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Full symmetry groups, Painlevé integrability and exact solutions of the nonisospectral BKP equation
Applied Mathematics and Computation 217 (2010) 1555–1560
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Full symmetry groups, Painlevé integrability and exact solutions of the nonisospectral BKP equation Huan-ping Zhang a, Biao Li a,c, Yong Chen a,b,* a b c
Nonlinear Science Center, Ningbo University, Ningbo 315211, China Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China MM Key Lab, Chinese Academy of Sciences, Beijing 100080, China
a r t i c l e
i n f o
Keywords: Symmetry group Nonisospectral BKP equation Painlevé analysis Solitons
a b s t r a c t Based on the generalized symmetry group method presented by Lou and Ma [Lou and Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A: Math. Gen. 38 (2005) L129], firstly, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. Secondly, the nonisospectral BKP can be proved to be Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification, some solutions are obtained by using the standard truncated Painlevé expansion. Finally, based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The nonisospectral soliton equations are important physical models, because some of them can describe the waves in a certain type of nonuniform media [1–3]. There are many methods for finding solutions of nonisospectral soliton equations, such as Darboux transformation [4], IST [1,2,5] and so on. In recent years, the study of symmetries [6–8], symmetry groups [9], symmetry reductions [10,11] and group invariant solutions of nonlinear partial differential equations (PDEs) has become one of the most exciting and extremely active areas of research [12–16]. Some powerful methods to obtain the similarity reductions of a given PDE have been developed by mathematicians and physicist, such as, the Lie approach [6,9] and the direct method presented by Clarkson and Kruskal (CK) [10]. Most recently, Lou et al. develop a new symmetry group method, named generalized symmetry group method, in a series of papers [17–21]. By the new symmetry group method, both the Lie point symmetry groups and the full symmetry group can be obtained for some PDEs [22]. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches for some nonlinear PDEs. Here we use the generalized symmetry group method and the standard WTC approach with the Kruskal’s simplification [23–27] to investigate (2 + 1)-dimensional nonisospectral BKP equation [28]:
9ut þ yðuxxxxx þ 15uuxxx þ 15ux uxx þ 45u2 ux 5uxxy 15uuy 15ux @ 1 uy 5@ 1 uyy Þ 3xuy 3uxx 3u2 3ux @ 1 u 6@ 1 uy ¼ 0;
ð1Þ
where u ¼ uðx; y; tÞ. The nonisospectral BKP equation had been researched, for example, Deng obtained the soliton solutions for the nonisospectral BKP equation are derived through Hirota method and Pfaffian technique [29]. * Corresponding author. Address: Nonlinear Science Center, Ningbo University, Ningbo 315211, China. E-mail address: [email protected] (Y. Chen). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.044
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This paper is arranged as follows: In Section 2, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. In Section 3, we give the proof of the Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification [23–27] and to obtain some exact solutions by using Painlevé expansion. Based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed. In Section 4, we give the conclusion of the article.
2. The full symmetry group of a (2 + 1)-dimensional nonisospectral BKP equation In order to obtain the full symmetry group of the the nonisospectral BKP equation, firstly, we let
u ¼ vx:
ð2Þ
Substituting Eqs. (2) into (1), then Eq. (1) becomes:
ð15v x v xxxx þ 15v xx v xxx þ 45v 2x v xx þ v xxxxxx 15v x v xy 15v xx v y 5v yy 5v xxxy Þy þ 9v xt 3xv xy 3v xxx 3v 2x 3v xx v 6v y ¼ 0; where Let
ð3Þ
v ¼ v ðx; y; tÞ.
