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Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev–Petviashvili equation
Applied Mathematics and Computation 216 (2010) 3154–3161
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Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev–Petviashvili equation Yong-Qi Wu Mathematics and Computational Science School, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, PR China
a r t i c l e
i n f o
Keywords: (3+1)-Dimensional Kadomtsev–Petviashvili equation Hirota bilinear method Asymptotic behavior Periodic wave solution Soliton solution
a b s t r a c t In this paper, the one- and two-periodic wave solutions for the (3+1)-dimensional Kadomtsev–Petviashvili equation are presented by means of the Hirota’s bilinear method and the Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In the past decades, much effort has been spent on the construction of exact solutions of nonlinear equations for their important role in understanding the nonlinear problems, and there are several systematic approaches to obtain them [1–9]. Besides, the nonlinearization method of Lax pairs (or constrained flow) [10–16], the homogeneous balance method [17–19], the hyperbolic function method [20–23], the Jacobian elliptic function expansion method [24,25] have also been applied to construct exact solutions of nonlinear evolution equations. Therefore it is an interesting work to find out the new type solution to a nonlinear evolution equations. In this paper, we consider the following (3+1)-dimensional Kadomtsev–Petviashvili equation,
uxt þ 6u2x þ 6uuxx uxxxx uyy uzz ¼ 0;
ð1:1Þ
which is introduced to describe the dynamics of solitons and nonlinear waves in the fields of plasma physics, fluid dynamics, etc. [26]. Eq. (1.1) has been widely studied by many authors and various different exact solutions have been presented [27– 33]. For instance, Wang and Lou have revealed some special type exact solutions in Ref. [27], Bai et al. constructed several new families of exact solutions after applying a generalized variable-coefficient algebraic method to Eq. (1.1) [28], Chen et al. obtained some exact solutions for Eq. (1.1) by using a new generalized transformation in homogeneous balance method [29]. EI-Sayed et al. studied Eq. (1.1) by considering the decomposition scheme and obtained some exact solutions [30]. Hu obtained some traveling wave solutions by applying algebraic method [31]. Liu and Liu derived the similarity solutions by using the compatibility method [32]. Zhao used the extended mapping method to derive a new family of the exact solutions of Eq. (1.1) [33]. The bilinear Bäcklund transformation and some new explicit solutions of Eq. (1.1) are also derived in our recent work [34]. This paper tries to extend the Hirota’s bilinear method and the Riemann theta function [35–39] to (3+1)-dimensional Kadomtsev–Petviashvili equation, the one- and two-periodic wave solutions for Eq. (1.1) are presented and the rigorous proofs are given that the soliton solutions can be obtained from the periodic wave solutions in an appropriate limiting procedure. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.030
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The Hirota bilinear operator is defined as [1,2] l m n r n r 0 0 0 0 0 Dlx Dm y Dz Dt aðx; y; z; tÞ bðx; y; z; tÞ ð@ x @ x0 Þ ð@ y @ y0 Þ ð@ z @ z0 Þ ð@ t @ t Þ aðx; y; z; tÞbðx ; y ; z ; t Þjx0 ¼x;y0 ¼y;z0 ¼z;t 0 ¼t ;
ð1:2Þ
where the l; m; n; r 2 N, to deal with bilinear equation, we use the property of the Hirota’s operator. From the definition, we have the relation
Dnx expðpxÞ expðp0 xÞ ¼ ðp p0 Þn exp½ðp þ p0 Þx;
ð1:3Þ
or more generally, let P(Dx, Dy, . . . , Dt) be a polynomial of Dx, Dy, . . . , Dt then 0
PðDx ; Dy ; . . . ; Dt Þ expðpx þ ly þ þ xtÞ expðp0 x þ l y þ þ x0 tÞ 0
0
0
0
0
¼ Pðp p ; l l ; . . . ; x x Þ exp½ðp þ p Þx þ ðl þ l Þy þ þ ðx þ x0 Þt:
ð1:4Þ
With the dependent variable transformation
uðx; y; z; tÞ ¼ 2@ 2 ln f ðx; y; z; tÞ=@x2 ;
ð1:5Þ
the bilinear form of the (3+1)-dimensional Kadomtsev–Petviashvili equation appropriate to the periodic problem is written as
FðDx ; Dy ; Dz ; Dt Þf f ðDx Dt D4x D2y D2z þ cÞf f ¼ 0;
ð1:6Þ
where c is an integration constant generally dependent on time. 2. One-periodic wave solution and asymptotic behavior 2.1. One-periodic wave solution First we consider the case of one-periodic wave solution of Eq. (1.1). We take the one-dimensional Riemann theta function in the form [40] 1 X
f ¼ #ðg; sÞ ¼
2
expð2ping þ pin sÞ;
i¼
pffiffiffiffiffiffiffi 1;
ð2:1Þ
n¼1
with g = px + ly + kz + xt + g0, where s is a complex constant satisfying the condition Ims > 0 and p, l, k and x represent the wave number and frequency, respectively, and g0 is a phase constant. Substituting (2.1) into (1.6) and using the property of Eq. (1.4), we have
Ff f ¼
1 X
2
0
02
FðDx ; Dy ; Dz ; Dt Þ expð2ping þ pin sÞ expð2pin g þ pin
sÞ
n;n0 ¼1
¼
1 X
2
FðDx ; Dy ; Dz ; Dt Þ expð2ping þ pin sÞ expð2piðm nÞg þ piðm nÞ2 sÞ
n;m¼1
¼
1 X
F½2pið2n mÞp; . . . ; 2pið2n mÞx expf2pimg þ pi½n2 þ ðn mÞ2 sg ¼
n;m¼1
1 X
e F ðmÞ expð2pimgÞ;
ð2:2Þ
m¼1
0 where the new summation index m = n + n has been introduced and e F ðmÞ is defined by
e F ðmÞ ¼
1 X
F½2pið2n mÞp; 2pið2n mÞl; 2pið2n mÞk; 2pið2n mÞx expfpi½n2 þ ðn mÞ2 sg:
ð2:3Þ
n¼1 0
In Eq. (2.3), by shifting summation index n by unity as n = n + 1, we have the relation
e F ðmÞ ¼
1 X
F½2pið2n0 ðm 2ÞÞp; . . . ; 2pið2n0 ðm 2ÞÞx expfpi½n02 þ ðm 2 n0 Þ2 sg expf2piðm 1Þsg
n0 ¼1
¼e F ðm 2Þ expf2piðm 1Þsg:
ð2:4Þ
From relation (2.4) we can conclude that, if e F ð0Þ and e F ð1Þ are zero, then all the other e F ðmÞ are also zero. On the other hand we recall that we have at least two unknown parameters, namely, arbitrary integration constant c and the nonlinear frequency x even if all the other parameters such as wavevectors p, l, k and amplitude parameter s are taken as arbitrarily given paramF ð0Þ ¼ 0 and e F ð1Þ ¼ 0, eters. Therefore, by solving integration constant c and nonlinear dispersion relation x from the Eqs. e we have exact periodic wave solution to the bilinear equation (1.6).
