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An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation
Applied Mathematics and Computation 219 (2013) 5837–5848
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An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation Yufeng Zhang a,⇑, Zhong Han a, Hon-Wah Tam b a b
Center of Nonlinear Equations, College of Sciences, China University of Mining and Technology, Xuzhou 221116, PR China Department of Computer Science, Hong Kong Baptist University, Hong Kong, PR China
a r t i c l e
i n f o
a b s t r a c t A new integrable hierarchy of evolution equations is obtained by making use of a Lie algebra and Tu–Ma scheme, from which a new generalized Broer–Kaup (gBK) equation is produced. Then two kinds of Darboux transformations, the bilinear presentation, the bilinear Bäcklund transformation and the new Lax pair of the gBK equation are generated, respectively, by employing the Bell polynomials. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Integrable hierarchy Bäcklund transformation Bell polynomials
1. Introduction Tu [1] once employed Lie algebras as a tool for generating nonlinear-equation hierarchies and proposed a powerful approach for producing Hamiltonian structures of soliton hierarchies. Ma [2] called the method the Tu–Ma scheme. By following the way one has obtained some interesting integrable hierarchies of evolution type and their Hamiltonian structures, Darboux transformations, and some other properties [2–8]. Of course, the way for generating equation hierarchies is not unique. For example, papers [9–13] adopted the Lenard operators to generate nonlinear equations. Once an integrable hierarchy is obtained, the single nonlinear equation or coupled nonlinear equations are followed to present by reductions. For example, the well-known nonlinear Schrödinger equation can be derived from the reduction of the AKNS hierarchy. The KdV equation can be obtained by reduction of the AKNS hierarchy as well. Study of the algebraic properties of such the equations is an important aspect in soliton theory. Li et al. [14,15] proposed some ways for generating Darboux transformations of nonlinear soliton equations by starting from isospectral problems. In the paper, we shall introduce a Lie algebra based on the Lie algebra A1 . Then we adopt the Tu–Ma scheme to derive a new integrable soliton hierarchy which can reduce to a coupled generalized Broer–Kaup (gBK) equation, whose two kinds of Darboux transformations are obtained. Finally, the bilinear representation, the bilinear Bäcklund transformation and a new Lax pair of the gBK equation are given, respectively, with the help of the Bell polynomials. 2. An integrable hierarchy and its Hamiltonian structure A simple subalgebra of the Lie algebra A1 presents
a ¼ ff1 ; f2 ; f3 g;
ð1Þ
where
f1 ¼
1 0 ; 0 1
f2 ¼
0 1 ; 0 0
f3 ¼
0 0 : 1 0
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Zhang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.086
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
along with the commutative relations
½f1 ; f2 ¼ 2f 2 ;
½f1 ; f3 ¼ 2f 3 ;
½f2 ; f3 ¼ f1 :
Make a linear transformation as follows
f : A1 ! A1 ;
ð2Þ
with that
t1 ¼ f2 2f 3 ;
t 2 ¼ 2f 3 ;
t3 ¼ f1 :
It is easy to see that the commutation operations among t 1 ; t 2 and t3 are as follows
½t 1 ; t2 ¼ 2t3 ;
½t 1 ; t 3 ¼ 2t 1 4t 2 ;
½t 2 ; t 3 ¼ 2t2 :
ð3Þ
Denote by b ¼ ft 1 ; t 2 ; t 3 g , it is easy to verify that it forms a Lie algebra with the commutators 3. A loop algebra of b can be defined as
~ ¼ ft1 ðnÞ; t2 ðnÞ; t3 ðnÞg; b
ð4Þ
where
ti ðnÞ ¼ t i kn ;
½t i ðmÞ; t j ðnÞ ¼ ½t i ; t j kmþn ;
1 6 i < j 6 3; m; n 2 Z:
With the help of (4), let us introduce an isospectral problem as follows
v
ux ¼ U u; U ¼ t3 ð1Þ þ t3 ð0Þ wt2 ð0Þ þ t1 ð0Þ; 2
where
v ¼ v ðx; tÞ;
ð5Þ
w ¼ wðx; tÞ are derivative potential functions. If let
ut ¼ V u; V ¼
1 X ðam t 1 ðmÞ þ bm t2 ðmÞ þ cm t 3 ðmÞÞ;
ð6Þ
m¼0
then the stationary zero curvature equation
@ V ¼ ½U; V @x
ð7Þ
admits the following recursive relations
8 1 1 > < bmþ1 ¼ 2 ðbm;x 2am;x Þ þ 2 v bm þ wcm ; 1 1 amþ1 ¼ 2 am;x þ 2 v am cm ; > : cm;x ¼ 2wam þ 2bm :
ð8Þ
Eq. (8) is local. In fact, if set a0 ¼ b0 ¼ 0; c0 ¼ a – 0, then we have from Eq. (8) that
a1 ¼ a;
b1 ¼ aw;
a
a
a2 ¼ a3 ¼ b3 ¼
a
2
v;
b2 ¼
2
a
c1 ¼ 0; ðwx þ wv Þ;
a
c2 ¼
a 2
w;
4
ðv x v 2 Þ w; 2
4
ðwxx þ ðwv Þx þ v wx þ wv 2 Þ þ ðv x þ w2 Þ; . . . 2
a
c3 ¼
2
wx þ aðwv þ v Þ;
a
Denote that
V ðnÞ þ ¼
n X ðam t1 ðn mÞ þ bm t2 ðn mÞ þ cm t 3 ðn mÞÞ ¼ kn V V ðnÞ ; m¼0
then Eq. (7) can be decomposed into the following ðnÞ ðnÞ ðnÞ V ðnÞ þ;x þ ½U; V þ ¼ V ;x ½U; V :
By calculation of the right-hand side of Eq. (9), we have ðnÞ V ðnÞ þ;x þ ½U; V þ ¼ 2anþ1 t 1 ð0Þ þ ð4anþ1 2bnþ1 Þt 2 ð0Þ: ðnÞ Choose a modified term Dn of V ðnÞ by V ðnÞ ¼ V ðnÞ þ as Dn ¼ anþ1 t 3 ð0Þ, and denote V þ þ Dn , we find that
ðnÞ V ðnÞ ¼ ð2bnþ1 þ 2wanþ1 Þt 2 ð0Þ anþ1;x t 3 ð0Þ: x þ ½U; V
Thus, the compatibility condition of the following isospectral problems
ð9Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
ux ¼ U u; ut ¼ V ðnÞ u
ð10Þ
gives rise to an integrable soliton hierarchy
utn ¼
v w
¼
2anþ1;x 2bnþ1 2wanþ1
tn
:
ð11Þ
The Lax matrices U and V in Eqs. (5) and (6) can be written as
U¼
!
k þ v2
1
2w 2
kv
;
V¼
c
a
2a þ 2b c
2
;
ð12Þ
P P P where a ¼ mP0 am km ; b ¼ mP0 bm km ; c ¼ mP0 cm km . We remark that this spectral problem (12) introduced by using the loop algebra (4) can be transformed into the spectral problem presented in [12] which has more potentials. In what follows, we want to investigate the Hamiltonian structure of system (11). Denote ha; bi ¼ traceðabÞ ¼ ~ one infers that trðabÞ; a; b 2 b,
@U ¼ c; V; @v
@U V; ¼ 2a; @w
@U V; ¼ 2c: @k
Substituting into the trace identity proposed by Tu [1] yields that
c d @ ð2cÞ ¼ kc kc : du @k 2a
ð13Þ
Comparing the coefficients of kn1 of both sides in (13) reads
cn d : ð2cnþ1 Þ ¼ ðn þ cÞ du 2an
ð14Þ
Applying the initial values in Eq. (8), one gets c ¼ 0. Thus, we have
cn
¼
2an
dHn ; du
Hn ¼
2cnþ1 : n
ð15Þ
The Hamiltonian form of the inregrable soliton hierarchy (11) is given by
utn ¼
v w
¼
tn
2anþ1;x 2bnþ1 2wanþ1
¼
2anþ1;x cnþ1;x
¼
0
@
@
0
cnþ1 2anþ1
¼J
cnþ1 2anþ1
¼ JL
cn 2an
¼ JL
dHn ; du ð16Þ
where J is a Hamiltonian operator, and L is a recurrence operator which satisfies that
cnþ1 2anþ1
cn ¼L ; 2an
L¼
1 1 @ 2
v @ þ 12
1 ð@ 1 w@ þ w þ 2 1 ð @Þ 2
v
2
2Þ
! ;
@ here @ ¼ @x ; @ 1 @ ¼ @@ 1 ¼ 1. Let a ¼ 2; n ¼ 2; t2 ¼ t, the integrable hierarchy (11) reduces to a generalized Broer–Kaup (gBK) equation as follows
v t ¼ v xx 2vv x 2wx ; wt ¼ wxx 2ðwv Þx 2v x ;
ð17Þ
whose Lax pair matrices present that
8 > > > > /x ¼ U/; > < > > > > > :V ¼
/t ¼ V/;
U¼
k þ v2
1
2w 2 k v2
! ;
2k2 þ 12 ðv x v 2 Þ
2k v
4kð1 þ wÞ þ 2v þ 2wx þ 2wv
2k2 þ 12 ðv 2 v x Þ
That is, /xt ¼ /tx leads to
U t V x þ ½U; V ¼ 0; which gives rise to the gBK Eq. (17).
