Periodic Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional Kadomtsev Petviashvili Equation

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1 Commun. Theor. Phys. (Beijing, China) 50 (2008) pp c Chinese Physical Society Vol. 50, No. 5, November 15, 2008 Periodic Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional Kadomtsev Petviashvili Equation ZHANG Huan, 1 TIAN Bo, 1,2,3, ZHANG Hai-Qiang, 1 GENG Tao, 1 MENG Xiang-Hua, 1 LIU Wen-Jun, 1 and CAI Ke-Jie 1 1 School of Science, P.O. Box 122, Beijing University of Posts and Telecommunications, Beijing , China 2 State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing , China 3 Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing , China (Received January 8, 2008; Revised May 20, 2008) Abstract For describing various complex nonlinear phenomena in the realistic world, the higher-dimensional nonlinear evolution equations appear more attractive in many fields of physical and engineering sciences. In this paper, by virtue of the Hirota bilinear method and Riemann theta functions, the periodic wave solutions for the (2+1)-dimensional Boussinesq equation and (3+1)-dimensional Kadomtsev Petviashvili (KP) equation are obtained. Furthermore, it is shown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions. PACS numbers: Yv, Ge, Jr, Wz Key words: periodic wave solutions, (2+1)-dimensional Boussinesq equation, (3+1)-dimensional KP equation, Hirota bilinear method 1 Introduction In the past decades, nonlinear evolution equations (NLEEs) have been widely used to describe complex phenomena in various sciences, such as fluid dynamics, plasma physics, field theory, optics, and condensed matter physics. 1 4] Currently, more and more attention has been paid to the higher-dimensional NLEEs because in higher-dimensional nonlinear wave fields there exist richer phenomena and interaction structures. 2 4] For instance, the (2+1)-dimensional Boussinesq equation, which can be used to describe the propagation of gravity waves on water surface, has been investigated in Refs. 5] 8]. The soliton resonant phenomena and interactions taking place in this (2+1)-dimensional model have been presented in Ref. 8]. Another typical example is the (2+1)- dimensional Kadomtsev Petviashvili (KP) equation 9 11] emerging from generalization of the Korteweg-de Vries equation ] The (2+1)-dimensional KP equation has been proposed as a model for surface waves and internal waves in straits or channels of varying depth and width. Furthermore, it is noted that the (2+1)-dimensional KP equation has been generalized to the (3+1)-dimensional