v ¼ a þ bVðn; g; sÞ;
ð4Þ
where a; b; n; g and s are functions of {x, y, t}. Restricting Vðn; g; sÞ V, and satisfies the same form as the nonisospectral BKP equations (3) but with new independent variables, i.e.,
ð15V n V nnnn þ 15V nn V nnn þ 45V 2n V nn þ V nnnnnn 15V n V ng 15V nn V g 5V gg 5V nnng Þg þ 9V ns 3nV ng 3V nnn 3V 2n 3V nn V 6V g ¼ 0:
ð5Þ
Substituting Eqs. (4) into (3), then eliminate V nnnnnn by using Eq. (5), from that, the remained determining equations of the functions n; g; s; a; b can be got by vanishing the coefficients of V and its derivatives, then we find out the general solution of the determining equations by tedious calculations. The result reads 1
3
3 5
5 2
2 5
s ¼ s0 ; g ¼ ðd s0t y þ g0 Þ ; b¼
2 ds5 3 2 n ¼ 10t d3 s50t y5 þ g0 x þ n1 ðy; tÞ; y5 1
ð6Þ
2 d 3 2 d s0t y5 þ g0 s50t ; 1 y5 1 2
ð7Þ
3 3 8 1 x2 g0 x 3 3 4 3 2 3 2 1 2 1 2 5 5 5s 5s 5 g s5 þ 3 þ 6 3 y s þ g þ g g þ n s d y y d d 0tt 0t 0tt 0 0 0t 0t 0t 0t 0 30 d3 s5 y25 þ g y s5 d3 s5 y25 þ g y25 8 10 4 2 15 0 0 0t 0t 0t 8 2 9 3 > > > > 3 2 Z y 3 < = 4 6 1 7 y5 n1y ðy; tÞd 1 3 d n1t ðy; tÞ 6 7 3 2 þ dy þ c0 ; þ 1 2 1 4 5 > 3 s5 d3 s5 y5 þ g 5 3 35 2 2 3 y5 > > > : ; 0 0t 0t d s y5 þ g y5
a¼
0t
where n0 n0 ðtÞ; g0 g0 ðtÞ;
n1 ðy; tÞ ¼
ð8Þ
0
s0 s0 ðtÞ; c0 c0 ðtÞ are arbitrary functions of time t and
3 2 12 1 4 2 ð9ds0tt y5 þ 30y5 g0t d3 þ 8n0 s0t Þ d3 s50t y5 þ g0 ; 8st
while the constants d possess discrete values determined by
d ¼ 1;
1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 þ i 10 þ 2 5 ; 4
1þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 i 10 2 5 ; 4
1þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 þ i 10 2 5 ; 4
1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 i 10 þ 2 5 : 4 ð9Þ
In summary, the following theorem holds: Theorem. If V Vðx; y; tÞ is a solution of the Eq. (3), then so is
v ¼ a þ bVðn; g; sÞ;
ð10Þ
where a; b; n; g; s are given by Eqs. (6)–(8), discrete value of the d are given by Eq. (9). The relationship is constructed between the new solutions and the old ones of Eq. (3). Thus from the theorem of Eq. (3) with Eq. (2), we also can obtain the relationship between the new solutions and the old ones of Eq. (1).
H.-p. Zhang et al. / Applied Mathematics and Computation 217 (2010) 1555–1560
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From the symmetry group theorem, for the nonisospectral BKP equation, the symmetry group is divided into five sectors which correspond to
d ¼ 1;
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 5 þ i 10 þ 2 5 d¼ ; 4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1 þ 5 i 10 2 5 ; d¼ 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pffiffiffi pffiffiffi 1 þ 5 þ i 10 2 5 d¼ ; 4 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 þ 5 i 10 þ 2 5 ; d¼ 4 1 þ
of theorem respectively. That is to say, the full symmetry group, GCBKP , expressed by theorem for the complex the nonisospectral BKP equation is the product of the usual Lie point symmetry group S ðd ¼ 1Þ and the discrete group D5
GCBKP ¼ D5 S; D5 fI; R1 ; R2 ; R3 ; R4 g;
ð11Þ ð12Þ
where I is the identity transformation, and
