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

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.

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 consider 3+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 −4_𝑥𝑡+𝑢_{𝑥𝑥𝑥𝑧}+4𝑢_𝑥𝑢_𝑥𝑧+2𝑢_𝑥𝑥𝑢_𝑧+3𝑢_𝑦𝑦=0. (1) 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 𝜁=𝑥+𝑑𝑧,𝑢=2ln𝑓_𝜁, (2) we transform (1) to the bilinear form −4𝐷_𝜁𝐷_𝑡+𝑑𝐷⁴_𝜁+3𝐷²_𝑦+𝑐𝑓⋅𝑓=0, (3) where 𝑑 is an arbitrary positive constant and Hirota bilinear operators 𝐷_𝜁, 𝐷_𝑦, and 𝐷_𝑡 are defined by 𝐷^𝑚_𝜁𝐷^𝑚′_𝑡𝑓()()=𝜕𝜁,𝑡⋅𝑔𝜁,𝑡_𝜁−𝜕_𝜁′^𝑚𝜕_𝑡−𝜕_𝑡′^𝑚′∣_{𝜁′=𝜁,𝑡′=𝑡,} (4) which have a nice property when acting on exponential functions 𝐷^𝑚_𝜁𝐷^𝑚′_𝑡𝑒^𝜂1⋅𝑒^𝜂2=𝑘₁−𝑘₂^𝑚𝑤₁−𝑤₂^𝑚′⋅𝑒^𝜂1+𝜂2, (5) with 𝜂_𝑗=𝑘_𝑗𝜁+𝑙_𝑗𝑦+𝑤_𝑗𝑡, 𝑗=1,2.

2.1. N Exponential Wave Solutions

Let us consider the special case 𝑐=0 in (3) and introduce N-wave testing function 𝑓=^𝑁_𝑗=1𝜖_𝑗𝑓_𝑗=^𝑁_𝑗=1𝜖_𝑗𝑒^{𝑘𝑗𝜁+𝑙𝑗𝑦+𝑤𝑗𝑡},1≤𝑗≤𝑁, (6) where 𝜀_𝑗, 𝑘_𝑗, 𝑙_𝑗, and 𝑤_𝑗 are arbitrary constants. Substituting (6) into (3) and using (5) give −4𝑘^𝑁𝑁_{𝑗=1𝑗=1𝑖}−𝑘_𝑗𝑤_𝑖−𝑤_𝑗+𝑑𝑘_𝑖−𝑘_𝑗⁴+3𝑙_𝑖−𝑙_𝑗²𝑓_𝑖⋅𝑓_𝑗=0. (7) If the following condition −4𝑘_𝑖−𝑘_𝑗𝑤_𝑖−𝑤_𝑗+𝑑𝑘_𝑖−𝑘_𝑗⁴+3𝑙_𝑖−𝑙_𝑗²𝑓_𝑖⋅𝑓_𝑗=0 (8) is satisfied, then any linear combination of the 𝑁 exponential wave solutions 𝑒^𝜂𝑗, 1≤𝑗≤𝑁 solves (3). By inspection, a solution to (8) is 𝑤_𝑗=𝑑𝑘³_𝑗,𝑙_𝑗=±√𝑑𝑘²_𝑗,1≤𝑗≤𝑁. (9) Therefore, (1) has the following N-wave solution 𝑢=2ln𝑓_𝜁,𝑓=^𝑁_𝑗=1𝜖_𝑗𝑓_𝑗=^𝑁_𝑗=1𝜖_𝑗𝑒^{𝑘𝑗√(𝑥+𝑑𝑧)±𝑑𝑘2𝑗𝑦+𝑑𝑘3𝑗𝑡},1≤𝑗≤𝑁, (10) where 𝜖_𝑗, 𝑘_𝑗 are arbitrary constants. Especially assume that 𝜖₁=𝜖₂=𝜖₃=1, 𝑘₁=−𝑘₂, and 𝑘₃=0 in (10); then, we get 𝑓=1+𝑒^{𝑘1(𝑥+𝑑𝑧)+𝑑𝑘31𝑡}+𝑒^{−𝑘1(𝑥+𝑑𝑧)−𝑑𝑘31𝑡}𝑒^{±√𝑑𝑘21𝑦},𝑢=2ln𝑓_𝜁=2𝑘₁sinh𝑘₁(𝑥+𝑑𝑧)+𝑑𝑘³₁𝑡𝑒^{±√𝑑𝑘21𝑦}(1/2)+cosh𝑘₁(𝑥+𝑑𝑧)+𝑑𝑘³₁𝑡𝑒^{±√𝑑𝑘21𝑦}. (11)