case. The (3+1)-dimensional KP equation, 11,12,14,15] which has direct application in plasma physics, has also been studied to figure problems of dynamics of threedimensional wave structures per se, and the nonlinear self-influence of waves in a plasma, namely, the wave of collapse of sonic waves and the self-focusing of the beams of the fast magnetosonic waves propagating in a magnetized plasma. Variable-coefficient extensions of the (2+1)- and (3+1)-dimensional KP equations can be found, respectively, in Refs. 2] 4]. On the other hand, the studies of exact solutions for the NLEEs are of importance. 1 4] In order to explore different kinds of solutions, a vast variety of methods have been developed including inverse scattering theory, 16] Darboux transformation, 17] algebro-geometric method, 18] Hirota bilinear method, 19] and so on. Among them, the Hirota bilinear method has been providing a powerful and effective tool for finding exact soliton solutions of various NLEEs. Recently, this method has been developed for deriving Wronskian and Pffafian solutions, 20,21] especially periodic wave solutions ] This paper is devoted to discussing the (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation by the Hirota bilinear method in terms of Riemann theta functions. 28,29] It is shown that the periodic wave solutions can be reduced to classical soliton solutions under certain limits. Moreover, all parameters appearing in the solutions are free variables, whereas usual periodic solutions involve Riemann constants that are difficult to determine and need to make complicated Abel transformation on the Riemann surface. Based on the symbolic computation, 1 4] the structure of this paper is organized as follows. The one- and twoperiodic wave solutions for the (2+1)-dimensional Boussinesq equation are obtained in Sec. 2, respectively. According to the above procedure of solving the Boussinesq equation, the one- and two-periodic wave solutions for the (3+1)-dimensional KP equation are presented in Sec. 3. Section 4 is our conclusion. The project supported by the National Natural Science Foundation of China under Grant Nos and , the Key Project of the Ministry of Education under Grant No , the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE , Beijing University of Aeronautics and Astronautics, the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901, and the Specialized Research Fund for the Doctoral Program of Higher Education of the Ministry of Education under Grant No Corresponding author, gaoyt@public.bta.net.cn