R1 R2 R3 R4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 i 10 þ 2 5 : v ðx; y; tÞ ! v 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pffiffiffi pffiffiffi 1 þ 5 þ i 10 þ 2 5 : v ðx; y; tÞ ! v 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1 þ 5 i 10 2 5 : v ðx; y; tÞ ! v 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1 þ 5 þ i 10 2 5 : v ðx; y; tÞ ! v 4 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi pffiffiffi 5 þ i 10 þ 2 5 x; y; t ; 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ! pffiffiffi pffiffiffi 1 þ 5 i 10 þ 2 5 x; y; t ; 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi pffiffiffi 1 þ 5 þ i 10 2 5 x; y; t ; 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi pffiffiffi 1 þ 5 i 10 2 5 x; y; t : 4 1 þ
From theorem, by restricting ðf f ðtÞ; g gðtÞ; h hðtÞÞ; p pðtÞÞ
s ¼ t þ f ; n0 ¼ g; g0 ¼ h; c0 ¼ p; then Eq. (10) can be written as
v ¼ V þ rðVÞ;
! 5 3 3 9 1 1 xh 15 3 9 1 1 Vh 3 1 xf 1 5 5 5 rðVÞ ¼ V t g þ y h þ yg t V y þ ygtt þ y f þ xg t þ 2 þ y ht V x þ ygttt þ Vgt þ 2 þ xg tt þ 2 2 8 2 2 y5 4 16 2 2 y5 8 15 y45
1 x2 h 3 xht 3 1 45 3 p y5 htt þ 1 ; þ þ y5 ft þ 30 y75 4 y25 2 16 y5
The equivalent vector expression of the above symmetry is
(
! ) @ 1 xf 3 1 @ 9 1 @ 3 @ @ 9 1 3 @ 5f þ y yg xg þ yg þ g yg Vg xg þ þ þ þ t @x 15 y45 2 @V 8 tt 2 t @x 2 t @y @t 16 ttt 2 t 8 tt @V ( ! ! ) 1 xh 15 3 @ 5 3 @ 1 Vh 1 x2 h 3 xht 45 3 @ p @ 5h 5h 5h 1 y þ y y þ þ þ þ t tt 2 y25 4 @x 2 @y 2 y25 30 y75 4 y25 16 @V y5 @V 1
C ¼ y5 f
r1 ðf Þ þ r2 ðgÞ þ r3 ðhÞ þ r4 ðpÞ; which is exactly the same as that we obtained by the standard Lie approach. The commutation relations for the Kac–Moody–Virasoro algebra among r1 ðf Þ;
r2 ðgÞ; r3 ðhÞ and r4 ðpÞ are as follows:
½r1 ðf1 Þ; r1 ðf2 Þ ¼ 0; ½r1 ðf Þ; r4 ðpÞ ¼ 0; ½r3 ðhÞ; r4 ðpÞ ¼ 0; ½r4 ðp1 Þ; r4 ðp2 Þ ¼ 0; 1 1 3 ½r1 ðf Þ; r2 ðgÞ ¼ r1 gft g t f ; ½r1 ðf Þ; r3 ðhÞ ¼ r4 fht ft h ; 5 2 2 3 ½r2 ðg 1 Þ; r2 ðg 2 Þ ¼ r2 ðg 1 g 2t g 2 g 1t Þ; ½r2 ðgÞ; r3 ðhÞ ¼ r3 ght g t h ; 5 1 15 r1 ðh1 h2t h2 h1t Þ: ½r2 ðgÞ; r4 ðpÞ ¼ r4 gpt þ g t p ; ½r3 ðh1 Þ; r3 ðh2 Þ ¼ 5 4
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We can use the relationship form this theorem to obtain new solutions from old solutions. So in the next section, we prove the nonisospectral BKP equation is Painlevé integrability and obtain a solution by using the standard truncated Painlevé expansion. 3. Painlevé integrability of the (2 + 1)-dimensional the nonisospectral BKP equation and some solution from the truncated Painlevé expansion Painlevé analysis is one of the most powerful method to prove the integrability of a model developed by WTC (Weiss– Tabor–Canvela) [23–27]. If one needs only to prove the Painlevé property of a model, one may use the Kruskal’s simplification for WTC method. Furthermore, the Painlevé analysis can also be used to find some exact solutions no matter whether the model is integrable or not. At first, the Painlevé expansion may have the form:
v¼
1 X
v j f ja ;
ð13Þ
j¼0
where the arbitrary function f f ðx; y; tÞ may have different forms in different approaches, v j v j ðx; y; tÞ ðj ¼ 0; 1; 2 . . . 1Þ. Using any one possible form, the final conclusion will be exactly the same. In order to give out a complete treatment, it is convenient by using the Kruskal’s simplification, i.e.,
f ¼ x þ wðy; tÞ;
ð14Þ
with wðy; tÞ w being an arbitrary function of y and t. By substituting v ¼ v 0 f a into Eq. (3), comparing the leading order terms for f ! 0, we get two possible branch:
a ¼ 1;
ð15Þ
v 0 ¼ 2f x ;
ð16Þ
or
v 0 ¼ 4f x :
ð17Þ
For the first branch, by equating the coefficients of f 6
5
4
3
j1
, the polynomial equation in j is derived as:
2
j 21j þ 145j 375j þ 214j þ 396j 360 ¼ 0:
ð18Þ
Using Eq. (18), the resonances are found to be
j ¼ 1; 1; 2; 3; 6; 10: For j ¼ 1; 2 and 3, consequently
v4 v5
v 1 ; v 2 and v 3 are arbitrary functions. For j ¼ 4 and 5, we obtain
1 3xwy 9wt þ 5yw2y ; ¼ 90y 1 ð5ywyy þ 12wy Þ: ¼ 360y
ð19Þ ð20Þ
For j ¼ 6; v 6 is arbitrary function. For j ¼ 7; 8 and 9, we get
v7 ¼
1 3xwy 9wt þ 5yw2y þ 5y2 wy wyy ; 2 8640y
ð21Þ
1 720yw3y x 2160yw2y wt þ 600y2 w4y 180ywyy 315wy 1296xwt wy 2 907200y þ1944w2t þ 216x2 w2y 100y2 wyyy ;
ð22Þ
1 75y3 w2y wyy 108yxwyt 360y2 wy wyt þ 162ywtt 309xw2y y þ 927wy wt y 604800y3 445w3y y2 18x2 wy þ 54xwt þ 18yx2 wyy þ 180y2 wyy wt þ 60y2 xwy wyy :
ð23Þ
v8 ¼
v9 ¼
Then for j ¼ 10; v 10 is arbitrary function. Up to now, we prove that all the resonance conditions are satisfied for the first branch. It is concluded that the Eq. (3) passes the P-test in the first branch. For the second branch, we can get the polynomial equation in j is 6
5
4
3
2
j 21j þ 115j 15j 836j þ 36j þ 720 ¼ 0; then we get the resonances
ð24Þ
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j ¼ 1; 2; 1; 5; 6; 12; we can also prove that all the resonance conditions are satisfied for the second branch. In all, it is concluded that the Eq. (3) passes the P-test and hence it is expected to be integrable whether in the first branch or second branch. Then we use Painlevé expansion to obtain some solutions of Eq. (3). Because in the first branch, when j ¼ 1; v 1 is a resonance point, let
2f x ¼ 2½lnðf Þx ; f f ¼ aðtÞ þ expðpðtÞx þ qðtÞy þ wðtÞÞ:
v¼
ð25Þ ð26Þ
Substituting Eqs. (25) and (26) into Eq. (3), collecting the coefficient of expðpðtÞx þ qðtÞy þ wðtÞÞ; x and y, then we can get
qðtÞ ¼ 3pt ðtÞ;
wðtÞ ¼ ln½aðtÞ þ
1 3
Z
t
p2 ðtÞdt;
ð27Þ
aðtÞ are arbitrary function of t, pðtÞ should be satisfied
45p2t ðtÞ 27pðtÞptt ðtÞ p6 ðtÞ þ 15p3 ðtÞpt ðtÞ ¼ 0:
ð28Þ
So we get the solution of Eq. (3),
v¼
Rt
2pðtÞe
Rt
1þe
pðtÞxþ3pt ðtÞyþ13
pðtÞxþ3pt ðtÞyþ13
p2 ðtÞdt
p2 ðtÞdt
:
Then we also obtain the solution of Eq. (1), 2
u ¼ p2 ðtÞsech
Z 1 p2 ðtÞdt : xpðtÞ þ 3pt ðtÞy þ 3
ð29Þ
If we let
pðtÞ ¼
k1 ðtÞ þ q1 ðtÞ ; 2
3 k1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 9c1 6t
3 q1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 9c2 6t
then Eq. (28) become one-soliton solution of the nonisospectral BKP, which had been given in [29]. If we give out the appropriate form of pðtÞ, we can obtain more abundant solutions of Eq. (1). By theorem, we also can obtain the new solution of Eq. (3),
2pðsÞe
pðsÞnþ3ps ðsÞgþ13
v ¼ a þ b
1
1 þ eðpðsÞnþ3ps ðsÞgþ3
where a; b; n; g and
Rs
Rs
p2 ðsÞds
p2 ðsÞdsÞ
;
ð30Þ
s are determined by Eqs. (6)–(8), pðsÞ is satisfied
2
45ps ðsÞ 27pðsÞpss ðsÞ p6 ðsÞ þ 15p3 ðsÞps ðsÞ ¼ 0: In the second branch, if we let f have the same form as Eq. (25), we cannot obtain a solution of Eq. (3), which contains three variables x; y and t. So we let
f ¼ xpðtÞ þ qðy; tÞ:
ð31Þ
Substituting Eqs. (17) and (31) into Eq. (3), collecting the coefficient of x and y, then we get
qðy; tÞ ¼
7 3 7 y7 F 1 ðtÞ þ pt ðtÞy þ F 2 ðtÞ; 3 3
ð32Þ
where pðtÞ; F 1 ðtÞ and F 2 ðtÞ is arbitrary function of t, then we can obtain the solution of Eq. (3)
v¼
12pðtÞ 3
3xpðtÞ þ 7y7 F 1 ðtÞ þ 7pt ðtÞy þ 3F 2 ðtÞ
;
then the solution of Eq. (1) is
36pðtÞ2 u¼ 2 : 3 3xpðtÞ þ 7y7 F 1 ðtÞ þ 7pt ðtÞy þ 3F 2 ðtÞ We can also get a new solution of Eq. (3) by theorem.