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From Eqs. (1.6) and (2.3), the Eqs. e F ð0Þ ¼ 0 and e F ð1Þ ¼ 0 are explicitly written as 1 X n¼1 1 X
2
2
2
½16p2 n2 px 256p4 n4 p4 þ 16p2 n2 l þ 16p2 n2 k þ c expð2pin sÞ ¼ 0; 2
ð2:5Þ 2
½4p2 ð2n 1Þ2 px 16p4 ð2n 1Þ4 p4 þ 4p2 ð2n 1Þ2 l þ 4p2 ð2n 1Þ2 k þ c expfpi½n2 þ ðn 1Þ2 sg ¼ 0; ð2:6Þ
n¼1
respectively. Introducing the quantities
A0 ðsÞ ¼ A1 ðsÞ ¼ A2 ðsÞ ¼ B0 ðsÞ ¼ B1 ðsÞ ¼ B2 ðsÞ ¼
1 X
2
expð2pin sÞ;
n¼1 1 X n¼1 1 X n¼1 1 X n¼1 1 X n¼1 1 X
ð2:7Þ 2
ð2:8Þ
2
ð2:9Þ
ð4nÞ2 expð2pin sÞ; ð4nÞ4 expð2pin sÞ; expfpi½n2 þ ðn 1Þ2 sg;
ð2:10Þ
ð4n 2Þ2 expfpi½n2 þ ðn 1Þ2 sg;
ð2:11Þ
ð4n 2Þ4 expfpi½n2 þ ðn 1Þ2 sg:
ð2:12Þ
n¼1
Eqs. (2.5) and (2.6) can be written compactly as 2
2
ð2:13Þ
2
2
ð2:14Þ
A1 ðsÞp2 px A2 ðsÞðppÞ4 þ A1 ðsÞp2 ðl þ k Þ þ cA0 ðsÞ ¼ 0; B1 ðsÞp2 px B2 ðsÞðppÞ4 þ B1 ðsÞp2 ðl þ k Þ þ cB0 ðsÞ ¼ 0: From these two equations, quantities x and c are determined to be 2
2
l þk A0 ðsÞB2 ðsÞ B0 ðsÞA2 ðsÞ 2 3 pp; A0 ðsÞB1 ðsÞ B0 ðsÞA1 ðsÞ p A2 ðsÞB1 ðsÞ B2 ðsÞA1 ðsÞ c¼ ðppÞ4 : A0 ðsÞB1 ðsÞ B0 ðsÞA1 ðsÞ
x¼
ð2:15Þ ð2:16Þ
Therefore, expression (1.5) is the exact one-periodic wave solution of the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) with f given by Eq. (2.1) and x by Eq. (2.15). 2.2. Asymptotic behavior of the one-periodic wave solution It is interesting that the one-soliton solution can be obtained from the one-periodic wave solution in an appropriate limiting procedure. To illustrate this we introduce a quantity
q ¼ expðpisÞ;
ð2:17Þ
and take a limit q ? 0 (or Ims ? 1). Theorem 1. In the condition q ? 0 (or Ims ? 1), the one-periodic solution (2.1) of Eq. (1.1) tends to one-soliton solution via Eq. (1.5)
! ~ þx ~t þg ~2 ~x þ ~ly þ kz ~0 p 2 p ; uðx; y; z; tÞ ¼ sech 2 2
ð2:18Þ
~ ¼ ~l2 þp~k~2 þ p ~3 holds. where the relation x Proof. By using q ¼ expðpisÞ, the quantifies defined in Eqs. (2.7)–(2.12) are then expanded in powers of q as
A0 ¼ 1 þ 2q2 þ 2q8 þ ; 2
8
ð2:19Þ
A1 ¼ 32q þ 128q þ ;
ð2:20Þ
A2 ¼ 512q2 þ 8192q8 þ ;
ð2:21Þ
B0 ¼ 2q þ 2q5 þ 2q13 þ ;
ð2:22Þ
B1 ¼ 8q þ 72q5 þ 200q13 þ ;
ð2:23Þ
B2 ¼ 32q þ 2592q5 þ 20000q13 þ :
ð2:24Þ
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Substituting these expressions into Eqs. (2.15) and (2.16) yields 2
2
2
2
l þk 1 30q2 þ 81q4 þ 2 3 l þk pp ! 4 4p2 p3 ; 1 6q2 þ 9q4 þ p p 1 15q4 þ 20q6 þ c ¼ 384q2 ðppÞ4 ! 0; q ! 0: 1 6q2 þ 9q4 þ
x¼
q ! 0;
ð2:25Þ ð2:26Þ
By introducing the quantities
~ ¼ 2pip; p
~l ¼ 2pil;
~ ¼ 2pik; k
~ ¼ 2pix; g ~ 0 ¼ 2pi g0 þ x
s
;
2
~ þx ~t þg ~0 ; g~ ¼ p~x þ ~ly þ kz
ð2:27Þ
the one-periodic wave solution (2.1) reduces in the limit of q ? 0 (or Ims ? 1) to
f ¼ 1 þ expð2pig þ pisÞ þ expð2pig þ pisÞ þ expð4pig þ 2pisÞ þ expð4pig þ 2pisÞ þ ~ þx ~t þg ~x þ ~ly þ kz ~ 0 Þ; ¼ 1 þ eg~ þ q2 ðeg~ þ e2g~ Þ þ q6 ðe2g~ þ e3g~ Þ þ ! 1 þ eg~ ¼ 1 þ expðp
ð2:28Þ
which is transformed into the one-soliton solution to the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) with Eq. (1.5) as
! ~ þx ~t þg ~2 ~x þ ~ly þ kz ~0 p 2 p ; uðx; y; z; tÞ ¼ sech 2 2 ~2 þ~ k2 ~ p
~ ¼l where the relation x
ð2:29Þ
~3 has been used, which is a direct result of (2.25) and (2.27). Thus the proof is completed. þp
h
3. Two-periodic wave solution and asymptotic behavior In what follows, we consider the two-periodic wave solution to the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1), which is a two-dimensional generalization of one-periodic wave solution. 3.1. Construction of two-periodic wave solution Without loss of generality, we consider multidimensional Riemann theta function [40] 1 X
f ¼ #ðg1 ; . . . ; gN ; sÞ ¼
exp 2pi
N X
n1 ;...;nN ¼1
nj gj þ pi
j¼1
N X
!
pffiffiffiffiffiffiffi
sjk nj nk ; i ¼ 1;
ð3:1Þ
j;k¼1
with
gj ¼ pj x þ lj y þ kj z þ xj t þ g0j ðj ¼ 1; . . . ; NÞ;
ð3:2Þ
where pj, lj, kj, xj and g0j have the same meaning as in the one-periodic wave case. The term sjk (j – k) represents the effect of interaction between periodic waves and is assumed to satisfy the conditions sjk = skj, Imsjk > 0, (j, k = 1, . . . , N) and the complex matrix (sjk)N N is of positive definite imaginary part. Substituting (3.1) into (1.6) and using the property of Eq. (1.4), we have the analogous result to Eq. (2.2).
Ff f ¼
1 X
e F ðm1 ; . . . ; mN Þ exp 2pi
m1 ;...;mN ¼1
N X
! mj gj ;
ð3:3Þ
j¼1
where
"
1 X
e F ðm1 ; . . . ; mN Þ ¼
F 2p i
n1 ;...;nN ¼1
(
exp
pi
N X j¼1
N X
ð2nj mj Þpj ; 2pi
N N N X X X ð2nj mj Þlj ; 2pi ð2nj mj Þkj ; 2pi ð2nj mj Þxj j¼1
j¼1
)
½nj sjk nk þ ðnj mj Þsjk ðnk mk Þ :
#
j¼1
ð3:4Þ
j;k¼1
In Eq. (3.4), by shifting the hth summation index nh by unity, we obtain the relation corresponding to (2.4) (h = 1, . . . , N).
