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Multiple periodic-soliton solutions to Kadomtsev–Petviashvili equation
Applied Mathematics and Computation 218 (2011) 368–375
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Multiple periodic-soliton solutions to Kadomtsev–Petviashvili equation Long Wei Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Zhejiang 310018, China
a r t i c l e
i n f o
Keywords: Extended homoclinic test technique Hirota’s bilinear method Periodic-soliton solutions Singular periodic-soliton solutions Kadomtsev–Petviashvili equation
a b s t r a c t Based on the extended homoclinic test technique, we introduce two new ansätz functions to construct multiple periodic-soliton solutions of Kadomtsev–Petviashvili (KP) equation by the Hirota’s bilinear method. Some entirely new periodic-soliton solutions are obtained. The obtained results show that there exist multiple-periodic solitary waves in the different directions for the KP equation, which differ from complexiton. The employed approach is powerful and can be also applied to solve other nonlinear differential equations. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In recent years, more and more attention has been paid to the study of (2 + 1)-dimensional integrable systems. Several celebrated examples of multi-dimensional integrable systems have been found in fields ranging from fluid dynamics, nonlinear optics, particle physics and general relativity to differential and algebraic geometry, topology and so on. The special significance of integrable systems is that they combine tractability with nonlinearity. Hence these systems enable one to explore nonlinear phenomena in multi-dimensions while working with explicit solutions. To seek more exact solutions of nonlinear problems, many powerful methods have been presented such as Bäcklund transformation [1], Darboux transformation [2], the extended Jacobian elliptic function expansion method [3], the tanh method [4–8], the sine–cosine method [9–11], the homogeneous balance method [12], the direct reduction method [13], transformed rational function method [14–16] and many other techniques. Recently, Dai et al. introduced a new technique, extended homoclinic test technique, which is used seeking periodic-soliton solutions of integrable equations and the authors obtained some new periodic-soliton solutions for some integrable equations [17–19]. This paper is concerned with the Kadomtsev–Petviashvili (KP) equation
ðut 6uux þ uxxx Þx þ 3uyy ¼ 0;
ð1Þ
which was introduced in 1970 [20] and is both physical and mathematical interest. It is the generalization of the well-known Korteweg–de-Vries (KdV) equation [20–25]
ut 6uux þ uxxx ¼ 0
ð2Þ
and, similar to Eq. (1), Eq. (1) is also completely integrable. Kadomtsev and Petviashvili [20] discovered Eq. (1) when they relaxed the restriction that the waves be strictly one-dimensional. The KP equation is used to model shallow-water waves with weakly nonlinear restoring forces. It is also used to model waves in ferromagnetic media. Note that the KP Eq. (1) is the generalization of the KdV Eq. (2) from (1 + 1) to (2 + 1) dimensions, we can reduce the obtained solutions of the KP equation to those of the KdV equation by setting the coefficients of y to be zero (see Section 3). This shows us a different approach to seek exact solutions of Eq. (2). E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.072
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
369
In this work, based on the idea of extended homoclinic test technique, we introduce two new ansätz functions to construct many new exact solutions of the KP equation (1) by the Hirota bilinear method [23], and some new periodic-soliton solutions are obtained. Our results show that there exist multiple-periodic solitary waves in the different directions for the KP equation. Here, we should mention a kind of special solution—complexiton, which is expressed by combinations of trigonometric and hyperbolic functions [26–28]. This kind of special solution is also periodic solitary wave solution. But our results suggest that there exist doubly periodic solitary wave solutions for some integrable problems, which differ from complexiton. The presented solutions maybe shed light on the richness of solutions to higher dimensional soliton equations [29] and are expected to understand complete integrability of nonlinear differential equations, though they are far from complete. 2. Exact periodic-soliton solutions for KP equation The KP equaiton (1) has the bilinear form
u ¼ 2ðln f Þxx ;
ð3Þ
ðDt Dx þ D4x þ 3D2y Þf f ¼ 0;
ð4Þ
where the bilinear operator D is defined as
@ @ @x @x0
n Dm x Dt f g ¼
m
@ @ @t @t 0
n
f ðx; y; tÞ gðx0 ; y0 ; t 0 Þ
ð5Þ
: ðx;y;tÞ¼ðx0 ;y0 ;t 0 Þ
Eq. (4) can be rewritten as
fftx ft fx þ ffxxxx 4f x fxxx þ 3fxx2 þ 3ffyy 3fy2 ¼ 0:
ð6Þ
In what follows, we seek periodic-soliton solutions for KP equation by new ansatzs. 2.1. Single-periodic soliton solutions According to the extended homoclinic test technique, we seek the solution of the form n
f ¼ aen þ be
þ c sin g þ d cos g;
ð7Þ
where n = kx + ly + at, g = Kx + Ly + bt, and a, b, c, d, k, l, K, L, a, b are all parameters to be determined later. Substituting Eq. (7) into Eq. (4) and equating corresponding coefficients of ein(i = 1, 0, 1), cosg and sing to zero yields an algebraic system of a, b, c, d, k, l, K, L, a and b as follows: 2
4
2
2
2
