b1=2Θ32(0)Θ42(0),b2=2Θ2′′(0)Θ2(0),b3=2(Θ3′′(0)Θ3(0)+Θ4′′(0)Θ4(0)),b4=2Θ2(4)(0)Θ2(0),c1=2b1(b2+b3),c2=32b22+b4.
Next, we will calculate analytically the quasiperiodic solutions using the Hirota derivatives in (14).Case 1.
In the bilinear equation (3), we suppose f(x,y,z,t) to be
(16)f(x,y,z,t)=Θ3(ξ),ξ=k1ζ+l1y+w1t,
where k1,  l1, and w1 are arbitrary constants. Substituting (16) into (3) and setting the coefficients of the terms Θ32(ξ), Θ42(ξ) to be zero, we obtain a system of algebraic equations
(17)(-4k1w1+3l12)b2+dk14c2+c=0,(4k1w1-3l12)b1-dk14c1=0.
Solving (17), we find
(18)w1=3l12b1+dk14c14k1b1,c=-(3l12b2+dk14c1)+b2b1(3l12b1+dk14c1),
where b1, b2 and c1, c2 are given by (15). We have thus constructed a kind of quasiperiodic solution for (1)
(19)u=2ln(Θ3(k1ζ+l1z+w1t))ζ.
In the following, we further analyze the asymptotic property of (19).Proposition 1.
If the vector(w1,c)T is given by (18) and supposing ξ′=2πiξ+πiτ, k′=2πik, and Imτ→∞, then one has
(20)f=Θ3(ξ)=1+eπiτ[e2πiξ+e-2πiξ]+e4πiτ[e4πiξ+e-4πiξ]+⋯=1+eξ′+e2πiτe-ξ′+e2πiτe2ξ′…≈1+eξ′,asImτ→∞.
From (2), one gets the solution of (1) as follows:
(21)u=2(lnf)ζ⟶4kπieξ′1+eξ′=k′secξ′2e(ξ′/2).Case 2.
Assumef(x,y,z,t) in (3) to be
(22)f(x,y,z,t)=Θ4(ξ),ξ=k2ζ+l2y+w2t,
where k1, l1, and w1 are arbitrary constants. Setting the coefficients of the terms Θ32(ξ), Θ42(ξ) to be zero, we have
(23)(4k2w2-3l22)b2-dk24c1=0,(-4k2w2+3l22)b2+dk24c2+c=0.
Solving (23) yields
(24)w2=3l22b1+dk24c14k2b1,c=-(3l22b2+dk24c1)+b2b1(3l22b1+dk24c1),
where b1, b2, and c1, c2 are given by (15). Thus, the second kind of quasiperiodic solution of (1) is constructed as follows:
(25)u=2ln(Θ4(k2ζ+l2z+w2t))ζ.Proposition 2.
If the vector(w2,c)T is given by (24) and supposing ξ′=2πiξ+πiτ, k′=2πik, and Imτ→∞, then one has
(26)f=Θ4(ξ)=1-eπiτ[e2πiξ+e-2πiξ]+e4πiτ[e4πiξ+e-4πiξ]+⋯=1-eξ′-e2πiτe-ξ′+e2πiτe2ξ′…≈1-eξ′,asImτ⟶∞.
Using the variable transformation (2), one gets the solution of (1) as follows:
(27)u=2(lnf)ζ⟶-4kπieξ′1-eξ′=-k′cscξ′2e(ξ^′/2).


## 3. Discrete KP Equation

The discrete analogue of KP equation [21, 22] (or a two-dimensional analogue of the discrete KdV equation) is
(28)∇tu+v1+u=χ∇xE+yu,∇yu+v1+u=E+yv-E-xE+tvδy(1+E+yu),
where u and v are functions of x,y, and t,χ is a constant, ∇ and E are the difference and shift operators defined by
(29)∇zf(z)=1δz[f(z+δz2)-f(z-δz2)],E±zf(z)=f(z±δz2).