2 1170 ZHANG Huan, TIAN Bo, ZHANG Hai-Qiang, GENG Tao, MENG Xiang-Hua, LIU Wen-Jun, and CAI Ke-Jie Vol Periodic Wave Solutions for (2+1)-Dimensional Boussinesq Equation and Their Reductions The (2+1)-dimensional Boussinesq equation can be written in the form u tt u xx + 3(u 2 ) xx u xxxx u yy 0, (1) where the suffices refer to differentiation with respect to time t and the two space variables x and y. It was first derived in Ref. 5] by combining the classical Boussinesq equation with the weak dependence on the second spatial dimension, and has been further studied in Refs. 6] 8] and 27]. By the dependent variable transformation u u 0 2(lnf) xx, (2) equation (1) can be reduced to the bilinear form D 2 t (1 6u 0 )D 2 x D 4 x D 2 y + c](f f) 0, (3) where c c(t) is an integration function, u 0 is a constant solution of Eq. (1), and D x, D y, and D t are the bilinear derivative operators 19] defined by ( Dx m Dy n Dt k f g x ) m ( x y ) n ( y t ) k t f(x, y, t) g(x, y, t ) x x,y y,t t. The bilinear derivative operators have good property when acting on exponential functions, or more generally D n x e kx e kx (k k ) n e (k+k )x, G(D x, D y, D t ) e kx+ρy+ωt e (k x+ρ y+ω t) G(k k, ρ ρ, ω ω ) e (k+k )x+(ρ+ρ )y+(ω+ω )t. (4) Afterwards, we consider the Riemann theta function solution of Eq. (1) f n Z N e πi τn,n +2πi ξ,n, (5) where n (n 1, n 2,...,n N ) T, (T denotes the transpose of the vector), ξ (ξ 1, ξ 2,...,ξ N ) and τ is a symmetric matrix with Im τ > 0, ξ j k j x + ρ j y + ω j t + ξ (0) j, j (1, 2,...,N). 2.1 One-Periodic Wave Solution and Its Reduction Considering the case N 1, then equation (5) becomes f e 2πinξ+πin2τ. (6) Substituting Eq. (6) into Eq. (3) gives Gf f G(D x, D y, D t ) m m n+mm m It can be noted that Ḡ(m ) m e 2πinξ+πin2 τ m e 2πimξ+πim2 τ G(D x, D y, D t ) e 2πinξ+πin2τ e 2πimξ+πim2 τ G 2πi(n m)k, 2πi(n m)ρ, 2πi(n m)ω ] e 2πi(n+m)ξ+πi(n2 +m 2 )τ { Ḡ(m ) e 2πimξ. nn +1 G 2πi( m )k, 2πi( m )ρ, 2πi( m )ω ] e πi(n2 +(n m ) 2 )τ } e 2πim ξ G 2πi( m )k, 2πi( m )ρ, 2πi( m )ω ] e πi(n2 +(n m ) 2 )τ G 2πi( (m 2))k, 2πi( (m 2))ρ, 2πi( (m 2))ω ] exp { πi(n ) 2 + (n (m 2)) 2] τ } exp { 2πi(m 1)τ } Ḡ(m 2) exp {2πi(m 1)τ}, which implies that if Ḡ(0) Ḡ(1) 0, then Ḡ(m ) 0, m Z. (7) In this way, we may let Ḡ(0) 16π 2 n 2 ω (1 6u 0 )π 2 n 2 k π 2 n 2 ρ 2 256π 4 n 4 k 4 + c ] e 2πin2τ 0, (8) Ḡ(1) 4π 2 ( 1) 2 ω 2 + 4(1 6u 0 )π 2 ( 1) 2 k 2 + 4π 2 ( 1) 2 ρ 2 16π 4 ( 1) 4 k 4 + c ]