( ) N X e e F ðm1 ; . . . ; mN Þ ¼ F ðm1 ; . . . ; mh1 ; mh 2; mhþ1 ; . . . ; mN Þ exp 2pi shj mj 2shh :
ð3:5Þ
j¼1
If relations
e F ðm1 ; . . . ; mN Þ ¼ 0;
ð3:6Þ
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Y.-Q. Wu / Applied Mathematics and Computation 216 (2010) 3154–3161
hold for all combinations of m1 = 0, 1, m2 = 0, 1, . . . , mN = 0, 1 then all e F 0s become zero and expression (3.1) gives the N-periodic wave solution to the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1). Notice the number of equations of the type (3.6) be 2N and the total number of unknown parameters as integration constant c, nonlinear frequency x1, . . . , xN and interaction term sjk(1 6 j, k 6 N, j – k) 1 þ N þ C 2N ¼ 12 ðN 2 þ N þ 2Þ. For N = 1, 2, the number of equations corresponds to that of unknowns, implying the existence of the one- and two-periodic wave solutions. In the case N = 1, we have just discussed. For N = 2,we have the conclusion from Eqs. (3.4) and (1.6)
"
1 X
e F ðm1 ; m2 Þ ¼
F 2pi
n1 ;n2 ¼1
2 2 2 2 X X X X ð2nj mj Þpj ; 2pi ð2nj mj Þlj ; 2pi ð2nj mj Þkj ; 2pi ð2nj mj Þxj j¼1
expfpi
2 X
j¼1
j¼1
#
j¼1
½nj sjk nk þ ðnj mj Þsjk ðnk mk Þg
j;k¼1 1 X
¼
4p2 ½ð2n1 m1 Þp1 þ ð2n2 m2 Þp2 ½ð2n1 m1 Þx1 þ ð2n2 m2 Þx2 16p4 ½ð2n1 m1 Þp1
n1 ;n2 ¼1
þ 2n2 m2 Þp2 4 þ 4p2 ½ð2n1 m1 Þl1 þ ð2n2 m2 Þl2 2 þ 4p2 ½ð2n1 m1 Þk1 o n2 þðn m Þ2 n2 þðn m Þ2 n n þðn m Þðn m Þ þ 2n2 m2 Þk2 2 þ c k11 1 1 k22 2 2 k31 2 1 1 2 2 ¼ 0;
ð3:7Þ
k2 ¼ epis22 ;
ð3:8Þ
where
k1 ¼ epis11 ;
k3 ¼ e2pis12 ;
specially, 1 X
e F ð0; 0Þ ¼
n 4p2 ½ð2n1 Þp1 þ ð2n2 Þp2 ½ð2n1 Þx1 þ ð2n2 Þx2 16p4 ½ð2n1 Þp1 þ ð2n2 Þp2 4 þ 4p2 ½ð2n1 Þl1 þ ð2n2 Þl2 2
n1 ;n2 ¼1
o 2n2 2n2 1 n2 ¼ 0; þ 4p2 ½ð2n1 Þk1 þ ð2n2 Þk2 2 þ c k1 1 k2 2 k2n 3 1 X
e F ð1; 0Þ ¼
n
ð3:9Þ
4p2 ½ð2n1 1Þp1 þ ð2n2 Þp2 ½ð2n1 1Þx1 þ ð2n2 Þx2 16p4 ½ð2n1 1Þp1 þ ð2n2 Þp2 4
n1 ;n2 ¼1
o n2 þðn 1Þ2 2n2 n n þðn 1Þn2 þ 4p2 ½ð2n1 1Þl1 þ ð2n2 Þl2 2 þ 4p2 ½ð2n1 1Þk1 þ ð2n2 Þk2 2 þ c k11 1 k2 2 k31 2 1 ¼ 0; 1 X
e F ð0; 1Þ ¼
n
ð3:10Þ
4p2 ½ð2n1 Þp1 þ ð2n2 1Þp2 ½ð2n1 Þx1 þ ð2n2 1Þx2 16p4 ½ð2n1 Þp1 þ ð2n2 1Þp2 4
n1 ;n2 ¼1
o 2n2 n2 þðn 1Þ2 n n þn ðn 1Þ þ 4p2 ½ð2n1 Þl1 þ ð2n2 1Þl2 2 þ 4p2 ½ð2n1 Þk1 þ ð2n2 1Þk2 2 þ c k1 1 k22 2 k31 2 1 2 ¼ 0; 1 X
e F ð1; 1Þ ¼
ð3:11Þ
4p2 ½ð2n1 1Þp1 þ ð2n2 1Þp2 ½ð2n1 1Þx1 þ ð2n2 1Þx2 16p4 ½ð2n1 1Þp1
n1 ;n2 ¼1
þ 2n2 1Þp2 4 þ 4p2 ½ð2n1 1Þl1 þ ð2n2 1Þl2 2 þ 4p2 ½ð2n1 1Þk1 o n2 þðn 1Þ2 n2 þðn 1Þ2 n n þðn 1Þðn2 1Þ þ 2n2 1Þk2 2 þ c k11 1 k22 2 k31 2 1 ¼ 0;
ð3:12Þ
so, from the relations
e F ð0; 0Þ ¼ 0;
e F ð0; 1Þ ¼ 0;
e F ð1; 0Þ ¼ 0;
e F ð1; 1Þ ¼ 0;
ð3:13Þ
we can solve the integration constant c, two kinds of nonlinear dispersion relations and the interaction term s12. Thus expression (1.5) is the exact two-periodic wave solution of the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) with f given by Eq. (3.1) (N = 2). 3.2. Asymptotic behavior of the two-periodic wave solution It is noticed that the two-soliton solution can also be obtained from the two-periodic wave solution in an appropriate limiting procedure. Theorem 2. Suppose that 1 < r1 < 2, 1 < r2 < 2 are two constants, satisfying jk1 jr1 ! 0 and jk2 jr2 ! 0, then the periodic solution (3.1) of Eq. (1.1) tends to soliton solution via Eq. (1.5)
Y.-Q. Wu / Applied Mathematics and Computation 216 (2010) 3154–3161
3159
uðx; y; z; tÞ ¼ 2@ 2x ln f ~22 exp g ~1 þ p ~2 Þ2 expðg ~21 exp g ~1 þ p ~ 2 þ ðp ~1 þ g ~ 2 þ AÞ p ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g ~1 þ p ~2 exp g ~ 2 þ ðp ~1 þ p ~2 Þ expðg ~1 þ g ~ 2 þ AÞ 2 ~1 exp g p þ2 ; ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g
¼ 2
ð3:14Þ
where the relations
~1 ¼ x
~2 ~l2 þ k 1 1 ~31 ; þp ~1 p
~2 ¼ x
~2 ~l2 þ k 2 2 ~32 ; þp ~2 p
ð3:15Þ
and
eA ¼