In fact, using the dependent variable transformation
(30)n=a(xδx+yδy),l=b(yδy+tδt),m=-c(xδx+tδt),u=ζn,lmζn,l+bm+cζn,l+bmζn,lm+c-1,v=1-ζn,lmζn+a,l+bm+cζn+a,l+bmζn,lm+c,χ=a(b-c)b(c-a)δxδt
and substituting (30) into (28), we have the bilinear form of discrete KP equation as follows:
(31)a(b-c)ζn+a,lmζn,l+bm+c+b(c-a)ζn,l+bmζn+a,lm+c+c(a-b)ζn,lm+cζn+a,l+bm=0,
where a, b, and c are the difference intervals for independent variables n, l, and m, respectively. Some operator solutions of (31) have been addressed by the transformation groups methods [23, 24]. This famous three-dimensional difference equation is interesting and important because it is emerging in the context of quantum integrable systems [25, 26] as the model-independent functional relations for eigenvalues of quantum transfer matrices. Moreover, some typical soliton equations can be obtained by performing a scaling continuum limit for appropriate combinations of parameters and variables, such as continuous Korteweg-de Vries (KdV) equation, KP equation, two-dimensional Toda lattice (2DTL) equation, Sine-Gordon (SG) equation, and Benjamin-Ono equation.


### 3.1. The Solutions in Terms of One-Theta Function

Suppose that(32)ζn+(a/2),l+(b/2)m+(c/2)=Θi(k1n+k2l+k3m+k1a+k2b+k3c2,τ)=Θi(ξ,τ),i=1,2,3,4,
where
(33)ξ=k1n+k2l+k3m+k1a+k2b+k3c2,k1, k2, and k3 are arbitrary constants, and Θi(ξ,τ) is defined by (12). Similarly, we can obtain
(34)ζn+a,lm=Θi(ξ-α,τ),ζn,l+bm+c=Θi(ξ+α,τ),ζn,l+bm=Θi(ξ-β,τ),ζn+a,lm+c=Θi(ξ+β,τ),ζn,lm+c=Θi(ξ-γ,τ),ζn+a,l+bm=Θi(ξ+γ,τ),
with
(35)α=k2b+k3c-k1a2,β=k1a+k3c-k2b2,γ=k1a+k2b-k3c2.
Substituting (34) into (31) and using the following identities of the theta functions
(36)Θ1(x+y)Θ1(x-y)=Θ2-2(0)[Θ12(x)Θ22(y)-Θ22(x)Θ12(y)],Θ2(x+y)Θ2(x-y)=Θ2-1(0)[Θ22(x)Θ22(y)-Θ12(x)Θ12(y)],Θ3(x+y)Θ3(x-y)=Θ2-2(0)[Θ22(x)Θ32(y)+Θ12(x)Θ42(y)],Θ4(x+y)Θ4(x-y)=Θ2-2(0)[Θ12(x)Θ32(y)+Θ22(x)Θ42(y)],
yield
(37)a(b-c)Θ22(α)+b(c-a)Θ22(β)+c(a-b)Θ22(γ)=0,a(b-c)Θ12(α)+b(c-a)×Θ12(β)+c(a-b)Θ12(γ)=0,hhhhhhhhhhhhhhhwheni=1,2;a(b-c)Θ32(α)+b(c-a)Θ32(β)+c(a-b)Θ32(γ)=0,a(b-c)Θ42(α)+b(c-a)Θ42(β)+c(a-b)Θ42(γ)=0,hhhhhhhhwheni=3,4.
Here, we use Θi(ξ)=Θi(ξ,τ) for simplicity. From (37), we can find that if α=β=γ, that is, k1a=k2b=k3c, which follows from (35), then (31) holds. In virtue of the variable transformation (30), the solution of (28) is expressed by one-theta function as follows:
(38)u=(Θi[k1(n+abl+acm)]×Θi[k1(n+abl+acm)+2k1a])×(Θi2[k1(n+abl+acm)+k1a])-1-1,v=1-(Θi[k1(n+abl+acm)]×Θi[k1(n+abl+acm)+3k1a])×(Θi[k1(n+abl+acm)+2k1a]×Θi[k1(n+abl+acm)+k1a]),i=1,2,3,4.Remark 3 (see [27]).
Ifζn,lm=Θi(ξ-(3k1a/2),τ) is a solution of (31), then R0((n/a)+(l/b)+(m/c))R1(n/a)R2(l/b)R3(m/c)Θi(ξ-(3k1a/2),τ) is a solution too, where Ri, (i=0,1,2,3), are arbitrary functions.Remark 4 (see [27]).
By means of the transformation,ζn,lm=[(b(c-a))/(c(a-b))](nl/ab)[(b(c-a))/(a(b-c))](lm/bc)ζn,lm; then, [(b(c-a))/(c(a-b))](nl/ab)[(b(c-a))/(a(b-c))](lm/bc)Θi(ξ-(3k1a/2),τ) satisfies bilinear equation
(39)ζn+a,lmζn,l+bm+c+ζn,l+bmζn+a,lm+c+ζn,lm+cζn+a,l+bm=0.Proposition 5.