2.2. Quasiperiodic Solutions

The Riemann theta functions of genus one [20] are defined by Θ₁(𝜉,𝜏)=−𝑖^∞_𝑚=−∞(−1)^𝑚𝑒^{𝜋𝑖𝜉(2𝑚+1)+𝜋𝑖𝜏(𝑚+(1/2))2},Θ₂(𝜉,𝜏)=^∞_𝑚=−∞𝑒^{𝜋𝑖𝜉(2𝑚+1)+𝜋𝑖𝜏(𝑚+(1/2))2},Θ₃(𝜉,𝜏)=^∞_𝑚=−∞𝑒^{2𝑚𝜋𝑖𝜉+𝜋𝑖𝜏𝑚2},Θ₄(𝜉,𝜏)=^∞_𝑚=−∞(−1)^𝑚𝑒^{2𝑚𝜋𝑖𝜉+𝜋𝑖𝜏𝑚2}, (12) respectively, where 𝜉=𝑘𝜁+𝑙𝑦+𝑤𝑡. Letting Θ_𝑘(𝑥)=Θ_𝑘(𝑥,𝜏) (𝑘=1,2,3,4) and using the identities Θ₃𝑥+𝑦Θ₃𝑥−𝑦Θ²₂(0)=Θ²₃(𝑥)Θ²₂𝑦+Θ²₄(𝑥)Θ²₁𝑦,Θ₄𝑥+𝑦Θ₄𝑥−𝑦Θ²₂(0)=Θ²₄(𝑥)Θ²₂𝑦+Θ²₃(𝑥)Θ²₁𝑦, (13) we can obtain the Hirota derivatives of Θ₃(𝑥), Θ₄(𝑥): 𝐷²_𝑥Θ₃(𝑥)⋅Θ₃(𝑥)=𝜕²_𝑦Θ₃𝑥+𝑦Θ₃𝑥−𝑦∣_𝑦=0=𝑏₁Θ²₄(𝑥)+𝑏₂Θ²₃(𝑥),𝐷⁴_𝑥Θ₃(𝑥)⋅Θ₃(𝑥)=𝜕⁴_𝑦Θ₃𝑥+𝑦Θ₃𝑥−𝑦∣_𝑦=0=𝑐₁Θ²₄(𝑥)+𝑐₂Θ²₃(𝑥),𝐷²_𝑥Θ₄(𝑥)⋅Θ₄(𝑥)=𝜕²_𝑦Θ₄𝑥+𝑦Θ₄𝑥−𝑦∣_𝑦=0=𝑏₂Θ²₄(𝑥)−𝑏₁Θ²₃(𝑥),𝐷⁴_𝑥Θ₄(𝑥)⋅Θ₄(𝑥)=𝜕⁴_𝑦Θ₃𝑥+𝑦Θ₃𝑥−𝑦∣_𝑦=0=𝑐₂Θ²₄(𝑥)−𝑐₁Θ²₃(𝑥), (14) where 𝑏₁=2Θ²₃(0)Θ²₄(0),𝑏₂Θ=2^2′′(0)Θ₂(0),𝑏₃Θ=2^3′′(0)Θ₃(0)+Θ₄^′′(0)Θ₄(0),𝑏₄Θ=2²⁽⁴⁾(0)Θ₂(0),𝑐₁=2𝑏₁𝑏₂+𝑏₃,𝑐₂=32𝑏²₂+𝑏₄. (15) 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 𝑓𝑥,𝑦,𝑧,𝑡=Θ₃(𝜉),𝜉=𝑘₁𝜁+𝑙₁𝑦+𝑤₁𝑡, (16) where 𝑘₁,  𝑙₁, and 𝑤₁ are arbitrary constants. Substituting (16) into (3) and setting the coefficients of the terms Θ²₃(𝜉), Θ²₄(𝜉) to be zero, we obtain a system of algebraic equations −4𝑘₁𝑤₁+3𝑙²₁𝑏₂+𝑑𝑘⁴₁𝑐₂+𝑐=0,4𝑘₁𝑤₁−3𝑙²₁𝑏₁−𝑑𝑘⁴₁𝑐₁=0. (17) Solving (17), we find 𝑤₁=3𝑙²₁𝑏₁+𝑑𝑘⁴₁𝑐₁4𝑘₁𝑏₁,𝑐=−3𝑙²₁𝑏₂+𝑑𝑘⁴₁𝑐₁+𝑏₂𝑏₁3𝑙²₁𝑏₁+𝑑𝑘⁴₁𝑐₁, (18) where 𝑏₁, 𝑏₂ and 𝑐₁, 𝑐₂ are given by (15). We have thus constructed a kind of quasiperiodic solution for (1) 𝑢=2lnΘ₃𝑘₁𝜁+𝑙₁𝑧+𝑤₁𝑡_𝜁. (19) 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 𝑓=Θ₃(𝜉)=1+𝑒^𝜋𝑖𝜏𝑒^{2𝜋𝑖𝜉}+𝑒^{−2𝜋𝑖𝜉}+𝑒^{4𝜋𝑖𝜏}𝑒^{4𝜋𝑖𝜉}+𝑒^{−4𝜋𝑖𝜉}+⋯=1+𝑒^𝜉′+𝑒^{2𝜋𝑖𝜏}𝑒^−𝜉′+𝑒^{2𝜋𝑖𝜏}𝑒^2𝜉′…≈1+𝑒^𝜉′,𝑎𝑠𝐼𝑚𝜏→∞. (20) From (2), one gets the solution of (1) as follows: 𝑢=2ln𝑓_𝜁⟶4𝑘𝜋𝑖𝑒^𝜉′1+𝑒^𝜉′=𝑘^′sec𝜉^′2𝑒^(𝜉′/2). (21)