~1 k ~ 2 Þ2 ~1 k ~2 ; x ~1 x ~ 2 Þ ðp ~1 x ~ 2Þ ~1 p ~2 Þðx ~1 p ~2 Þ4 ð~l1 ~l2 Þ2 ðk ~1 p ~2 ; ~l1 ~l2 ; k ðp Fðp ; ¼ 4 2 2 ~ ~ ~ ~ ~ ~ ~ ~ ~1 þx ~ 2 Þ ðp ~1 þx ~ 2Þ ~1 þ p ~1 þ p ~2 Þðx ~1 þ p ~2 Þ ðl1 þ l2 Þ ðk1 þ k2 Þ ~2 ; l1 þ l2 ; k1 þ k2 ; x Fðp ðp
ð3:16Þ
hold. Proof. We expand the two-periodic wave solution (3.1) (N = 2) in the following form by using the quantities
~ ¼ 2pik ; x ~ j ¼ 2pixj ~j ¼ 2pipj ; ~lj ¼ 2pilj ; k p j j ~ ~ ~ ~ ~ ~ gj ¼ pj x þ lj y þ kj z þ xj t þ g0j ; j ¼ 1; 2;
g~ 0j ¼ 2pig0j þ pisjj ; j ¼ 1; 2;
ð3:17Þ ð3:18Þ
f ¼ #ðg1 ; g2 ; sÞ ¼ 1 þ expð2pig1 þ pis11 Þ þ expð2pig1 þ pis11 Þ þ expð2pig2 þ pis22 Þ þ expð2pig2 þ pis22 Þ þ expð2piðg1 þ g2 Þ þ piðs11 þ 2s12 þ s22 ÞÞ þ expð2piðg1 þ g2 Þ þ piðs11 þ 2s12 þ s22 ÞÞ þ ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ 2pis12 Þ þ k21 expðg ~ 1 Þ þ k22 expðg ~ 2 Þ þ k21 k22 expðg ~1 g ~ 2 þ 2pis12 Þ þ ¼ 1 þ exp g r1 r2 ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ; ~ 1 þ exp g jk1 j ! 0; jk2 j ! 0 ; ð3:19Þ ! 1 þ exp g where
A ¼ 2pis12 :
ð3:20Þ
In the following, we verify the formulae (3.15) and (3.16) hold, to this end, we expand each function in Eqs. (3.9)–(3.12) into a series with k1 and k2. It is slightly tedious, but this process is easily carried out by using symbolic computation software MATHEMATICA or MAPLE. Actually, we only need to make the first order expansions with k1 and k2 to show the asymptotic relations (3.15) and (3.16). Here we consider their second order expansion to see deeper relations among parameters for the two-periodic solution and two-soliton solution. From
h i 2 2 e F ð0; 0Þ ¼ c þ 2 16p2 p1 x1 16p4 ð2p1 Þ4 þ 16p2 l1 þ 16p2 k1 þ c k21 h i 2 2 ¼ þ2 16p2 p2 x2 16p4 ð2p2 Þ4 þ 16p2 l2 þ 16p2 k2 þ c k22 þ o ks11 ks22 ¼ 0;
ð3:21Þ
where s1 + s2 P 4, we obtain that
c ¼ 0:
ð3:22Þ
h i 2 2 e F ð1; 0Þ ¼ 2 4p2 p1 x1 16p4 p41 þ 4p2 l1 þ 4p2 k1 þ c k1 þ o ks11 ks22 ¼ 0;
ð3:23Þ
From
where s1 + s2 P 3, and using c = 0, we obtain the nonlinear dispersion relation
~1 ¼ x
~l2 þ k ~2 1 1 ~31 : þp ~ p1
ð3:24Þ
From
h i 2 2 e F ð0; 1Þ ¼ 2 4p2 p2 x2 16p4 p42 þ 4p2 l2 þ 4p2 k2 þ c k2 þ o ks11 ks22 ¼ 0;
ð3:25Þ
where s1 + s2 P 3, and using c = 0, we obtain another nonlinear dispersion relation
~2 ¼ x
~2 ~l2 þ k 2 2 ~32 : þp ~2 p
ð3:26Þ
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Y.-Q. Wu / Applied Mathematics and Computation 216 (2010) 3154–3161
From
nh i e F ð1; 1Þ ¼ 2 4p2 ðp1 þ p2 Þðx1 þ x2 Þ 16p4 ðp1 þ p2 Þ4 þ 4p2 ðl1 þ l2 Þ2 þ 4p2 ðk1 þ k2 Þ2 þ c k3 h io þ 4p2 ðp1 p2 Þðx1 x2 Þ 16p4 ðp1 p2 Þ4 þ 4p2 ðl1 l2 Þ2 þ 4p2 ðk1 k2 Þ2 þ c k1 k2 þ o ks11 ks22 ¼ 0; ð3:27Þ r1
r2
where s1 + s2 P 6, we know that the coefficient of k1k2 must be zero in the conditions jk1 j ! 0; jk2 j ! 0. By using c = 0, we obtain that
k3 ¼
4p2 ðp1 p2 Þðx1 x2 Þ 16p4 ðp1 p2 Þ4 þ 4p2 ðl1 l2 Þ2 þ 4p2 ðk1 k2 Þ2
4p2 ðp1 þ p2 Þðx1 þ x2 Þ 16p4 ðp1 þ p2 Þ4 þ 4p2 ðl1 þ l2 Þ2 þ 4p2 ðk1 þ k2 Þ2 ~1 k ~2 Þ2 ~1 x ~ 2 Þ ðp ~1 p ~2 Þðx ~1 p ~2 Þ4 ð~l1 ~l2 Þ2 ðk ðp ¼ : 4 2 ~1 þ k ~2 Þ2 ~1 þx ~ 2 Þ ðp ~1 þ p ~2 Þðx ~1 þ p ~2 Þ ð~l1 þ ~l2 Þ ðk ðp
ð3:28Þ
Thus, we have
eA ¼ k3 ¼
~1 k ~2 Þ2 ~1 k ~2 ; x ~1 x ~ 2 Þ ðp ~1 x ~ 2Þ ~1 p ~2 Þðx ~1 p ~2 Þ4 ð~l1 ~l2 Þ2 ðk ~1 p ~2 ; ~l1 ~l2 ; k ðp Fðp ¼ : 4 2 2 ~ ~ ~ ~ ~ ~ ~ ~ ~1 þx ~ 2 Þ ðp ~1 þx ~ 2Þ ~1 þ p ~1 þ p ~2 Þðx ~1 þ p ~2 Þ ðl1 þ l2 Þ ðk1 þ k2 Þ ~2 ; l1 þ l2 ; k1 þ k2 ; x Fðp ðp
ð3:29Þ
Finally, by using (1.5) and (3.19), the two-soliton solution of the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) is obtained
~22 exp g ~1 þ p ~2 Þ2 expðg ~21 exp g ~1 þ p ~ 2 þ ðp ~1 þ g ~ 2 þ AÞ p ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g ~1 þ p ~2 exp g ~ 2 þ ðp ~1 þ p ~2 Þ expðg ~1 þ g ~ 2 þ AÞ 2 ~1 exp g p þ 2½ ; ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g
uðx; y; z; tÞ ¼ 2
thus, the proof is completed.