3 No. 5 Periodic Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional 1171 Denoting e πi(2 +1)τ 0. δ 1 (n) e 2πin2τ, δ 2 (n) e πi(2 +1)τ, a 11 a 12 b 1 b 2 δ 1 (n), a 21 then equations (8) and (9) can be written as Solving this system, we obtain Finally, we get the one-periodic wave solution 4π 2 ( 1) 2 δ 2 (n), a 22 16(1 6u0 )π 2 n 2 k π 2 n 2 ρ 2 256π 4 n 4 k 4] δ 1 (n), 16π 2 n 2 δ 1 (n), δ 2 (n), 4(1 6u0 )π 2 ( 1) 2 k 2 + 4π 2 ( 1) 2 ρ 2 16π 4 ( 1) 4 k 4] δ 2 (n), a 11 ω 2 + a 12 c + b 1 0, a 21 ω 2 + a 22 c + b 2 0. ω 2 b 2a 12 b 1 a 22 a 11 a 22 a 12 a 21, c b 1a 21 b 2 a 11 a 11 a 22 a 12 a 21. (10) u u 0 2(lnf) xx, (11) where f and ω are given by Eqs. (6) and (10), respectively. Figure 1 gives the plot of Eq. (1) with k 0.1, ρ 2, τ 3i. (9) Fig. 1 (a) One-periodic wave solution (11) with k 0.1, ρ 2, τ 3i; (b) Along y-axis; (c) Along x-axis. The classical one-soliton solution of the (2+1)- dimensional Boussinesq equation can be obtained as a limit of the periodic wave solution (11). For this purpose, we rewrite f as f 1 + α( e 2πiξ + e 2πiξ ) + α 4 ( e 4πiξ + e 4πiξ ) +, with α e πiτ. Setting u 0 0, ξ (ξ /2πi) (τ/2), k 2πik, ρ 2πiρ, and ω 2πiω, we get f 1 + e k x+ρ y+ω t + α 2 e ξ + α 2 e 2ξ + α 6 e 2ξ e k x+ρ y+ω t, as α 0. Thus, the one-periodic wave solution (11) converges to the one-soliton solution u 2(lnf) xx, f 1 + e k x+ρ y+ω t, (12) if only we can prove that (ω ) 2 (k ) 2 + (k ) 4 + (ρ ) 2. (13) Fig. 2 One-soliton solution with k 0.5, ρ 0.5, and t 0. In fact, we can get that a 11 32π 2 (α 2 + 4α 8 + ), a α 2 +, a 21 8π 2 (α + 9α 5 + ), a 22 2(α + α 5 + ),

4 1172 ZHANG Huan, TIAN Bo, ZHANG Hai-Qiang, GENG Tao, MENG Xiang-Hua, LIU Wen-Jun, and CAI Ke-Jie Vol. 50 b 1 32π 2 α 2 (k 2 16π 2 k 4 + ρ 2 ), b 2 8π 2 α(k 2 4π 2 k 4 + ρ 2 ) + 72π 2 (k 2 36π 2 k 4 + ρ 2 )α 5 +, which lead to b 2 a 12 b 1 a 22 8π 2 α(k 2 4π 2 k 4 + ρ 2 ) + o(α 2 ), a 11 a 22 a 12 a 21 8π 2 α + o(α 2 ). Therefore, we have It is easy to obtain ɛ ±1, as α 0, which implies Eq. (13). Figure 2 gives the plot of the one-soliton solution with k 0.5, ρ 0.5, and t Two-Periodic Wave Solution and Its Reduction ω ɛ k 2 4π 2 k 4 + ρ 2, Substituting Eq. (5) into Eq. (3), we have Gf f G(D x, D y, D t ) e 2πi ξ,n +πi τn,n e 2πi ξ,m +πi τm,m n,m Z 2 ( ) 2πi n m, k, 2πi n m, ρ, 2πi n m, ω e 2πi ξ,m+n +πi( τm,m + τn,n ) n,m Z 2 G n+mm Ḡ(m 1, m 2) which implies if m Z 2 G ( 2πi m, k, 2πi m, ρ, 2πi m, ω ) exp { πi ( τ(n m ), n m + τn, n )} exp { 2πi ξ, m } m Z 2 Ḡ(m 1, m 2) e 2πi ξ,m n j n j +δ jl, l1,2. G ( 2πi m, k, 2πi m, ρ, 2πi m, ω ) e πi( τ(n m ),n m + τn,n ) 2πi G 2πi j (m j 2δ jl ) ) k j, 2πi j1 j (m j 2δ jl ) ) ] ω j j1 { exp πi 2 j,k1 + ( (m j 2δ jl n ) ( j) + δ jl τjk (m k 2δ kl n ) ]} j) + δ kl {Ḡ( ) m 1 2, m 2 e 2πi(m 1 1)τ 11+2πim 2 τ 12, l 1, Ḡ ( m 1, m 2 2 ) e 2πi(m 2 1)τ 22+2πim 1 τ 12, l 2, Ḡ(0, 