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A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation
Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
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A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation Zhaqilao College of Mathematics Science, Inner Mongolia Normal University, Huhhot 010022, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 July 2011 Accepted 9 October 2011 Available online 25 October 2011
A new generalized AKNS hierarchy is presented starting from a 4 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Darboux transformation Soliton solution Generalized AKNS hierarchy Bi-Hamiltonian structure
1. Introduction It is well known that the Darboux transformation (DT) method plays an important role in solving many nonlinear evolution equations [1–13] such as Korteweg-de Vries (KdV), nonlinear Schrödinger, sine–Gordon, and other nonlinear differential equations which have great application in physics. But the main difficulty of the DT method lies in finding an appropriate spectral problem for a given nonlinear evolution equation. Hence, it is interesting for us to search for as many new spectral problems and corresponding nonlinear evolution equations as possible in soliton theory. In 1989, Tu [14] proposed a so-called trace identity to generate integrable evolution equations form the spectral problem. According to this method, many integrable evolution equations, their soliton hierarchies, and Hamiltonian structures are successively obtained [14–18]. We note that most of these well-known soliton hierarchies are related to the 2 2 zero-trace matrix spectral problems [19–22]. It is more important to investigate their mathematical structures and integrable properties, such as Hamiltonian structures [23], constrained flows [24], Darboux transformations (DT) [3], soliton solutions [4–13], algebra–geometric solutions [25], etc. [26]. Some 3 3 and 4 4 matrix AKNS spectral problems have been studied in many papers [27,28], but they have not considered their Darboux transformations and explicit solutions. In this paper, we are interested in bi-Hamiltonian structure, Darboux transformation and multiple soliton solutions for a generalized AKNS hierarchy associated with the following 4 4 matrix spectral problem
0 /x ¼ U/;
u1
u2
Bu B 4 U¼B @ u3
k
u3 C C C; k u4 A u1 k
0
0 u2
0
1
k
0
which is introduced by a matrix Lie algebra [24]. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.10.010
ð1Þ
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The outline of the present paper is as follows. In Section 2, a new generalized AKNS hierarchy is derived. It is shown that the new hierarchy is integrable in Liouville’s sense and possesses bi-Hamiltonian structure. Most importantly, some important nonlinear equations are produced from the generalized AKNS hierarchy, which including the special coupled nonlinear equation, the coupled KdV equation [29], the KdV equation, the coupled mKdV equation [30] and the mKdV equation, etc. In Section 3, the Darboux transformation for the generalized AKNS hierarchy is constructed by considering the gauge transformation of its Lax pair. In Section 4, some multiple soliton solutions of the generalized AKNS hierarchy are derived by the application of the Darboux transformation. In Section 5, as a reduction, a Darboux transformation of the mKdV equation and its new positon [30], negaton and complexiton [31,32] solutions are also obtained. The conclusion is given in Section 6.