ð3:30Þ
h
4. Conclusion In summary, we have obtained the one-periodic wave solution, two-periodic wave solution and soliton solutions for the (3+1)-dimensional Kadomtsev–Petviashvili equation by means of the Hirota bilinear method and the Riemann theta function, which belong to the cases when N = 1 and N = 2. This method is valid for quite a few nonlinear evolution equations. For general case when N > 2, the results can be extended, but there are still certain numerical difficulties in the calculation and other aspects, which will be considered in our future work. Acknowledgement Supported by the Science Research Foundation of Zhanjiang Normal University (L0803). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev–Petviashvili equation Yong-Qi Wu Mathematics and Computational Science School, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, PR China
a r t i c l e
i n f o
Keywords: (3+1)-Dimensional Kadomtsev–Petviashvili equation Hirota bilinear method Asymptotic behavior Periodic wave solution Soliton solution
a b s t r a c t In this paper, the one- and two-periodic wave solutions for the (3+1)-dimensional Kadomtsev–Petviashvili equation are presented by means of the Hirota’s bilinear method and the Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In the past decades, much effort has been spent on the construction of exact solutions of nonlinear equations for their important role in understanding the nonlinear problems, and there are several systematic approaches to obtain them [1–9]. Besides, the nonlinearization method of Lax pairs (or constrained flow) [10–16], the homogeneous balance method [17–19], the hyperbolic function method [20–23], the Jacobian elliptic function expansion method [24,25] have also been applied to construct exact solutions of nonlinear evolution equations. Therefore it is an interesting work to find out the new type solution to a nonlinear evolution equations. In this paper, we consider the following (3+1)-dimensional Kadomtsev–Petviashvili equation,
uxt þ 6u2x þ 6uuxx uxxxx uyy uzz ¼ 0;
ð1:1Þ
which is introduced to describe the dynamics of solitons and nonlinear waves in the fields of plasma physics, fluid dynamics, etc. [26]. Eq. (1.1) has been widely studied by many authors and various different exact solutions have been presented [27– 33]. For instance, Wang and Lou have revealed some special type exact solutions in Ref. [27], Bai et al. constructed several new families of exact solutions after applying a generalized variable-coefficient algebraic method to Eq. (1.1) [28], Chen et al. obtained some exact solutions for Eq. (1.1) by using a new generalized transformation in homogeneous balance method [29]. EI-Sayed et al. studied Eq. (1.1) by considering the decomposition scheme and obtained some exact solutions [30]. Hu obtained some traveling wave solutions by applying algebraic method [31]. Liu and Liu derived the similarity solutions by using the compatibility method [32]. Zhao used the extended mapping method to derive a new family of the exact solutions of Eq. (1.1) [33]. The bilinear Bäcklund transformation and some new explicit solutions of Eq. (1.1) are also derived in our recent work [34]. This paper tries to extend the Hirota’s bilinear method and the Riemann theta function [35–39] to (3+1)-dimensional Kadomtsev–Petviashvili equation, the one- and two-periodic wave solutions for Eq. (1.1) are presented and the rigorous proofs are given that the soliton solutions can be obtained from the periodic wave solutions in an appropriate limiting procedure. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.030
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The Hirota bilinear operator is defined as [1,2] l m n r n r 0 0 0 0 0 Dlx Dm y Dz Dt aðx; y; z; tÞ bðx; y; z; tÞ ð@ x @ x0 Þ ð@ y @ y0 Þ ð@ z @ z0 Þ ð@ t @ t Þ aðx; y; z; tÞbðx ; y ; z ; t Þjx0 ¼x;y0 ¼y;z0 ¼z;t 0 ¼t ;
ð1:2Þ
where the l; m; n; r 2 N, to deal with bilinear equation, we use the property of the Hirota’s operator. From the definition, we have the relation
Dnx expðpxÞ expðp0 xÞ ¼ ðp p0 Þn exp½ðp þ p0 Þx;
ð1:3Þ
or more generally, let P(Dx, Dy, . . . , Dt) be a polynomial of Dx, Dy, . . . , Dt then 0
PðDx ; Dy ; . . . ; Dt Þ expðpx þ ly þ þ xtÞ expðp0 x þ l y þ þ x0 tÞ 0
0
0
0
0
¼ Pðp p ; l l ; . . . ; x x Þ exp½ðp þ p Þx þ ðl þ l Þy þ þ ðx þ x0 Þt:
ð1:4Þ
With the dependent variable transformation
uðx; y; z; tÞ ¼ 2@ 2 ln f ðx; y; z; tÞ=@x2 ;
ð1:5Þ
the bilinear form of the (3+1)-dimensional Kadomtsev–Petviashvili equation appropriate to the periodic problem is written as
FðDx ; Dy ; Dz ; Dt Þf f ðDx Dt D4x D2y D2z þ cÞf f ¼ 0;
ð1:6Þ
where c is an integration constant generally dependent on time. 2. One-periodic wave solution and asymptotic behavior 2.1. One-periodic wave solution First we consider the case of one-periodic wave solution of Eq. (1.1). We take the one-dimensional Riemann theta function in the form [40] 1 X
f ¼ #ðg; sÞ ¼
2
expð2ping þ pin sÞ;
i¼
pffiffiffiffiffiffiffi 1;
ð2:1Þ
n¼1
with g = px + ly + kz + xt + g0, where s is a complex constant satisfying the condition Ims > 0 and p, l, k and x represent the wave number and frequency, respectively, and g0 is a phase constant. Substituting (2.1) into (1.6) and using the property of Eq. (1.4), we have
Ff f ¼
1 X
2
0
02
FðDx ; Dy ; Dz ; Dt Þ expð2ping þ pin sÞ expð2pin g þ pin
sÞ
n;n0 ¼1
¼
1 X
2
FðDx ; Dy ; Dz ; Dt Þ expð2ping þ pin sÞ expð2piðm nÞg þ piðm nÞ2 sÞ
n;m¼1
¼
1 X
F½2pið2n mÞp; . . . ; 2pið2n mÞx expf2pimg þ pi½n2 þ ðn mÞ2 sg ¼
n;m¼1
1 X
e F ðmÞ expð2pimgÞ;
ð2:2Þ
m¼1
0 where the new summation index m = n + n has been introduced and e F ðmÞ is defined by
e F ðmÞ ¼
1 X
F½2pið2n mÞp; 2pið2n mÞl; 2pið2n mÞk; 2pið2n mÞx expfpi½n2 þ ðn mÞ2 sg:
ð2:3Þ
n¼1 0
In Eq. (2.3), by shifting summation index n by unity as n = n + 1, we have the relation
e F ðmÞ ¼
1 X
F½2pið2n0 ðm 2ÞÞp; . . . ; 2pið2n0 ðm 2ÞÞx expfpi½n02 þ ðm 2 n0 Þ2 sg expf2piðm 1Þsg
n0 ¼1
¼e F ðm 2Þ expf2piðm 1Þsg:
ð2:4Þ
From relation (2.4) we can conclude that, if e F ð0Þ and e F ð1Þ are zero, then all the other e F ðmÞ are also zero. On the other hand we recall that we have at least two unknown parameters, namely, arbitrary integration constant c and the nonlinear frequency x even if all the other parameters such as wavevectors p, l, k and amplitude parameter s are taken as arbitrarily given paramF ð0Þ ¼ 0 and e F ð1Þ ¼ 0, eters. Therefore, by solving integration constant c and nonlinear dispersion relation x from the Eqs. e we have exact periodic wave solution to the bilinear equation (1.6).