12abl þ 16ak b þ 4abka c2 bK d Kb 3c2 L2 þ 4c2 K 4 3d L2 þ 4d K 4 ¼ 0; 3
2
2
4
3acL2 4dK ak 6ak cK 2 þ 3cal þ 6aldL þ dKaa acbK þ ak c þ caka 3
þ akdb þ cK 4 a þ 4ak dK ¼ 0; 2
2
2
2
3dal 6alcL 6ak dK 3adL þ 4cK 3 ak þ daka akcb adbK cKaa 4
4
3
þ ak d þ dK a 4ak cK ¼ 0; 2
2
2
2
4
4
3bdL 6bk dK þ 3dbl þ bk d þ dK b bdbetaK þ dbka þ bkcb þ cKba 3
þ 4bk cK 4cK 3 bk þ 6blcL ¼ 0; 2
2
3
2
3
4
3cbl 6bk cK 2 þ 4dK bk 3bcL 4bk dK dKba þ bk c þ cK 4 b bcbK þ cbka bkdb 6bldL ¼ 0: Solving this set of algebraic equations with the aid of Maple, we obtain the following sets of solutions. Case 1.1: 2
a¼
L2 ðc2 þ d Þ 4
2
4bðk þ L Þ
2
;
K ¼ 0;
a¼
4
3ðL2 l Þ k ; k
b¼
6lL ; k
ð8Þ
where b, c, d, k, l and L are free parameters. Substituting Eq. (8) into Eq. (7), we get that 2
f ¼
L2 ðc2 þ d Þ 4
4bðk þ L2 Þ
n
en þ be
þ c sin g þ d cos g;
ð9Þ
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L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
where 4
n ¼ kx þ ly
2
k þ 3l 3L2 t; k
g ¼ Ly
6lL t: k
Next, from (9) and (3), it follows that there exists periodic soliton solution of Eq. (1). L2 ðc2 þd2 Þ When 4bðk ¼ b, from Eqs. (9) and (3), we obtain periodic soliton solutions of Eq. (1) 4 þL2 Þ 2
4bk ½ðd cos g c sin gÞ cosh n þ 2b
u1;2 ¼
ð2b cosh n þ d cos g c sin gÞ2
ð10Þ
;
where 3
n ¼ kx þ ly k þ
! 2 2 3 3l 12b k t; 2 2 k c2 þ d 4b
2kbðky 6ltÞ
ffi: g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ð11Þ
c2 þ d 4b
The above solutions are periodic in the y-direction, meanwhile they are bell-like wave in x-y-direction. Case 1.2: 2
2
ðl K 4 Þðc2 þ d Þ
a¼
2
4bl
a¼
k ¼ 0;
;
2
6lL ; K
b¼
3L2 þ K 4 þ 3l ; K
ð12Þ
where b, c, d, k, l, K and L are free parameters. Substituting Eq. (12) into Eq. (7), we have 2
f ¼
2
ðK 4 l Þðc2 þ d Þ 2
4bl
n
en þ be
þ c sin g þ d cos g;
ð13Þ
where
n ¼ ly when
4
2
2
6lL t; K
Þðc2 þd2 Þ
ðK l 4bl2
K 4 þ 3l 3L2 t; K
g ¼ Kx þ Ly þ
¼ b, from Eqs. (13) and (3), we obtain periodic soliton solutions of Eq. (1) 2
2
u3;4 ¼
ð14Þ
2K ½2bðc sin g þ d cos gÞ sinh n þ c2 þ d ð2b sinh n þ c sin g þ d cos gÞ
ð15Þ
2
and 2
u5;6 ¼
2K 2 ½2bðc sin g d cos gÞ sinh n þ c2 þ d ð2b sinh n þ c sin g d cos gÞ2
where n, g are defined by (14) and K ¼ direction. Case 1.3:
a ¼ ðc
2 þd2 Þ½ðKlkLÞ2 K 2 ðk2 þK 2 Þ2
4b½ðKlkLÞ2 þk2 ðk2 þK 2 Þ2 5
2
3 2
K a ¼ 3L k6lLKk kþ2k 2 þK 2
b¼
l ðc2 þd þ4b Þ . c2 þd2
4
ð16Þ
;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2
These solutions are periodic in x-y-direction and solitary wave in y-
;
2
3kl þ3kK 4
ð17Þ
;
4 2 3 2 KK 5 þ3KL2 3k Kþ2k K þ6klL3l k2 þK 2
;
where b, c, d, k, l, K and L are free parameters. Substituting Eq. (17) into Eq. (7), we obtain that 2
f ¼
2
ðc2 þ d Þ½ðKl kLÞ2 K 2 ðk þ K 2 Þ2 2
2
4b½ðKl kLÞ2 þ k ðk þ K 2 Þ2
n
en þ be
þ c sin g þ d cos g;
ð18Þ
where
n ¼ kx þ ly þ 3kL
2
6lLKk5 þ2k3 K 2 3kl2 þ3kK 4 k2 þK 2 2
5
2 3
K g ¼ Kx þ Ly 3KL þ6klLK kþ2k 2 þK 2
If we take
ðc2 þd2 Þ½ðKlkLÞ2 K 2 ðk2 þK 2 Þ2 4b½ðKlkLÞ2 þk2 ðk2 þK 2 Þ2
2
u7;8 ¼
2
2
t;
2
3l Kþ3Kk4
ð19Þ
t:
¼ b, from Eqs. (18) and (3), we obtain periodic-soliton solutions of Eq. (1)
h i 2 2R ðK 2 k Þðc sin g þ d cos gÞ cosh n þ 2kKðc cos g d sin gÞ sinhðnÞ
2 2
2ðc þ d ÞK k R
ðR cosh n þ c sin g þ d cos gÞ2
ðR cosh n þ c sin g þ d cos gÞ2
;
ð20Þ
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L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
where n, g are defined by (19) and R ¼ Case 1.4: 2
a¼
3
l ¼ kLðk KKþK Þ ;
b ¼ 0; 3
2
2
b ¼ 4K
2
4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðc2 þd2 Þ½ðKlkLÞ2 K 2 ðk2 þK 2 Þ2 . ðKlkLÞ2 þk2 ðk2 þK 2 Þ2
3L2 K
;
ð21Þ
2
þK Þ 4k K þ3kL K6KLðk : 2
Substituting Eq. (21) into Eq. (7), we get
f ¼ aen þ c sin g þ d cos g;
ð22Þ
where 2
3
n ¼ kx þ kLðk KKþK Þ y 4k 4
3 2
K þ3kL2 6KLðk2 þK 2 Þ t; K2
ð23Þ
2
g ¼ Kx þ Ly þ 4K K3L t; where a, c, d, k, K and L are free parameters. Thus, we obtain periodic-soliton solutions of Eq. (1) 2
u9;10 ¼
2aen ½ðK 2 k Þðc sin g þ d cos gÞ þ 2kKðc cos g d sin gÞ ðaen þ c sin g þ d cos gÞ
2
2
þ
2K 2 ðc2 þ d Þ ðaen þ c sin g þ d cos gÞ2
;
ð24Þ
where n, g are defined by (23). Case 1.5:
b ¼ 0; b¼
2
2
2
L ¼ K½lðkk þK Þ ;
a ¼ 6k K
2
þ3K 4 k4 3l2 k
;
2 2 2 4 2 4 2 K½2k K þ3K þ3lk2þ3k 6lðk þK Þ :
ð25Þ
Substituting Eq. (25) into Eq. (7), we have
f ¼ aen þ c sin g þ d cos g;
ð26Þ
where
n ¼ kx þ ly þ 6k
g ¼ Kx þ
2 2
K þ3K 4 k4 3l2 k
K½lðk2 þK 2 Þ y k
t;
K½2k2 K 2 þ3K 4 þ3l2 þ3k4 6lðk2 þK 2 Þ t; k2
ð27Þ
where a, c, d, k, l and K are free parameters. Thus, we obtain periodic-soliton solutions u11,12 to Eq. (1) of the same form as u9,10, but now the n and g are defined by (27). We should note that u7 u12 are both periodic and solitary wave in x-y-direction. 2.2. Double-periodic soliton solutions Next, we seek solutions with double-periodic soliton structures. To the end, we set f of the form
f ¼ a0 þ en ða1 sin g þ b1 cos gÞ þ c1 e2n þ el ða2 sin q þ b2 cos qÞ þ c2 e2l ; where n = k1x + l1y + a1t, l = k2x + l2y + a2t, g = K1x + L1y + b1t, q = K2x + L2y + b2t, and ai, bi, ci, ki, li, Ki, Li, ai, bi (i = 1, 2) are all parameters to be determined later. Now, to roughly illustrate our idea and simply the computation, we assume that
a0 ¼ b1 ¼ c1 ¼ a2 ¼ c2 ¼ 0; that is, the function f is given by l
f ¼ aen sin g þ be cos q:
ð28Þ
We proceed as above, substituting Eq. (28) into Eq. (4) and equating corresponding coefficients of en, el, cosg, sing, cosq and sinq to zero, we obtain an algebraic system of a, b, ki, li, Ki, Li, ai and bi and solve the system with the aid of Maple, we get the following sets of solutions.