Case 2. Assume 𝑓(𝑥,𝑦,𝑧,𝑡) in (3) to be 𝑓𝑥,𝑦,𝑧,𝑡=Θ₄(𝜉),𝜉=𝑘₂𝜁+𝑙₂𝑦+𝑤₂𝑡, (22) where 𝑘₁, 𝑙₁, and 𝑤₁ are arbitrary constants. Setting the coefficients of the terms Θ²₃(𝜉), Θ²₄(𝜉) to be zero, we have 4𝑘₂𝑤₂−3𝑙²₂𝑏₂−𝑑𝑘⁴₂𝑐₁=0,−4𝑘₂𝑤₂+3𝑙²₂𝑏₂+𝑑𝑘⁴₂𝑐₂+𝑐=0. (23) Solving (23) yields 𝑤₂=3𝑙²₂𝑏₁+𝑑𝑘⁴₂𝑐₁4𝑘₂𝑏₁,𝑐=−3𝑙²₂𝑏₂+𝑑𝑘⁴₂𝑐₁+𝑏₂𝑏₁3𝑙²₂𝑏₁+𝑑𝑘⁴₂𝑐₁, (24) where 𝑏₁, 𝑏₂, and 𝑐₁, 𝑐₂ are given by (15). Thus, the second kind of quasiperiodic solution of (1) is constructed as follows: 𝑢=2lnΘ₄𝑘₂𝜁+𝑙₂𝑧+𝑤₂𝑡_𝜁. (25)

Proposition 2. If the vector (𝑤₂,𝑐)^𝑇 is given by (24) and supposing 𝜉^′=2𝜋𝑖𝜉+𝜋𝑖𝜏, 𝑘^′=2𝜋𝑖𝑘, and 𝐼𝑚𝜏→∞, then one has 𝑓=Θ₄(𝜉)=1−𝑒^𝜋𝑖𝜏𝑒^{2𝜋𝑖𝜉}+𝑒^{−2𝜋𝑖𝜉}+𝑒^{4𝜋𝑖𝜏}𝑒^{4𝜋𝑖𝜉}+𝑒^{−4𝜋𝑖𝜉}+⋯=1−𝑒^𝜉′−𝑒^{2𝜋𝑖𝜏}𝑒^−𝜉′+𝑒^{2𝜋𝑖𝜏}𝑒^2𝜉′…≈1−𝑒^𝜉′,𝑎𝑠𝐼𝑚𝜏⟶∞. (26) Using the variable transformation (2), one gets the solution of (1) as follows: 𝑢=2ln𝑓_𝜁⟶−4𝑘𝜋𝑖𝑒^𝜉′1−𝑒^𝜉′=−𝑘^′csc𝜉^′2𝑒^(𝜉′/2). (27)