The Riemann theta functionΘ3(ξ,τ) defined by (12) has the periodic properties
(40)Θ3(ξ+1+τ,τ)=exp(-2iπξ-iπτ)Θ3(ξ,τ).
So, the meromorphic functions F(ξ,τ) defined by
(41)(I)F(ξ,τ)=Θ3(ξ,τ)Θ3(ξ+2h,τ)Θ32(ξ+h,τ),(∀h,ξ∈C)(II)F(ξ,τ)=Θ3(ξ,τ)Θ3(ξ+3h,τ)Θ3(ξ+2h,τ)Θ3(ξ+h,τ),(∀h,ξ∈C)
are all double periodic functions with two fundamental periods 1 and τ. Because the other three Riemann theta functions in (12) are the deformations of Θ3(ξ,τ), they can be proved to be periodic in a similar way.Proof.
By using definitionΘ3(ξ,τ) in (12), it is easy to see that
(42)Θ3(ξ+1+τ,τ)=∑m=-∞∞‍e2mπi(ξ+1+τ)+πiτm2=e-2πiξe-πiτΘ3(ξ,τ),Θ3(ξ+h+1+τ,τ)=∑m=-∞∞‍e2mπi(ξ+h+1+τ)+πiτm2=e-2πi(ξ+h)e-πiτΘ3(ξ+h,τ).
Therefore, in both (I) and (II) cases, it holds that
(43)F(ξ+1+τ,τ)=F(ξ,τ),ξ∈C.
That is to say, (38) is periodic wave solution of (28).


### 3.2. The Solutions in Terms of Two-Theta Function

In order to look for its solutions in terms of two-theta functions, we suppose(44)ζn+(a/2),l+(b/2)m+(c/2)=Θi(ξ,τ1)Θj(ξ,τ2)=Θi(k1a+k2b+k3c2k1n+k2l+k3mhhh+k1a+k2b+k3c2,τ1)×Θj(k1a+k2b+k3c2k1n+k2l+k3mh+k1a+k2b+k3c2,τ2),hhhhhhhhhhh∀i,j=1,2,3,4.
Therefore,
(45)ζn+a,lm=Θi(ξ-α,τ1)Θj(ξ-α,τ2),ζn,l+bm+c=Θi(ξ+α,τ1)Θj(ξ+α,τ2),ζn,l+bm=Θi(ξ-β,τ1)Θj(ξ-β,τ2),ζn+a,lm+c=Θi(ξ+β,τ1)Θj(ξ+β,τ2),ζn,lm+c=Θi(ξ-γ,τ1)Θj(ξ-γ,τ2),ζn+a,l+bm=Θi(ξ+γ,τ1)Θj(ξ+γ,τ2),
where α, β, and γ are the same as in (35).Case 1.
Ifi=1,j=2, we substitute (45) into (31) to get
(46)a(b-c)Θ1(ξ-α,τ1)Θ2(ξ-α,τ2)×Θ1(ξ+α,τ1)Θ2(ξ+α,τ2)+b(c-a)×Θ1(ξ-β,τ1)Θ2(ξ-β,τ2)Θ1(ξ+β,τ1)×Θ2(ξ+β,τ2)+c(a-b)Θ1(ξ-γ,τ1)×Θ2(ξ-γ,τ2)Θ1(ξ+γ,τ1)Θ2(ξ+γ,τ2)=0.
By using the first two identities of the theta functions (36), the above equation is equivalent to
(47)Θ12(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ22(α,τ1)Θ22(α,τ2)hhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ22(β,τ2)hhhhhhhhhhhh+ϵ3Θ22(γ,τ1)Θ22(γ,τ2)]-Θ12(ξ,τ1)Θ12(ξ,τ2)[ϵ1Θ22(α,τ1)Θ12(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ12(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ22(γ,τ1)Θ12(γ,τ2)]-Θ22(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ12(α,τ1)Θ22(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ22(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(α,τ1)Θ22(α,τ2)]+Θ22(ξ,τ1)Θ12(ξ,τ2)[ε1Θ12(α,τ1)Θ12(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ12(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(γ,τ1)Θ12(γ,τ2)]=0.