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L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
Case 2.1:
l1 ¼ K 22 K 21 ðk1 k2 Þ2 þ l2 þ L1 ðkK11k2 Þ ; h i L2 ¼ K 2 2ðk2 k1 Þ þ KL11 ; 2
2
2
1 k2 Þ a1 ¼ a2 þ 4ðk32 k31 Þ þ 6L1 ½K 1 K 2Kþðk þ 1
b1 ¼ b2 ¼
4K 41 3L21 K1
4K 32
3ðk1 k2 Þð4K 21 K 22 L21 þ4k1 k2 K 21 Þ K 21
;
ð29Þ
; 2
3K 2 ðL1 þ2KK1 2k2 2K 1 k1 Þ : 1
From Eqs. (29), (28) and (3), we obtain double-periodic soliton solutions of Eq. (1):
nþl ðk1 k2 Þ2 K 21 K 22 sin g cos q þ 2ðk1 k2 ÞK 2 sin g sin q u13 ¼ 2 abe
1 2 þ 2ðk1 k2 ÞK 1 cos g cos q þ 2K 1 K 2 cos g sin q a2 e2n K 21 b e2l K 22 : l ðaen sin g þ be cos qÞ2
ð30Þ
where
g ¼ K 1 x þ L1 y þ b1 t; l ¼ k2 x þ l2 y þ a2 t; q ¼ K 2 x þ L2 y þ b2 t
n ¼ k1 x þ l1 y þ a1 t;
with free parameters a, b, k1, k2, l2, K1, K2, L1, a2 and l1, L2, a1, b1, b2 are given by (29). Especially, taking 2
l1 ¼ k1 K 21 ;
a1 ¼ 12k1 K 21 4k31 ; L1 ¼ 2k1 K 1 ; b1 ¼ 12k21 K 1 þ 4K 31 ; l2 ¼ k22 K 22 ; a2 ¼ 12k2 K 22 4k32 ; 2
b2 ¼ 12k2 K 2 þ 4K 32 ;
L2 ¼ 2k2 K 2 ; we have
h i 2 2 2 2 2 k xþ k2 K 2 yþ4k2 ð3K 22 k22 Þt f ¼ aek1 xþðk1 K 1 Þyþ4k1 ð3K 1 k1 Þt sin K 1 x 2k1 y 4tK 21 þ 12tk1 þ be 2 ð 2 2 Þ h i 2 cos K 2 x 2k2 y 4tK 22 þ 12tk2 ;
ð31Þ
then substituting the above f into (3) we can obtain a double-periodic soliton solution to Eq. (1). Case 2.2:
l1 ¼ K 21 K 22 þ ðk1 k2 Þ2 þ l2 þ L1 ðkK11k2 Þ ; h i L2 ¼ K 2 2ðk1 k2 Þ þ KL11 ; 2
2
2
1 k2 Þ a1 ¼ a2 þ 4ðk32 k31 Þ þ 6L1 ½K 2 K 1Kðk þ 1
b1 ¼
4K 41 3L21 K1
3ðk1 k2 Þð4K 21 K 22 L21 þ4k1 k2 K 21 Þ K 21
;
ð32Þ
; 2
b2 ¼ 4K 32 3K 2 ðL1 2KK1 2k2 þ2K 1 k1 Þ : 1
Substituting (32) into Eq. (28) and from (3) we obtain another double-periodic soliton solution u14 of Eq. (1) with the form of (30), but l1, L2, a1, b1, b2 are given by (32) and k1, k2, l2, K1, K2, L1, a2 are free parameters. Case 2.3:
ðL1 4k2 K 1 ÞK 2 k1 ¼ k2 ; l1 ¼ l2 ; L2 ¼ ; K1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 K 1 4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 ; K2 ¼ K1 6 8k22 K 1 L1 þ 2l2 K 1 L1 k2 L21 2 2 a1 ¼ a2 8k2 6l2 þ 3K 1 þ 8k2 þ ; K 21 b1 ¼ b2 ¼
4K 41 3L21 ; K 1 2 K 2 16K 1 k2 L1 3L21 þ 8K 21 l2 þ 4K 41 32K 21 k2 K 21
ð33Þ :
373
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
Substituting Eq. (33) into Eq. (28), we get
f ¼ a1 en sin g þ b2 el cos q ¼ a1 sin K 1 x þ L1 y
4K 41
þ
3L21
! t
k2 xl2 yþ a2 8k2 ð6l2 þ3K 21 þ8k32 Þþ
e
ð
6 8k2 K 1 L1 þ2l2 K 1 L1 k2 L2 2 1 K2 1
Þ
t
K1 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !3 2 2 K 1 ð4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 Þ L1 4k2 K 1 16K 1 k2 L1 3L21 þ 8K 21 l2 þ 4K 41 32K 21 k2 5 4 þ b2 cos xþ yþ t K1 K1 K 21 ek2 xþl2 yþa2 t ;
ð34Þ
where k1, k2, l2, K1, L1, and a2 are free parameters. Thus, we obtain double-periodic soliton solution of Eq. (1):
n h 2 2 u15 ¼ 2 a21 K 31 e2n þ b2 4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 e2l þ 2a1 b2 2k2 K 21 cos g cos q þ K 31 k2 L1 þ l2 K 1 sin g cos q o.n
2 o K 1 a1 en sin g þ b2 el cos q ; ð35Þ þ 2k2 P sin g sin q K 1 P cos g sin qeða1 þa2 Þt where P ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 K 1 4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 ; n; g; l; q are defined by 4
2
g ¼ K 1 x þ L1 y 4KK1 1þ3L1 t;
n ¼ k2 x l2 y þ a1 t;
ð36Þ
l ¼ k2 x þ l2 y þ a2 t; q ¼ K 2 x þ L2 y þ b2 t and K2, L2, a1, b2 are given by (33). Case 2.4: 2
K2
l1 ¼ l2 ¼ ð4k2 þ 42 þ kK2 L22 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðK 22 8k22 Þ ; L1 ¼ ðL2 þ4kK22K 2 ÞK 1 ; K 1 ¼ 2
k1 ¼ k2 ;
a1 ¼ a2 12k2 K 22 þ 128k32 þ 3K 2 L2 þ 6k2 L2 ð8kK 22 K 2 þL2 Þ ;
ð37Þ
2
b1 ¼ b2 ¼
K 1 ð2K 42 þ64k22 K 22 þ24k2 K 2 L2 þ3L22 Þ K 22
4K 42 3L22 K2
;
:
Substituting Eq. (37) into Eq. (28), then from (3) we obtain double-periodic soliton solutions of Eq. (1):
u16;17 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a1 b2 16k2 3K 22 sin g cos q 4k2 2K 22 16k2 cos g cos q þ 2K 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2K 22 16k2 cos g sin q 8k2 K 2 sin g sin q eða1 þa2 Þt
. n
2 2 2 a1 e sin g b2 el cos q ; þ a21 K 22 8k2 e2n þ 2b2 K 22 e2l
ð38Þ
where n, g, l and q are defined as follows:
h i K2 2 3 n ¼ k2 x 4k2 þ 42 þ kK2 L22 y þ a2 12k2 K 22 þ 128k2 þ 3K 2 L2 þ 6k2 L2 ð8kK 22 K 2 þL2 Þ t; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 2 2ðK 22 8k22 Þ 64k K 2K þ24k K L þ3L 2K2 g¼ x þ L2 þ4k y 2 2 2 K2 2 2 2 2 t ; 2 K2 2
l ¼ k2 x þ
2 ð4k2
þ
K 22 4 4
þ
k2 L2 Þy K2
þ a2 t;
2
2 q ¼ K 2 x þ L2 y þ 4K 2K3L t; 2
and k2, K2, L2, a2 are free parameters.