! :
ð18Þ
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3. Two kinds of Darboux transformations of the gBK equation Ma [16] investigated in detail the Darboux transformations for a Lax pair integrable system in 2n-dimensions in which the spectral problem can be able to give the spectral problem presented in the paper through a propewr gauge transformation. Based on this and Li, work, in the section we shall investigate the Darboux transformations of the gBK equation. Consider a Darboux transformation
¼ T/; /
ð19Þ
and / satisfying a new spectral problem and require /
/; x ¼ U /
ð20Þ
has the same form as U expect replacing w and v by w and v , respectively. It is easy to and the spectral problem 5, where U see that T meets
T x þ TU ¼ UT:
ð21Þ
In what follows, we discuss two kinds of forms of matrix T. 3.1. The first Darboux transformation Assume that
T ¼ a0
k þ a1
b1
c1
d1
;
ð22Þ
where a0 ; a1 ; b1 ; c1 ; d1 are functions in x and t. Inserting (22) into (21) reads that
a0;x
k þ a1
b1
c1
d1
¼ a0
þ a0
a1;x
b1;x
c1;x
d1;x
þ a0
ðk þ v2 Þðk þ a1 Þ þ c1 2Þðk þ a1 Þ þ ðk v2 Þc1 ð2w
ðk þ a1 Þðk þ v2Þ ð2w þ 2Þb1 ðk þ v2Þc1 ð2w þ 2Þd1 ! ðk þ v2 Þb1 þ d1 : 2Þb1 þ ðk v2 Þd1 ð2w
k þ a1 þ ðk v2Þb1 c1 þ ðk v2Þd1
! ð23Þ
Comparing the coefficients of kj ðj ¼ 2; 1; 0Þ in Eq. (23), we find that the case of j ¼ 2 is trivial. As for the case of j ¼ 1, we get that
a0;x þ
v 2
a0 ¼
v 2
a0 ;
ð24Þ
a0 þ a0 b1 ¼ a0 b1 ;
ð25Þ
2Þ þ a0 c1 ; a0 c1 ¼ a0 ð2w
ð26Þ
a0 d1 ¼ a0 d1 :
ð27Þ
The case of j ¼ 0 leads to the following equations
a0;x a1 þ a0 a1;x þ
1 1 v a0 a1 ð2w þ 2Þa0 b1 ¼ v a0 a1 þ a0 c1 ; 2 2
a0;x b1 þ a0 b1;x þ a0 a1 a0;x c1 þ a0 c1;x þ
1 1 v a0 b1 ¼ a0 d1 þ v a0 b1 ; 2 2
1 1 v a0 c1 ðð2w þ 2Þa0 d1 ¼ ð2w þ 2Þa0 a1 v c1 a0 ; 2 2
a0;x d1 þ a0 d1;x þ a0 c1
1 1 v a0 d1 ¼ ð2w þ 2Þa0 b1 v a0 d1 : 2 2
ð28Þ
ð29Þ
ð30Þ
ð31Þ
Substituting (24) into (28)–(31) gives
a1;x ¼ ð2w þ 2Þb1 þ c1 ;
ð32Þ
b1;x ¼ v b1 þ d1 a1 ;
ð33Þ
þ 2Þa1 v c1 ; c1;x ¼ ð2w þ 2Þd1 ð2w
ð34Þ
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þ 2Þb1 : d1;x ¼ ðv v Þd1 c1 ð2w
ð35Þ
Inserting (25) and (26) into (32)–(35), one infers that
w; a1;x ¼ w
ð36Þ
v 2d1 þ 2a1 ¼ 0;
ð37Þ
x ¼ 2ðw wÞd 1 ðw þ 1Þð2v v Þ; w
ð38Þ
d1;x ¼ ðv v Þd1 :
ð39Þ
¼ 0, we see that det U ¼ det U of (5) and (20) are 2 2 matrices, and trU ¼ trU ¼ 0. There exists a Since the solutions U and U constant k ¼ k1 and a solution / ¼ ð/1 ; /2 ÞT of the spectral problem (5) which satisfy that
a0 ðk1 þ a1 Þ/1 þ a0 b1 /2 ¼ 0; a0 c1 /1 þ a0 d1 /2 ¼ 0;
that is,
b1
/2 ¼ k1 þ a1 ; /1
c1 þ
/2 d1 ¼ 0: /1
Combining (25) and (26), we get
a1 ¼ k1 þ
¼wþ w
1 /2 ; 2 /1
ð40Þ
1 /2 : 2 /1 x
ð41Þ
From (37), we obtain
v ¼ v
d1;x ; d1
ð42Þ
here d1 satisfies
d1 ¼
1 1 1/ v a1 ¼ v þ k1 2 : 2 2 2 /1
ð43Þ
and v . After verifications we see that (36) and (38) hold automatically. Eqs. (41) and (42) manifest the new solutions w 3.2. The second Darboux transformation Suppose T has the following form
T ¼ d0
a2
b2
c2
k þ d2
;
ð44Þ
where d0 ; a2 ; b2 ; c2 ; d2 are all functions in x and t. Inserting (44) into (21) yields that
d0;x
a2
b2
c2
k þ d2
¼ d0
þ d0
a2;x
b2;x
c2;x
d2;x
ðk þ v2 Þa2 þ c2 þ 2Þa2 þ ðk v2 Þc2 ð2w
ðk þ v2Þa2 ð2w þ 2Þb2 a2 þ ðk v2Þb2 þ d0 v ðk þ 2Þc2 ð2w þ 2Þðk þ d2 Þ c2 þ ðk v2Þðk þ d2 Þ ! ðk þ v2 Þb2 þ k þ d2 : þ 2Þb2 þ ðk v2 Þðk þ d2 Þ ð2w
! ð45Þ
Similar to the discussion as above, comparing the coefficients of kj ðj ¼ 2; 1; 0Þ of both sides in Eq. (45), we get a few cases. For the case of j ¼ 2, it is obviously trivial. The case j ¼ 1 shows
d0 ða2 Þ ¼ d0 a2 ; d0 b2 ¼ d0 ð1 b2 Þ;
ð46Þ
d0 ðc2 2w 2Þ ¼ d0 c2 ;
ð47Þ
d0;x þ d0 ð
v þ d2 Þ ¼ d0 þ d2 : 2 2
v
ð48Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
The case j ¼ 0 leads to the following equations
d0;x a2 þ d0 a2;x þ d0
1 v a2 ð2w þ 2Þb2 2
¼ d0
1 v a2 þ c2 ; 2
ð49Þ
1 1 d0;x b2 þ d0 b2;x þ d0 a2 v b2 ¼ d0 v b2 þ d2 ; 2 2 d0;x c2 þ d0 c2;x þ d0
ð50Þ
1 1 v c2 ð2w þ 2Þd2 ¼ d0 ð2w þ 2Þa2 v c2 ; 2 2
ð51Þ
1 1 þ 2Þb2 v d2 ; d0;x d2 þ d0 d2;x þ d0 c2 v d2 ¼ d0 ð2w 2 2
ð52Þ
Eqs. (46)–(48) are reduced to
1 ; 2
ð53Þ
c2 ¼ w 1;
ð54Þ
b2 ¼
d0;x ¼
1 ðv v Þd0 : 2
ð55Þ
By employing (53)–(55), Eqs. (49)–(52) are reduced to again
1 1 a2;x ¼ ðv v Þ þ ðv v Þa2 ; 2 2 a2 d2 ¼
ð56Þ
1 v ; 2
ð57Þ
þ 2Þa2 þ ðw þ 1Þðv v Þ þ wx ¼ 0; ð2w þ 2Þd2 þ ð2w
ð58Þ
d2;x ¼ w w:
ð59Þ T
On the other hand, there exists a constant k ¼ k2 and a solution w ¼ ðw1 ; w2 Þ of Eq. (5), which satisfies that
d0 a2 w1 þ d0 b2 w2 ¼ 0;
d0 c2 w1 þ d0 ðk2 þ d2 Þw2 ¼ 0;
ð60Þ
that is,
a2 ¼
1 w2 ; 2 w1
ð61Þ
d2 ¼
1 w1 w k2 : 2 w2
ð62Þ
From (57) and (59), we have a new solution to the gBK Eq. (17) as follows
(
v ¼ ww
2 1
w ww12 þ 2k2 ;
¼ w d2;x ; w
ð63Þ
where d2 is given by (62). It can be directly verified that Eqs. (56) and (58) are satisfied automatically. Next, we shall prove again that the transformations (22) and (44) map equation
/t ¼ V/;
ð64Þ
where V is presented in (18), into equation
t ¼ V /; /
ð65Þ
has the same form as V in (64) except replacing w; v ; wx ; v x by w; xt ¼ / tx , that v ; w x; v x . The compatibility condition / here V t V þ ½U; V ¼ 0 holds so that ðw; v Þ is a new solution of the gBK equation. For the sake, we should prove is, U
T t þ TV ¼ VT:
ð66Þ
Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
5843
v Þ given by Proposition. If ðw; v Þ is a solution of Eq. (17), ð/1 ; /2 ÞT is a solution of (18) along with k ¼ k1 , then the functions ðw; 41 and 42 is a new solution of the gBK equation. Proof. It is enough to prove (66) holds. Substituting the transformation (22) into (66) along with
V¼
!