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4. Conclusions In summary, making use of the generalized symmetry group method and symbolic computation, the fully symmetry transformation groups for the nonisospectral BKP equation are given. The full symmetry group of the nonisospectral BKP equation is a product of one discrete group ðD5 Þ and one infinite dimensional Kac–Moody–Virasoro type Lie group with four arbitrary functions. The relationship is constructed between the new solutions and the old ones of equation, which is a relationship of group invariant solution. If we have obtained a solution of the nonisospectral BKP equation by other method, we can use this relationship to obtain another solution. Then we prove the nonisospectral BKP can pass the Painlevé test and hence it is expected to be integrable. So we obtain a solution of the nonisospectral BKP by the standard truncated Painlevé expansion, then we get new general solution by the relationship, which is the known one-soliton solution [29]. It is necessary to point out that the general solution contain all its group invariant solutions, so it is enough to submit it to the relationship for one time only. Acknowledgements We would like to thank Prof. Senyue Lou, Dr. Yuqi Li, Zhongzhou Dong, Xiaorui Hu, Jia Wang and Wangchuan Ye for their enthusiastic guidance and helpful discussions. The work is supported by the National Natural Science Foundation of China (Grant Nos. 10735030, 10747141 and 90718041), Shanghai Leading Academic Discipline Project (No. B412), Program for Changjiang Scholars and Innovative Research Team in University (IRT0734) Zhejiang Provincial Natural Science Foundations of China (Grant No. 605408), Ningbo Natural Science Foundation (Grant Nos. 2007A610049 and 2006A610093) and K.C. Wong Magna Fund in Ningbo University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Full symmetry groups, Painlevé integrability and exact solutions of the nonisospectral BKP equation Huan-ping Zhang a, Biao Li a,c, Yong Chen a,b,* a b c
Nonlinear Science Center, Ningbo University, Ningbo 315211, China Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China MM Key Lab, Chinese Academy of Sciences, Beijing 100080, China
a r t i c l e
i n f o
Keywords: Symmetry group Nonisospectral BKP equation Painlevé analysis Solitons
a b s t r a c t Based on the generalized symmetry group method presented by Lou and Ma [Lou and Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A: Math. Gen. 38 (2005) L129], firstly, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. Secondly, the nonisospectral BKP can be proved to be Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification, some solutions are obtained by using the standard truncated Painlevé expansion. Finally, based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The nonisospectral soliton equations are important physical models, because some of them can describe the waves in a certain type of nonuniform media [1–3]. There are many methods for finding solutions of nonisospectral soliton equations, such as Darboux transformation [4], IST [1,2,5] and so on. In recent years, the study of symmetries [6–8], symmetry groups [9], symmetry reductions [10,11] and group invariant solutions of nonlinear partial differential equations (PDEs) has become one of the most exciting and extremely active areas of research [12–16]. Some powerful methods to obtain the similarity reductions of a given PDE have been developed by mathematicians and physicist, such as, the Lie approach [6,9] and the direct method presented by Clarkson and Kruskal (CK) [10]. Most recently, Lou et al. develop a new symmetry group method, named generalized symmetry group method, in a series of papers [17–21]. By the new symmetry group method, both the Lie point symmetry groups and the full symmetry group can be obtained for some PDEs [22]. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches for some nonlinear PDEs. Here we use the generalized symmetry group method and the standard WTC approach with the Kruskal’s simplification [23–27] to investigate (2 + 1)-dimensional nonisospectral BKP equation [28]:
9ut þ yðuxxxxx þ 15uuxxx þ 15ux uxx þ 45u2 ux 5uxxy 15uuy 15ux @ 1 uy 5@ 1 uyy Þ 3xuy 3uxx 3u2 3ux @ 1 u 6@ 1 uy ¼ 0;
ð1Þ
where u ¼ uðx; y; tÞ. The nonisospectral BKP equation had been researched, for example, Deng obtained the soliton solutions for the nonisospectral BKP equation are derived through Hirota method and Pfaffian technique [29]. * Corresponding author. Address: Nonlinear Science Center, Ningbo University, Ningbo 315211, China. E-mail address: [email protected] (Y. Chen). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.044
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This paper is arranged as follows: In Section 2, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. In Section 3, we give the proof of the Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification [23–27] and to obtain some exact solutions by using Painlevé expansion. Based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed. In Section 4, we give the conclusion of the article.
2. The full symmetry group of a (2 + 1)-dimensional nonisospectral BKP equation In order to obtain the full symmetry group of the the nonisospectral BKP equation, firstly, we let
u ¼ vx:
ð2Þ
Substituting Eqs. (2) into (1), then Eq. (1) becomes:
ð15v x v xxxx þ 15v xx v xxx þ 45v 2x v xx þ v xxxxxx 15v x v xy 15v xx v y 5v yy 5v xxxy Þy þ 9v xt 3xv xy 3v xxx 3v 2x 3v xx v 6v y ¼ 0; where Let
ð3Þ
v ¼ v ðx; y; tÞ.