2. A generalized AKNS hierarchy and its bi-Hamiltonian structure To obtain the generalized AKNS hierarchy and its bi-Hamiltonian structure with the matrix spectral problem (1), we first consider the stationary zero-curvature equation
V x ½U; V ¼ 0;
V ¼ ðV ij Þ44 ;
ð½U; V ¼ UV VUÞ:
ð2Þ
Let
V 21 ¼ V 34 ¼ A;
V 24 ¼ V 31 ¼ B;
V 11 ¼ V 44 ¼ F;
V 22 ¼ V 33 ¼ F;
A¼
X
aj kj ;
B¼
jP0
X
bj kj ;
V 13 ¼ V 24 ¼ C;
V 12 ¼ V 43 ¼ D;
V 14 ¼ V 41 ¼ H; X
C¼
jP0
cj kj ;
D¼
jP0
ð3Þ
V 23 ¼ V 32 ¼ H;
X
dj kj :
ð4Þ
jP0
Substituting (3) and (4) into (2), we have
F ¼ @ 1 ðu1 A þ u2 B u3 C u4 DÞ; kA ¼ u3 H þ u4 F 12 Ax ;
H ¼ @ 1 ðu2 A þ u1 B u4 C u3 DÞ;
kB ¼ u4 H þ u3 F Bx ;
kC ¼ u1 H þ u2 F þ 12 C x ;
ð5Þ kD ¼ u2 H þ u1 F þ 12 Dx :
ð6Þ
Thus from Eqs. (5) and (6) we obtain the Lenard gradient sequence
kðA; B; C; DÞT ¼ LðA; B; C; DÞT ;
ð7Þ
where
0 B B L¼B @
12 @ þ L11
L12
L13
L14
L12
12 @ þ L11
L14
L13
L31
L32
L32
1 @ 2
L31
L33
1 C C C; A
L34 1 @ 2
L34
L33
with
L11 ¼ u3 @ 1 u2 þ u4 @ 1 u1 ; 1
1
L33 ¼ u1 @ 1 u4 þ u2 @ 1 u3 ;
L13 ¼ u3 @ u4 u4 @ u3 ; 1
1
L32 ¼ u1 @ u1 þ u2 @ u2 ;
1
1
L12 ¼ u3 @ 1 u1 þ u4 @ 1 u2 ;
L14 ¼ u3 @ u3 u4 @ u4 ; 1
1
L34 ¼ u1 @ u3 u2 @ u4 ;
L31 ¼ u1 @ 1 u2 þ u2 @ 1 u1 ; @@ 1 ¼ @ 1 @ ¼ 1:
Substituting (4) into (7) and comparing coefficients for the same power of k gives rise to the recursion relation
LGj ¼ Gjþ1 ;
Gj ¼ ðaj ; bj ; cj ; dj ÞT ;
j P 0:
We take the initial value
G0 ¼ ðbu4 ; bu3 ; bu2 ; bu1 ÞT ; then we work out Gj, which is uniquely determined by (8), the first few terms are as follows
0
12 bu4;x
B 1 bu C 3;x C B G1 ¼ B 1 2 C; @ 2 bu2;x A 1 bu1;x 2
0b
1 G2 ¼
1 2u1 u23 þ u24 þ u4;xx 2 C 2u2 u3 þ u24 þ u3;xx C C; 2u3 u21 þ u22 þ u2;xx C A 2 b 2 4u u u 2u u þ u þ u 1 2 3 4 1;xx 1 2 4
4u2 u3 u4 4 Bb B 4 4u1 u3 u4 B B b 4u u u @4 1 2 4
ð8Þ
Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
2321
0 G3 ¼
1 b ð6u2 ðu4 u3;x þ u3 u4;x Þ þ 6u1 ðu3 u3;x þ u4 u4;x Þ u4;xxx Þ 8 B b C B 8 ð6u1 ðu4 u3;x þ u3 u4;x Þ þ 6u2 ðu3 u3;x þ u4 u4;x Þ u3;xxx Þ C B C: B b ð6u ðu u þ u u Þ þ 6u ðu u þ u u Þ u C @ 8 2 4 1;x 3 2;x 1 3 1;x 4 2;x 2;xxx Þ A 8b ð6u1 ðu4 u1;x þ u3 u2;x Þ þ 6u2 ðu3 u1;x þ u4 u2;x Þ u1;xxx Þ