ð39Þ
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L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
Case 2.5:
k1 ¼ k2 ; L2 ¼ K2 ¼
l1 ¼ ðL1 þ4k2 K 1 ÞK 2 ; K1
l2 ;
ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 ð4k22 K 1 þ2k2 L1 þK 31 2l2 K 1 Þ
; K1 6L 8k2 K 2l K þk L 2 a1 ¼ a2 þ 8k2 6l2 3K 1 8k22 1 ð 2 1 K 2 2 1 2 1 Þ ;
ð40Þ
1
b1 ¼
4K 41 3L21 K1
b2 ¼
;
K 2 ð16K 1 k2 L1 þ3L21 þ8K 21 l2 4K 41 þ32K 21 k22 Þ K 21
:
Substituting Eq. (40) into Eq. (28), we get
! 6L1 ð8k2 K 1 2l2 K 1 þk2 L1 Þ 2 4 2 k2 xl2 yþ a2 þ8k2 ð6l2 3K 21 8k22 Þ t 2 4K þ 3L K 1 1 1 f ¼ a1 en sin g þ b2 el cos q ¼ a1 sin K 1 x þ L1 y t e K1 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !3 2 2 K 1 ð4k2 K 1 þ 2k2 L1 þ K 31 2l2 K 1 Þ L1 þ 4k2 K 1 16K 1 k2 L1 þ 3L21 þ 8K 21 l2 4K 41 þ 32K 21 k2 5 4 þ b2 cos xþ y t K1 K1 K 21 ek2 xþl2 yþa2 t ;
ð41Þ
where k1, k2, l2, K1, L1, and a2 are free parameters. Thus, we obtain double-periodic soliton solution of Eq. (1)
u18 ¼ 2 ln a1 en sin g þ b2 el cos q ;
ð42Þ
where n, g, l and q are defined by 4
2
g ¼ K 1 x þ L1 y 4KK1 1þ3L1 t; l ¼ k2 x þ l2 y þ a2 t; q ¼ K 2 x þ L2 y þ b2 t; n ¼ k2 x l2 y þ a1 t;
ð43Þ
and K2, L2, a1, b2 are given by (40). Remark 1 1. We point out that many singular periodic soliton solutions are included in the obtained results if we chose the free parameters properly. 2. In sub Section 2.2 we omit some single-periodic soliton solutions to KP Eq. (1). We have verified that the above equations u1—u18 are solutions to the original KP Eq. (1). 3. Discussion In this work, two extended ansätzs are employed to seek some new exact mutiple periodic-soliton solutions to KP Eq. (1) and some new solutions are obtained. The results show that there exist multiple periodic solitary waves in the different directions for the (2 + 1)-dimensional KP equation. As mentioned in the introduction above, we can get some exact solutions of the KdV Eq. (2) by reducing the obtained solutions of the KP Eq. (1). For instance, let the coefficients of y in the solution u13 be zero, that is,
l2 ¼ 0;
L1 ¼ 0;
K 2 ð2k2 2k1 þ
L1 Þ ¼ 0; K1
K 22 K 21 ðk1 k2 Þ2 þ l2 þ
L1 ðk1 k2 Þ ¼ 0; K1
then we have the following periodic solution 2
u¼
2K 21 ða2 þ b Þ ½a sinðK 1 x þ
4K 31 tÞ
þ b cosðK 1 x þ 4K 31 tÞ2
and double-soliton solution
u¼
2aðk1 k2 Þ2 en ½aen 2bðsinh g þ cosh gÞel l
ðaen sinh g þ be Þ2
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
375
to the KdV Eq. (2), where
n ¼ k1 x þ ½a2 4ðk1 k2 Þ3 t; g ¼ ðk1 k2 Þx 4ðk1 k2 Þ3 t; l ¼ k2 x þ a2 t and a, b, K1, k1, k2, a2 are free parameters. This gives us a new approach to find exact solutions of the KdV Eq. (2). In this paper, the obtained solutions of KP equation maybe shed light on the richness of solutions of higher dimensional soliton equations and is expected to help understand complete integrability of nonlinear differential equations. But the physical signification of these strange soliton solutions is unclear and worthy to be investigated. Acknowledgement The author would like to express his sincere gratitude to the anonymous referee for many valuable comments and suggestions which served to improve the article. Moveover, Eq. (31) has been showed by the referee. The research of the author is partly supported by Subjects research and development foundation of Hangzhou Dianzi University under Grant No. ZX100204004. References [1] M. Wadati, H. Sanuki, K. Konno, Relationships among inverse method Backlund transformation and infinite number of conservation laws, Prog. Theor. Phys. 53 (1975) 419–436. [2] V.A. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin, Heidelberg, 1991. [3] Z.T. Fu, S.K. Liu, S.D. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A 290 (2001) 72–76. [4] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 98 (1996) 288–296. [5] W. Malfliet, W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54 (1996) 563–568. [6] W. Malfliet, W. Hereman, The tanh method: II. Perturbation technique for conservative systems, Phys. Scr. 54 (1996) 569–575. [7] A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput. 15 (3) (2004) 713–723. [8] A.M. Wazwaz, The tanh method: exact solutions of the sine–Gordon and the Sinh–Gordon equations, Appl. Math. Comput. 167 (2) (2005) 1196–1210. [9] C.T. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–82. [10] Z.Y. Yan, H.Q. Zhang, New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Phys. Lett. A 252 (1999) 291–296. [11] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [12] M.L. Wang, Y.B. Zhou, Z.B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [13] P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq equation, J. Math. Phys. 30 (1989) 2201–2213. [14] W.X. Ma, J.H. Lee, A transformed rational function method and exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation, Chaos Solitons Fractals 42 (2009) 1356–1363. [15] L. Wei, Exact solutions to a combined sinh–cosh-Gordon equation, Commun. Theor. Phys. 54 (2010) 599–602. [16] L. Wei, A function transformation method and exact solutions to a generalized sinh–Gordon equation, Comput. Math. Appl. 60 (2010) 3003–3011. [17] Z.D. Dai, S.L. Li, Q.Y. Dai, J. Huang, Singular periodic soliton solutions and resonance for the Kadomtsev–Petviashvili equation, Chaos Solitons Fractals 34 (4) (2007) 1148–1153. [18] Z.D. Dai, Z.J. Liu, D.L. Li, Exact periodic solitary-wave solution for KdV equation, Chin. Phys. Lett. A 25 (5) (2008) 1151–1153. [19] Z.D. Dai, J. Huang, M.R. Jiang, S.H. Wang, Homoclinic orbits and periodic solitons for Boussinesq equation with even constraint, Chaos Solitons Fractals 26 (2005) 1189–1194. [20] B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (16) (1970) 539–541. [21] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28 (8) (1987) 1732– 