v ¼ a þ bVðn; g; sÞ;
ð4Þ
where a; b; n; g and s are functions of {x, y, t}. Restricting Vðn; g; sÞ V, and satisfies the same form as the nonisospectral BKP equations (3) but with new independent variables, i.e.,
ð15V n V nnnn þ 15V nn V nnn þ 45V 2n V nn þ V nnnnnn 15V n V ng 15V nn V g 5V gg 5V nnng Þg þ 9V ns 3nV ng 3V nnn 3V 2n 3V nn V 6V g ¼ 0:
ð5Þ
Substituting Eqs. (4) into (3), then eliminate V nnnnnn by using Eq. (5), from that, the remained determining equations of the functions n; g; s; a; b can be got by vanishing the coefficients of V and its derivatives, then we find out the general solution of the determining equations by tedious calculations. The result reads 1
3
3 5
5 2
2 5
s ¼ s0 ; g ¼ ðd s0t y þ g0 Þ ; b¼
2 ds5 3 2 n ¼ 10t d3 s50t y5 þ g0 x þ n1 ðy; tÞ; y5 1
ð6Þ
2 d 3 2 d s0t y5 þ g0 s50t ; 1 y5 1 2
ð7Þ
3 3 8 1 x2 g0 x 3 3 4 3 2 3 2 1 2 1 2 5 5 5s 5s 5 g s5 þ 3 þ 6 3 y s þ g þ g g þ n s d y y d d 0tt 0t 0tt 0 0 0t 0t 0t 0t 0 30 d3 s5 y25 þ g y s5 d3 s5 y25 þ g y25 8 10 4 2 15 0 0 0t 0t 0t 8 2 9 3 > > > > 3 2 Z y 3 < = 4 6 1 7 y5 n1y ðy; tÞd 1 3 d n1t ðy; tÞ 6 7 3 2 þ dy þ c0 ; þ 1 2 1 4 5 > 3 s5 d3 s5 y5 þ g 5 3 35 2 2 3 y5 > > > : ; 0 0t 0t d s y5 þ g y5
a¼
0t
where n0 n0 ðtÞ; g0 g0 ðtÞ;
n1 ðy; tÞ ¼
ð8Þ
0
s0 s0 ðtÞ; c0 c0 ðtÞ are arbitrary functions of time t and
3 2 12 1 4 2 ð9ds0tt y5 þ 30y5 g0t d3 þ 8n0 s0t Þ d3 s50t y5 þ g0 ; 8st
while the constants d possess discrete values determined by
d ¼ 1;
1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 þ i 10 þ 2 5 ; 4
1þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 i 10 2 5 ; 4
1þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 þ i 10 2 5 ; 4
1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 i 10 þ 2 5 : 4 ð9Þ
In summary, the following theorem holds: Theorem. If V Vðx; y; tÞ is a solution of the Eq. (3), then so is
v ¼ a þ bVðn; g; sÞ;
ð10Þ
where a; b; n; g; s are given by Eqs. (6)–(8), discrete value of the d are given by Eq. (9). The relationship is constructed between the new solutions and the old ones of Eq. (3). Thus from the theorem of Eq. (3) with Eq. (2), we also can obtain the relationship between the new solutions and the old ones of Eq. (1).
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From the symmetry group theorem, for the nonisospectral BKP equation, the symmetry group is divided into five sectors which correspond to
d ¼ 1;
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 5 þ i 10 þ 2 5 d¼ ; 4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1 þ 5 i 10 2 5 ; d¼ 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pffiffiffi pffiffiffi 1 þ 5 þ i 10 2 5 d¼ ; 4 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 þ 5 i 10 þ 2 5 ; d¼ 4 1 þ
of theorem respectively. That is to say, the full symmetry group, GCBKP , expressed by theorem for the complex the nonisospectral BKP equation is the product of the usual Lie point symmetry group S ðd ¼ 1Þ and the discrete group D5
GCBKP ¼ D5 S; D5 fI; R1 ; R2 ; R3 ; R4 g;
ð11Þ ð12Þ
where I is the identity transformation, and
R1 R2 R3 R4
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 5 i 10 þ 2 5 : v ðx; y; tÞ ! v 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p pffiffiffi pffiffiffi 1 þ 5 þ i 10 þ 2 5 : v ðx; y; tÞ ! v 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1 þ 5 i 10 2 5 : v ðx; y; tÞ ! v 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 1 þ 5 þ i 10 2 5 : v ðx; y; tÞ ! v 4 1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi pffiffiffi 5 þ i 10 þ 2 5 x; y; t ; 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ! pffiffiffi pffiffiffi 1 þ 5 i 10 þ 2 5 x; y; t ; 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi pffiffiffi 1 þ 5 þ i 10 2 5 x; y; t ; 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffi pffiffiffi 1 þ 5 i 10 2 5 x; y; t : 4 1 þ
From theorem, by restricting ðf f ðtÞ; g gðtÞ; h hðtÞÞ; p pðtÞÞ
s ¼ t þ f ; n0 ¼ g; g0 ¼ h; c0 ¼ p; then Eq. (10) can be written as
v ¼ V þ rðVÞ;