Assuming the auxiliary problem
V ðnÞ ¼ ðkn VÞþ ;
/tn ¼ V ðnÞ /;
n P 0;
ð9Þ
where the symbol + denotes the choice of non-negative power of k. Then the compatibility condition of (1) and (9) gives rise to the zero-curvature representation ðnÞ U tn V ðnÞ ¼ 0; x þ ½U; V
ð10Þ
which is equivalent to
1 u1 Bu C B 2C B C ¼ X n ¼ JGn ¼ KGn1 ; @ u3 A 0
u4
ð11Þ
tn
where
0
0
B 0 B J¼B @ 0
0
0 2
1
2 0C C C; 2 0 0 A
2
0 0
0
0
K ¼ JL and L is given by Eq. (7). When n = 2, we have the auxiliary problem
0 /t2 ¼ V ð2Þ /;
B B B V ð2Þ ¼ B B @
ð2Þ
ð2Þ
bk2 M 5
M1
ð2Þ
bk2 þ M 5
ð2Þ
ð2Þ
M6
M4 M3
ð2Þ
M 6
ð2Þ
M3
1
ð2Þ
ð2Þ
M6
ð2Þ
bk2 þ M5
ð2Þ
ð2Þ
M1
ð2Þ
M2
M 6
ð2Þ
M2
ð2Þ
M4
ð2Þ
bk2 M 5
C C C C; C A
ð12Þ
where
b ð2Þ M 1 ¼ bu1 k þ u1;x ; 2 b ð2Þ M 4 ¼ bu4 k u4;x ; 2
b b ð2Þ ð2Þ M 2 ¼ bu2 k þ u2;x ; M 3 ¼ bu3 k u3;x ; 2 2 b b ð2Þ ð2Þ M 5 ¼ ðu1 u4 þ u2 u3 Þ; M6 ¼ ðu1 u3 þ u2 u4 Þ: 2 2
Substituting (1) and (12) into (10), we have a new coupled integrable equation
8 u1;t2 > > > > u > 3;t 2 > > : u4;t2
¼ 2b 4u1 u2 u3 2u4 u21 þ u22 þ u1;xx ; ¼ 2b 4u1 u2 u4 2u3 u21 þ u22 þ u2;xx ; ¼ 2b 4u1 u3 u4 2u2 u23 þ u24 þ u3;xx ; ¼ 2b 4u2 u3 u4 2u1 u23 þ u24 þ u4;xx :
ð13Þ
When u2 = u3 = 0, b = 1, Eq. (13) is reduced to a special coupled nonlinear equation
1 u1;t2 ¼ u1;xx þ u4 u21 ; 2
u4;t2 ¼
1 u4;xx u1 u24 : 2
ð14Þ
When n = 3, we have the auxiliary problem
0 /t3 ¼ V ð3Þ /;
ð3Þ
V 11
B ð2Þ B M k þ M ð3Þ B 5 V ð3Þ ¼ B 4ð2Þ B M k þ M ð3Þ 6 @ 3 ð2Þ ð3Þ M6 k þ M4
ð2Þ
ð3Þ
M1 k þ M2 ð3Þ
ð2Þ
ð3Þ
M 6 k M 4
ð2Þ
ð3Þ
M3 k þ M6
M2 k þ M3 M6 k þ M4
V 11 ð2Þ
ð3Þ
V 11
ð2Þ
ð3Þ
M1 k þ M2
M6 k þ M4 M2 k þ M3
ð3Þ
ð2Þ
ð2Þ
ð3Þ
ð2Þ
ð3Þ
ð2Þ
ð3Þ
M4 k þ M5 ð3Þ
ð3Þ
V 11
1 C C C C; C A
ð15Þ
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
where ð3Þ
ð2Þ
ð3Þ
ð3Þ
V 11 ¼ bk3 M 5 k þ M 1 ; M 1 ¼ 4b ðu4 u1;x u3 u2;x þ u2 u3;x þ u1 u4;x Þ; ð3Þ ð3Þ M2 ¼ 4b 4u1 u2 u3 2u21 u4 2u22 u4 þ u1;xx ; M 3 ¼ 4b 2u21 u3 2u22 u3 4u1 u2 u4 þ u2;xx ; ð3Þ ð3Þ M4 ¼ 4b ðu3 u1;x þ u4 u2;x u1 u3;x u2 u4;x Þ; M5 ¼ 4b ð4u2 u3 u4 2u1 u23 þ u24 þ u4;xx Þ; ð3Þ