2k21 þ 12 ðv x v 2 Þ
2k1 v
ð4 þ 4wÞk1 þ 2v þ 2wx þ 2wv
2k21 v2x þ 12 v 2
2k21 þ 12 ðv x v 2 Þ
2k1 v
1 þ 2v þ 2w x þ 2w v ð4 þ 4wÞk
2k21 v2x þ 12 v 2
ð67Þ
and
V ¼
! ð68Þ
yields that
a0;t
k1 þ a1
b1
c1
d1
þ a0
a1;t
b1;t
c1;t
d1;t
þ a0
a11 a21
a12 a22
¼ a0
b11
b12
b21
b22
;
ð69Þ
where
1 1 a11 ðk1 þ a1 Þ 2k21 þ v x v 2 þ b1 ½ð4 þ 4wÞk1 þ 2v þ 2wx þ 2wv ; 2 2 vx v2 a12 ¼ ð2k1 v Þðk1 þ a1 Þ þ b1 2k21 þ ; 2 2 vx v2 þ d1 ½ð4 þ 4wÞk1 þ 2v þ 2wx þ 2wv ; a21 ¼ c1 2k21 þ 2 2 vx v2 a22 ¼ ð2k1 v Þc1 þ d1 2k21 þ ; 2 2 v x v 2 c1 ð2k1 þ v Þ; b11 ¼ ðk1 þ a1 Þ 2k21 þ 2 2 v x v 2 b12 ¼ b1 2k21 þ ð2k1 þ v Þd1 ; 2 2 v v 2 1 þ 2v þ 2w x þ 2w v þ c1 2k21 x þ ; b21 ¼ ðk1 þ a1 Þ½ð4 þ 4wÞk 2 2 v v 2 1 þ 2v þ 2w x þ 2w v þ d1 2k21 x þ b22 ¼ b1 ½ð4 þ 4wÞk : 2 2 Comparing the coefficients of kj ðj ¼ 3; 2; 1; 0Þ on both sides in Eq. (69), we get that
2a0 c1 ; a0;t þ a0 w þ 4a0 ð1 þ wÞb1 ¼ a0 w
ð70Þ
a0 v 2a0 a1 ¼ 2a0 d1 ;
ð71Þ
1: 2a0 c1 ¼ 4a0 ð1 þ wÞb
ð72Þ
The case of j ¼ 0 leads to
v c1 Þ; a0;t a1 þ a0 a1;t þ a0 ða1 w þ 2wx b1 Þ þ 2a0 v ð1 þ wÞb1 ¼ a0 ða1 w
ð73Þ
1 v d1 Þ; a0;t b1 þ a0 b1;t þ a0 ðv a1 b1 wÞ ¼ a0 ðwb
ð74Þ
wc 1 ; x þ 2a1 v ð1 þ wÞ a0;t c1 þ a0 c1;t þ a0 ½wc1 þ 2wx d1 þ 2v ð1 þ wÞd1 ¼ a0 ½2a1 w
ð75Þ
x b1 þ 2v ð1 þ wÞb 1 þ wd 1 : a0;t d1 þ a0 d1;t þ a0 ðv c1 wd1 Þ ¼ a0 ½2w
ð76Þ
Eq. (71) is just right Eq. (37). Eq. (72) is the same with Eq. (26). By using Eqs. (25) and (26), we see that Eqs. (73)–(76) can be reduced to
1 1 v a0 þ a0 a1 ðv x v 2 Þ; a0;t a1 þ a0 a1;t þ ðv x v 2 Þa0 a1 ðv þ wx þ wv Þa0 ¼ ð1 þ wÞ 2 2
1 1 1 2 v v x v d1 a0 ; a0;t þ v a1 þ ðv x v 2 Þ a0 ¼ 2 4 4
ð77Þ
ð78Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
1 0;t þ w x a0 þ ð1 þ wÞð v x v 2 Þa0 þ ð2v þ 2wx þ 2wv Þa0 d1 ð1 þ wÞa 2 1 x þ w v Þa0 a1 þ ð1 þ wÞð v 2 v x Þa0 ; ¼ 2ðv þ w 2
ð79Þ
1 1 0 þ a0 d1 ðv 2 v x Þ ¼ ðv þ w x þ w v Þa0 a0 d1 ðv x v 2 Þ: a0;t d1 þ a0 d1;t v ð1 þ wÞa 2 2
ð80Þ
From (24) and (39) we get that
M a0 ¼ pffiffiffiffiffi ; d1
ð81Þ
where M is a constant. In terms of Eq. (18) with k ¼ k1 , one infers that
8 2
/2 > < //2 ¼ //2 þ 2 k1 v2 /1 þ 2w 2; 1 x 1
2 > : /2 ¼ ð4 þ 4wÞk þ 2v þ 2w þ 2wv ð4k2 þ v v 2 Þ /2 þ ð2k þ v Þ /2 : 1 x x 1 1 /1 /1 /1
ð82Þ
t
By using (24)–(26), (71), (36)–(39) and (81), (82), we can verify that (77) holds. Similarly, Eqs. (78)–(80)can be proved. Through taking the above similar ways, the results for the transformations 44 can also be proved, here we omit them. h Remark. By the Darboux transformation (19) along with (22), we obtained the new solutions (41) and (42) of the gBK Eq. of Eq. (17) is obviously different from its original solution ðv ; wÞ. Obviously, v ¼ 0; w ¼ 0 is a set of ; wÞ (17). The solution ðv trivial solutions of (17), it follows from (41) and (42) that a set of new solutions of Eq. (17) presents that
v ¼
/2;x /1 /2 /1;x ; 2ð2k1 /1 /2 Þ/1
¼ w
1 /2 ; 2 /1 x
where the functions /1 ; /2 satisfy the Eq. (18) when take v ¼ 0; w ¼ 0. Similarly, it is easy to see that the new solution (63) of Eq. (17) differs from its original solution v ; w. 4. Integrable properties of the gBK equation In this section, we shall use the Bell polynomials to investigate the bilinear representation, the bilinear Bäcklund transformations and the new Lax pair of the gBK equation. We first briefly recall some known results on the Bell polynomials. The recursion equation
Y nþ1 ¼
n X ckn Y nk ykþ1
ð83Þ
k¼0
for n ¼ 0; 1; 2; . . . along with the initial condition
Y0 ¼ 1 is called the Bell polynomials [17]. Denote k ¼ ðk1 ; k2 ; . . . ; kn Þ; ki P 0; i ¼ 1; 2; . . . ; n k! ¼ k1 !k2 ! . . . kn !; jkj ¼ k1 þ þ kn ; kkk ¼ k1 þ 2k2 þ þ nkn . There holds for n P 1:
Y n ðy1 ; . . . ; yn Þ ¼
y kn X n! y k1 y k2 1 2 ... n : k! 1! 2! n! kkk¼n
and
set
ð84Þ
A simple form of the Bell polynomials presents [18]:
Y nt ðyÞ ¼ Y n ðyt ; . . . ; ynt Þ ¼ ey @ nt ey ; eat
1 2!
ð85Þ @ kt y.
2
where y ¼ a0 a1 t þ a2 t þ . . . ; ykt ¼ Recently, Lambert et al. [19,20] proposed a general generalization of the Bell polynomials. Set nk P 0; k ¼ 1; 2 . . . ; l denote arbitrary integers, f ¼ f ðx1 ; . . . ; xl Þ be a c1 multi-variable function, the following
Y n1 x1 ;...;nl xl ðf Þ ef @ nx11 . . . @ nxll ef
ð86Þ
is called multi-dimensional Bell polynomials, or Y-polynomials. As for the case of f ¼ f ðx; tÞ, the related two-dimensional Bell polynomials are inferred as follows
Y x ðf Þ ¼ fx ;
Y 2x ðf Þ ¼ f2x þ fx2 ;
Y 2x;t ¼ f2x;t þ f2x ft þ 2f x;t fx þ
Y 3x ðf Þ ¼ f3x þ 3f x f2x þ fx3 ;
fx2 ft ; . . .
Y x;t ðf Þ ¼ fx;t þ fx ft ;
Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
5845
With the help of (86), the multi-dimensional binary Bell polynomials (also denoted by !polynomials) can be defined as
(
!n1 x1 ;...;nl xl ðv ; wÞ ¼ Y n1 x1 ;...;nl xl ðf Þjfr
1 x1 ;...;r l xl
¼
v r x ;...;r x ;
r1 þ rl
wr1 x1 ;...;rl xl ;
r 1 þ þ r l xl
1 1
l l
is odd; is even;
ð87Þ
where k ¼ 0; 1; . . . ; l. The first few lowest order Bell polynomials are presented as follows
(
!x ðv Þ ¼ v x ; !2x ðv ; wÞ ¼ w2x þ v 2x ; !x;t ðv ; wÞ ¼ wx;t þ v x v t ; !3x ðv ; wÞ ¼ v 3x þ 3v x w2x þ v 2x ; . . .
ð88Þ
A beautiful relation between the binary Bell polynomials !n1 x1 ;...;nl xl ðv ; wÞ and the standard Hirota bilinear operator Dnx11 Dnxll F G is given by [18]:
!n1 x1 ;...;nl xl ðv ¼ ln F=G; w ¼ ln FGÞ ¼ ðFGÞ1 Dnx11 Dnxll F G:
ð89Þ
specially, when F ¼ G, 89 reduces to
( G2 Dnx11 . . . Dnxll G G ¼ !n1 x1 ;...;nl xl ð0; q ¼ 2 ln GÞ ¼
0; n1 þ þ nl
is odd;
Pn1 x1 ;...;nl xl ðqÞ; n1 þ . . . þ nl
is even:
ð90Þ
The first few Ppolynomials are given by 88 as follows
(
P2x ðqÞ ¼ q2x ; Px;t ðqÞ ¼ qx;t ; P4x ðqÞ ¼ q4x þ 3q22x ; P6x ðqÞ ¼ q6x þ 15q2x q4x þ 15q22x ; . . .