3. Discrete KP Equation

The discrete analogue of KP equation [21, 22] (or a two-dimensional analogue of the discrete KdV equation) is ∇_𝑡𝑢+𝑣1+𝑢=𝜒∇_𝑥𝐸_+𝑦𝑢,∇_𝑦𝑢+𝑣=𝐸1+𝑢^+𝑦𝑣−𝐸_−𝑥𝐸_+𝑡𝑣𝛿𝑦1+𝐸_+𝑦𝑢, (28) 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 ∇_𝑧𝑓(𝑧)=1𝑓𝛿𝑧𝑧+𝛿𝑧2−𝑓𝑧−𝛿𝑧2,𝐸_±𝑧𝑓(𝑧)=𝑓𝑧±𝛿𝑧2. (29) In fact, using the dependent variable transformation 𝑥𝑛=𝑎+𝑦𝛿𝑥,𝑦𝛿𝑦𝑙=𝑏+𝑡𝛿𝑦,𝑥𝛿𝑡𝑚=−𝑐+𝑡𝛿𝑥,𝜁𝛿𝑡𝑢=^{𝑚𝑛,𝑙}𝜁^𝑚+𝑐_{𝑛,𝑙+𝑏}𝜁^𝑚_{𝑛,𝑙+𝑏}𝜁^𝑚+𝑐_𝑛,𝑙𝜁−1,𝑣=1−^{𝑚𝑛,𝑙}𝜁^𝑚+𝑐_{𝑛+𝑎,𝑙+𝑏}𝜁^𝑚_{𝑛+𝑎,𝑙+𝑏}𝜁^𝑚+𝑐_𝑛,𝑙,𝑎()𝜒=𝑏−𝑐𝑏(𝑐−𝑎)𝛿𝑥𝛿𝑡 (30) and substituting (30) into (28), we have the bilinear form of discrete KP equation as follows: 𝑎(𝑏−𝑐)𝜁^𝑚_{𝑛+𝑎,𝑙}𝜁^𝑚+𝑐_{𝑛,𝑙+𝑏}+𝑏(𝑐−𝑎)𝜁^𝑚_{𝑛,𝑙+𝑏}𝜁^𝑚+𝑐_{𝑛+𝑎,𝑙}+𝑐(𝑎−𝑏)𝜁^𝑚+𝑐_𝑛,𝑙𝜁^𝑚_{𝑛+𝑎,𝑙+𝑏}=0, (31) 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 𝜁^{𝑚+(𝑐/2)}_{𝑛+(𝑎/2),𝑙+(𝑏/2)}=Θ_𝑖𝑘₁𝑛+𝑘₂𝑙+𝑘₃𝑘𝑚+¹𝑎+𝑘₂𝑏+𝑘₃𝑐2,𝜏=Θ_𝑖(𝜉,𝜏),𝑖=1,2,3,4, (32) where 𝜉=𝑘₁𝑛+𝑘₂𝑙+𝑘₃𝑘𝑚+¹𝑎+𝑘₂𝑏+𝑘₃𝑐2, (33)𝑘₁, 𝑘₂, and 𝑘₃ are arbitrary constants, and Θ_𝑖(𝜉,𝜏) is defined by (12). Similarly, we can obtain 𝜁^𝑚_{𝑛+𝑎,𝑙}=Θ_𝑖(𝜉−𝛼,𝜏),𝜁^𝑚+𝑐_{𝑛,𝑙+𝑏}=Θ_𝑖(),𝜁𝜉+𝛼,𝜏_{𝑚𝑛,𝑙+𝑏}=Θ_𝑖𝜉−𝛽,𝜏,𝜁^𝑚+𝑐_{𝑛+𝑎,𝑙}=Θ_𝑖,𝜁𝜉+𝛽,𝜏_{𝑚+𝑐𝑛,𝑙}=Θ_𝑖𝜉−𝛾,𝜏,𝜁^𝑚_{𝑛+𝑎,𝑙+𝑏}=Θ_𝑖𝜉+𝛾,𝜏, (34) with 𝑘𝛼=²𝑏+𝑘₃𝑐−𝑘₁𝑎2𝑘,𝛽=¹𝑎+𝑘₃𝑐−𝑘₂𝑏2,𝑘𝛾=¹𝑎+𝑘₂𝑏−𝑘₃𝑐2. (35) Substituting (34) into (31) and using the following identities of the theta functions Θ₁𝑥+𝑦Θ₁𝑥−𝑦=Θ₂⁻²(0)Θ²₁(𝑥)Θ²₂𝑦−Θ²₂(𝑥)Θ²₁𝑦,Θ₂𝑥+𝑦Θ₂𝑥−𝑦=Θ₂⁻¹(0)Θ²₂(𝑥)Θ²₂𝑦−Θ²₁(𝑥)Θ²₁𝑦,Θ₃𝑥+𝑦Θ₃𝑥−𝑦=Θ₂⁻²(0)Θ²₂(𝑥)Θ²₃𝑦+Θ²₁(𝑥)Θ²₄𝑦,Θ₄𝑥+𝑦Θ₄𝑥−𝑦=Θ₂⁻²(0)Θ²₁(𝑥)Θ²₃𝑦+Θ²₂(𝑥)Θ²₄𝑦, (36) yield 𝑎(𝑏−𝑐)Θ²₂(𝛼)+𝑏(𝑐−𝑎)Θ²₂𝛽+𝑐(𝑎−𝑏)Θ²₂𝛾𝑎()Θ=0,𝑏−𝑐²₁(𝛼)+𝑏(𝑐−𝑎)×Θ²₁𝛽+𝑐(𝑎−𝑏)Θ²₁𝛾=0,when𝑎()Θ𝑖=1,2;𝑏−𝑐²₃(𝛼)+𝑏(𝑐−𝑎)Θ²₃𝛽+𝑐(𝑎−𝑏)Θ²₃𝛾𝑎()Θ=0,𝑏−𝑐²₄(𝛼)+𝑏(𝑐−𝑎)Θ²₄𝛽+𝑐(𝑎−𝑏)Θ²₄𝛾=0,when𝑖=3,4. (37) Here, we use Θ_𝑖(𝜉)=Θ_𝑖(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: 𝑢=Θ_𝑖𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚×Θ_𝑖𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎×Θ²_𝑖𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎⁻¹Θ−1,𝑣=1−_𝑖𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚×Θ_𝑖𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+3𝑘₁𝑎×Θ_𝑖𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎×Θ_𝑖𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝑖=1,2,3,4. (38)