0) Ḡ(0, 1) Ḡ(1, 0) Ḡ(1, 1) 0, j (m j 2δ jl ) ) ρ j, j1 (n j + δ jl )τ jk (n k + δ kl ) then Ḡ(m 1, m 2) 0 and equation (2) is an exact solution of the (2+1)-dimensional Boussinesq equation. Then, we have by denoting a j1 a j3 b j δ j (n) 4π 2 ( 1 m j 1 )2 δ j (n), A a j2 24π 2 m j, k 2 δ j (n), a j4 ω1 2 ω2 2 u 0 c b, (14) 4π 2 ( 2 m j 2 )2 δ j (n), δ j (n), ( 4π 2 m j, k 2 4π 2 m j, ρ π 4 m j, k 4 )δ j (n), e πi τ(n mj ),(n m j ) +πi τn,n, m 1 (0, 0), m 2 (0, 1), m 3 (1, 0), m 4 (1, 1),

5 No. 5 Periodic Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional 1173 A (a jk ) 4 4, b (b 1, b 2, b 3, b 4 ) T, k 1, 2, 3, 4. With the help of Eq. (14), we derive ω1 2 1, ω2 2 2, u 0 3, (15) where A and 1, 2, and 3 are produced from by replacing its first, second and third columns with b, respectively. Finally, we obtain the two-periodic wave solution u u 0 2(lnf) xx, (16) where f and ω 1, ω 2, and u 0 are defined by Eqs. (5) and (6), respectively. Similarly, the two-soliton solution for the (2+1)- dimensional Boussinesq equation can be derived from the two-periodic solution (16). We rewrite f as f 1 + ( e 2πiξ 1 + e 2πiξ 1 ) e πiτ 11 + e πiτ 22 ( e 2πiξ 2 + e 2πiξ 2 ) + e πi(τ 11+2τ 12 +τ 22 ) ( e 2πi(ξ 1+ξ 2 ) + e 2πi(ξ 1+ξ 2 ) ). Setting ξ 1 2πiξ 1 +πiτ 11, ξ 2 2πiξ 2 +πiτ 22, τ 12 i τ ( τ is a real constant), we get where f 1 + e ξ 1 + e ξ 2 + e ξ 1 +ξ 2 +2πiτ 12 + α 2 1 e ξ 1 + α 2 2 e ξ 2 + α 2 1 α 2 2 e ξ 1 ξ 2 +2πiτ e ξ 1 + e ξ 2 + e ξ 1 +ξ 2 2π τ, as α 1, α 2 0, α 1 e πiτ 11, α 2 e πiτ 22, ξ j k jx + ρ jy + ω jt + πiτ jj, e 2π τ (ω 1 ω 2) 2 (k 1 k 2) 2 (ρ 1 ρ 2) 2 (k 1 k 2) 4 (ω 1 +ω 2 )2 (k 1 +k 2 )2 (ρ 1 +ρ 2 )2 (k 1 +k 2 )4, ω j ɛ j (k j )2 + (k j )4 + (ρ j )2, ɛ j ±1, j 1, 2, as α 1, α 2 0, which is consistent with the result of Eq. (8). 3 Periodic Wave Solutions for the (3+1)-Dimensional KP Equation and Their Reductions The (3+1)-dimensional KP equation 11,12,14,15] reads (u t 6uu x + u xxx ) x + 3u yy + 3u zz 0, (17) where u u(x, y, z, t) is a function of the spatial x, y, z, and temporal t. It can be transformed into the following bilinear form (D t D x 6u 0 D 2 x + 3D 2 y + 3D 2 z + D 4 x + c)(f f) 0, (18) by the dependent variable transformation u u 0 2(lnf) xx, (19) where c c(t) is an integration function and u 0 is a constant solution of Eq. (17). In order to get periodic wave solutions, we choose f n Z N e πi τn,n +2πi ξ,n, (20) where n (n 1, n 2,..., n N ) T, ξ (ξ 1, ξ 2,...,ξ N ) and τ is a symmetric matrix with Im τ > 0, ξ j k j (x + ρ j y + σ j z + ω j t) + r j, j (1, 2,...,N). 