ð91Þ
Fan, et al. [21,22] extended the bell polynomials to investigate the integrable properties of some variable-coefficient nonlinear partial differential equations, including the bilinear Bäcklund transformations, Lax pairs, and so on. Furthermore, Fan [23,24] extended the Bell polynomials to the case of the super-Bell polynomials and obtained some interesting results. Based on the method and theory mentioned as above, we want to investigate the bilinear Bäcklund transformation and the new Lax pair of the gBK Eq. (17), Set v ¼ c1 V x ; w ¼ c2 W 2x ; c1 and c2 are constants to be determined. Substituting into the gBK Eq. (17) yields
(
c1 V x;t c1 V 3x þ 2c21 V x V 2x þ 2c2 W 3x ¼ 0; c2 W 2x;t þ c2 W 4x þ 2c1 c2 ðW 2x V x Þx þ 2c1 V 2x ¼ 0:
ð92Þ
Integrating once time with respect to x gives
(
c1 V t c1 V 2x þ c21 ðV x Þ2 þ 2c2 W 2x ¼ 0; c2 W x;t þ c2 W 3x þ 2c1 c2 W 2x V x þ 2c1 V x ¼ 0:
ð93Þ
The first equation in (93) can be cast into
c1 c1 V t þ 2c2 W V þ c21 V 2x ¼ 0: 2c2 2x
ð94Þ
¼ W 2cc1 V; V ¼ v , then W ¼ w þ 2cc1 v . Thus, Eq. (94) becomes that Set w 2 2
2x þ c21 v 2x ¼ 0: c1 v t þ 2c2 w
ð95Þ
The second equation in (93) is cast into
c c c þ 1 v þ 1 v þ 1 v c2 w þ c2 w þ 2c1 c2 w v x þ 2c1 v x ¼ 0: 2c2 2c2 2c2 x;t 3x 2x
ð96Þ
Set 2c2 ¼ c21 ¼ 1, then Eqs. (95) and (96) reduce to, respectively
v t þ w 2x þ v 2x ¼ 0;
ð97Þ
3x þ v 3x þ 2w 2x v x þ 2v 2x v x þ 4v x ¼ 0; x;t þ v x;t þ w w
ð98Þ
which can be expressed by the Bell polynomials as follows
¼ 0; !t ðv Þ þ !2x ðv ; wÞ þ !3x ðv ; wÞ þ 4!x ðv Þ ¼ 0: !x;t ðv ; wÞ ¼ lnðF=GÞ; w ¼ lnðFGÞ, we get the bilinear representation of the gBK equation Let v
ð99Þ ð100Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
(
ðDt þ D2x ÞF G ¼ 0;
ð101Þ
ðDx Dt þ D3x þ 4Dx ÞF G ¼ 0: For the sake of simplicity, we rewrite Eqs. (99) and (100) in the following
v t þ w2x þ v 2x ¼ 0;
ð102Þ
wx;t þ v 3x þ 2w2x v x þ 4v x ¼ 0:
ð103Þ
; w are solutions to Eqs. (102) and (103). Thus, we have that From (97), (98), we know v
wÞ2x þ ðv v Þx ðv þ v Þx ¼ 0; ðv v Þt þ ðw
ð104Þ
wÞx;t þ ðv v Þ3x þ ðv x v x Þðw 2x þ w2x Þ þ ðw 2x w2x Þðv x þ v x Þ þ 4ðv v Þx ¼ 0: ðw
ð105Þ
Set
(
v ¼ lnðF=GÞ; ~ v 2 ¼ lnðF=FÞ;
~ ~ ~ ~ ~ GÞ; ¼ lnðF~GÞ; w ¼ lnðFGÞ; v ¼ lnðF= w v 1 ¼ lnðG=GÞ; w1 ¼ lnðGGÞ; ~ ~ ~ ~ ~ w2 ¼ lnðFFÞ; v 3 ¼ lnðF=GÞ; w3 ¼ lnðFGÞ; v 4 ¼ lnðG=FÞ; w4 ¼ lnðGFÞ;
ð106Þ
we find that
v v ¼ v 2 v 1 ;
w ¼ v 1 þ v 2 ¼ v 3 þ v 4 ¼ w1 w2 þ 2v 3 ; w
þ w ¼ w1 þ w2 ; w
v 2 v 1 ¼ 2w3 w1 w2 ; v 4 ¼ v 1 v ;
v þ v ¼ v 3 v 4 ¼ 2v 3 v 1 v 2 ; v 3 ¼ v 2 þ v ; w3 ¼ w2 v ; w4 ¼ w1 þ v : ð107Þ
With the help of (108), Eq. (105) becomes
ðv 2 v 1 Þt þ ðw1 w2 Þ2x þ 2v 3;2x þ 2v 3;x ðv 2 v 1 Þx ðv 22;x v 21;x Þ ¼ 0; that is,
ðv 2 v 1 Þt þ w1;2x þ v 21;x ðw2;2x þ v 22;x Þ þ 2v 3;2x þ 2v 3;x ðv 2 v 1 Þx ¼ 0: Set
ð108Þ
v 3;2x þ v 3;x ðv 2 v 1 Þx ¼ 0, we get
v 3;x ¼ cev
v :
1 2
ð109Þ
Thus, Eq. (108) can be written as
v 2;t v 1;t þ w1;2x þ v 21;x ðw2;2x þ v 22;x Þ ¼ 0; which can be written as by using the Bell polynomials
!t ðv 2 Þ !2x ðv 2 ; w2 Þ !t ðv 1 Þ þ !2x ðv 1 ; w1 Þ ¼ 0:
ð110Þ
Under the constraint (109), Eq. (110) can be decomposed into
!t ðv 1 Þ !2x ðv 1 ; w1 Þ ¼ 0;
!t ðv 2 Þ !2x ðv 2 ; w2 Þ ¼ 0:
ð111Þ
By using (108) and (111), Eq. (105) becomes
w3;3x þ w3;2x ðv 2 v 1 Þx þ v 3;x ðv 1 þ v 2 Þ2x þ 2ðv 2 v 1 Þx ¼ 0:
ð112Þ
Eq. (109) can lead to
v 3;2x ¼ ðv 1 v 2 Þx v 3;x ; thus, Eq. (112) can be cast into
w3;2x
v 3;x
þ ðv 1 þ v 2 Þ2x þ
2ðv 2 v 1 Þx
x
v 3;x
¼ 0;
ð113Þ
again by using (107), we get
w3;2x
v 3;x
x
þ ðv 1 þ v 2 Þ2x 2
v 3;2x ¼ 0: v 23;x
ð114Þ
Integrating with respect to x yields that
w3;2x þ v 23;x þ v 3;x v 4;x av 3;x þ 2 ¼ 0; here a is an integration constant, which can be expressed by the Bell polynomials
ð115Þ
Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
!2x ðv 3 ; w3 Þ þ !x ðv 3 Þ!x ðv 4 Þ a!x ðv 3 Þ þ 2 ¼ 0:
5847
ð116Þ
By employing (107), we have from Eq. (116) that
~ F ¼ 0: ðD2x aDx þ 2ÞF~ G þ Dx G
ð117Þ
Eq. (112) can be written as
ðDt D2x ÞF~ F ¼ 0;
~ G ¼ 0: ðDt D2x ÞG
ð118Þ
Through using (107), Eq. (109) is cast into
~ Dx F~ G ¼ cF G:
ð119Þ
Therefore, Eqs. (117)–(119) constitute the bilinear Bäcklund transformation of the gBK Eq. (17). In what follows, we discuss the new Lax pair of the gBK equation with the help of the Bell polynomials. Introducing the new variables v i ¼ ln wi ; i ¼ 1; 2; 3; 4. Eq. (109) is equivalent to
w3;x ¼ c
w1 w3 : w2
ð120Þ
Due to w2 ¼ ev 2 ; w3 ¼ ev 3 , Eq. (120) becomes that
w3;x ¼ cev w1 :
ð121Þ
Since
!2x ðv 3 ; w3 Þ ¼ ðw v Þ2x þ
w3;2x ; w3
w4;x ¼
w1;x vx; w1
Eq. (115) is cast into
½ðw v Þ2x þ 2
w3 @ þ cev ðln w21 Þ acev ¼ 0: @x w1
ð122Þ
Hence, Eqs. (121) and (122) constitute the space part of the Lax pair of the gBK equation. In order to deduce the time part of the Lax pair, we have from (107) that w1 v 1 ¼ w v . Hence, the first equation in Eq. (112) becomes
v 1;t v 1;2x ðw v Þ2x v 21;x ¼ 0; that is,
w1;t w1 þ w1;2x w1 w21;x ðw v Þ2x w21 w21;x ¼ 0:
ð123Þ
Since
v 2;t ¼
w3;t w3;2x w3;x v t ; !2x ðv 2 ; w2 Þ ¼ w2x þ 2v x þ v 2x ; w3 w3 w3
the second equation in Eq. (112) becomes
w3;t w3;2x w3;x v t w2x þ 2v x v 2x ¼ 0; w3 w3 w3 that is,
@ 2 ðw Þ 2w3;2x þ 4v x w3;x ¼ 0: @t 3
ð124Þ
Eqs. (123) and (124) are just the time part of the Lax pais of the gBK equation. 5. Conclusion In the paper we derived the gBK Eq. (17) by reduction of the new integrable system (16). Furthermore, the Darboux transformations and the bilinear Bäcklund transformation of the gBK Eq. (17) were obtained, respectively. We have known the fact that Darboux transformations and Bäcklund transformations are powerful tools for investigating solutions to nonlinear equations. As long as see solutions are presented, a series of new solutions could be produced. While the approaches for seeking see solutions have many ways, such as the rational-function method for constructing traveling wave solutions, the multiple exp-function method for multiple wave solutions [25,26], and so on. Recently, the linear supperposition principle method [27,28] for constructing linear-subspaces of solutions was proposed, which provides us a new way for investigating exact solutions of equations. With the help of these ways mentioned as above, we could obtain some exact solutions of the gBK equation. This problem will be treated in another paper.