Therefore,
(48)ϵ1Θ22(α,τ1)Θ22(α,τ2)+ϵ2Θ22(β,τ1)Θ22(β,τ2)+ϵ3Θ22(γ,τ1)Θ22(γ,τ2)=0,ϵ1Θ22(α,τ1)Θ12(α,τ2)+ϵ2Θ22(β,τ1)Θ12(β,τ2)+ϵ3Θ22(γ,τ1)Θ12(γ,τ2)=0,ϵ1Θ12(α,τ1)Θ22(α,τ2)+ϵ2Θ12(β,τ1)Θ22(β,τ2)+ϵ3Θ12(γ,τ1)Θ22(γ,τ2)=0,ϵ1Θ12(α,τ1)Θ12(α,τ2)+ϵ2Θ12(β,τ1)Θ12(β,τ2)+ϵ3Θ12(γ,τ1)Θ12(γ,τ2)=0.
Here, ϵ1=a(b-c), ϵ2=b(c-a), and ϵ3=c(a-b). To get special solutions of (48), we can assume that α=β=γ; that is, k1a=k2b=k3c. From the variable transformation (30), the solution of (28) in terms of two-theta functions are
(49)u=(Θ1[k1(n+abl+acm),τ1]×Θ2[k1(n+abl+acm),τ2]×Θ1[k1(n+abl+acm)+2k1a,τ1]×Θ2[k1(n+abl+acm)+2k1a,τ2])×(Θ12[k1(n+abl+acm)+k1a,τ1]×Θ22[k1(n+abl+acm)+k1a,τ2])-1-1,v=1-(Θ1[k1(n+abl+acm),τ1]×Θ2[k1(n+abl+acm),τ2]×Θ1[k1(n+abl+acm)+3k1a,τ1]×Θ2[k1(n+abl+acm)+3k1a,τ2])×(Θ1[k1(n+abl+acm)+2k1a,τ1]×Θ2[k1(n+abl+acm)+2k1a,τ2]×Θ1[k1(n+abl+acm)+k1a,τ1]×Θ2[k1(n+abl+acm)+k1a,τ2])-1.Case 2.
Ifi=2, j=3 and substituting (45) into (31), then we are led to
(50)a(b-c)Θ2(ξ-α,τ1)Θ3(ξ-α,τ2)Θ2(ξ+α,τ1)×Θ3(ξ+α,τ2)+b(c-a)Θ2(ξ-β,τ1)×Θ3(ξ-β,τ2)Θ2(ξ+β,τ1)Θ3(ξ+β,τ2)+c(a-b)Θ2(ξ-γ,τ1)Θ3(ξ-γ,τ2)×Θ2(ξ+γ,τ1)Θ3(ξ+γ,τ2)=0.
From the identities of the theta functions (36), the last equation is changed into
(51)Θ22(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ22(α,τ1)Θ32(α,τ2)hhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ32(β,τ2)hhhhhhhhhhhh+ϵ3Θ22(γ,τ1)Θ32(γ,τ2)]-Θ12(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ12(α,τ1)Θ32(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ32(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(γ,τ1)Θ32(γ,τ2)]+Θ22(ξ,τ1)Θ12(ξ,τ2)[ϵ1Θ22(α,τ1)Θ42(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ42(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ22(α,τ1)Θ42(α,τ2)]-Θ12(ξ,τ1)Θ12(ξ,τ2)[ϵ1Θ12(α,τ1)Θ42(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ42(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(γ,τ1)Θ42(γ,τ2)]=0.