Remark 3 (see [27]). If 𝜁^𝑚_𝑛,𝑙=Θ_𝑖(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)Θ_𝑖(𝜉−(3𝑘₁𝑎/2),𝜏) is a solution too, where 𝑅_𝑖, (𝑖=0,1,2,3), are arbitrary functions.

Remark 4 (see [27]). By means of the transformation, 𝜁^𝑚_𝑛,𝑙=[(𝑏(𝑐−𝑎))/(𝑐(𝑎−𝑏))]^{(𝑛𝑙/𝑎𝑏)}[(𝑏(𝑐−𝑎))/(𝑎(𝑏−𝑐))]^{(𝑙𝑚/𝑏𝑐)}𝜁^𝑚_𝑛,𝑙; then, [(𝑏(𝑐−𝑎))/(𝑐(𝑎−𝑏))]^{(𝑛𝑙/𝑎𝑏)}[(𝑏(𝑐−𝑎))/(𝑎(𝑏−𝑐))]^{(𝑙𝑚/𝑏𝑐)}Θ_𝑖(𝜉−(3𝑘₁𝑎/2),𝜏) satisfies bilinear equation 𝜁^𝑚_{𝑛+𝑎,𝑙}𝜁^𝑚+𝑐_{𝑛,𝑙+𝑏}+𝜁^𝑚_{𝑛,𝑙+𝑏}𝜁^𝑚+𝑐_{𝑛+𝑎,𝑙}+𝜁^𝑚+𝑐_𝑛,𝑙𝜁^𝑚_{𝑛+𝑎,𝑙+𝑏}=0. (39)