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
However, it is remarkable that since the gBK Eq. (17) is a new generalized equation of the known original BK equation, we conclude that its solutions are not the same with the BK equation at all. Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (2012 LWB51). 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]
G.Z. Tu, The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys. 30 (2) (1989) 330–338. W.X. Ma, A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction, Chin. J. Contemp. Math. 13 (1) (1992) 79–89. W.X. Ma, The algebraic structure related to L-A-B triad representations of integrable system, Chin. Sci. Bull. 37 (14) (1992) 1249–1253. W.X. Ma, The Lie algebra structures of time-dependent symmetries of evolution equations, Acta Math. Sci. 14 (4) (1994) 388–392. X.B. Hu, A powerful approach to generate new integrable systems, J. Phys. A 27 (1994) 2497–2514. E.G. Fan, A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equation, J. Math. Phys. 42 (2001) 4327– 4344. E.G. Fan, Integrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian strutcure, finiti-dimensional integrable systems and Nfold Darboux transformation, J. Math. Phys. 47 (2000) 7769–7782. Y.F. Zhang, H.Q. Zhang, A method for integrable coupling of TD hierarchy, J. Math. Phys. 43 (1) (2002) 466–472. X.G. Geng, W.X. Ma, A mulitpotential generalization of the nonlinear diffusion equation, J. Phys. Soc. Jpn. 69 (4) (2000) 985–986. X.G. Geng, H.H. Dai, Decomposition and straightening out of the coupled mixed nonlinear Schroding flows, Phys. Lett. A 20 (2004) 311–321. Zhijun Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys. 47 (2006) 112701. 09. Zhijun Qiao, Integrable hierarchy, 3 3 constrained systems, and parametric solutions, Acta Appl. Math. 83 (2004) 199–220. Zhijun Qiao, S. Li, A new integrable hierarchy, parametric solutions and traveling wave solutions, Math. Phys. Anal. Geomet. 7 (2004) 289–308. YiShen Li, Jin E. Zhang, Darboux transformations of classical Boussinesq system and its multi-soliton solutions, Phys. Lett. A 284 (2001) 253–258. Yishen Li, Wen-Xiu Ma, Jin E. Zhang, Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A 275 (2000) 60–66. W.X. Ma, Darboux transformations for a Lax integrable system in 2n-dimensions, Lett. Math. Phys. 39 (1997) 33–49. E.T. Bell, Exponential polynomials, Ann. Math. 35 (1934) 258–277. F. Lambert, I. Loris, J. Springael, Classical Darboux transformations and the KP hierarchy, Inverse Probl. 17 (2001) 1067–1074. C. Gilson, F. Lambert, J. Nimmo, R. Willox, On the combinatorics of the Hirota–Doperators, Proc. R. Soc. Lond. A 452 (1996) 223–234. F. Lmabert, J. Springael, Soliton equations and simple combinatorics, Acta Appl. Math. 102 (2008) 147–178. E.G. Fan, The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials, Phys. Lett. A 375 (3) (2011) 493–497. E.G. Fan, K.W. Chow, Darboux covariant Lax pairs and infinite conservation laws of the (2 + 1)-dimensional breaking soliton equation, J. Math. Phys. 52 (2011) 023504. pp. 1–10. E.G. Fan, New bilinear Backlund transformation and Lax pair for the supersymmetric two-Boson equation, Stud. Appl. Math. 127 (2011) 284–301. E.G. Fan, Super extension of Bell polynomials with applications to supersymmetric equation, J. Math. Phys. 53 (2012) 013503. pp. 1–8. W.X. Ma et al, Nonlinear continuous integrable Hamiltonian couplings, Appl. Math. Comput. 217 (2011) 7238–7277. E.G. Fan, Quasi-periodic waves and asymptotic property for the asymmetrical Nizhnik–Norikov–Veselov equation, J. Phys. A 42 (095206) (2009) 1751– 8113. W.X. Ma, E.G. Fan, Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl. 61 (2011) 950–959. W.X. Ma et al, Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218 (2012) 7174–7183.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation Yufeng Zhang a,⇑, Zhong Han a, Hon-Wah Tam b a b
Center of Nonlinear Equations, College of Sciences, China University of Mining and Technology, Xuzhou 221116, PR China Department of Computer Science, Hong Kong Baptist University, Hong Kong, PR China
a r t i c l e
i n f o
a b s t r a c t A new integrable hierarchy of evolution equations is obtained by making use of a Lie algebra and Tu–Ma scheme, from which a new generalized Broer–Kaup (gBK) equation is produced. Then two kinds of Darboux transformations, the bilinear presentation, the bilinear Bäcklund transformation and the new Lax pair of the gBK equation are generated, respectively, by employing the Bell polynomials. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Integrable hierarchy Bäcklund transformation Bell polynomials
1. Introduction Tu [1] once employed Lie algebras as a tool for generating nonlinear-equation hierarchies and proposed a powerful approach for producing Hamiltonian structures of soliton hierarchies. Ma [2] called the method the Tu–Ma scheme. By following the way one has obtained some interesting integrable hierarchies of evolution type and their Hamiltonian structures, Darboux transformations, and some other properties [2–8]. Of course, the way for generating equation hierarchies is not unique. For example, papers [9–13] adopted the Lenard operators to generate nonlinear equations. Once an integrable hierarchy is obtained, the single nonlinear equation or coupled nonlinear equations are followed to present by reductions. For example, the well-known nonlinear Schrödinger equation can be derived from the reduction of the AKNS hierarchy. The KdV equation can be obtained by reduction of the AKNS hierarchy as well. Study of the algebraic properties of such the equations is an important aspect in soliton theory. Li et al. [14,15] proposed some ways for generating Darboux transformations of nonlinear soliton equations by starting from isospectral problems. In the paper, we shall introduce a Lie algebra based on the Lie algebra A1 . Then we adopt the Tu–Ma scheme to derive a new integrable soliton hierarchy which can reduce to a coupled generalized Broer–Kaup (gBK) equation, whose two kinds of Darboux transformations are obtained. Finally, the bilinear representation, the bilinear Bäcklund transformation and a new Lax pair of the gBK equation are given, respectively, with the help of the Bell polynomials. 2. An integrable hierarchy and its Hamiltonian structure A simple subalgebra of the Lie algebra A1 presents
a ¼ ff1 ; f2 ; f3 g;
ð1Þ
where
f1 ¼
1 0 ; 0 1
f2 ¼
0 1 ; 0 0
f3 ¼
0 0 : 1 0
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Zhang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.086
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
along with the commutative relations
½f1 ; f2 ¼ 2f 2 ;
½f1 ; f3 ¼ 2f 3 ;
½f2 ; f3 ¼ f1 :
Make a linear transformation as follows
f : A1 ! A1 ;
ð2Þ
with that
t1 ¼ f2 2f 3 ;
t 2 ¼ 2f 3 ;
t3 ¼ f1 :
It is easy to see that the commutation operations among t 1 ; t 2 and t3 are as follows
½t 1 ; t2 ¼ 2t3 ;
½t 1 ; t 3 ¼ 2t 1 4t 2 ;
½t 2 ; t 3 ¼ 2t2 :
ð3Þ
Denote by b ¼ ft 1 ; t 2 ; t 3 g , it is easy to verify that it forms a Lie algebra with the commutators 3. A loop algebra of b can be defined as
~ ¼ ft1 ðnÞ; t2 ðnÞ; t3 ðnÞg; b
ð4Þ
where
ti ðnÞ ¼ t i kn ;
½t i ðmÞ; t j ðnÞ ¼ ½t i ; t j kmþn ;
1 6 i < j 6 3; m; n 2 Z:
With the help of (4), let us introduce an isospectral problem as follows
v
ux ¼ U u; U ¼ t3 ð1Þ þ t3 ð0Þ wt2 ð0Þ þ t1 ð0Þ; 2
where
v ¼ v ðx; tÞ;
ð5Þ
w ¼ wðx; tÞ are derivative potential functions. If let
ut ¼ V u; V ¼
1 X ðam t 1 ðmÞ þ bm t2 ðmÞ þ cm t 3 ðmÞÞ;
ð6Þ
m¼0
then the stationary zero curvature equation
@ V ¼ ½U; V @x
ð7Þ
admits the following recursive relations
8 1 1 > < bmþ1 ¼ 2 ðbm;x 2am;x Þ þ 2 v bm þ wcm ; 1 1 amþ1 ¼ 2 am;x þ 2 v am cm ; > : cm;x ¼ 2wam þ 2bm :
ð8Þ
Eq. (8) is local. In fact, if set a0 ¼ b0 ¼ 0; c0 ¼ a – 0, then we have from Eq. (8) that
a1 ¼ a;
b1 ¼ aw;
a
a
a2 ¼ a3 ¼ b3 ¼
a
2
v;
b2 ¼
2
a
c1 ¼ 0; ðwx þ wv Þ;
a
c2 ¼
a 2
w;
4
ðv x v 2 Þ w; 2
4
ðwxx þ ðwv Þx þ v wx þ wv 2 Þ þ ðv x þ w2 Þ; . . . 2
a
c3 ¼
2
wx þ aðwv þ v Þ;
a
Denote that
V ðnÞ þ ¼
n X ðam t1 ðn mÞ þ bm t2 ðn mÞ þ cm t 3 ðn mÞÞ ¼ kn V V ðnÞ ; m¼0
then Eq. (7) can be decomposed into the following ðnÞ ðnÞ ðnÞ V ðnÞ þ;x þ ½U; V þ ¼ V ;x ½U; V :
By calculation of the right-hand side of Eq. (9), we have ðnÞ V ðnÞ þ;x þ ½U; V þ ¼ 2anþ1 t 1 ð0Þ þ ð4anþ1 2bnþ1 Þt 2 ð0Þ: ðnÞ Choose a modified term Dn of V ðnÞ by V ðnÞ ¼ V ðnÞ þ as Dn ¼ anþ1 t 3 ð0Þ, and denote V þ þ Dn , we find that
ðnÞ V ðnÞ ¼ ð2bnþ1 þ 2wanþ1 Þt 2 ð0Þ anþ1;x t 3 ð0Þ: x þ ½U; V
Thus, the compatibility condition of the following isospectral problems
ð9Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
ux ¼ U u; ut ¼ V ðnÞ u
ð10Þ
gives rise to an integrable soliton hierarchy
utn ¼
v w
¼
2anþ1;x 2bnþ1 2wanþ1
tn
:
ð11Þ
The Lax matrices U and V in Eqs. (5) and (6) can be written as
U¼
!
k þ v2
1
2w 2
kv
;
V¼
c
a
2a þ 2b c
2
;
ð12Þ
P P P where a ¼ mP0 am km ; b ¼ mP0 bm km ; c ¼ mP0 cm km . We remark that this spectral problem (12) introduced by using the loop algebra (4) can be transformed into the spectral problem presented in [12] which has more potentials. In what follows, we want to investigate the Hamiltonian structure of system (11). Denote ha; bi ¼ traceðabÞ ¼ ~ one infers that trðabÞ; a; b 2 b,
@U ¼ c; V; @v
@U V; ¼ 2a; @w
@U V; ¼ 2c: @k
Substituting into the trace identity proposed by Tu [1] yields that
c d @ ð2cÞ ¼ kc kc : du @k 2a
ð13Þ
Comparing the coefficients of kn1 of both sides in (13) reads
cn d : ð2cnþ1 Þ ¼ ðn þ cÞ du 2an
ð14Þ
Applying the initial values in Eq. (8), one gets c ¼ 0. Thus, we have
cn
¼
2an
dHn ; du
Hn ¼
2cnþ1 : n
ð15Þ
The Hamiltonian form of the inregrable soliton hierarchy (11) is given by
utn ¼
v w
¼
tn
2anþ1;x 2bnþ1 2wanþ1
¼
2anþ1;x cnþ1;x
¼
0
@
@
0
cnþ1 2anþ1
¼J
cnþ1 2anþ1
¼ JL
cn 2an
¼ JL
dHn ; du ð16Þ
where J is a Hamiltonian operator, and L is a recurrence operator which satisfies that
cnþ1 2anþ1
cn ¼L ; 2an
L¼
1 1 @ 2
v @ þ 12
1 ð@ 1 w@ þ w þ 2 1 ð @Þ 2
v
2
2Þ
! ;
@ here @ ¼ @x ; @ 1 @ ¼ @@ 1 ¼ 1. Let a ¼ 2; n ¼ 2; t2 ¼ t, the integrable hierarchy (11) reduces to a generalized Broer–Kaup (gBK) equation as follows
v t ¼ v xx 2vv x 2wx ; wt ¼ wxx 2ðwv Þx 2v x ;
ð17Þ
whose Lax pair matrices present that
8 > > > > /x ¼ U/; > < > > > > > :V ¼
/t ¼ V/;
U¼
k þ v2
1
2w 2 k v2
! ;
2k2 þ 12 ðv x v 2 Þ
2k v
4kð1 þ wÞ þ 2v þ 2wx þ 2wv
2k2 þ 12 ðv 2 v x Þ
That is, /xt ¼ /tx leads to
U t V x þ ½U; V ¼ 0; which gives rise to the gBK Eq. (17).