Therefore,
(52)ϵ1Θ22(α,τ1)Θ32(α,τ2)+ϵ2Θ22(β,τ1)Θ32(β,τ2)+ϵ3Θ22(γ,τ1)Θ32(γ,τ2)=0,ϵ1Θ12(α,τ1)Θ32(α,τ2)+ϵ2Θ12(β,τ1)Θ32(β,τ2)+ϵ3Θ12(γ,τ1)Θ32(γ,τ2)=0,ϵ1Θ22(α,τ1)Θ42(α,τ2)+ϵ2Θ22(β,τ1)Θ42(β,τ2)+ϵ3Θ22(α,τ1)Θ42(α,τ2)=0,ϵ1Θ12(α,τ1)Θ42(α,τ2)+ϵ2Θ12(β,τ1)Θ42(β,τ2)+ϵ3Θ12(γ,τ1)Θ42(γ,τ2)=0,
with ϵ1=a(b-c), ϵ2=b(c-a), and ϵ3=c(a-b). To get the special solutions of (52), we can assume that α=β=γ; that is, k1a=k2b=k3c. From the variable transformation (30), the solution of (28) in terms of two-theta functions are
(53)u=(Θ2[k1(n+abl+acm),τ1]×Θ3[k1(n+abl+acm),τ2]×Θ2[k1(n+abl+acm)+2k1a,τ1]×Θ3[k1(n+abl+acm)+2k1a,τ2])×(Θ22[k1(n+abl+acm)+k1a,τ1]×Θ32[k1(n+abl+acm)+k1a,τ2])-1-1,v=1-(Θ2[k1(n+abl+acm),τ1]×Θ3[k1(n+abl+acm),τ2]×Θ2[k1(n+abl+acm)+3k1a,τ1]×Θ3[k1(n+abl+acm)+3k1a,τ2])×Θ2([k1(n+abl+acm)+2k1a,τ1]×Θ3[k1(n+abl+acm)+2k1a,τ2]×Θ2[k1(n+abl+acm)+k1a,τ1]×Θ3[k1(n+abl+acm)+k1a,τ2])-1.Proposition 6.
The Riemann theta functionsΘ2(ξ,τ), Θ3(ξ,τ) defined by (12) have the periodic properties
(54)Θ2(ξ+2+τ,τ)=exp(-2iπξ-iπτ)Θ2(ξ,τ),Θ3(ξ+1+τ,τ)=exp(-2iπξ-iπτ)Θ3(ξ,τ).
So, the meromorphic functions F1(ξ,τ), F2(ξ,τ) defined by
(55)F1(ξ,τ)=Θ2(ξ,τ)Θ2(ξ+2h,τ)Θ22(ξ+h,τ),F2(ξ,τ)=Θ3(ξ,τ)Θ3(ξ+3h,τ)Θ3(ξ+2h,τ)Θ3(ξ+h,τ)hhhhhhhhhhhh(∀h,ξ∈C)
are all periodic functions; moreover, F1(ξ,τ)*F2(ξ,τ) is also periodic functions.From Proposition6, we verify that (53) is periodic wave solution of (28). In a similar way, the other cases of two-theta functions solutions of (28) can also be proved to be periodic.


## 3.1. The Solutions in Terms of One-Theta Function

Suppose that(32)ζn+(a/2),l+(b/2)m+(c/2)=Θi(k1n+k2l+k3m+k1a+k2b+k3c2,τ)=Θi(ξ,τ),i=1,2,3,4,
where
(33)ξ=k1n+k2l+k3m+k1a+k2b+k3c2,k1, k2, and k3 are arbitrary constants, and Θi(ξ,τ) is defined by (12). Similarly, we can obtain
(34)ζn+a,lm=Θi(ξ-α,τ),ζn,l+bm+c=Θi(ξ+α,τ),ζn,l+bm=Θi(ξ-β,τ),ζn+a,lm+c=Θi(ξ+β,τ),ζn,lm+c=Θi(ξ-γ,τ),ζn+a,l+bm=Θi(ξ+γ,τ),
with
(35)α=k2b+k3c-k1a2,β=k1a+k3c-k2b2,γ=k1a+k2b-k3c2.
Substituting (34) into (31) and using the following identities of the theta functions
(36)Θ1(x+y)Θ1(x-y)=Θ2-2(0)[Θ12(x)Θ22(y)-Θ22(x)Θ12(y)],Θ2(x+y)Θ2(x-y)=Θ2-1(0)[Θ22(x)Θ22(y)-Θ12(x)Θ12(y)],Θ3(x+y)Θ3(x-y)=Θ2-2(0)[Θ22(x)Θ32(y)+Θ12(x)Θ42(y)],Θ4(x+y)Θ4(x-y)=Θ2-2(0)[Θ12(x)Θ32(y)+Θ22(x)Θ42(y)],
yield
(37)a(b-c)Θ22(α)+b(c-a)Θ22(β)+c(a-b)Θ22(γ)=0,a(b-c)Θ12(α)+b(c-a)×Θ12(β)+c(a-b)Θ12(γ)=0,hhhhhhhhhhhhhhhwheni=1,2;a(b-c)Θ32(α)+b(c-a)Θ32(β)+c(a-b)Θ32(γ)=0,a(b-c)Θ42(α)+b(c-a)Θ42(β)+c(a-b)Θ42(γ)=0,hhhhhhhhwheni=3,4.