! 5 3 3 9 1 1 xh 15 3 9 1 1 Vh 3 1 xf 1 5 5 5 rðVÞ ¼ V t g þ y h þ yg t V y þ ygtt þ y f þ xg t þ 2 þ y ht V x þ ygttt þ Vgt þ 2 þ xg tt þ 2 2 8 2 2 y5 4 16 2 2 y5 8 15 y45
1 x2 h 3 xht 3 1 45 3 p y5 htt þ 1 ; þ þ y5 ft þ 30 y75 4 y25 2 16 y5
The equivalent vector expression of the above symmetry is
(
! ) @ 1 xf 3 1 @ 9 1 @ 3 @ @ 9 1 3 @ 5f þ y yg xg þ yg þ g yg Vg xg þ þ þ þ t @x 15 y45 2 @V 8 tt 2 t @x 2 t @y @t 16 ttt 2 t 8 tt @V ( ! ! ) 1 xh 15 3 @ 5 3 @ 1 Vh 1 x2 h 3 xht 45 3 @ p @ 5h 5h 5h 1 y þ y y þ þ þ þ t tt 2 y25 4 @x 2 @y 2 y25 30 y75 4 y25 16 @V y5 @V 1
C ¼ y5 f
r1 ðf Þ þ r2 ðgÞ þ r3 ðhÞ þ r4 ðpÞ; which is exactly the same as that we obtained by the standard Lie approach. The commutation relations for the Kac–Moody–Virasoro algebra among r1 ðf Þ;
r2 ðgÞ; r3 ðhÞ and r4 ðpÞ are as follows:
½r1 ðf1 Þ; r1 ðf2 Þ ¼ 0; ½r1 ðf Þ; r4 ðpÞ ¼ 0; ½r3 ðhÞ; r4 ðpÞ ¼ 0; ½r4 ðp1 Þ; r4 ðp2 Þ ¼ 0; 1 1 3 ½r1 ðf Þ; r2 ðgÞ ¼ r1 gft g t f ; ½r1 ðf Þ; r3 ðhÞ ¼ r4 fht ft h ; 5 2 2 3 ½r2 ðg 1 Þ; r2 ðg 2 Þ ¼ r2 ðg 1 g 2t g 2 g 1t Þ; ½r2 ðgÞ; r3 ðhÞ ¼ r3 ght g t h ; 5 1 15 r1 ðh1 h2t h2 h1t Þ: ½r2 ðgÞ; r4 ðpÞ ¼ r4 gpt þ g t p ; ½r3 ðh1 Þ; r3 ðh2 Þ ¼ 5 4
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We can use the relationship form this theorem to obtain new solutions from old solutions. So in the next section, we prove the nonisospectral BKP equation is Painlevé integrability and obtain a solution by using the standard truncated Painlevé expansion. 3. Painlevé integrability of the (2 + 1)-dimensional the nonisospectral BKP equation and some solution from the truncated Painlevé expansion Painlevé analysis is one of the most powerful method to prove the integrability of a model developed by WTC (Weiss– Tabor–Canvela) [23–27]. If one needs only to prove the Painlevé property of a model, one may use the Kruskal’s simplification for WTC method. Furthermore, the Painlevé analysis can also be used to find some exact solutions no matter whether the model is integrable or not. At first, the Painlevé expansion may have the form:
v¼
1 X
v j f ja ;
ð13Þ
j¼0
where the arbitrary function f f ðx; y; tÞ may have different forms in different approaches, v j v j ðx; y; tÞ ðj ¼ 0; 1; 2 . . . 1Þ. Using any one possible form, the final conclusion will be exactly the same. In order to give out a complete treatment, it is convenient by using the Kruskal’s simplification, i.e.,
f ¼ x þ wðy; tÞ;
ð14Þ
with wðy; tÞ w being an arbitrary function of y and t. By substituting v ¼ v 0 f a into Eq. (3), comparing the leading order terms for f ! 0, we get two possible branch:
a ¼ 1;
ð15Þ
v 0 ¼ 2f x ;
ð16Þ
or
v 0 ¼ 4f x :
ð17Þ
For the first branch, by equating the coefficients of f 6
5
4
3
j1
, the polynomial equation in j is derived as:
2
j 21j þ 145j 375j þ 214j þ 396j 360 ¼ 0:
ð18Þ
Using Eq. (18), the resonances are found to be
j ¼ 1; 1; 2; 3; 6; 10: For j ¼ 1; 2 and 3, consequently
v4 v5
v 1 ; v 2 and v 3 are arbitrary functions. For j ¼ 4 and 5, we obtain
1 3xwy 9wt þ 5yw2y ; ¼ 90y 1 ð5ywyy þ 12wy Þ: ¼ 360y
ð19Þ ð20Þ
For j ¼ 6; v 6 is arbitrary function. For j ¼ 7; 8 and 9, we get
v7 ¼
1 3xwy 9wt þ 5yw2y þ 5y2 wy wyy ; 2 8640y
ð21Þ
1 720yw3y x 2160yw2y wt þ 600y2 w4y 180ywyy 315wy 1296xwt wy 2 907200y þ1944w2t þ 216x2 w2y 100y2 wyyy ;
ð22Þ
1 75y3 w2y wyy 108yxwyt 360y2 wy wyt þ 162ywtt 309xw2y y þ 927wy wt y 604800y3 445w3y y2 18x2 wy þ 54xwt þ 18yx2 wyy þ 180y2 wyy wt þ 60y2 xwy wyy :
ð23Þ
v8 ¼
v9 ¼
Then for j ¼ 10; v 10 is arbitrary function. Up to now, we prove that all the resonance conditions are satisfied for the first branch. It is concluded that the Eq. (3) passes the P-test in the first branch. For the second branch, we can get the polynomial equation in j is 6
5
4
3
2
j 21j þ 115j 15j 836j þ 36j þ 720 ¼ 0; then we get the resonances
ð24Þ
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j ¼ 1; 2; 1; 5; 6; 12; we can also prove that all the resonance conditions are satisfied for the second branch. In all, it is concluded that the Eq. (3) passes the P-test and hence it is expected to be integrable whether in the first branch or second branch. Then we use Painlevé expansion to obtain some solutions of Eq. (3). Because in the first branch, when j ¼ 1; v 1 is a resonance point, let