Proposition 5. The Riemann theta function Θ₃(𝜉,𝜏) defined by (12) has the periodic properties Θ₃(𝜉+1+𝜏,𝜏)=exp(−2𝑖𝜋𝜉−𝑖𝜋𝜏)Θ₃(𝜉,𝜏). (40) So, the meromorphic functions 𝐹(𝜉,𝜏) defined by (𝐼)𝐹()=Θ𝜉,𝜏³(𝜉,𝜏)Θ₃(𝜉+2ℎ,𝜏)Θ²₃(),()()𝐹()=Θ𝜉+ℎ,𝜏∀ℎ,𝜉∈𝐶𝐼𝐼𝜉,𝜏³(𝜉,𝜏)Θ₃(𝜉+3ℎ,𝜏)Θ₃(𝜉+2ℎ,𝜏)Θ₃(),()𝜉+ℎ,𝜏∀ℎ,𝜉∈𝐶 (41) 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 definition Θ₃(𝜉,𝜏) in (12), it is easy to see that Θ₃(𝜉+1+𝜏,𝜏)=^∞_𝑚=−∞𝑒^{2𝑚𝜋𝑖(𝜉+1+𝜏)+𝜋𝑖𝜏𝑚2}=𝑒^{−2𝜋𝑖𝜉}𝑒^{−𝜋𝑖𝜏}Θ₃(),Θ𝜉,𝜏₃(𝜉+ℎ+1+𝜏,𝜏)=^∞_𝑚=−∞𝑒^{2𝑚𝜋𝑖(𝜉+ℎ+1+𝜏)+𝜋𝑖𝜏𝑚2}=𝑒^{−2𝜋𝑖(𝜉+ℎ)}𝑒^{−𝜋𝑖𝜏}Θ₃(𝜉+ℎ,𝜏). (42) Therefore, in both (I) and (II) cases, it holds that 𝐹(𝜉+1+𝜏,𝜏)=𝐹(𝜉,𝜏),𝜉∈𝐶. (43) 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 𝜁^{𝑚+(𝑐/2)}_{𝑛+(𝑎/2),𝑙+(𝑏/2)}=Θ_𝑖𝜉,𝜏₁Θ_𝑗𝜉,𝜏₂=Θ_𝑖𝑘₁𝑛+𝑘₂𝑙+𝑘₃𝑚+𝑘₁𝑎+𝑘₂𝑏+𝑘₃𝑐2,𝜏₁×Θ_𝑗𝑘₁𝑛+𝑘₂𝑙+𝑘₃𝑚+𝑘₁𝑎+𝑘₂𝑏+𝑘₃𝑐2,𝜏₂,∀𝑖,𝑗=1,2,3,4. (44) Therefore, 𝜁^𝑚_{𝑛+𝑎,𝑙}=Θ_𝑖𝜉−𝛼,𝜏₁Θ_𝑗𝜉−𝛼,𝜏₂,𝜁^𝑚+𝑐_{𝑛,𝑙+𝑏}=Θ_𝑖𝜉+𝛼,𝜏₁Θ_𝑗𝜉+𝛼,𝜏₂,𝜁^𝑚_{𝑛,𝑙+𝑏}=Θ_𝑖𝜉−𝛽,𝜏₁Θ_𝑗𝜉−𝛽,𝜏₂,𝜁^𝑚+𝑐_{𝑛+𝑎,𝑙}=Θ_𝑖𝜉+𝛽,𝜏₁Θ_𝑗𝜉+𝛽,𝜏₂,𝜁^𝑚+𝑐_𝑛,𝑙=Θ_𝑖𝜉−𝛾,𝜏₁Θ_𝑗𝜉−𝛾,𝜏₂,𝜁^𝑚_{𝑛+𝑎,𝑙+𝑏}=Θ_𝑖𝜉+𝛾,𝜏₁Θ_𝑗𝜉+𝛾,𝜏₂, (45) where 𝛼, 𝛽, and 𝛾 are the same as in (35).