Here, we use Θi(ξ)=Θi(ξ,τ) for simplicity. From (37), we can find that if α=β=γ, that is, k1a=k2b=k3c, which follows from (35), then (31) holds. In virtue of the variable transformation (30), the solution of (28) is expressed by one-theta function as follows:
(38)u=(Θi[k1(n+abl+acm)]×Θi[k1(n+abl+acm)+2k1a])×(Θi2[k1(n+abl+acm)+k1a])-1-1,v=1-(Θi[k1(n+abl+acm)]×Θi[k1(n+abl+acm)+3k1a])×(Θi[k1(n+abl+acm)+2k1a]×Θi[k1(n+abl+acm)+k1a]),i=1,2,3,4.Remark 3 (see [27]).
Ifζn,lm=Θi(ξ-(3k1a/2),τ) is a solution of (31), then R0((n/a)+(l/b)+(m/c))R1(n/a)R2(l/b)R3(m/c)Θi(ξ-(3k1a/2),τ) is a solution too, where Ri, (i=0,1,2,3), are arbitrary functions.Remark 4 (see [27]).
By means of the transformation,ζn,lm=[(b(c-a))/(c(a-b))](nl/ab)[(b(c-a))/(a(b-c))](lm/bc)ζn,lm; then, [(b(c-a))/(c(a-b))](nl/ab)[(b(c-a))/(a(b-c))](lm/bc)Θi(ξ-(3k1a/2),τ) satisfies bilinear equation
(39)ζn+a,lmζn,l+bm+c+ζn,l+bmζn+a,lm+c+ζn,lm+cζn+a,l+bm=0.Proposition 5.
The Riemann theta functionΘ3(ξ,τ) defined by (12) has the periodic properties
(40)Θ3(ξ+1+τ,τ)=exp(-2iπξ-iπτ)Θ3(ξ,τ).
So, the meromorphic functions F(ξ,τ) defined by
(41)(I)F(ξ,τ)=Θ3(ξ,τ)Θ3(ξ+2h,τ)Θ32(ξ+h,τ),(∀h,ξ∈C)(II)F(ξ,τ)=Θ3(ξ,τ)Θ3(ξ+3h,τ)Θ3(ξ+2h,τ)Θ3(ξ+h,τ),(∀h,ξ∈C)
are all double periodic functions with two fundamental periods 1 and τ. Because the other three Riemann theta functions in (12) are the deformations of Θ3(ξ,τ), they can be proved to be periodic in a similar way.Proof.
By using definitionΘ3(ξ,τ) in (12), it is easy to see that
(42)Θ3(ξ+1+τ,τ)=∑m=-∞∞‍e2mπi(ξ+1+τ)+πiτm2=e-2πiξe-πiτΘ3(ξ,τ),Θ3(ξ+h+1+τ,τ)=∑m=-∞∞‍e2mπi(ξ+h+1+τ)+πiτm2=e-2πi(ξ+h)e-πiτΘ3(ξ+h,τ).
Therefore, in both (I) and (II) cases, it holds that
(43)F(ξ+1+τ,τ)=F(ξ,τ),ξ∈C.
That is to say, (38) is periodic wave solution of (28).


## 3.2. The Solutions in Terms of Two-Theta Function

In order to look for its solutions in terms of two-theta functions, we suppose(44)ζn+(a/2),l+(b/2)m+(c/2)=Θi(ξ,τ1)Θj(ξ,τ2)=Θi(k1a+k2b+k3c2k1n+k2l+k3mhhh+k1a+k2b+k3c2,τ1)×Θj(k1a+k2b+k3c2k1n+k2l+k3mh+k1a+k2b+k3c2,τ2),hhhhhhhhhhh∀i,j=1,2,3,4.
Therefore,
(45)ζn+a,lm=Θi(ξ-α,τ1)Θj(ξ-α,τ2),ζn,l+bm+c=Θi(ξ+α,τ1)Θj(ξ+α,τ2),ζn,l+bm=Θi(ξ-β,τ1)Θj(ξ-β,τ2),ζn+a,lm+c=Θi(ξ+β,τ1)Θj(ξ+β,τ2),ζn,lm+c=Θi(ξ-γ,τ1)Θj(ξ-γ,τ2),ζn+a,l+bm=Θi(ξ+γ,τ1)Θj(ξ+γ,τ2),
where α, β, and γ are the same as in (35).Case 1.