1742. [22] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations, J. Math. Phys. 28 (9) (1987) 2094–2101. [23] R. Hirota, Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (18) (1971) 1192–1194. [24] W.X. Ma, Y. You, Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions, Trans. Amer. Math. Soc 357 (2005) 1753–1778. [25] W.X. Ma, Wronskians, generalized Wronskians and solutions to the Korteweg–de Vries equation, Chaos Solitons Fractals 19 (2004) 163–170. [26] W.X. Ma, Complexiton solutions to the Korteweg–de Vries equation, Phys. Lett. A 301 (2002) 35–44. [27] W.X. Ma, K. Maruno, Complexiton solutions of the Toda lattice equation, Phys. A 343 (2004) 219–237. [28] W.X. Ma, Complexiton solutions to integrable equations, Nonlinear Anal. 63 (2005) e2461–e2471. [29] W.X. Ma, Diversity of exact solutions to a restricted Boiti–Leon–Pempinelli dispersive long-wave system, Phys. Lett. A 319 (2003) 325–333.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Multiple periodic-soliton solutions to Kadomtsev–Petviashvili equation Long Wei Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Zhejiang 310018, China
a r t i c l e
i n f o
Keywords: Extended homoclinic test technique Hirota’s bilinear method Periodic-soliton solutions Singular periodic-soliton solutions Kadomtsev–Petviashvili equation
a b s t r a c t Based on the extended homoclinic test technique, we introduce two new ansätz functions to construct multiple periodic-soliton solutions of Kadomtsev–Petviashvili (KP) equation by the Hirota’s bilinear method. Some entirely new periodic-soliton solutions are obtained. The obtained results show that there exist multiple-periodic solitary waves in the different directions for the KP equation, which differ from complexiton. The employed approach is powerful and can be also applied to solve other nonlinear differential equations. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In recent years, more and more attention has been paid to the study of (2 + 1)-dimensional integrable systems. Several celebrated examples of multi-dimensional integrable systems have been found in fields ranging from fluid dynamics, nonlinear optics, particle physics and general relativity to differential and algebraic geometry, topology and so on. The special significance of integrable systems is that they combine tractability with nonlinearity. Hence these systems enable one to explore nonlinear phenomena in multi-dimensions while working with explicit solutions. To seek more exact solutions of nonlinear problems, many powerful methods have been presented such as Bäcklund transformation [1], Darboux transformation [2], the extended Jacobian elliptic function expansion method [3], the tanh method [4–8], the sine–cosine method [9–11], the homogeneous balance method [12], the direct reduction method [13], transformed rational function method [14–16] and many other techniques. Recently, Dai et al. introduced a new technique, extended homoclinic test technique, which is used seeking periodic-soliton solutions of integrable equations and the authors obtained some new periodic-soliton solutions for some integrable equations [17–19]. This paper is concerned with the Kadomtsev–Petviashvili (KP) equation
ðut 6uux þ uxxx Þx þ 3uyy ¼ 0;
ð1Þ
which was introduced in 1970 [20] and is both physical and mathematical interest. It is the generalization of the well-known Korteweg–de-Vries (KdV) equation [20–25]
ut 6uux þ uxxx ¼ 0
ð2Þ
and, similar to Eq. (1), Eq. (1) is also completely integrable. Kadomtsev and Petviashvili [20] discovered Eq. (1) when they relaxed the restriction that the waves be strictly one-dimensional. The KP equation is used to model shallow-water waves with weakly nonlinear restoring forces. It is also used to model waves in ferromagnetic media. Note that the KP Eq. (1) is the generalization of the KdV Eq. (2) from (1 + 1) to (2 + 1) dimensions, we can reduce the obtained solutions of the KP equation to those of the KdV equation by setting the coefficients of y to be zero (see Section 3). This shows us a different approach to seek exact solutions of Eq. (2). E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.072
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
369
In this work, based on the idea of extended homoclinic test technique, we introduce two new ansätz functions to construct many new exact solutions of the KP equation (1) by the Hirota bilinear method [23], and some new periodic-soliton solutions are obtained. Our results show that there exist multiple-periodic solitary waves in the different directions for the KP equation. Here, we should mention a kind of special solution—complexiton, which is expressed by combinations of trigonometric and hyperbolic functions [26–28]. This kind of special solution is also periodic solitary wave solution. But our results suggest that there exist doubly periodic solitary wave solutions for some integrable problems, which differ from complexiton. The presented solutions maybe shed light on the richness of solutions to higher dimensional soliton equations [29] and are expected to understand complete integrability of nonlinear differential equations, though they are far from complete. 2. Exact periodic-soliton solutions for KP equation The KP equaiton (1) has the bilinear form
u ¼ 2ðln f Þxx ;
ð3Þ
ðDt Dx þ D4x þ 3D2y Þf f ¼ 0;
ð4Þ
where the bilinear operator D is defined as
@ @ @x @x0
n Dm x Dt f g ¼
m
@ @ @t @t 0
n
f ðx; y; tÞ gðx0 ; y0 ; t 0 Þ
ð5Þ
: ðx;y;tÞ¼ðx0 ;y0 ;t 0 Þ
Eq. (4) can be rewritten as
fftx ft fx þ ffxxxx 4f x fxxx þ 3fxx2 þ 3ffyy 3fy2 ¼ 0:
ð6Þ
In what follows, we seek periodic-soliton solutions for KP equation by new ansatzs. 2.1. Single-periodic soliton solutions According to the extended homoclinic test technique, we seek the solution of the form n