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From Eqs. (1.6) and (2.3), the Eqs. e F ð0Þ ¼ 0 and e F ð1Þ ¼ 0 are explicitly written as 1 X n¼1 1 X
2
2
2
½16p2 n2 px 256p4 n4 p4 þ 16p2 n2 l þ 16p2 n2 k þ c expð2pin sÞ ¼ 0; 2
ð2:5Þ 2
½4p2 ð2n 1Þ2 px 16p4 ð2n 1Þ4 p4 þ 4p2 ð2n 1Þ2 l þ 4p2 ð2n 1Þ2 k þ c expfpi½n2 þ ðn 1Þ2 sg ¼ 0; ð2:6Þ
n¼1
respectively. Introducing the quantities
A0 ðsÞ ¼ A1 ðsÞ ¼ A2 ðsÞ ¼ B0 ðsÞ ¼ B1 ðsÞ ¼ B2 ðsÞ ¼
1 X
2
expð2pin sÞ;
n¼1 1 X n¼1 1 X n¼1 1 X n¼1 1 X n¼1 1 X
ð2:7Þ 2
ð2:8Þ
2
ð2:9Þ
ð4nÞ2 expð2pin sÞ; ð4nÞ4 expð2pin sÞ; expfpi½n2 þ ðn 1Þ2 sg;
ð2:10Þ
ð4n 2Þ2 expfpi½n2 þ ðn 1Þ2 sg;
ð2:11Þ
ð4n 2Þ4 expfpi½n2 þ ðn 1Þ2 sg:
ð2:12Þ
n¼1
Eqs. (2.5) and (2.6) can be written compactly as 2
2
ð2:13Þ
2
2
ð2:14Þ
A1 ðsÞp2 px A2 ðsÞðppÞ4 þ A1 ðsÞp2 ðl þ k Þ þ cA0 ðsÞ ¼ 0; B1 ðsÞp2 px B2 ðsÞðppÞ4 þ B1 ðsÞp2 ðl þ k Þ þ cB0 ðsÞ ¼ 0: From these two equations, quantities x and c are determined to be 2
2
l þk A0 ðsÞB2 ðsÞ B0 ðsÞA2 ðsÞ 2 3 pp; A0 ðsÞB1 ðsÞ B0 ðsÞA1 ðsÞ p A2 ðsÞB1 ðsÞ B2 ðsÞA1 ðsÞ c¼ ðppÞ4 : A0 ðsÞB1 ðsÞ B0 ðsÞA1 ðsÞ
x¼
ð2:15Þ ð2:16Þ
Therefore, expression (1.5) is the exact one-periodic wave solution of the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) with f given by Eq. (2.1) and x by Eq. (2.15). 2.2. Asymptotic behavior of the one-periodic wave solution It is interesting that the one-soliton solution can be obtained from the one-periodic wave solution in an appropriate limiting procedure. To illustrate this we introduce a quantity
q ¼ expðpisÞ;
ð2:17Þ
and take a limit q ? 0 (or Ims ? 1). Theorem 1. In the condition q ? 0 (or Ims ? 1), the one-periodic solution (2.1) of Eq. (1.1) tends to one-soliton solution via Eq. (1.5)
! ~ þx ~t þg ~2 ~x þ ~ly þ kz ~0 p 2 p ; uðx; y; z; tÞ ¼ sech 2 2
ð2:18Þ
~ ¼ ~l2 þp~k~2 þ p ~3 holds. where the relation x Proof. By using q ¼ expðpisÞ, the quantifies defined in Eqs. (2.7)–(2.12) are then expanded in powers of q as
A0 ¼ 1 þ 2q2 þ 2q8 þ ; 2
8
ð2:19Þ
A1 ¼ 32q þ 128q þ ;
ð2:20Þ
A2 ¼ 512q2 þ 8192q8 þ ;
ð2:21Þ
B0 ¼ 2q þ 2q5 þ 2q13 þ ;
ð2:22Þ
B1 ¼ 8q þ 72q5 þ 200q13 þ ;
ð2:23Þ
B2 ¼ 32q þ 2592q5 þ 20000q13 þ :
ð2:24Þ
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Substituting these expressions into Eqs. (2.15) and (2.16) yields 2
2
2
2
l þk 1 30q2 þ 81q4 þ 2 3 l þk pp ! 4 4p2 p3 ; 1 6q2 þ 9q4 þ p p 1 15q4 þ 20q6 þ c ¼ 384q2 ðppÞ4 ! 0; q ! 0: 1 6q2 þ 9q4 þ
x¼
q ! 0;
ð2:25Þ ð2:26Þ
By introducing the quantities
~ ¼ 2pip; p
~l ¼ 2pil;
~ ¼ 2pik; k
~ ¼ 2pix; g ~ 0 ¼ 2pi g0 þ x
s
;
2
~ þx ~t þg ~0 ; g~ ¼ p~x þ ~ly þ kz
ð2:27Þ
the one-periodic wave solution (2.1) reduces in the limit of q ? 0 (or Ims ? 1) to
f ¼ 1 þ expð2pig þ pisÞ þ expð2pig þ pisÞ þ expð4pig þ 2pisÞ þ expð4pig þ 2pisÞ þ ~ þx ~t þg ~x þ ~ly þ kz ~ 0 Þ; ¼ 1 þ eg~ þ q2 ðeg~ þ e2g~ Þ þ q6 ðe2g~ þ e3g~ Þ þ ! 1 þ eg~ ¼ 1 þ expðp
ð2:28Þ
which is transformed into the one-soliton solution to the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) with Eq. (1.5) as
! ~ þx ~t þg ~2 ~x þ ~ly þ kz ~0 p 2 p ; uðx; y; z; tÞ ¼ sech 2 2 ~2 þ~ k2 ~ p
~ ¼l where the relation x
ð2:29Þ
~3 has been used, which is a direct result of (2.25) and (2.27). Thus the proof is completed. þp
h
3. Two-periodic wave solution and asymptotic behavior In what follows, we consider the two-periodic wave solution to the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1), which is a two-dimensional generalization of one-periodic wave solution. 3.1. Construction of two-periodic wave solution Without loss of generality, we consider multidimensional Riemann theta function [40] 1 X
f ¼ #ðg1 ; . . . ; gN ; sÞ ¼
exp 2pi
N X
n1 ;...;nN ¼1
nj gj þ pi
j¼1
N X
!
pffiffiffiffiffiffiffi
sjk nj nk ; i ¼ 1;
ð3:1Þ
j;k¼1
with
gj ¼ pj x þ lj y þ kj z þ xj t þ g0j ðj ¼ 1; . . . ; NÞ;
ð3:2Þ
where pj, lj, kj, xj and g0j have the same meaning as in the one-periodic wave case. The term sjk (j – k) represents the effect of interaction between periodic waves and is assumed to satisfy the conditions sjk = skj, Imsjk > 0, (j, k = 1, . . . , N) and the complex matrix (sjk)N N is of positive definite imaginary part. Substituting (3.1) into (1.6) and using the property of Eq. (1.4), we have the analogous result to Eq. (2.2).
Ff f ¼
1 X
e F ðm1 ; . . . ; mN Þ exp 2pi
m1 ;...;mN ¼1
N X
! mj gj ;
ð3:3Þ
j¼1
where
"
1 X
e F ðm1 ; . . . ; mN Þ ¼
F 2p i
n1 ;...;nN ¼1
(
exp
pi
N X j¼1
N X
ð2nj mj Þpj ; 2pi
N N N X X X ð2nj mj Þlj ; 2pi ð2nj mj Þkj ; 2pi ð2nj mj Þxj j¼1
j¼1
)
½nj sjk nk þ ðnj mj Þsjk ðnk mk Þ :
#
j¼1
ð3:4Þ
j;k¼1
In Eq. (3.4), by shifting the hth summation index nh by unity, we obtain the relation corresponding to (2.4) (h = 1, . . . , N).