Case 1. If 𝑖=1,𝑗=2, we substitute (45) into (31) to get 𝑎(𝑏−𝑐)Θ₁𝜉−𝛼,𝜏₁Θ₂𝜉−𝛼,𝜏₂×Θ₁𝜉+𝛼,𝜏₁Θ₂𝜉+𝛼,𝜏₂+𝑏(𝑐−𝑎)×Θ₁𝜉−𝛽,𝜏₁Θ₂𝜉−𝛽,𝜏₂Θ₁𝜉+𝛽,𝜏₁×Θ₂𝜉+𝛽,𝜏₂+𝑐(𝑎−𝑏)Θ₁𝜉−𝛾,𝜏₁×Θ₂𝜉−𝛾,𝜏₂Θ₁𝜉+𝛾,𝜏₁Θ₂𝜉+𝛾,𝜏₂=0. (46) By using the first two identities of the theta functions (36), the above equation is equivalent to Θ²₁𝜉,𝜏₁Θ²₂𝜉,𝜏₂𝜖₁Θ²₂𝛼,𝜏₁Θ²₂𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₂𝛽,𝜏₂+𝜖₃Θ²₂𝛾,𝜏₁Θ²₂𝛾,𝜏₂−Θ²₁𝜉,𝜏₁Θ²₁𝜉,𝜏₂𝜖₁Θ²₂𝛼,𝜏₁Θ²₁𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₁𝛽,𝜏₂+𝜖₃Θ²₂𝛾,𝜏₁Θ²₁𝛾,𝜏₂−Θ²₂𝜉,𝜏₁Θ²₂𝜉,𝜏₂𝜖₁Θ²₁𝛼,𝜏₁Θ²₂𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₂𝛽,𝜏₂+𝜖₃Θ²₁𝛼,𝜏₁Θ²₂𝛼,𝜏₂+Θ²₂𝜉,𝜏₁Θ²₁𝜉,𝜏₂𝜀₁Θ²₁𝛼,𝜏₁Θ²₁𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₁𝛽,𝜏₂+𝜖₃Θ²₁𝛾,𝜏₁Θ²₁𝛾,𝜏₂=0. (47) Therefore, 𝜖₁Θ²₂𝛼,𝜏₁Θ²₂𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₂𝛽,𝜏₂+𝜖₃Θ²₂𝛾,𝜏₁Θ²₂𝛾,𝜏₂𝜖=0,₁Θ²₂𝛼,𝜏₁Θ²₁𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₁𝛽,𝜏₂+𝜖₃Θ²₂𝛾,𝜏₁Θ²₁𝛾,𝜏₂𝜖=0,₁Θ²₁𝛼,𝜏₁Θ²₂𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₂𝛽,𝜏₂+𝜖₃Θ²₁𝛾,𝜏₁Θ²₂𝛾,𝜏₂𝜖=0,₁Θ²₁𝛼,𝜏₁Θ²₁𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₁𝛽,𝜏₂+𝜖₃Θ²₁𝛾,𝜏₁Θ²₁𝛾,𝜏₂=0. (48) 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 𝑢=Θ₁𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₁×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₂×Θ₁𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₁×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₂×Θ²₁𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₁×Θ²₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₂⁻¹Θ−1,𝑣=1−₁𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₁×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₂×Θ₁𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+3𝑘₁𝑎,𝜏₁×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+3𝑘₁𝑎,𝜏₂×Θ₁𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₁×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₂×Θ₁𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₁×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₂⁻¹. (49)