f ¼ aen þ be
þ c sin g þ d cos g;
ð7Þ
where n = kx + ly + at, g = Kx + Ly + bt, and a, b, c, d, k, l, K, L, a, b are all parameters to be determined later. Substituting Eq. (7) into Eq. (4) and equating corresponding coefficients of ein(i = 1, 0, 1), cosg and sing to zero yields an algebraic system of a, b, c, d, k, l, K, L, a and b as follows: 2
4
2
2
2
12abl þ 16ak b þ 4abka c2 bK d Kb 3c2 L2 þ 4c2 K 4 3d L2 þ 4d K 4 ¼ 0; 3
2
2
4
3acL2 4dK ak 6ak cK 2 þ 3cal þ 6aldL þ dKaa acbK þ ak c þ caka 3
þ akdb þ cK 4 a þ 4ak dK ¼ 0; 2
2
2
2
3dal 6alcL 6ak dK 3adL þ 4cK 3 ak þ daka akcb adbK cKaa 4
4
3
þ ak d þ dK a 4ak cK ¼ 0; 2
2
2
2
4
4
3bdL 6bk dK þ 3dbl þ bk d þ dK b bdbetaK þ dbka þ bkcb þ cKba 3
þ 4bk cK 4cK 3 bk þ 6blcL ¼ 0; 2
2
3
2
3
4
3cbl 6bk cK 2 þ 4dK bk 3bcL 4bk dK dKba þ bk c þ cK 4 b bcbK þ cbka bkdb 6bldL ¼ 0: Solving this set of algebraic equations with the aid of Maple, we obtain the following sets of solutions. Case 1.1: 2
a¼
L2 ðc2 þ d Þ 4
2
4bðk þ L Þ
2
;
K ¼ 0;
a¼
4
3ðL2 l Þ k ; k
b¼
6lL ; k
ð8Þ
where b, c, d, k, l and L are free parameters. Substituting Eq. (8) into Eq. (7), we get that 2
f ¼
L2 ðc2 þ d Þ 4
4bðk þ L2 Þ
n
en þ be
þ c sin g þ d cos g;
ð9Þ
370
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
where 4
n ¼ kx þ ly
2
k þ 3l 3L2 t; k
g ¼ Ly
6lL t: k
Next, from (9) and (3), it follows that there exists periodic soliton solution of Eq. (1). L2 ðc2 þd2 Þ When 4bðk ¼ b, from Eqs. (9) and (3), we obtain periodic soliton solutions of Eq. (1) 4 þL2 Þ 2
4bk ½ðd cos g c sin gÞ cosh n þ 2b
u1;2 ¼
ð2b cosh n þ d cos g c sin gÞ2
ð10Þ
;
where 3
n ¼ kx þ ly k þ
! 2 2 3 3l 12b k t; 2 2 k c2 þ d 4b
2kbðky 6ltÞ
ffi: g ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
ð11Þ
c2 þ d 4b
The above solutions are periodic in the y-direction, meanwhile they are bell-like wave in x-y-direction. Case 1.2: 2
2
ðl K 4 Þðc2 þ d Þ
a¼
2
4bl
a¼
k ¼ 0;
;
2
6lL ; K
b¼
3L2 þ K 4 þ 3l ; K
ð12Þ
where b, c, d, k, l, K and L are free parameters. Substituting Eq. (12) into Eq. (7), we have 2
f ¼
2
ðK 4 l Þðc2 þ d Þ 2
4bl
n
en þ be
þ c sin g þ d cos g;
ð13Þ
where
n ¼ ly when
4
2
2
6lL t; K
Þðc2 þd2 Þ
ðK l 4bl2
K 4 þ 3l 3L2 t; K
g ¼ Kx þ Ly þ
¼ b, from Eqs. (13) and (3), we obtain periodic soliton solutions of Eq. (1) 2
2
u3;4 ¼
ð14Þ
2K ½2bðc sin g þ d cos gÞ sinh n þ c2 þ d ð2b sinh n þ c sin g þ d cos gÞ
ð15Þ
2
and 2
u5;6 ¼
2K 2 ½2bðc sin g d cos gÞ sinh n þ c2 þ d ð2b sinh n þ c sin g d cos gÞ2
where n, g are defined by (14) and K ¼ direction. Case 1.3:
a ¼ ðc
2 þd2 Þ½ðKlkLÞ2 K 2 ðk2 þK 2 Þ2
4b½ðKlkLÞ2 þk2 ðk2 þK 2 Þ2 5
2
3 2
K a ¼ 3L k6lLKk kþ2k 2 þK 2
b¼
l ðc2 þd þ4b Þ . c2 þd2
4
ð16Þ
;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2
These solutions are periodic in x-y-direction and solitary wave in y-
;
2
3kl þ3kK 4
ð17Þ
;
4 2 3 2 KK 5 þ3KL2 3k Kþ2k K þ6klL3l k2 þK 2
;
where b, c, d, k, l, K and L are free parameters. Substituting Eq. (17) into Eq. (7), we obtain that 2
f ¼
2
ðc2 þ d Þ½ðKl kLÞ2 K 2 ðk þ K 2 Þ2 2
2
4b½ðKl kLÞ2 þ k ðk þ K 2 Þ2
n
en þ be
þ c sin g þ d cos g;
ð18Þ
where
n ¼ kx þ ly þ 3kL
2
6lLKk5 þ2k3 K 2 3kl2 þ3kK 4 k2 þK 2 2
5
2 3
K g ¼ Kx þ Ly 3KL þ6klLK kþ2k 2 þK 2
If we take
ðc2 þd2 Þ½ðKlkLÞ2 K 2 ðk2 þK 2 Þ2 4b½ðKlkLÞ2 þk2 ðk2 þK 2 Þ2
2
u7;8 ¼
2
2
t;
2
3l Kþ3Kk4
ð19Þ
t:
¼ b, from Eqs. (18) and (3), we obtain periodic-soliton solutions of Eq. (1)
h i 2 2R ðK 2 k Þðc sin g þ d cos gÞ cosh n þ 2kKðc cos g d sin gÞ sinhðnÞ
2 2
2ðc þ d ÞK k R
ðR cosh n þ c sin g þ d cos gÞ2
ðR cosh n þ c sin g þ d cos gÞ2
;
ð20Þ
371
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
where n, g are defined by (19) and R ¼ Case 1.4: 2
a¼
3
l ¼ kLðk KKþK Þ ;
b ¼ 0; 3
2
2
b ¼ 4K
2
4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðc2 þd2 Þ½ðKlkLÞ2 K 2 ðk2 þK 2 Þ2 . ðKlkLÞ2 þk2 ðk2 þK 2 Þ2
3L2 K
;
ð21Þ
2
þK Þ 4k K þ3kL K6KLðk : 2
Substituting Eq. (21) into Eq. (7), we get
f ¼ aen þ c sin g þ d cos g;
ð22Þ
where 2
3
n ¼ kx þ kLðk KKþK Þ y 4k 4
3 2
K þ3kL2 6KLðk2 þK 2 Þ t; K2
ð23Þ
2
g ¼ Kx þ Ly þ 4K K3L t; where a, c, d, k, K and L are free parameters. Thus, we obtain periodic-soliton solutions of Eq. (1) 2
u9;10 ¼
2aen ½ðK 2 k Þðc sin g þ d cos gÞ þ 2kKðc cos g d sin gÞ ðaen þ c sin g þ d cos gÞ
2
2
þ
2K 2 ðc2 þ d Þ ðaen þ c sin g þ d cos gÞ2
;
ð24Þ
where n, g are defined by (23). Case 1.5:
b ¼ 0; b¼
2
2
2
L ¼ K½lðkk þK Þ ;
a ¼ 6k K
2
þ3K 4 k4 3l2 k
;
2 2 2 4 2 4 2 K½2k K þ3K þ3lk2þ3k 6lðk þK Þ :
ð25Þ
Substituting Eq. (25) into Eq. (7), we have
f ¼ aen þ c sin g þ d cos g;
ð26Þ
where
n ¼ kx þ ly þ 6k
g ¼ Kx þ
2 2
K þ3K 4 k4 3l2 k
K½lðk2 þK 2 Þ y k
t;
K½2k2 K 2 þ3K 4 þ3l2 þ3k4 6lðk2 þK 2 Þ t; k2
ð27Þ
where a, c, d, k, l and K are free parameters. Thus, we obtain periodic-soliton solutions u11,12 to Eq. (1) of the same form as u9,10, but now the n and g are defined by (27). We should note that u7 u12 are both periodic and solitary wave in x-y-direction. 2.2. Double-periodic soliton solutions Next, we seek solutions with double-periodic soliton structures. To the end, we set f of the form
f ¼ a0 þ en ða1 sin g þ b1 cos gÞ þ c1 e2n þ el ða2 sin q þ b2 cos qÞ þ c2 e2l ; where n = k1x + l1y + a1t, l = k2x + l2y + a2t, g = K1x + L1y + b1t, q = K2x + L2y + b2t, and ai, bi, ci, ki, li, Ki, Li, ai, bi (i = 1, 2) are all parameters to be determined later. Now, to roughly illustrate our idea and simply the computation, we assume that
a0 ¼ b1 ¼ c1 ¼ a2 ¼ c2 ¼ 0; that is, the function f is given by l
f ¼ aen sin g þ be cos q:
ð28Þ
We proceed as above, substituting Eq. (28) into Eq. (4) and equating corresponding coefficients of en, el, cosg, sing, cosq and sinq to zero, we obtain an algebraic system of a, b, ki, li, Ki, Li, ai and bi and solve the system with the aid of Maple, we get the following sets of solutions.