M6 ¼ 4b ð2u2 u23 4u1 u3 u4 2u2 u24 þ u3;xx Þ: Substituting (1) and (15) into (10), we have a new coupled integrable equation
8 u1;t3 > > > > u > 3;t3 > > : u4;t3
¼ 4b ð6u1 ðu4 u1;x þ u3 u2;x Þ þ 6u2 ðu3 u1;x þ u4 u2;x Þ u1;xxx Þ; ¼ 4b ð6u2 ðu4 u1;x þ u3 u2;x Þ þ 6u1 ðu3 u1;x þ u4 u2;x Þ u2;xxx Þ;
ð16Þ
¼ 4b ð6u1 ðu4 u3;x þ u3 u4;x Þ þ 6u2 ðu3 u3;x þ u4 u4;x Þ u3;xxx Þ; ¼ 4b ð6u2 ðu4 u3;x þ u3 u4;x Þ þ 6u1 ðu3 u3;x þ u4 u4;x Þ u4;xxx Þ:
(i) When u4 ¼ 1; u3 ¼ 12 ; b ¼ 4 and t3 = t, Eq. (16) is reduced to the coupled KdV equation [29]
u1;t ¼ 3u2 u1;x þ 3u1 u2;x 6u1 u1;x 6u2 u2;x u1;xxx ;
ð17Þ
u2;t ¼ 3u1 u1;x þ 3u2 u2;x 6u2 u1;x 6u1 u2;x u2;xxx : When u2 = u1 = u, Eq. (17) is reduced to the celebrated KdV equation [1–3]
ut þ 6uux þ uxxx ¼ 0:
ð18Þ
(ii) When u4 = u1, u3 = u2, and t3 = t, Eq. (16) is reduced to the coupled mKdV equation [30]
(
u1;t ¼ 4b 6u21 u1;x þ 6u22 u1;x þ 12u1 u2 u2;x þ u1;xxx ; u2;t ¼ 4b 6u21 u2;x þ 6u22 u2;x þ 12u1 u2 u1;x þ u2;xxx :
ð19Þ
When u1 = u, u2 = 0, Eq. (19) is reduced to the celebrated mKdV equation [3]
ut ¼
b ð6u2 ux þ uxxx Þ: 4
ð20Þ
To investigate the generalized Hamiltonian structure of the hierarchy (11), we take the Killing-Cartan form hA, Bi as tr(AB). Then by direct calculations, we conclude that
@U ¼ 4F; V; @k
@U ¼ 2A; V; @u1
@U V; ¼ 2B; @u2
@U V; ¼ 2C; @u3
@U V; ¼ 2D: @u4
ð21Þ
By using trace identity [14], we have
d d d d ; ; ; du1 du2 du3 du4
V;
@U @k
@ @U @U @U @U ¼ kc kc V; ; V; ; V; ; V; : @k @u1 @u2 @u3 @u4
ð22Þ
Substituting (4) and (21) into (22), we have
d d d d ; ; ; du1 du2 du3 du4
Z 4 ðu1 aj þ u2 bj u3 cj u4 dj Þdx ¼ ðc jÞð2aj ; 2bj ; 2cj ; 2dj Þ;
j P 0:
ð23Þ
To fix the constant c, we compare the coefficient of j = 0 in the above equation and get c = 0. Substituting c = 0 into (23), we arrive at
Z d d d d 2 Hj ¼ GTj ; Hj ¼ ðu1 aj þ u2 bj u3 cj u4 dj Þdx; ; ; ; du1 du2 du3 du4 j
j P 0:
ð24Þ
Therefore, we obtain the generalized bi-Hamiltonian structure of the hierarchy (11)
0
u1
0 dHn1 1
1
du1
Bu C B 2C B C @ u3 A u4
0 dHn 1 du
B dHn1 C B dH1n C C B B C B du C B du C ¼ K B dH 2 C ¼ J B dH2 C; B n1 C B nC @ du3 A @ du3 A tn
dHn1 du4
where K and J are given by (11).