3.1 One-Periodic Wave Solution and Its Reduction When N 1, equation (20) becomes f e 2πinξ+πin2τ. (21) Substituting Eq. (21) into Eq. (18) gives with Gf f G(D x, D y, D z, D t ) m m n+mm Ḡ(m ) m nn +1 m e 2πinξ+πin2τ m e 2πimξ+πim2 τ G(D x, D y,, D z, D t ) e 2πinξ+πin2τ e 2πimξ+πim2 τ G 2πi(n m)k, 2πi(n m)kρ, 2πi(n m)kσ, 2πi(n m)kω ] e 2πi(n+m)ξ+πi(n2 +m 2 )τ { G 2πi( m )k, 2πi( m )kρ, 2πi( m )kσ, 2πi( m )kω ] e πi(n2 +(n m ) 2 )τ } e 2πim ξ Ḡ(m ) e 2πim ξ G 2πi( m )k, 2πi( m )kρ, 2πi( m )kσ, 2πi( m )kω ] e πi(n2 +(n m ) 2 )τ G 2πi ( (m 2) ) k, 2πi ( (m 2) ) kρ, 2πi ( (m 2) ) kσ,

6 1174 ZHANG Huan, TIAN Bo, ZHANG Hai-Qiang, GENG Tao, MENG Xiang-Hua, LIU Wen-Jun, and CAI Ke-Jie Vol. 50 2πi ( (m 2) ) kω ] exp { πi (n ) 2 + (n (m 2)) 2] τ } exp { 2πi(m 1)τ } Ḡ(m 2) exp { 2πi(m 1)τ }. Then we get Ḡ(m ) 0, (m Z), provided that Ḡ(0) Ḡ(1) 0, where Ḡ(0) 16π 2 n 2 k 2 ω 48π 2 n 2 k 2 ρ 2 48π 2 n 2 k 2 σ u 0 π 2 n 2 k π 4 n 4 k 4 + c ] e 2πin2τ 0, (22) Denote Ḡ(1) α e πiτ, a 21 b 1 b 2 4π 2 ( 1) 2 k 2 ω 12π 2 ( 1) 2 k 2 ρ 2 12π 2 ( 1) 2 k 2 σ π 4 ( 1) 4 k u 0 π 2 ( 1) 2 k 2 + c ] e πi(2 +1)τ 0. (23) a 11 16π 2 k 2 n 2 α 2, a 12 4π 2 k 2 ( 1) 2 α 2 +1, a 22 α 2, α 2 +1, 96u0 π 2 n 2 k 2 48π 2 n 2 k 2 ρ 2 48π 2 n 2 k 2 σ π 4 n 4 k 4] α 2, 24u0 π 2 ( 1) 2 k 2 12π 2 k 2 ( 1) 2 ρ 2 12π 2 k 2 ( 1) 2 σ π 4 ( 1) 4 k 4] α Equations (22) and (23) can be written as a 11 ω + a 12 c b 1, a 21 ω + a 22 c b 2, from which we obtain ω b 1a 22 b 2 a 12, c b 2a 11 b 1 a 21. (24) a 11 a 22 a 12 a 21 a 11 a 22 a 12 a 21 Finally, we get the one-periodic wave solution u u 0 2(lnf) xx, (25) where f and ω are given by Eqs. (21) and (24), respectively. In what follows, we show that the classical one-soliton solution of the (3+1)-dimensional KP equation can be obtained as a limit of the periodic solution (25). For this purpose, we rewrite f as f 1 + α( e 2πiξ + e 2πiξ ) + α 4 ( e 4πiξ + e 4πiξ ) + with α e πiτ. Setting u 0 0, r r 0 (τ/2), k 2πik, and r 0 2πir 0, we get f 1 + e k (x+ρy+σz+ωt)+r 0 + α 2 e k (x+ρy+σz+ωt) r 0 +α 2 e 2k (x+ρy+σz+ωt)+2r 0 +α 6 e 2k (x+ρy+σz+ωt) 2r e k (x+ρy+σz+ωt)+r 0, as α 0. So the one-periodic solution (25) converges to the onesoliton solution u 2(lnf) xx, f 1 + e k (x+ρy+σz+ωt)+r 0, Gf f if only ω (k ) 2 3ρ 2 3σ 2, (26) can be verified. In fact, it is easy to get that a 11 32π 2 k 2 (α 2 + 4α 8 + ), a α 2 +, a 21 8π 2 k 2 (α + 9α 5 + ), a 22 2(α + α 5 + ), b 1 296u 0 π 2 k π 4 k 4 48π 2 k 2 ρ 2 48π 2 k 2 σ 2 ]α 2 + o(α 2 ), b 2 224u 0 π 2 k π 4 k 4 12π 2 k 2 ρ 2 12π 2 k 2 σ 2 ] + o(α 2 ). Therefore, we have b 1 a 22 b 2 a 12 32π 4 k 4 24π 2 k 2 ρ 2 24π 2 k 2 σ 2 ]α+o(α 2 ), a 11 a 22 a 12 a 21 8π 2 k 2 α + o(α 2 ), which leads to Eq. (26). 