( ) N X e e F ðm1 ; . . . ; mN Þ ¼ F ðm1 ; . . . ; mh1 ; mh 2; mhþ1 ; . . . ; mN Þ exp 2pi shj mj 2shh :
ð3:5Þ
j¼1
If relations
e F ðm1 ; . . . ; mN Þ ¼ 0;
ð3:6Þ
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Y.-Q. Wu / Applied Mathematics and Computation 216 (2010) 3154–3161
hold for all combinations of m1 = 0, 1, m2 = 0, 1, . . . , mN = 0, 1 then all e F 0s become zero and expression (3.1) gives the N-periodic wave solution to the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1). Notice the number of equations of the type (3.6) be 2N and the total number of unknown parameters as integration constant c, nonlinear frequency x1, . . . , xN and interaction term sjk(1 6 j, k 6 N, j – k) 1 þ N þ C 2N ¼ 12 ðN 2 þ N þ 2Þ. For N = 1, 2, the number of equations corresponds to that of unknowns, implying the existence of the one- and two-periodic wave solutions. In the case N = 1, we have just discussed. For N = 2,we have the conclusion from Eqs. (3.4) and (1.6)
"
1 X
e F ðm1 ; m2 Þ ¼
F 2pi
n1 ;n2 ¼1
2 2 2 2 X X X X ð2nj mj Þpj ; 2pi ð2nj mj Þlj ; 2pi ð2nj mj Þkj ; 2pi ð2nj mj Þxj j¼1
expfpi
2 X
j¼1
j¼1
#
j¼1
½nj sjk nk þ ðnj mj Þsjk ðnk mk Þg
j;k¼1 1 X
¼
4p2 ½ð2n1 m1 Þp1 þ ð2n2 m2 Þp2 ½ð2n1 m1 Þx1 þ ð2n2 m2 Þx2 16p4 ½ð2n1 m1 Þp1
n1 ;n2 ¼1
þ 2n2 m2 Þp2 4 þ 4p2 ½ð2n1 m1 Þl1 þ ð2n2 m2 Þl2 2 þ 4p2 ½ð2n1 m1 Þk1 o n2 þðn m Þ2 n2 þðn m Þ2 n n þðn m Þðn m Þ þ 2n2 m2 Þk2 2 þ c k11 1 1 k22 2 2 k31 2 1 1 2 2 ¼ 0;
ð3:7Þ
k2 ¼ epis22 ;
ð3:8Þ
where
k1 ¼ epis11 ;
k3 ¼ e2pis12 ;
specially, 1 X
e F ð0; 0Þ ¼
n 4p2 ½ð2n1 Þp1 þ ð2n2 Þp2 ½ð2n1 Þx1 þ ð2n2 Þx2 16p4 ½ð2n1 Þp1 þ ð2n2 Þp2 4 þ 4p2 ½ð2n1 Þl1 þ ð2n2 Þl2 2
n1 ;n2 ¼1
o 2n2 2n2 1 n2 ¼ 0; þ 4p2 ½ð2n1 Þk1 þ ð2n2 Þk2 2 þ c k1 1 k2 2 k2n 3 1 X
e F ð1; 0Þ ¼
n
ð3:9Þ
4p2 ½ð2n1 1Þp1 þ ð2n2 Þp2 ½ð2n1 1Þx1 þ ð2n2 Þx2 16p4 ½ð2n1 1Þp1 þ ð2n2 Þp2 4
n1 ;n2 ¼1
o n2 þðn 1Þ2 2n2 n n þðn 1Þn2 þ 4p2 ½ð2n1 1Þl1 þ ð2n2 Þl2 2 þ 4p2 ½ð2n1 1Þk1 þ ð2n2 Þk2 2 þ c k11 1 k2 2 k31 2 1 ¼ 0; 1 X
e F ð0; 1Þ ¼
n
ð3:10Þ
4p2 ½ð2n1 Þp1 þ ð2n2 1Þp2 ½ð2n1 Þx1 þ ð2n2 1Þx2 16p4 ½ð2n1 Þp1 þ ð2n2 1Þp2 4
n1 ;n2 ¼1
o 2n2 n2 þðn 1Þ2 n n þn ðn 1Þ þ 4p2 ½ð2n1 Þl1 þ ð2n2 1Þl2 2 þ 4p2 ½ð2n1 Þk1 þ ð2n2 1Þk2 2 þ c k1 1 k22 2 k31 2 1 2 ¼ 0; 1 X
e F ð1; 1Þ ¼
ð3:11Þ
4p2 ½ð2n1 1Þp1 þ ð2n2 1Þp2 ½ð2n1 1Þx1 þ ð2n2 1Þx2 16p4 ½ð2n1 1Þp1
n1 ;n2 ¼1
þ 2n2 1Þp2 4 þ 4p2 ½ð2n1 1Þl1 þ ð2n2 1Þl2 2 þ 4p2 ½ð2n1 1Þk1 o n2 þðn 1Þ2 n2 þðn 1Þ2 n n þðn 1Þðn2 1Þ þ 2n2 1Þk2 2 þ c k11 1 k22 2 k31 2 1 ¼ 0;
ð3:12Þ
so, from the relations
e F ð0; 0Þ ¼ 0;
e F ð0; 1Þ ¼ 0;
e F ð1; 0Þ ¼ 0;
e F ð1; 1Þ ¼ 0;
ð3:13Þ
we can solve the integration constant c, two kinds of nonlinear dispersion relations and the interaction term s12. Thus expression (1.5) is the exact two-periodic wave solution of the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) with f given by Eq. (3.1) (N = 2). 3.2. Asymptotic behavior of the two-periodic wave solution It is noticed that the two-soliton solution can also be obtained from the two-periodic wave solution in an appropriate limiting procedure. Theorem 2. Suppose that 1 < r1 < 2, 1 < r2 < 2 are two constants, satisfying jk1 jr1 ! 0 and jk2 jr2 ! 0, then the periodic solution (3.1) of Eq. (1.1) tends to soliton solution via Eq. (1.5)
Y.-Q. Wu / Applied Mathematics and Computation 216 (2010) 3154–3161
3159
uðx; y; z; tÞ ¼ 2@ 2x ln f ~22 exp g ~1 þ p ~2 Þ2 expðg ~21 exp g ~1 þ p ~ 2 þ ðp ~1 þ g ~ 2 þ AÞ p ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g ~1 þ p ~2 exp g ~ 2 þ ðp ~1 þ p ~2 Þ expðg ~1 þ g ~ 2 þ AÞ 2 ~1 exp g p þ2 ; ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g
¼ 2
ð3:14Þ
where the relations
~1 ¼ x
~2 ~l2 þ k 1 1 ~31 ; þp ~1 p
~2 ¼ x
~2 ~l2 þ k 2 2 ~32 ; þp ~2 p
ð3:15Þ
and
eA ¼
~1 k ~ 2 Þ2 ~1 k ~2 ; x ~1 x ~ 2 Þ ðp ~1 x ~ 2Þ ~1 p ~2 Þðx ~1 p ~2 Þ4 ð~l1 ~l2 Þ2 ðk ~1 p ~2 ; ~l1 ~l2 ; k ðp Fðp ; ¼ 4 2 2 ~ ~ ~ ~ ~ ~ ~ ~ ~1 þx ~ 2 Þ ðp ~1 þx ~ 2Þ ~1 þ p ~1 þ p ~2 Þðx ~1 þ p ~2 Þ ðl1 þ l2 Þ ðk1 þ k2 Þ ~2 ; l1 þ l2 ; k1 þ k2 ; x Fðp ðp
ð3:16Þ
hold. Proof. We expand the two-periodic wave solution (3.1) (N = 2) in the following form by using the quantities