dHn du4
ð25Þ
Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
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3. Darboux transformation In this section, we proceed to construct the Darboux transformation of the generalized AKNS hierarchy (11). We assume that / = (/1, /2, /3, /4)T is a solution of Lax pair (1) and (9). Now we introduce a gauge transformation for the Lax pair (1) and (9)
¼ T/; /
ð26Þ
¼ ð/ 1; / 2; / 3; / 4 ÞT and T is a Darboux matrix, from which Lax pair (1) and (9) are transformed into new matrix specwhere / as follows: tral problems of /
x ¼ U /; / U ¼ ðT x þ TUÞT 1 ; t ¼ V ðnÞ /; / V ðnÞ ¼ ðT t þ TUÞT 1 :
ð27Þ ð28Þ
With the help of the Darboux matrix method [12], we can obtain the following propositions. Proposition 1. The matrix U determined by the second expression of (27) has the same form as U:
0
k Bu B 4 U¼B 3 @u 0
1 0 3 C k 0 u C C 4 A 0 k u 2 u 1 k u 1 u
2 u
and the transformation formulae from original potentials u1, u2, u3 and u4 into new ones are given by
1 ¼ u1 2b; u
2 ¼ u2 2c; u
3 ¼ u3 þ 2f ; u
4 ¼ u4 þ 2s; u
ð29Þ
10 1 0 1 0 /1 kþa b c d /1 C B B/ B C kþh g f C CB /2 C B 2 C B s CB C; B C¼B @ /3 A @ f g kþh s A@ /3 A 4 /4 d c b kþa /
ð30Þ
with
a¼
Da ; D
ð1Þ f1 ð2Þ f D ¼ 1ð1Þ f4 ð2Þ f4
b¼
Db ; D
ð1Þ
f3
ð2Þ
f3
ð1Þ
f2
ð2Þ
f2
f2 f2 f3 f3
ð1Þ ð2Þ ð1Þ ð2Þ
c¼
Dc ; D
d¼
Dd ; D
s¼
Ds ; D
h¼
Dh ; D
g¼
Dg ; D
f ¼
Df ; D
ð1Þ f4 ð2Þ f4 ð1Þ f1 ð2Þ f1
ð31Þ
ð32Þ
T ð1Þ ð2Þ ð1Þ ð2Þ and Da, Db, Dc, Dd, are produced from D by replacing its1st, 2nd, 3rd, 4th column with k1 f1 ; k2 f1 ; k1 f4 ; k2 f4 , respec
T ð1Þ ð2Þ ð1Þ ð2Þ tively. Ds, Dh, Dg, Df are produced from D by replacing its1st, 2nd, 3rd, 4th column with k1 f2 ; k2 f2 ; k1 f3 ; k2 f3 , respec
T
T ð1Þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ tively. Where f1 ; f2 ; f3 ; f4 and f1 ; f2 ; f3 ; f4 are solutions of the Lax pair (1), (9) with k = k1 and k = k2, respectively. Proof. Let T1 = T⁄/detT, where it is a known formula of the inverse matrix,
ðT x þ TUÞT ¼ ðfsl ðkÞÞ44 ;
ð33Þ
⁄
⁄
where T stands for the adjugate matrix of T, and every element of the matrix T is an algebraic cofactors of the matrix T. A direct calculation shows that fsl(k) are polynomials of k. Resorting to Eqs. (31) and (32), it can be verified that all kj(j = 1, 2) are roots of fsl(k)(s, l = 1, 2, 3, 4). Hence, Eq. (33) can be written as
ðT x þ TUÞT ¼ ðdet TÞPðkÞ;
ð34Þ
where
0
k
B ð0Þ Bp B PðkÞ ¼ B 21 B pð0Þ @ 31 0