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L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
Case 2.1:
l1 ¼ K 22 K 21 ðk1 k2 Þ2 þ l2 þ L1 ðkK11k2 Þ ; h i L2 ¼ K 2 2ðk2 k1 Þ þ KL11 ; 2
2
2
1 k2 Þ a1 ¼ a2 þ 4ðk32 k31 Þ þ 6L1 ½K 1 K 2Kþðk þ 1
b1 ¼ b2 ¼
4K 41 3L21 K1
4K 32
3ðk1 k2 Þð4K 21 K 22 L21 þ4k1 k2 K 21 Þ K 21
;
ð29Þ
; 2
3K 2 ðL1 þ2KK1 2k2 2K 1 k1 Þ : 1
From Eqs. (29), (28) and (3), we obtain double-periodic soliton solutions of Eq. (1):
nþl ðk1 k2 Þ2 K 21 K 22 sin g cos q þ 2ðk1 k2 ÞK 2 sin g sin q u13 ¼ 2 abe
1 2 þ 2ðk1 k2 ÞK 1 cos g cos q þ 2K 1 K 2 cos g sin q a2 e2n K 21 b e2l K 22 : l ðaen sin g þ be cos qÞ2
ð30Þ
where
g ¼ K 1 x þ L1 y þ b1 t; l ¼ k2 x þ l2 y þ a2 t; q ¼ K 2 x þ L2 y þ b2 t
n ¼ k1 x þ l1 y þ a1 t;
with free parameters a, b, k1, k2, l2, K1, K2, L1, a2 and l1, L2, a1, b1, b2 are given by (29). Especially, taking 2
l1 ¼ k1 K 21 ;
a1 ¼ 12k1 K 21 4k31 ; L1 ¼ 2k1 K 1 ; b1 ¼ 12k21 K 1 þ 4K 31 ; l2 ¼ k22 K 22 ; a2 ¼ 12k2 K 22 4k32 ; 2
b2 ¼ 12k2 K 2 þ 4K 32 ;
L2 ¼ 2k2 K 2 ; we have
h i 2 2 2 2 2 k xþ k2 K 2 yþ4k2 ð3K 22 k22 Þt f ¼ aek1 xþðk1 K 1 Þyþ4k1 ð3K 1 k1 Þt sin K 1 x 2k1 y 4tK 21 þ 12tk1 þ be 2 ð 2 2 Þ h i 2 cos K 2 x 2k2 y 4tK 22 þ 12tk2 ;
ð31Þ
then substituting the above f into (3) we can obtain a double-periodic soliton solution to Eq. (1). Case 2.2:
l1 ¼ K 21 K 22 þ ðk1 k2 Þ2 þ l2 þ L1 ðkK11k2 Þ ; h i L2 ¼ K 2 2ðk1 k2 Þ þ KL11 ; 2
2
2
1 k2 Þ a1 ¼ a2 þ 4ðk32 k31 Þ þ 6L1 ½K 2 K 1Kðk þ 1
b1 ¼
4K 41 3L21 K1
3ðk1 k2 Þð4K 21 K 22 L21 þ4k1 k2 K 21 Þ K 21
;
ð32Þ
; 2
b2 ¼ 4K 32 3K 2 ðL1 2KK1 2k2 þ2K 1 k1 Þ : 1
Substituting (32) into Eq. (28) and from (3) we obtain another double-periodic soliton solution u14 of Eq. (1) with the form of (30), but l1, L2, a1, b1, b2 are given by (32) and k1, k2, l2, K1, K2, L1, a2 are free parameters. Case 2.3:
ðL1 4k2 K 1 ÞK 2 k1 ¼ k2 ; l1 ¼ l2 ; L2 ¼ ; K1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 K 1 4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 ; K2 ¼ K1 6 8k22 K 1 L1 þ 2l2 K 1 L1 k2 L21 2 2 a1 ¼ a2 8k2 6l2 þ 3K 1 þ 8k2 þ ; K 21 b1 ¼ b2 ¼
4K 41 3L21 ; K 1 2 K 2 16K 1 k2 L1 3L21 þ 8K 21 l2 þ 4K 41 32K 21 k2 K 21
ð33Þ :
373
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
Substituting Eq. (33) into Eq. (28), we get
f ¼ a1 en sin g þ b2 el cos q ¼ a1 sin K 1 x þ L1 y
4K 41
þ
3L21
! t
k2 xl2 yþ a2 8k2 ð6l2 þ3K 21 þ8k32 Þþ
e
ð
6 8k2 K 1 L1 þ2l2 K 1 L1 k2 L2 2 1 K2 1
Þ
t
K1 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !3 2 2 K 1 ð4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 Þ L1 4k2 K 1 16K 1 k2 L1 3L21 þ 8K 21 l2 þ 4K 41 32K 21 k2 5 4 þ b2 cos xþ yþ t K1 K1 K 21 ek2 xþl2 yþa2 t ;
ð34Þ
where k1, k2, l2, K1, L1, and a2 are free parameters. Thus, we obtain double-periodic soliton solution of Eq. (1):
n h 2 2 u15 ¼ 2 a21 K 31 e2n þ b2 4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 e2l þ 2a1 b2 2k2 K 21 cos g cos q þ K 31 k2 L1 þ l2 K 1 sin g cos q o.n
2 o K 1 a1 en sin g þ b2 el cos q ; ð35Þ þ 2k2 P sin g sin q K 1 P cos g sin qeða1 þa2 Þt where P ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 K 1 4k2 K 1 þ 2l2 K 1 þ K 31 2k2 L1 ; n; g; l; q are defined by 4
2
g ¼ K 1 x þ L1 y 4KK1 1þ3L1 t;
n ¼ k2 x l2 y þ a1 t;
ð36Þ
l ¼ k2 x þ l2 y þ a2 t; q ¼ K 2 x þ L2 y þ b2 t and K2, L2, a1, b2 are given by (33). Case 2.4: 2
K2
l1 ¼ l2 ¼ ð4k2 þ 42 þ kK2 L22 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðK 22 8k22 Þ ; L1 ¼ ðL2 þ4kK22K 2 ÞK 1 ; K 1 ¼ 2
k1 ¼ k2 ;
a1 ¼ a2 12k2 K 22 þ 128k32 þ 3K 2 L2 þ 6k2 L2 ð8kK 22 K 2 þL2 Þ ;
ð37Þ
2
b1 ¼ b2 ¼
K 1 ð2K 42 þ64k22 K 22 þ24k2 K 2 L2 þ3L22 Þ K 22
4K 42 3L22 K2
;
:
Substituting Eq. (37) into Eq. (28), then from (3) we obtain double-periodic soliton solutions of Eq. (1):
u16;17 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a1 b2 16k2 3K 22 sin g cos q 4k2 2K 22 16k2 cos g cos q þ 2K 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2K 22 16k2 cos g sin q 8k2 K 2 sin g sin q eða1 þa2 Þt
. n
2 2 2 a1 e sin g b2 el cos q ; þ a21 K 22 8k2 e2n þ 2b2 K 22 e2l
ð38Þ
where n, g, l and q are defined as follows:
h i K2 2 3 n ¼ k2 x 4k2 þ 42 þ kK2 L22 y þ a2 12k2 K 22 þ 128k2 þ 3K 2 L2 þ 6k2 L2 ð8kK 22 K 2 þL2 Þ t; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 2 2ðK 22 8k22 Þ 64k K 2K þ24k K L þ3L 2K2 g¼ x þ L2 þ4k y 2 2 2 K2 2 2 2 2 t ; 2 K2 2
l ¼ k2 x þ
2 ð4k2
þ
K 22 4 4
þ
k2 L2 Þy K2
þ a2 t;
2
2 q ¼ K 2 x þ L2 y þ 4K 2K3L t; 2
and k2, K2, L2, a2 are free parameters.