2f x ¼ 2½lnðf Þx ; f f ¼ aðtÞ þ expðpðtÞx þ qðtÞy þ wðtÞÞ:
v¼
ð25Þ ð26Þ
Substituting Eqs. (25) and (26) into Eq. (3), collecting the coefficient of expðpðtÞx þ qðtÞy þ wðtÞÞ; x and y, then we can get
qðtÞ ¼ 3pt ðtÞ;
wðtÞ ¼ ln½aðtÞ þ
1 3
Z
t
p2 ðtÞdt;
ð27Þ
aðtÞ are arbitrary function of t, pðtÞ should be satisfied
45p2t ðtÞ 27pðtÞptt ðtÞ p6 ðtÞ þ 15p3 ðtÞpt ðtÞ ¼ 0:
ð28Þ
So we get the solution of Eq. (3),
v¼
Rt
2pðtÞe
Rt
1þe
pðtÞxþ3pt ðtÞyþ13
pðtÞxþ3pt ðtÞyþ13
p2 ðtÞdt
p2 ðtÞdt
:
Then we also obtain the solution of Eq. (1), 2
u ¼ p2 ðtÞsech
Z 1 p2 ðtÞdt : xpðtÞ þ 3pt ðtÞy þ 3
ð29Þ
If we let
pðtÞ ¼
k1 ðtÞ þ q1 ðtÞ ; 2
3 k1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 9c1 6t
3 q1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 9c2 6t
then Eq. (28) become one-soliton solution of the nonisospectral BKP, which had been given in [29]. If we give out the appropriate form of pðtÞ, we can obtain more abundant solutions of Eq. (1). By theorem, we also can obtain the new solution of Eq. (3),
2pðsÞe
pðsÞnþ3ps ðsÞgþ13
v ¼ a þ b
1
1 þ eðpðsÞnþ3ps ðsÞgþ3
where a; b; n; g and
Rs
Rs
p2 ðsÞds
p2 ðsÞdsÞ
;
ð30Þ
s are determined by Eqs. (6)–(8), pðsÞ is satisfied
2
45ps ðsÞ 27pðsÞpss ðsÞ p6 ðsÞ þ 15p3 ðsÞps ðsÞ ¼ 0: In the second branch, if we let f have the same form as Eq. (25), we cannot obtain a solution of Eq. (3), which contains three variables x; y and t. So we let
f ¼ xpðtÞ þ qðy; tÞ:
ð31Þ
Substituting Eqs. (17) and (31) into Eq. (3), collecting the coefficient of x and y, then we get
qðy; tÞ ¼
7 3 7 y7 F 1 ðtÞ þ pt ðtÞy þ F 2 ðtÞ; 3 3
ð32Þ
where pðtÞ; F 1 ðtÞ and F 2 ðtÞ is arbitrary function of t, then we can obtain the solution of Eq. (3)
v¼
12pðtÞ 3
3xpðtÞ þ 7y7 F 1 ðtÞ þ 7pt ðtÞy þ 3F 2 ðtÞ
;
then the solution of Eq. (1) is
36pðtÞ2 u¼ 2 : 3 3xpðtÞ þ 7y7 F 1 ðtÞ þ 7pt ðtÞy þ 3F 2 ðtÞ We can also get a new solution of Eq. (3) by theorem.
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4. Conclusions In summary, making use of the generalized symmetry group method and symbolic computation, the fully symmetry transformation groups for the nonisospectral BKP equation are given. The full symmetry group of the nonisospectral BKP equation is a product of one discrete group ðD5 Þ and one infinite dimensional Kac–Moody–Virasoro type Lie group with four arbitrary functions. The relationship is constructed between the new solutions and the old ones of equation, which is a relationship of group invariant solution. If we have obtained a solution of the nonisospectral BKP equation by other method, we can use this relationship to obtain another solution. Then we prove the nonisospectral BKP can pass the Painlevé test and hence it is expected to be integrable. So we obtain a solution of the nonisospectral BKP by the standard truncated Painlevé expansion, then we get new general solution by the relationship, which is the known one-soliton solution [29]. It is necessary to point out that the general solution contain all its group invariant solutions, so it is enough to submit it to the relationship for one time only. Acknowledgements We would like to thank Prof. Senyue Lou, Dr. Yuqi Li, Zhongzhou Dong, Xiaorui Hu, Jia Wang and Wangchuan Ye for their enthusiastic guidance and helpful discussions. The work is supported by the National Natural Science Foundation of China (Grant Nos. 10735030, 10747141 and 90718041), Shanghai Leading Academic Discipline Project (No. B412), Program for Changjiang Scholars and Innovative Research Team in University (IRT0734) Zhejiang Provincial Natural Science Foundations of China (Grant No. 605408), Ningbo Natural Science Foundation (Grant Nos. 2007A610049 and 2006A610093) and K.C. Wong Magna Fund in Ningbo University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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