Ifi=1,j=2, we substitute (45) into (31) to get
(46)a(b-c)Θ1(ξ-α,τ1)Θ2(ξ-α,τ2)×Θ1(ξ+α,τ1)Θ2(ξ+α,τ2)+b(c-a)×Θ1(ξ-β,τ1)Θ2(ξ-β,τ2)Θ1(ξ+β,τ1)×Θ2(ξ+β,τ2)+c(a-b)Θ1(ξ-γ,τ1)×Θ2(ξ-γ,τ2)Θ1(ξ+γ,τ1)Θ2(ξ+γ,τ2)=0.
By using the first two identities of the theta functions (36), the above equation is equivalent to
(47)Θ12(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ22(α,τ1)Θ22(α,τ2)hhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ22(β,τ2)hhhhhhhhhhhh+ϵ3Θ22(γ,τ1)Θ22(γ,τ2)]-Θ12(ξ,τ1)Θ12(ξ,τ2)[ϵ1Θ22(α,τ1)Θ12(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ12(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ22(γ,τ1)Θ12(γ,τ2)]-Θ22(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ12(α,τ1)Θ22(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ22(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(α,τ1)Θ22(α,τ2)]+Θ22(ξ,τ1)Θ12(ξ,τ2)[ε1Θ12(α,τ1)Θ12(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ12(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(γ,τ1)Θ12(γ,τ2)]=0.
Therefore,
(48)ϵ1Θ22(α,τ1)Θ22(α,τ2)+ϵ2Θ22(β,τ1)Θ22(β,τ2)+ϵ3Θ22(γ,τ1)Θ22(γ,τ2)=0,ϵ1Θ22(α,τ1)Θ12(α,τ2)+ϵ2Θ22(β,τ1)Θ12(β,τ2)+ϵ3Θ22(γ,τ1)Θ12(γ,τ2)=0,ϵ1Θ12(α,τ1)Θ22(α,τ2)+ϵ2Θ12(β,τ1)Θ22(β,τ2)+ϵ3Θ12(γ,τ1)Θ22(γ,τ2)=0,ϵ1Θ12(α,τ1)Θ12(α,τ2)+ϵ2Θ12(β,τ1)Θ12(β,τ2)+ϵ3Θ12(γ,τ1)Θ12(γ,τ2)=0.
Here, ϵ1=a(b-c), ϵ2=b(c-a), and ϵ3=c(a-b). To get special solutions of (48), we can assume that α=β=γ; that is, k1a=k2b=k3c. From the variable transformation (30), the solution of (28) in terms of two-theta functions are
(49)u=(Θ1[k1(n+abl+acm),τ1]×Θ2[k1(n+abl+acm),τ2]×Θ1[k1(n+abl+acm)+2k1a,τ1]×Θ2[k1(n+abl+acm)+2k1a,τ2])×(Θ12[k1(n+abl+acm)+k1a,τ1]×Θ22[k1(n+abl+acm)+k1a,τ2])-1-1,v=1-(Θ1[k1(n+abl+acm),τ1]×Θ2[k1(n+abl+acm),τ2]×Θ1[k1(n+abl+acm)+3k1a,τ1]×Θ2[k1(n+abl+acm)+3k1a,τ2])×(Θ1[k1(n+abl+acm)+2k1a,τ1]×Θ2[k1(n+abl+acm)+2k1a,τ2]×Θ1[k1(n+abl+acm)+k1a,τ1]×Θ2[k1(n+abl+acm)+k1a,τ2])-1.Case 2.
Ifi=2, j=3 and substituting (45) into (31), then we are led to
(50)a(b-c)Θ2(ξ-α,τ1)Θ3(ξ-α,τ2)Θ2(ξ+α,τ1)×Θ3(ξ+α,τ2)+b(c-a)Θ2(ξ-β,τ1)×Θ3(ξ-β,τ2)Θ2(ξ+β,τ1)Θ3(ξ+β,τ2)+c(a-b)Θ2(ξ-γ,τ1)Θ3(ξ-γ,τ2)×Θ2(ξ+γ,τ1)Θ3(ξ+γ,τ2)=0.