~ ¼ 2pik ; x ~ j ¼ 2pixj ~j ¼ 2pipj ; ~lj ¼ 2pilj ; k p j j ~ ~ ~ ~ ~ ~ gj ¼ pj x þ lj y þ kj z þ xj t þ g0j ; j ¼ 1; 2;
g~ 0j ¼ 2pig0j þ pisjj ; j ¼ 1; 2;
ð3:17Þ ð3:18Þ
f ¼ #ðg1 ; g2 ; sÞ ¼ 1 þ expð2pig1 þ pis11 Þ þ expð2pig1 þ pis11 Þ þ expð2pig2 þ pis22 Þ þ expð2pig2 þ pis22 Þ þ expð2piðg1 þ g2 Þ þ piðs11 þ 2s12 þ s22 ÞÞ þ expð2piðg1 þ g2 Þ þ piðs11 þ 2s12 þ s22 ÞÞ þ ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ 2pis12 Þ þ k21 expðg ~ 1 Þ þ k22 expðg ~ 2 Þ þ k21 k22 expðg ~1 g ~ 2 þ 2pis12 Þ þ ¼ 1 þ exp g r1 r2 ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ; ~ 1 þ exp g jk1 j ! 0; jk2 j ! 0 ; ð3:19Þ ! 1 þ exp g where
A ¼ 2pis12 :
ð3:20Þ
In the following, we verify the formulae (3.15) and (3.16) hold, to this end, we expand each function in Eqs. (3.9)–(3.12) into a series with k1 and k2. It is slightly tedious, but this process is easily carried out by using symbolic computation software MATHEMATICA or MAPLE. Actually, we only need to make the first order expansions with k1 and k2 to show the asymptotic relations (3.15) and (3.16). Here we consider their second order expansion to see deeper relations among parameters for the two-periodic solution and two-soliton solution. From
h i 2 2 e F ð0; 0Þ ¼ c þ 2 16p2 p1 x1 16p4 ð2p1 Þ4 þ 16p2 l1 þ 16p2 k1 þ c k21 h i 2 2 ¼ þ2 16p2 p2 x2 16p4 ð2p2 Þ4 þ 16p2 l2 þ 16p2 k2 þ c k22 þ o ks11 ks22 ¼ 0;
ð3:21Þ
where s1 + s2 P 4, we obtain that
c ¼ 0:
ð3:22Þ
h i 2 2 e F ð1; 0Þ ¼ 2 4p2 p1 x1 16p4 p41 þ 4p2 l1 þ 4p2 k1 þ c k1 þ o ks11 ks22 ¼ 0;
ð3:23Þ
From
where s1 + s2 P 3, and using c = 0, we obtain the nonlinear dispersion relation
~1 ¼ x
~l2 þ k ~2 1 1 ~31 : þp ~ p1
ð3:24Þ
From
h i 2 2 e F ð0; 1Þ ¼ 2 4p2 p2 x2 16p4 p42 þ 4p2 l2 þ 4p2 k2 þ c k2 þ o ks11 ks22 ¼ 0;
ð3:25Þ
where s1 + s2 P 3, and using c = 0, we obtain another nonlinear dispersion relation
~2 ¼ x
~2 ~l2 þ k 2 2 ~32 : þp ~2 p
ð3:26Þ
3160
Y.-Q. Wu / Applied Mathematics and Computation 216 (2010) 3154–3161
From
nh i e F ð1; 1Þ ¼ 2 4p2 ðp1 þ p2 Þðx1 þ x2 Þ 16p4 ðp1 þ p2 Þ4 þ 4p2 ðl1 þ l2 Þ2 þ 4p2 ðk1 þ k2 Þ2 þ c k3 h io þ 4p2 ðp1 p2 Þðx1 x2 Þ 16p4 ðp1 p2 Þ4 þ 4p2 ðl1 l2 Þ2 þ 4p2 ðk1 k2 Þ2 þ c k1 k2 þ o ks11 ks22 ¼ 0; ð3:27Þ r1
r2
where s1 + s2 P 6, we know that the coefficient of k1k2 must be zero in the conditions jk1 j ! 0; jk2 j ! 0. By using c = 0, we obtain that
k3 ¼
4p2 ðp1 p2 Þðx1 x2 Þ 16p4 ðp1 p2 Þ4 þ 4p2 ðl1 l2 Þ2 þ 4p2 ðk1 k2 Þ2
4p2 ðp1 þ p2 Þðx1 þ x2 Þ 16p4 ðp1 þ p2 Þ4 þ 4p2 ðl1 þ l2 Þ2 þ 4p2 ðk1 þ k2 Þ2 ~1 k ~2 Þ2 ~1 x ~ 2 Þ ðp ~1 p ~2 Þðx ~1 p ~2 Þ4 ð~l1 ~l2 Þ2 ðk ðp ¼ : 4 2 ~1 þ k ~2 Þ2 ~1 þx ~ 2 Þ ðp ~1 þ p ~2 Þðx ~1 þ p ~2 Þ ð~l1 þ ~l2 Þ ðk ðp
ð3:28Þ
Thus, we have
eA ¼ k3 ¼
~1 k ~2 Þ2 ~1 k ~2 ; x ~1 x ~ 2 Þ ðp ~1 x ~ 2Þ ~1 p ~2 Þðx ~1 p ~2 Þ4 ð~l1 ~l2 Þ2 ðk ~1 p ~2 ; ~l1 ~l2 ; k ðp Fðp ¼ : 4 2 2 ~ ~ ~ ~ ~ ~ ~ ~ ~1 þx ~ 2 Þ ðp ~1 þx ~ 2Þ ~1 þ p ~1 þ p ~2 Þðx ~1 þ p ~2 Þ ðl1 þ l2 Þ ðk1 þ k2 Þ ~2 ; l1 þ l2 ; k1 þ k2 ; x Fðp ðp
ð3:29Þ
Finally, by using (1.5) and (3.19), the two-soliton solution of the (3+1)-dimensional Kadomtsev–Petviashvili equation (1.1) is obtained
~22 exp g ~1 þ p ~2 Þ2 expðg ~21 exp g ~1 þ p ~ 2 þ ðp ~1 þ g ~ 2 þ AÞ p ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g ~1 þ p ~2 exp g ~ 2 þ ðp ~1 þ p ~2 Þ expðg ~1 þ g ~ 2 þ AÞ 2 ~1 exp g p þ 2½ ; ~ 1 þ exp g ~ 2 þ expðg ~1 þ g ~ 2 þ AÞ 1 þ exp g
uðx; y; z; tÞ ¼ 2
thus, the proof is completed.
ð3:30Þ
h
4. Conclusion In summary, we have obtained the one-periodic wave solution, two-periodic wave solution and soliton solutions for the (3+1)-dimensional Kadomtsev–Petviashvili equation by means of the Hirota bilinear method and the Riemann theta function, which belong to the cases when N = 1 and N = 2. This method is valid for quite a few nonlinear evolution equations. For general case when N > 2, the results can be extended, but there are still certain numerical difficulties in the calculation and other aspects, which will be considered in our future work. Acknowledgement Supported by the Science Research Foundation of Zhanjiang Normal University (L0803). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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