Case 2. If 𝑖=2, 𝑗=3 and substituting (45) into (31), then we are led to 𝑎(𝑏−𝑐)Θ₂𝜉−𝛼,𝜏₁Θ₃𝜉−𝛼,𝜏₂Θ₂𝜉+𝛼,𝜏₁×Θ₃𝜉+𝛼,𝜏₂+𝑏(𝑐−𝑎)Θ₂𝜉−𝛽,𝜏₁×Θ₃𝜉−𝛽,𝜏₂Θ₂𝜉+𝛽,𝜏₁Θ₃𝜉+𝛽,𝜏₂+𝑐(𝑎−𝑏)Θ₂𝜉−𝛾,𝜏₁Θ₃𝜉−𝛾,𝜏₂×Θ₂𝜉+𝛾,𝜏₁Θ₃𝜉+𝛾,𝜏₂=0. (50) From the identities of the theta functions (36), the last equation is changed into Θ²₂𝜉,𝜏₁Θ²₂𝜉,𝜏₂𝜖₁Θ²₂𝛼,𝜏₁Θ²₃𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₃𝛽,𝜏₂+𝜖₃Θ²₂𝛾,𝜏₁Θ²₃𝛾,𝜏₂−Θ²₁𝜉,𝜏₁Θ²₂𝜉,𝜏₂𝜖₁Θ²₁𝛼,𝜏₁Θ²₃𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₃𝛽,𝜏₂+𝜖₃Θ²₁𝛾,𝜏₁Θ²₃𝛾,𝜏₂+Θ²₂𝜉,𝜏₁Θ²₁𝜉,𝜏₂𝜖₁Θ²₂𝛼,𝜏₁Θ²₄𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₄𝛽,𝜏₂+𝜖₃Θ²₂𝛼,𝜏₁Θ²₄𝛼,𝜏₂−Θ²₁𝜉,𝜏₁Θ²₁𝜉,𝜏₂𝜖₁Θ²₁𝛼,𝜏₁Θ²₄𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₄𝛽,𝜏₂+𝜖₃Θ²₁𝛾,𝜏₁Θ²₄𝛾,𝜏₂=0. (51) Therefore, 𝜖₁Θ²₂𝛼,𝜏₁Θ²₃𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₃𝛽,𝜏₂+𝜖₃Θ²₂𝛾,𝜏₁Θ²₃𝛾,𝜏₂𝜖=0,₁Θ²₁𝛼,𝜏₁Θ²₃𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₃𝛽,𝜏₂+𝜖₃Θ²₁𝛾,𝜏₁Θ²₃𝛾,𝜏₂𝜖=0,₁Θ²₂𝛼,𝜏₁Θ²₄𝛼,𝜏₂+𝜖₂Θ²₂𝛽,𝜏₁Θ²₄𝛽,𝜏₂+𝜖₃Θ²₂𝛼,𝜏₁Θ²₄𝛼,𝜏₂𝜖=0,₁Θ²₁𝛼,𝜏₁Θ²₄𝛼,𝜏₂+𝜖₂Θ²₁𝛽,𝜏₁Θ²₄𝛽,𝜏₂+𝜖₃Θ²₁𝛾,𝜏₁Θ²₄𝛾,𝜏₂=0, (52) 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 𝑢=Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₁×Θ₃𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₂×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₁×Θ₃𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₂×Θ²₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₁×Θ²₃𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₂⁻¹Θ−1,𝑣=1−₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₁×Θ₃𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚,𝜏₂×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+3𝑘₁𝑎,𝜏₁×Θ₃𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+3𝑘₁𝑎,𝜏₂×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₁×Θ₃𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+2𝑘₁𝑎,𝜏₂×Θ₂𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₁×Θ₃𝑘₁𝑎𝑛+𝑏𝑎𝑙+𝑐𝑚+𝑘₁𝑎,𝜏₂⁻¹. (53)

Proposition 6. The Riemann theta functions Θ₂(𝜉,𝜏), Θ₃(𝜉,𝜏) defined by (12) have the periodic properties Θ₂(𝜉+2+𝜏,𝜏)=exp(−2𝑖𝜋𝜉−𝑖𝜋𝜏)Θ₂(),Θ𝜉,𝜏₃(𝜉+1+𝜏,𝜏)=exp(−2𝑖𝜋𝜉−𝑖𝜋𝜏)Θ₃(𝜉,𝜏). (54) So, the meromorphic functions 𝐹₁(𝜉,𝜏), 𝐹₂(𝜉,𝜏) defined by 𝐹₁()=Θ𝜉,𝜏²(𝜉,𝜏)Θ₂(𝜉+2ℎ,𝜏)Θ²₂(),𝐹𝜉+ℎ,𝜏₂()=Θ𝜉,𝜏³(𝜉,𝜏)Θ₃(𝜉+3ℎ,𝜏)Θ₃(𝜉+2ℎ,𝜏)Θ₃()()𝜉+ℎ,𝜏∀ℎ,𝜉∈𝐶 (55) are all periodic functions; moreover, 𝐹₁(𝜉,𝜏)∗𝐹₂(𝜉,𝜏) is also periodic functions.

From Proposition 6, 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.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant no. 11271246), the Natural Science Research Project of Henan Education Department (Grant no. 2011B110024), and the Research Fund for Luoyang Normal University (no. qnjj-2009-02).