! :
ð18Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
3. Two kinds of Darboux transformations of the gBK equation Ma [16] investigated in detail the Darboux transformations for a Lax pair integrable system in 2n-dimensions in which the spectral problem can be able to give the spectral problem presented in the paper through a propewr gauge transformation. Based on this and Li, work, in the section we shall investigate the Darboux transformations of the gBK equation. Consider a Darboux transformation
¼ T/; /
ð19Þ
and / satisfying a new spectral problem and require /
/; x ¼ U /
ð20Þ
has the same form as U expect replacing w and v by w and v , respectively. It is easy to and the spectral problem 5, where U see that T meets
T x þ TU ¼ UT:
ð21Þ
In what follows, we discuss two kinds of forms of matrix T. 3.1. The first Darboux transformation Assume that
T ¼ a0
k þ a1
b1
c1
d1
;
ð22Þ
where a0 ; a1 ; b1 ; c1 ; d1 are functions in x and t. Inserting (22) into (21) reads that
a0;x
k þ a1
b1
c1
d1
¼ a0
þ a0
a1;x
b1;x
c1;x
d1;x
þ a0
ðk þ v2 Þðk þ a1 Þ þ c1 2Þðk þ a1 Þ þ ðk v2 Þc1 ð2w
ðk þ a1 Þðk þ v2Þ ð2w þ 2Þb1 ðk þ v2Þc1 ð2w þ 2Þd1 ! ðk þ v2 Þb1 þ d1 : 2Þb1 þ ðk v2 Þd1 ð2w
k þ a1 þ ðk v2Þb1 c1 þ ðk v2Þd1
! ð23Þ
Comparing the coefficients of kj ðj ¼ 2; 1; 0Þ in Eq. (23), we find that the case of j ¼ 2 is trivial. As for the case of j ¼ 1, we get that
a0;x þ
v 2
a0 ¼
v 2
a0 ;
ð24Þ
a0 þ a0 b1 ¼ a0 b1 ;
ð25Þ
2Þ þ a0 c1 ; a0 c1 ¼ a0 ð2w
ð26Þ
a0 d1 ¼ a0 d1 :
ð27Þ
The case of j ¼ 0 leads to the following equations
a0;x a1 þ a0 a1;x þ
1 1 v a0 a1 ð2w þ 2Þa0 b1 ¼ v a0 a1 þ a0 c1 ; 2 2
a0;x b1 þ a0 b1;x þ a0 a1 a0;x c1 þ a0 c1;x þ
1 1 v a0 b1 ¼ a0 d1 þ v a0 b1 ; 2 2
1 1 v a0 c1 ðð2w þ 2Þa0 d1 ¼ ð2w þ 2Þa0 a1 v c1 a0 ; 2 2
a0;x d1 þ a0 d1;x þ a0 c1
1 1 v a0 d1 ¼ ð2w þ 2Þa0 b1 v a0 d1 : 2 2
ð28Þ
ð29Þ
ð30Þ
ð31Þ
Substituting (24) into (28)–(31) gives
a1;x ¼ ð2w þ 2Þb1 þ c1 ;
ð32Þ
b1;x ¼ v b1 þ d1 a1 ;
ð33Þ
þ 2Þa1 v c1 ; c1;x ¼ ð2w þ 2Þd1 ð2w
ð34Þ
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þ 2Þb1 : d1;x ¼ ðv v Þd1 c1 ð2w
ð35Þ
Inserting (25) and (26) into (32)–(35), one infers that
w; a1;x ¼ w
ð36Þ
v 2d1 þ 2a1 ¼ 0;
ð37Þ
x ¼ 2ðw wÞd 1 ðw þ 1Þð2v v Þ; w
ð38Þ
d1;x ¼ ðv v Þd1 :
ð39Þ
¼ 0, we see that det U ¼ det U of (5) and (20) are 2 2 matrices, and trU ¼ trU ¼ 0. There exists a Since the solutions U and U constant k ¼ k1 and a solution / ¼ ð/1 ; /2 ÞT of the spectral problem (5) which satisfy that
a0 ðk1 þ a1 Þ/1 þ a0 b1 /2 ¼ 0; a0 c1 /1 þ a0 d1 /2 ¼ 0;
that is,
b1
/2 ¼ k1 þ a1 ; /1
c1 þ
/2 d1 ¼ 0: /1
Combining (25) and (26), we get
a1 ¼ k1 þ
¼wþ w
1 /2 ; 2 /1
ð40Þ
1 /2 : 2 /1 x
ð41Þ
From (37), we obtain
v ¼ v
d1;x ; d1
ð42Þ
here d1 satisfies
d1 ¼
1 1 1/ v a1 ¼ v þ k1 2 : 2 2 2 /1
ð43Þ
and v . After verifications we see that (36) and (38) hold automatically. Eqs. (41) and (42) manifest the new solutions w 3.2. The second Darboux transformation Suppose T has the following form
T ¼ d0
a2
b2
c2
k þ d2
;
ð44Þ
where d0 ; a2 ; b2 ; c2 ; d2 are all functions in x and t. Inserting (44) into (21) yields that
d0;x
a2
b2
c2
k þ d2
¼ d0
þ d0
a2;x
b2;x
c2;x
d2;x
ðk þ v2 Þa2 þ c2 þ 2Þa2 þ ðk v2 Þc2 ð2w
ðk þ v2Þa2 ð2w þ 2Þb2 a2 þ ðk v2Þb2 þ d0 v ðk þ 2Þc2 ð2w þ 2Þðk þ d2 Þ c2 þ ðk v2Þðk þ d2 Þ ! ðk þ v2 Þb2 þ k þ d2 : þ 2Þb2 þ ðk v2 Þðk þ d2 Þ ð2w
! ð45Þ
Similar to the discussion as above, comparing the coefficients of kj ðj ¼ 2; 1; 0Þ of both sides in Eq. (45), we get a few cases. For the case of j ¼ 2, it is obviously trivial. The case j ¼ 1 shows
d0 ða2 Þ ¼ d0 a2 ; d0 b2 ¼ d0 ð1 b2 Þ;
ð46Þ
d0 ðc2 2w 2Þ ¼ d0 c2 ;
ð47Þ
d0;x þ d0 ð
v þ d2 Þ ¼ d0 þ d2 : 2 2
v
ð48Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
The case j ¼ 0 leads to the following equations
d0;x a2 þ d0 a2;x þ d0
1 v a2 ð2w þ 2Þb2 2
¼ d0
1 v a2 þ c2 ; 2
ð49Þ
1 1 d0;x b2 þ d0 b2;x þ d0 a2 v b2 ¼ d0 v b2 þ d2 ; 2 2 d0;x c2 þ d0 c2;x þ d0
ð50Þ
1 1 v c2 ð2w þ 2Þd2 ¼ d0 ð2w þ 2Þa2 v c2 ; 2 2
ð51Þ
1 1 þ 2Þb2 v d2 ; d0;x d2 þ d0 d2;x þ d0 c2 v d2 ¼ d0 ð2w 2 2
ð52Þ
Eqs. (46)–(48) are reduced to
1 ; 2
ð53Þ
c2 ¼ w 1;
ð54Þ
b2 ¼
d0;x ¼
1 ðv v Þd0 : 2
ð55Þ
By employing (53)–(55), Eqs. (49)–(52) are reduced to again
1 1 a2;x ¼ ðv v Þ þ ðv v Þa2 ; 2 2 a2 d2 ¼
ð56Þ
1 v ; 2
ð57Þ
þ 2Þa2 þ ðw þ 1Þðv v Þ þ wx ¼ 0; ð2w þ 2Þd2 þ ð2w
ð58Þ
d2;x ¼ w w:
ð59Þ T
On the other hand, there exists a constant k ¼ k2 and a solution w ¼ ðw1 ; w2 Þ of Eq. (5), which satisfies that
d0 a2 w1 þ d0 b2 w2 ¼ 0;
d0 c2 w1 þ d0 ðk2 þ d2 Þw2 ¼ 0;
ð60Þ
that is,
a2 ¼
1 w2 ; 2 w1
ð61Þ
d2 ¼
1 w1 w k2 : 2 w2
ð62Þ
From (57) and (59), we have a new solution to the gBK Eq. (17) as follows
(
v ¼ ww
2 1
w ww12 þ 2k2 ;
¼ w d2;x ; w
ð63Þ
where d2 is given by (62). It can be directly verified that Eqs. (56) and (58) are satisfied automatically. Next, we shall prove again that the transformations (22) and (44) map equation
/t ¼ V/;
ð64Þ
where V is presented in (18), into equation
t ¼ V /; /
ð65Þ
has the same form as V in (64) except replacing w; v ; wx ; v x by w; xt ¼ / tx , that v ; w x; v x . The compatibility condition / here V t V þ ½U; V ¼ 0 holds so that ðw; v Þ is a new solution of the gBK equation. For the sake, we should prove is, U
T t þ TV ¼ VT:
ð66Þ
Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
5843
v Þ given by Proposition. If ðw; v Þ is a solution of Eq. (17), ð/1 ; /2 ÞT is a solution of (18) along with k ¼ k1 , then the functions ðw; 41 and 42 is a new solution of the gBK equation. Proof. It is enough to prove (66) holds. Substituting the transformation (22) into (66) along with
V¼
!