ð39Þ
374
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
Case 2.5:
k1 ¼ k2 ; L2 ¼ K2 ¼
l1 ¼ ðL1 þ4k2 K 1 ÞK 2 ; K1
l2 ;
ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 ð4k22 K 1 þ2k2 L1 þK 31 2l2 K 1 Þ
; K1 6L 8k2 K 2l K þk L 2 a1 ¼ a2 þ 8k2 6l2 3K 1 8k22 1 ð 2 1 K 2 2 1 2 1 Þ ;
ð40Þ
1
b1 ¼
4K 41 3L21 K1
b2 ¼
;
K 2 ð16K 1 k2 L1 þ3L21 þ8K 21 l2 4K 41 þ32K 21 k22 Þ K 21
:
Substituting Eq. (40) into Eq. (28), we get
! 6L1 ð8k2 K 1 2l2 K 1 þk2 L1 Þ 2 4 2 k2 xl2 yþ a2 þ8k2 ð6l2 3K 21 8k22 Þ t 2 4K þ 3L K 1 1 1 f ¼ a1 en sin g þ b2 el cos q ¼ a1 sin K 1 x þ L1 y t e K1 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !3 2 2 K 1 ð4k2 K 1 þ 2k2 L1 þ K 31 2l2 K 1 Þ L1 þ 4k2 K 1 16K 1 k2 L1 þ 3L21 þ 8K 21 l2 4K 41 þ 32K 21 k2 5 4 þ b2 cos xþ y t K1 K1 K 21 ek2 xþl2 yþa2 t ;
ð41Þ
where k1, k2, l2, K1, L1, and a2 are free parameters. Thus, we obtain double-periodic soliton solution of Eq. (1)
u18 ¼ 2 ln a1 en sin g þ b2 el cos q ;
ð42Þ
where n, g, l and q are defined by 4
2
g ¼ K 1 x þ L1 y 4KK1 1þ3L1 t; l ¼ k2 x þ l2 y þ a2 t; q ¼ K 2 x þ L2 y þ b2 t; n ¼ k2 x l2 y þ a1 t;
ð43Þ
and K2, L2, a1, b2 are given by (40). Remark 1 1. We point out that many singular periodic soliton solutions are included in the obtained results if we chose the free parameters properly. 2. In sub Section 2.2 we omit some single-periodic soliton solutions to KP Eq. (1). We have verified that the above equations u1—u18 are solutions to the original KP Eq. (1). 3. Discussion In this work, two extended ansätzs are employed to seek some new exact mutiple periodic-soliton solutions to KP Eq. (1) and some new solutions are obtained. The results show that there exist multiple periodic solitary waves in the different directions for the (2 + 1)-dimensional KP equation. As mentioned in the introduction above, we can get some exact solutions of the KdV Eq. (2) by reducing the obtained solutions of the KP Eq. (1). For instance, let the coefficients of y in the solution u13 be zero, that is,
l2 ¼ 0;
L1 ¼ 0;
K 2 ð2k2 2k1 þ
L1 Þ ¼ 0; K1
K 22 K 21 ðk1 k2 Þ2 þ l2 þ
L1 ðk1 k2 Þ ¼ 0; K1
then we have the following periodic solution 2
u¼
2K 21 ða2 þ b Þ ½a sinðK 1 x þ
4K 31 tÞ
þ b cosðK 1 x þ 4K 31 tÞ2
and double-soliton solution
u¼
2aðk1 k2 Þ2 en ½aen 2bðsinh g þ cosh gÞel l
ðaen sinh g þ be Þ2
L. Wei / Applied Mathematics and Computation 218 (2011) 368–375
375
to the KdV Eq. (2), where
n ¼ k1 x þ ½a2 4ðk1 k2 Þ3 t; g ¼ ðk1 k2 Þx 4ðk1 k2 Þ3 t; l ¼ k2 x þ a2 t and a, b, K1, k1, k2, a2 are free parameters. This gives us a new approach to find exact solutions of the KdV Eq. (2). In this paper, the obtained solutions of KP equation maybe shed light on the richness of solutions of higher dimensional soliton equations and is expected to help understand complete integrability of nonlinear differential equations. But the physical signification of these strange soliton solutions is unclear and worthy to be investigated. Acknowledgement The author would like to express his sincere gratitude to the anonymous referee for many valuable comments and suggestions which served to improve the article. Moveover, Eq. (31) has been showed by the referee. The research of the author is partly supported by Subjects research and development foundation of Hangzhou Dianzi University under Grant No. ZX100204004. References [1] M. Wadati, H. Sanuki, K. Konno, Relationships among inverse method Backlund transformation and infinite number of conservation laws, Prog. Theor. Phys. 53 (1975) 419–436. [2] V.A. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin, Heidelberg, 1991. [3] Z.T. Fu, S.K. Liu, S.D. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A 290 (2001) 72–76. [4] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 98 (1996) 288–296. [5] W. Malfliet, W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54 (1996) 563–568. [6] W. Malfliet, W. Hereman, The tanh method: II. Perturbation technique for conservative systems, Phys. Scr. 54 (1996) 569–575. [7] A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput. 15 (3) (2004) 713–723. [8] A.M. Wazwaz, The tanh method: exact solutions of the sine–Gordon and the Sinh–Gordon equations, Appl. Math. Comput. 167 (2) (2005) 1196–1210. [9] C.T. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–82. [10] Z.Y. Yan, H.Q. Zhang, New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Phys. Lett. A 252 (1999) 291–296. [11] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [12] M.L. Wang, Y.B. Zhou, Z.B. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [13] P.A. Clarkson, M.D. Kruskal, New similarity reductions of the Boussinesq equation, J. Math. Phys. 30 (1989) 2201–2213. [14] W.X. Ma, J.H. Lee, A transformed rational function method and exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation, Chaos Solitons Fractals 42 (2009) 1356–1363. [15] L. Wei, Exact solutions to a combined sinh–cosh-Gordon equation, Commun. Theor. Phys. 54 (2010) 599–602. [16] L. Wei, A function transformation method and exact solutions to a generalized sinh–Gordon equation, Comput. Math. Appl. 60 (2010) 3003–3011. [17] Z.D. Dai, S.L. Li, Q.Y. Dai, J. Huang, Singular periodic soliton solutions and resonance for the Kadomtsev–Petviashvili equation, Chaos Solitons Fractals 34 (4) (2007) 1148–1153. [18] Z.D. Dai, Z.J. Liu, D.L. Li, Exact periodic solitary-wave solution for KdV equation, Chin. Phys. Lett. A 25 (5) (2008) 1151–1153. [19] Z.D. Dai, J. Huang, M.R. Jiang, S.H. Wang, Homoclinic orbits and periodic solitons for Boussinesq equation with even constraint, Chaos Solitons Fractals 26 (2005) 1189–1194. [20] B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (16) (1970) 539–541. [21] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28 (8) (1987) 1732– 1742. [22] J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations, J. Math. Phys. 28 (9) (1987) 2094–2101. [23] R. Hirota, Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (18) (1971) 1192–1194. [24] W.X. Ma, Y. You, Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions, Trans. Amer. Math. Soc 357 (2005) 1753–1778. [25] W.X. Ma, Wronskians, generalized Wronskians and solutions to the Korteweg–de Vries equation, Chaos Solitons Fractals 19 (2004) 163–170. [26] W.X. Ma, Complexiton solutions to the Korteweg–de Vries equation, Phys. Lett. A 301 (2002) 35–44. [27] W.X. Ma, K. Maruno, Complexiton solutions of the Toda lattice equation, Phys. A 343 (2004) 219–237. [28] W.X. Ma, Complexiton solutions to integrable equations, Nonlinear Anal. 63 (2005) e2461–e2471. [29] W.X. Ma, Diversity of exact solutions to a restricted Boiti–Leon–Pempinelli dispersive long-wave system, Phys. Lett. A 319 (2003) 325–333.