3.2 Two-Periodic Wave Solution and Its Reduction Our concern here is with the case N 2. Let ρ (k 1 ρ 1, k 2 ρ 2 ), σ (k 1 σ 1, k 2 σ 2 ), ω (k 1 ω 1, k 2 ω 2 ). Substituting Eq. (20) into Eq. (18), we have G(D x, D y, D z, D t ) e 2πi ξ,n +πi τn,n e 2πi ξ,m +πi τm,m n,m Z 2 ( ) 2πi n m, k, 2πi n m, ρ, 2πi n m, σ, 2πi n m, ω e 2πi ξ,m+n +πi( τm,m + τn,n ) n,m Z 2 G n+mm m Z 2 G ( 2πi m, k, 2πi m, ρ, 2πi m, σ, 2πi m, ω )

7 No. 5 Periodic Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional 1175 exp { πi ( τ(n m ), n m + τn, n )} exp { 2πi ξ, m } Ḡ(m 1, m 2) e 2πi ξ,m. m Z 2 Similarly, we obtain that Ḡ(m 1, m 2) G ( 2πi m, k, 2πi m, ρ, 2πi m, σ, 2πi m, ω ) e πi( τ(n m ),n m + τn,n ) n j n j +δ jl, l1,2 G 2πi j (m j 2δ jl ) ) k j, 2πi 2πi j1 j (m j 2δ jl ) ) k j σ j, 2πi j1 { exp πi j (m j 2δ jl ) ) k j ρ j, j1 j (m j 2δ jl ) ) ] k j ω j j1 2 (n j + δ jl )τ jk (n k + δ kl ) + ((m j 2δ jl n j) + δ jl )τ jk ((m k 2δ kl n k) + δ kl ) ]} j,k1 {Ḡ(m 1 2, m 2) e 2πi(m 1 1)τ 11+2πim 2 τ 12, l 1, Ḡ(m 1, m 2 2) e2πi(m 2 1)τ 22+2πim 1 τ 12, l 2, which implies if Ḡ(0, 0) Ḡ(0, 1) Ḡ(1, 0) Ḡ(1, 1) 0, then Ḡ(m 1, m 2 ) 0 and equation (19) is an exact solution of the (3+1)-dimensional KP equation. Denote a j1 4π 2 k 1 ( 1 m j 1 ) mj, k δ j (n), a j2 4π 2 k 2 ( 2 m j 2 ) mj, k δ j (n), a j3 b j δ j (n) 24π 2 m j, k 2 δ j (n), a j4 δ j (n), ( 16π 4 m j, k π 2 m j, ρ π 2 m j, σ 2) δ j (n), e πi τ(n mj ),(n m j ) +πi τn,n, m 1 (0, 0), m 2 (0, 1), m 3 (1, 0), m 4 (1, 1), A (a jk ) 4 4, b (b 1, b 2, b 3, b 4 ) T, j 1, 2, 3, 4. Then we have from which we get A ω 1 ω 2 u 0 c b, (27) ω 1 1, ω 2 2, u 0 3, (28) where A and 1, 2, 3 are produced from by replacing its first, second and third columns with b, respectively. Finally, we obtain the two-periodic wave solution u u 0 2(lnf) xx, (29) where f and ω 1, ω 2 are given by Eqs. (20) and (28), respectively. The same with the case N 1, the two-soliton solution of the (3+1)-dimensional KP equation can be derived from the two-periodic solution (29). We rewrite f as f 1 + ( e 2πiξ 1 + e 2πiξ 1 ) e πiτ 11 + e πiτ 22 ( e 2πiξ 2 + e 2πiξ 2 ) + e πi(τ 11+2τ 12 +τ 22 ) ( e 2πi(ξ 1+ξ 2 ) + e 2πi(ξ 1+ξ 2 ) ) Making the transformation r 1 r 1 + (1/2)τ 11, r 2 r 2 + (1/2)τ 22, ξ 1 2πiξ 1 + πiτ 11, ξ 2 2πiξ 2 + πiτ 22, and τ 12 iτ, (τ is a real constant), we get f 1 + e ξ 1 + e ξ 2 + e ξ 1 +ξ 2 +2πiτ 12 + α 2 1 e ξ 1 + α 2 2 e ξ 2 + α 2 1 α 2 2 e ξ 1 ξ 2 +2πiτ e ξ 1 + e ξ 2 + e ξ 1 +ξ 2 2πτ, as α 1, α 2 0, where α 1 e πiτ 11, α 2 e πiτ 22, ξ j 2πik j (x + ρ j y + σ j z + ω j t) + r j],

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