From the identities of the theta functions (36), the last equation is changed into
(51)Θ22(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ22(α,τ1)Θ32(α,τ2)hhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ32(β,τ2)hhhhhhhhhhhh+ϵ3Θ22(γ,τ1)Θ32(γ,τ2)]-Θ12(ξ,τ1)Θ22(ξ,τ2)[ϵ1Θ12(α,τ1)Θ32(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ32(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(γ,τ1)Θ32(γ,τ2)]+Θ22(ξ,τ1)Θ12(ξ,τ2)[ϵ1Θ22(α,τ1)Θ42(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ22(β,τ1)Θ42(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ22(α,τ1)Θ42(α,τ2)]-Θ12(ξ,τ1)Θ12(ξ,τ2)[ϵ1Θ12(α,τ1)Θ42(α,τ2)hhhhhhhhhhhhhhhhh+ϵ2Θ12(β,τ1)Θ42(β,τ2)hhhhhhhhhhhhhhhhh+ϵ3Θ12(γ,τ1)Θ42(γ,τ2)]=0.
Therefore,
(52)ϵ1Θ22(α,τ1)Θ32(α,τ2)+ϵ2Θ22(β,τ1)Θ32(β,τ2)+ϵ3Θ22(γ,τ1)Θ32(γ,τ2)=0,ϵ1Θ12(α,τ1)Θ32(α,τ2)+ϵ2Θ12(β,τ1)Θ32(β,τ2)+ϵ3Θ12(γ,τ1)Θ32(γ,τ2)=0,ϵ1Θ22(α,τ1)Θ42(α,τ2)+ϵ2Θ22(β,τ1)Θ42(β,τ2)+ϵ3Θ22(α,τ1)Θ42(α,τ2)=0,ϵ1Θ12(α,τ1)Θ42(α,τ2)+ϵ2Θ12(β,τ1)Θ42(β,τ2)+ϵ3Θ12(γ,τ1)Θ42(γ,τ2)=0,
with ϵ1=a(b-c), ϵ2=b(c-a), and ϵ3=c(a-b). To get the special solutions of (52), we can assume that α=β=γ; that is, k1a=k2b=k3c. From the variable transformation (30), the solution of (28) in terms of two-theta functions are
(53)u=(Θ2[k1(n+abl+acm),τ1]×Θ3[k1(n+abl+acm),τ2]×Θ2[k1(n+abl+acm)+2k1a,τ1]×Θ3[k1(n+abl+acm)+2k1a,τ2])×(Θ22[k1(n+abl+acm)+k1a,τ1]×Θ32[k1(n+abl+acm)+k1a,τ2])-1-1,v=1-(Θ2[k1(n+abl+acm),τ1]×Θ3[k1(n+abl+acm),τ2]×Θ2[k1(n+abl+acm)+3k1a,τ1]×Θ3[k1(n+abl+acm)+3k1a,τ2])×Θ2([k1(n+abl+acm)+2k1a,τ1]×Θ3[k1(n+abl+acm)+2k1a,τ2]×Θ2[k1(n+abl+acm)+k1a,τ1]×Θ3[k1(n+abl+acm)+k1a,τ2])-1.Proposition 6.
The Riemann theta functionsΘ2(ξ,τ), Θ3(ξ,τ) defined by (12) have the periodic properties
(54)Θ2(ξ+2+τ,τ)=exp(-2iπξ-iπτ)Θ2(ξ,τ),Θ3(ξ+1+τ,τ)=exp(-2iπξ-iπτ)Θ3(ξ,τ).
So, the meromorphic functions F1(ξ,τ), F2(ξ,τ) defined by
(55)F1(ξ,τ)=Θ2(ξ,τ)Θ2(ξ+2h,τ)Θ22(ξ+h,τ),F2(ξ,τ)=Θ3(ξ,τ)Θ3(ξ+3h,τ)Θ3(ξ+2h,τ)Θ3(ξ+h,τ)hhhhhhhhhhhh(∀h,ξ∈C)
are all periodic functions; moreover, F1(ξ,τ)*F2(ξ,τ) is also periodic functions.From Proposition6, we verify that (53) is periodic wave solution of (28). In a similar way, the other cases of two-theta functions solutions of (28) can also be proved to be periodic.


## 4. Conclusion

In this paper, based on the Riemann theta functions, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct some kinds of quasiperiodic wave solutions for(3+1)-dimensional potential-YTSF equation and discrete KP equation. This method is also suitable for other more general nonlinear evolution equations in mathematical physics. Moreover, for the (3+1)-dimensional potential-YTSF equation, we not only use linear superposition principle to generate N-wave solutions but also analyze the quasiperiodic wave solutions that tend to the soliton solutions under a small amplitude limit.



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*Source: 142027-2013-12-04.xml*