ð0Þ
ð0Þ
p12
p13
k
0
0
k
ð0Þ
p43
p42
ð0Þ
0
1
C ð0Þ p24 C C C; ð0Þ C p34 A k
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332 ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
with p12 ; p21 ; p13 ; p31 ; p24 ; p42 ; p34 and p43 are functions of (x, t) independent of k. Eq. (34) can be written as
T x þ TU ¼ PðkÞT:
ð35Þ
By comparing the coefficients of k in Eq. (35), we obtain ð0Þ
ð0Þ
p12 ¼ p43 ¼ u1 2b;
ð0Þ
ð0Þ
ð0Þ
p13 ¼ p42 ¼ u2 2c;
ð0Þ
p24 ¼ p31 ¼ u3 þ 2f ;
ð0Þ
ð0Þ
p21 ¼ p34 ¼ u4 þ 2s;
which implies U ¼ PðkÞ. The proof is completed. h Proposition 2. The matrix V ðnÞ determined by the second expression of (28) has the same form as V(n), in which original potentials 1 ; u 2 ; u 3 and u 4 according to the same transformation (29), (30). u1, u2, u3 and u4 are mapped into new potentials u For each integer n P 2, we can prove Proposition 2 similarly to the proof in Proposition 1. According to Propositions 1 and 2, the transformation (29), (30), maps the Lax pair (1) and (9) into another Lax pair (27) and (28) of the same type. Therefore, both of the Lax pair lead to the same system (11). We call the transformation u 1 ; u 2 ; u 3 ; u 4 Þ a Darboux transformation of the generalized AKNS hierarchy (11). We obtain the ð/; u1 ; u2 ; u3 ; u4 Þ ! ð/; following assertion: 1 ; u 2 ; u 3 ; u 4 Þ of the generalized AKNS Theorem 1. Every solution (u1, u2, u3, u4) of the system (11) is mapped into a new solution ðu hierarchy (11) under the Darboux transformation (29), (30).
4. Soliton solutions of the generalized AKNS hierarchy In this section, we will obtain some multi-soliton solutions of the whole generalized AKNS hierarchy (11) by applying the Darboux transformation (29), (30). If choose a trivial solution u1 = 0, u2 = 0, u3 = 0 and u4 = 0 of the generalized AKNS hierarchy (11), then the Lax pair (1) and (9) are written as
/1;x ¼ k/1 ;
/2;x ¼ k/2 ;
/1;t ¼ bkn /1 ;
/3;x ¼ k/3 ;
/2;t ¼ bkn /2 ;
/4;x ¼ k/4 ;
/3;t ¼ bkn /3 ;
ð36Þ
/4;t ¼ bkn /4 :
Eq. (36) has a solutions with k = kj (j = 1, 2, 3, . . .):
1 C 1j exp kj x þ bknj t C B
C C B B C n B ðjÞ C B C 2j exp kj x bkj t C B f2 C B C C¼B B
C; B ðjÞ C B C n B f3 C B C exp kj x bkj t C A B @ C B 3j @ ðjÞ
A f4 n C 4j exp kj x þ bkj t 0
ðjÞ f1
1
0
ð37Þ
where C1j, C2j, C3j, C4j are arbitrary constants and kj(j = 1, 2, 3, . . .) are spectral parameters. According to (29)–(32) and (37) with j = 1, 2, we have a new explicit solution for the generalized AKNS hierarchy (11) as follows:
u1½0 ¼ u1 2b1 ; u2½0 ¼ u2 2c1 ; u3½0 ¼ u3 þ 2f 1 ; u4½0 ¼ u4 þ 2s1 ;
ð38Þ
where b1 = b, c1 = c, f1 = f, s1 = s, a1 = a, d1 = d, g1 = g and h1 = h. Substituting (31), (32) and (37) (with j = 1, 2) into (38) with the zero seed (u1 = u2 = u3 = u4 = 0) and constant parameter selections
C 11 ¼ 1;
C 21 ¼ 2;
C 31 ¼
8 ; 3
C 41 ¼ 5;
C 12 ¼ 3;
C 22 ¼ 3;
C 32 ¼ 2;
one can obtain a new solution for the generalized AKNS hierarchy (11) as follows:
1 C 42 ¼ ; 5
b ¼ 2;
ð39Þ
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
6 expðn1 þ n2 Þð1935 expðn1 Þ þ 952 expðn2 ÞÞðk1 k2 Þ ; 3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 Þ 6 expðn1 þ n2 Þð1215 expðn1 Þ þ 616 expðn2 ÞÞðk1 k2 Þ ¼ ; 3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 Þ 15ð275 expðn1 Þ 126 expðn2 ÞÞðk1 k2 Þ ¼ ; 2ð3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 ÞÞ 5ð1275 expðn1 Þ þ 602 expðn2 ÞÞðk1 k2 Þ ¼ ; 2ð3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 ÞÞ
u1½0 ¼ u2½0 u3½0 u4½0
ð40Þ
where n1 ¼ 2 xk1 þ 2tkn1 ; n2 ¼ 2 xk2 þ 2tkn2 . Particularly, the solutions of the coupled integrable Eqs. (13) and (16) are obtained by (40) with n = 2 and n = 3 respectively. If different spectral parameters k1 and k2 are chosen, the solution determined by (40) may be of different properties. For example, when k1 = 1,k2 = 1, the functions (40) is a one-soliton solution of the generalized AKNS hierarchy (11) (see Fig. 1). After taking the Darboux transformation (29), (30) once again, we can obtain the explicit solution for the generalized AKNS hierarchy (11) as follows:
u1½1 ¼ u1½0 2b2 ;
u2½1 ¼ u2½0 2c2 ;
1 ð3Þ 0 f k3 þ a1 B 1½1 C B ð3Þ C B B f2½1 C B s1 C B B ð3Þ C ¼ B B f3½1 C @ f1 A @ d1 ð3Þ f4½1
u3½1 ¼ u3½0 þ 2f 2 ;
0
0
ð4Þ
c1
k3 þ h1
g1
g1
k3 þ h1
c1
b1
b1
c1
k4 þ h1
g1
g1
k4 þ h1
c1
b1
ð41Þ
1 0 1 f ð3Þ 1 B ð3Þ C C B f1 C CB f2 C CB ð3Þ C; C s1 AB @ f3 A ð3Þ k3 þ a1 f4
ð42Þ
1 0 1 f ð4Þ 1 B ð4Þ C C B f1 C CB f2 C CB ð4Þ C; C s1 AB @ f3 A ð4Þ k4 þ a1 f4
ð43Þ
d1
1
0 k4 þ a1 C B B ð4Þ C B B f2½1 C B s1 C B B ð4Þ C ¼ B B f3½1 C @ f1 A @ d1 ð4Þ f4½1 f1½1
b1
u4½1 ¼ u4½0 þ 2s2 ;
d1
with
u1 0
4
0.1
2
0.05
0 4
0 t
2 x
2 1 u2 0 0 1 2 4
0 t
2
0.05
0
0.1 0.05
x
2 4
0.05
0 2
0.1
4
0.1
0.5 u3 0 0 0.5
0
0.1 0.05
4 2 x
0 2 4
0.1
u4 0
0.5
0 t
1
0.05
4
0.1 0.05 0 t
2 x
0.05
0 2 4
Fig. 1. One soliton solution (40) with k1 = 1, k2 = 1, n = 2.