2k21 þ 12 ðv x v 2 Þ
2k1 v
ð4 þ 4wÞk1 þ 2v þ 2wx þ 2wv
2k21 v2x þ 12 v 2
2k21 þ 12 ðv x v 2 Þ
2k1 v
1 þ 2v þ 2w x þ 2w v ð4 þ 4wÞk
2k21 v2x þ 12 v 2
ð67Þ
and
V ¼
! ð68Þ
yields that
a0;t
k1 þ a1
b1
c1
d1
þ a0
a1;t
b1;t
c1;t
d1;t
þ a0
a11 a21
a12 a22
¼ a0
b11
b12
b21
b22
;
ð69Þ
where
1 1 a11 ðk1 þ a1 Þ 2k21 þ v x v 2 þ b1 ½ð4 þ 4wÞk1 þ 2v þ 2wx þ 2wv ; 2 2 vx v2 a12 ¼ ð2k1 v Þðk1 þ a1 Þ þ b1 2k21 þ ; 2 2 vx v2 þ d1 ½ð4 þ 4wÞk1 þ 2v þ 2wx þ 2wv ; a21 ¼ c1 2k21 þ 2 2 vx v2 a22 ¼ ð2k1 v Þc1 þ d1 2k21 þ ; 2 2 v x v 2 c1 ð2k1 þ v Þ; b11 ¼ ðk1 þ a1 Þ 2k21 þ 2 2 v x v 2 b12 ¼ b1 2k21 þ ð2k1 þ v Þd1 ; 2 2 v v 2 1 þ 2v þ 2w x þ 2w v þ c1 2k21 x þ ; b21 ¼ ðk1 þ a1 Þ½ð4 þ 4wÞk 2 2 v v 2 1 þ 2v þ 2w x þ 2w v þ d1 2k21 x þ b22 ¼ b1 ½ð4 þ 4wÞk : 2 2 Comparing the coefficients of kj ðj ¼ 3; 2; 1; 0Þ on both sides in Eq. (69), we get that
2a0 c1 ; a0;t þ a0 w þ 4a0 ð1 þ wÞb1 ¼ a0 w
ð70Þ
a0 v 2a0 a1 ¼ 2a0 d1 ;
ð71Þ
1: 2a0 c1 ¼ 4a0 ð1 þ wÞb
ð72Þ
The case of j ¼ 0 leads to
v c1 Þ; a0;t a1 þ a0 a1;t þ a0 ða1 w þ 2wx b1 Þ þ 2a0 v ð1 þ wÞb1 ¼ a0 ða1 w
ð73Þ
1 v d1 Þ; a0;t b1 þ a0 b1;t þ a0 ðv a1 b1 wÞ ¼ a0 ðwb
ð74Þ
wc 1 ; x þ 2a1 v ð1 þ wÞ a0;t c1 þ a0 c1;t þ a0 ½wc1 þ 2wx d1 þ 2v ð1 þ wÞd1 ¼ a0 ½2a1 w
ð75Þ
x b1 þ 2v ð1 þ wÞb 1 þ wd 1 : a0;t d1 þ a0 d1;t þ a0 ðv c1 wd1 Þ ¼ a0 ½2w
ð76Þ
Eq. (71) is just right Eq. (37). Eq. (72) is the same with Eq. (26). By using Eqs. (25) and (26), we see that Eqs. (73)–(76) can be reduced to
1 1 v a0 þ a0 a1 ðv x v 2 Þ; a0;t a1 þ a0 a1;t þ ðv x v 2 Þa0 a1 ðv þ wx þ wv Þa0 ¼ ð1 þ wÞ 2 2
1 1 1 2 v v x v d1 a0 ; a0;t þ v a1 þ ðv x v 2 Þ a0 ¼ 2 4 4
ð77Þ
ð78Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
1 0;t þ w x a0 þ ð1 þ wÞð v x v 2 Þa0 þ ð2v þ 2wx þ 2wv Þa0 d1 ð1 þ wÞa 2 1 x þ w v Þa0 a1 þ ð1 þ wÞð v 2 v x Þa0 ; ¼ 2ðv þ w 2
ð79Þ
1 1 0 þ a0 d1 ðv 2 v x Þ ¼ ðv þ w x þ w v Þa0 a0 d1 ðv x v 2 Þ: a0;t d1 þ a0 d1;t v ð1 þ wÞa 2 2
ð80Þ
From (24) and (39) we get that
M a0 ¼ pffiffiffiffiffi ; d1
ð81Þ
where M is a constant. In terms of Eq. (18) with k ¼ k1 , one infers that
8 2
/2 > < //2 ¼ //2 þ 2 k1 v2 /1 þ 2w 2; 1 x 1
2 > : /2 ¼ ð4 þ 4wÞk þ 2v þ 2w þ 2wv ð4k2 þ v v 2 Þ /2 þ ð2k þ v Þ /2 : 1 x x 1 1 /1 /1 /1
ð82Þ
t
By using (24)–(26), (71), (36)–(39) and (81), (82), we can verify that (77) holds. Similarly, Eqs. (78)–(80)can be proved. Through taking the above similar ways, the results for the transformations 44 can also be proved, here we omit them. h Remark. By the Darboux transformation (19) along with (22), we obtained the new solutions (41) and (42) of the gBK Eq. of Eq. (17) is obviously different from its original solution ðv ; wÞ. Obviously, v ¼ 0; w ¼ 0 is a set of ; wÞ (17). The solution ðv trivial solutions of (17), it follows from (41) and (42) that a set of new solutions of Eq. (17) presents that
v ¼
/2;x /1 /2 /1;x ; 2ð2k1 /1 /2 Þ/1
¼ w
1 /2 ; 2 /1 x
where the functions /1 ; /2 satisfy the Eq. (18) when take v ¼ 0; w ¼ 0. Similarly, it is easy to see that the new solution (63) of Eq. (17) differs from its original solution v ; w. 4. Integrable properties of the gBK equation In this section, we shall use the Bell polynomials to investigate the bilinear representation, the bilinear Bäcklund transformations and the new Lax pair of the gBK equation. We first briefly recall some known results on the Bell polynomials. The recursion equation
Y nþ1 ¼
n X ckn Y nk ykþ1
ð83Þ
k¼0
for n ¼ 0; 1; 2; . . . along with the initial condition
Y0 ¼ 1 is called the Bell polynomials [17]. Denote k ¼ ðk1 ; k2 ; . . . ; kn Þ; ki P 0; i ¼ 1; 2; . . . ; n k! ¼ k1 !k2 ! . . . kn !; jkj ¼ k1 þ þ kn ; kkk ¼ k1 þ 2k2 þ þ nkn . There holds for n P 1:
Y n ðy1 ; . . . ; yn Þ ¼
y kn X n! y k1 y k2 1 2 ... n : k! 1! 2! n! kkk¼n
and
set
ð84Þ
A simple form of the Bell polynomials presents [18]:
Y nt ðyÞ ¼ Y n ðyt ; . . . ; ynt Þ ¼ ey @ nt ey ; eat
1 2!
ð85Þ @ kt y.
2
where y ¼ a0 a1 t þ a2 t þ . . . ; ykt ¼ Recently, Lambert et al. [19,20] proposed a general generalization of the Bell polynomials. Set nk P 0; k ¼ 1; 2 . . . ; l denote arbitrary integers, f ¼ f ðx1 ; . . . ; xl Þ be a c1 multi-variable function, the following
Y n1 x1 ;...;nl xl ðf Þ ef @ nx11 . . . @ nxll ef
ð86Þ
is called multi-dimensional Bell polynomials, or Y-polynomials. As for the case of f ¼ f ðx; tÞ, the related two-dimensional Bell polynomials are inferred as follows
Y x ðf Þ ¼ fx ;
Y 2x ðf Þ ¼ f2x þ fx2 ;
Y 2x;t ¼ f2x;t þ f2x ft þ 2f x;t fx þ
Y 3x ðf Þ ¼ f3x þ 3f x f2x þ fx3 ;
fx2 ft ; . . .
Y x;t ðf Þ ¼ fx;t þ fx ft ;
Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
5845
With the help of (86), the multi-dimensional binary Bell polynomials (also denoted by !polynomials) can be defined as
(
!n1 x1 ;...;nl xl ðv ; wÞ ¼ Y n1 x1 ;...;nl xl ðf Þjfr
1 x1 ;...;r l xl
¼
v r x ;...;r x ;
r1 þ rl
wr1 x1 ;...;rl xl ;
r 1 þ þ r l xl
1 1
l l
is odd; is even;
ð87Þ
where k ¼ 0; 1; . . . ; l. The first few lowest order Bell polynomials are presented as follows
(
!x ðv Þ ¼ v x ; !2x ðv ; wÞ ¼ w2x þ v 2x ; !x;t ðv ; wÞ ¼ wx;t þ v x v t ; !3x ðv ; wÞ ¼ v 3x þ 3v x w2x þ v 2x ; . . .
ð88Þ
A beautiful relation between the binary Bell polynomials !n1 x1 ;...;nl xl ðv ; wÞ and the standard Hirota bilinear operator Dnx11 Dnxll F G is given by [18]:
!n1 x1 ;...;nl xl ðv ¼ ln F=G; w ¼ ln FGÞ ¼ ðFGÞ1 Dnx11 Dnxll F G:
ð89Þ
specially, when F ¼ G, 89 reduces to
( G2 Dnx11 . . . Dnxll G G ¼ !n1 x1 ;...;nl xl ð0; q ¼ 2 ln GÞ ¼
0; n1 þ þ nl
is odd;
Pn1 x1 ;...;nl xl ðqÞ; n1 þ . . . þ nl
is even:
ð90Þ
The first few Ppolynomials are given by 88 as follows
(
P2x ðqÞ ¼ q2x ; Px;t ðqÞ ¼ qx;t ; P4x ðqÞ ¼ q4x þ 3q22x ; P6x ðqÞ ¼ q6x þ 15q2x q4x þ 15q22x ; . . .
ð91Þ
Fan, et al. [21,22] extended the bell polynomials to investigate the integrable properties of some variable-coefficient nonlinear partial differential equations, including the bilinear Bäcklund transformations, Lax pairs, and so on. Furthermore, Fan [23,24] extended the Bell polynomials to the case of the super-Bell polynomials and obtained some interesting results. Based on the method and theory mentioned as above, we want to investigate the bilinear Bäcklund transformation and the new Lax pair of the gBK Eq. (17), Set v ¼ c1 V x ; w ¼ c2 W 2x ; c1 and c2 are constants to be determined. Substituting into the gBK Eq. (17) yields
(
c1 V x;t c1 V 3x þ 2c21 V x V 2x þ 2c2 W 3x ¼ 0; c2 W 2x;t þ c2 W 4x þ 2c1 c2 ðW 2x V x Þx þ 2c1 V 2x ¼ 0:
ð92Þ
Integrating once time with respect to x gives
(
c1 V t c1 V 2x þ c21 ðV x Þ2 þ 2c2 W 2x ¼ 0; c2 W x;t þ c2 W 3x þ 2c1 c2 W 2x V x þ 2c1 V x ¼ 0:
ð93Þ
The first equation in (93) can be cast into
c1 c1 V t þ 2c2 W V þ c21 V 2x ¼ 0: 2c2 2x
ð94Þ
¼ W 2cc1 V; V ¼ v , then W ¼ w þ 2cc1 v . Thus, Eq. (94) becomes that Set w 2 2
2x þ c21 v 2x ¼ 0: c1 v t þ 2c2 w
ð95Þ
The second equation in (93) is cast into
c c c þ 1 v þ 1 v þ 1 v c2 w þ c2 w þ 2c1 c2 w v x þ 2c1 v x ¼ 0: 2c2 2c2 2c2 x;t 3x 2x
ð96Þ
Set 2c2 ¼ c21 ¼ 1, then Eqs. (95) and (96) reduce to, respectively
v t þ w 2x þ v 2x ¼ 0;
ð97Þ
3x þ v 3x þ 2w 2x v x þ 2v 2x v x þ 4v x ¼ 0; x;t þ v x;t þ w w
ð98Þ
which can be expressed by the Bell polynomials as follows
¼ 0; !t ðv Þ þ !2x ðv ; wÞ þ !3x ðv ; wÞ þ 4!x ðv Þ ¼ 0: !x;t ðv ; wÞ ¼ lnðF=GÞ; w ¼ lnðFGÞ, we get the bilinear representation of the gBK equation Let v
ð99Þ ð100Þ
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Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
(
ðDt þ D2x ÞF G ¼ 0;
ð101Þ
ðDx Dt þ D3x þ 4Dx ÞF G ¼ 0: For the sake of simplicity, we rewrite Eqs. (99) and (100) in the following
v t þ w2x þ v 2x ¼ 0;
ð102Þ
wx;t þ v 3x þ 2w2x v x þ 4v x ¼ 0:
ð103Þ
; w are solutions to Eqs. (102) and (103). Thus, we have that From (97), (98), we know v
wÞ2x þ ðv v Þx ðv þ v Þx ¼ 0; ðv v Þt þ ðw
ð104Þ
wÞx;t þ ðv v Þ3x þ ðv x v x Þðw 2x þ w2x Þ þ ðw 2x w2x Þðv x þ v x Þ þ 4ðv v Þx ¼ 0: ðw
ð105Þ
Set
(
v ¼ lnðF=GÞ; ~ v 2 ¼ lnðF=FÞ;
~ ~ ~ ~ ~ GÞ; ¼ lnðF~GÞ; w ¼ lnðFGÞ; v ¼ lnðF= w v 1 ¼ lnðG=GÞ; w1 ¼ lnðGGÞ; ~ ~ ~ ~ ~ w2 ¼ lnðFFÞ; v 3 ¼ lnðF=GÞ; w3 ¼ lnðFGÞ; v 4 ¼ lnðG=FÞ; w4 ¼ lnðGFÞ;
ð106Þ
we find that
v v ¼ v 2 v 1 ;
w ¼ v 1 þ v 2 ¼ v 3 þ v 4 ¼ w1 w2 þ 2v 3 ; w
þ w ¼ w1 þ w2 ; w
v 2 v 1 ¼ 2w3 w1 w2 ; v 4 ¼ v 1 v ;
v þ v ¼ v 3 v 4 ¼ 2v 3 v 1 v 2 ; v 3 ¼ v 2 þ v ; w3 ¼ w2 v ; w4 ¼ w1 þ v : ð107Þ
With the help of (108), Eq. (105) becomes
ðv 2 v 1 Þt þ ðw1 w2 Þ2x þ 2v 3;2x þ 2v 3;x ðv 2 v 1 Þx ðv 22;x v 21;x Þ ¼ 0; that is,
ðv 2 v 1 Þt þ w1;2x þ v 21;x ðw2;2x þ v 22;x Þ þ 2v 3;2x þ 2v 3;x ðv 2 v 1 Þx ¼ 0: Set
ð108Þ
v 3;2x þ v 3;x ðv 2 v 1 Þx ¼ 0, we get
v 3;x ¼ cev
v :
1 2
ð109Þ
Thus, Eq. (108) can be written as
v 2;t v 1;t þ w1;2x þ v 21;x ðw2;2x þ v 22;x Þ ¼ 0; which can be written as by using the Bell polynomials
!t ðv 2 Þ !2x ðv 2 ; w2 Þ !t ðv 1 Þ þ !2x ðv 1 ; w1 Þ ¼ 0:
ð110Þ
Under the constraint (109), Eq. (110) can be decomposed into
!t ðv 1 Þ !2x ðv 1 ; w1 Þ ¼ 0;
!t ðv 2 Þ !2x ðv 2 ; w2 Þ ¼ 0:
ð111Þ
By using (108) and (111), Eq. (105) becomes
w3;3x þ w3;2x ðv 2 v 1 Þx þ v 3;x ðv 1 þ v 2 Þ2x þ 2ðv 2 v 1 Þx ¼ 0:
ð112Þ
Eq. (109) can lead to
v 3;2x ¼ ðv 1 v 2 Þx v 3;x ; thus, Eq. (112) can be cast into
w3;2x
v 3;x
þ ðv 1 þ v 2 Þ2x þ
2ðv 2 v 1 Þx
x
v 3;x
¼ 0;
ð113Þ
again by using (107), we get
w3;2x
v 3;x
x
þ ðv 1 þ v 2 Þ2x 2
v 3;2x ¼ 0: v 23;x
ð114Þ
Integrating with respect to x yields that
w3;2x þ v 23;x þ v 3;x v 4;x av 3;x þ 2 ¼ 0; here a is an integration constant, which can be expressed by the Bell polynomials
ð115Þ
Y. Zhang et al. / Applied Mathematics and Computation 219 (2013) 5837–5848
!2x ðv 3 ; w3 Þ þ !x ðv 3 Þ!x ðv 4 Þ a!x ðv 3 Þ þ 2 ¼ 0:
5847
ð116Þ
By employing (107), we have from Eq. (116) that
~ F ¼ 0: ðD2x aDx þ 2ÞF~ G þ Dx G
ð117Þ
Eq. (112) can be written as
ðDt D2x ÞF~ F ¼ 0;
~ G ¼ 0: ðDt D2x ÞG
ð118Þ
Through using (107), Eq. (109) is cast into
~ Dx F~ G ¼ cF G:
ð119Þ
Therefore, Eqs. (117)–(119) constitute the bilinear Bäcklund transformation of the gBK Eq. (17). In what follows, we discuss the new Lax pair of the gBK equation with the help of the Bell polynomials. Introducing the new variables v i ¼ ln wi ; i ¼ 1; 2; 3; 4. Eq. (109) is equivalent to
w3;x ¼ c
w1 w3 : w2
ð120Þ
Due to w2 ¼ ev 2 ; w3 ¼ ev 3 , Eq. (120) becomes that
w3;x ¼ cev w1 :
ð121Þ
Since
!2x ðv 3 ; w3 Þ ¼ ðw v Þ2x þ
w3;2x ; w3
w4;x ¼
w1;x vx; w1
Eq. (115) is cast into
½ðw v Þ2x þ 2
w3 @ þ cev ðln w21 Þ acev ¼ 0: @x w1
ð122Þ
Hence, Eqs. (121) and (122) constitute the space part of the Lax pair of the gBK equation. In order to deduce the time part of the Lax pair, we have from (107) that w1 v 1 ¼ w v . Hence, the first equation in Eq. (112) becomes
v 1;t v 1;2x ðw v Þ2x v 21;x ¼ 0; that is,
w1;t w1 þ w1;2x w1 w21;x ðw v Þ2x w21 w21;x ¼ 0:
ð123Þ
Since
v 2;t ¼
w3;t w3;2x w3;x v t ; !2x ðv 2 ; w2 Þ ¼ w2x þ 2v x þ v 2x ; w3 w3 w3
the second equation in Eq. (112) becomes
w3;t w3;2x w3;x v t w2x þ 2v x v 2x ¼ 0; w3 w3 w3 that is,
@ 2 ðw Þ 2w3;2x þ 4v x w3;x ¼ 0: @t 3
ð124Þ
Eqs. (123) and (124) are just the time part of the Lax pais of the gBK equation. 5. Conclusion In the paper we derived the gBK Eq. (17) by reduction of the new integrable system (16). Furthermore, the Darboux transformations and the bilinear Bäcklund transformation of the gBK Eq. (17) were obtained, respectively. We have known the fact that Darboux transformations and Bäcklund transformations are powerful tools for investigating solutions to nonlinear equations. As long as see solutions are presented, a series of new solutions could be produced. While the approaches for seeking see solutions have many ways, such as the rational-function method for constructing traveling wave solutions, the multiple exp-function method for multiple wave solutions [25,26], and so on. Recently, the linear supperposition principle method [27,28] for constructing linear-subspaces of solutions was proposed, which provides us a new way for investigating exact solutions of equations. With the help of these ways mentioned as above, we could obtain some exact solutions of the gBK equation. This problem will be treated in another paper.
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However, it is remarkable that since the gBK Eq. (17) is a new generalized equation of the known original BK equation, we conclude that its solutions are not the same with the BK equation at all. Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (2012 LWB51). 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]
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