0.1
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
a2 ¼
D½1a 2 D½1
s2 ¼
D½1s 2 D½1
D½1
; ;
f ð3Þ 1½1 ð4Þ f1½1 ¼ ð3Þ f4½1 ð4Þ f4½1
b2 ¼ h2 ¼
D½1b
2
D½1 D½1h
2
D½1
f2½1
ð3Þ
f3½1
f2½1
ð4Þ
f3½1
f3½1
ð3Þ
f2½1
ð4Þ
f2½1
f3½1
; ;
c2 ¼
D½1c 2 D½1
g2 ¼
D½1g 2 D½1
; ;
d2 ¼ f2 ¼
D½1d
2
D½1 D½1f
2
D½1
; ð44Þ ;
ð3Þ f4½1 ð4Þ f4½1 ð3Þ f1½1 ð4Þ f1½1
ð3Þ ð4Þ ð3Þ ð4Þ
ð45Þ
and D½1a2 ; D½1b2 ; D½1c2 ; D½1d2 , are produced from D[1] by replacing its 1st, 2nd, 3rd, 4th column with
T ð3Þ ð4Þ ð3Þ ð4Þ k3 f1½1 ; k4 f1½1 ; k3 f4½1 ; k4 f4½1 , respectively. D½1s2 ; D½1h2 ; D½1g2 ; D½1f2 are produced from D[1] by replacing its 1st, 2nd,
T
T
T ð3Þ ð4Þ ð3Þ ð4Þ ð3Þ ð3Þ ð3Þ ð3Þ ð4Þ ð4Þ ð4Þ ð4Þ and f1 ; f2 ; f3 ; f4 3rd, 4th column with k3 f2½1 ; k4 f2½1 ; k3 f3½1 ; k4 f3½1 , respectively. Where f1 ; f2 ; f3 ; f4 are solutions of the Lax pair (1), (9) with k = k3 and k = k4, respectively. Substituting (40), (42)–(45) into (41) with the zero seed (u1 = u2 = u3 = u4 = 0) and constant parameter selections
C 11 ¼ 1;
C 21 ¼ 2;
C 31 ¼
1 C 42 ¼ ; 5
k1 ¼ 1;
C 43 ¼ 7;
C 14 ¼ 8;
8 ; 3
k2 ¼ 1;
C 41 ¼ 5; b ¼ 2;
C 24 ¼ 3;
C 12 ¼ 3;
C 13 ¼ 2;
C 34 ¼ 6;
C 22 ¼ 3;
C 32 ¼ 2;
8 C 33 ¼ ; 3 3 3 k3 ¼ ; k4 ¼ ; 4 4
ð46Þ
C 23 ¼ 1;
C 44 ¼ 9;
one can obtain two-soliton solution for the generalized AKNS hierarchy (11), and their graphs are shown in Fig. 2. Iterating the above method, we can obtain a series of multiple soliton solutions of the generalized AKNS hierarchy (11).
5. Reduction of the Darboux transformation and analytical solutions of the mKdV equation In this section, we will discuss a reduction of the Darboux transformation (29), (30), by which the Darboux transformations of the coupled mKdV Eq. (19) and the mKdV Eq. (20) are given. Most importantly, some new nonsingular positon, negaton solutions and complexiton solutions for the mKdV Eq. (20) are given out both analytically and graphically by means of the iterative Darboux transformation.
8 u1 1 6 4 2 0
0.1 0.05 0 t
4
2 x
0
2
2
x
0.05
0 2
0.1
4
0.05 0 t 0 x
0 t
4
0.1
2
0.1 0.05
0.05 4
0 u3 1 1 2 3 4
0 u2 1 1 2 3
0.05
2 4 6
0.1
0.1
0 1 u4 1 2 3 4 4
0.1 0.05 0 t
2 0 x
0.05 2 4
Fig. 2. Two-solution (41) with (42)–(46) and u1 = u2 = u3 = u4 = 0, n = 2.
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
2327
Under the condition u3 = u2, u4 = u1, s = b, g = d, f = c, h = a, Darboux transformation (29), (30) is reduced to the Darboux transformation of the coupled mKdV Eq. (19) as follows
2 ¼ u2 2c; u
1 ¼ u1 2b; u
ð47Þ
10 1 0 1 0 /1 kþa b c d /1 B/ C B B C k a d c C CB /2 C B 2 C B b CB C; B C¼B @ /3 A @ c d k a b A@ /3 A 4 /4 d c b kþa /
ð48Þ
with
a¼
Da ; D
b¼
ð1Þ f1 ð1Þ f D ¼ 2ð1Þ f3 ð1Þ f4
Db ; D
c¼
ð1Þ
f3
ð1Þ
f4
ð1Þ
f1
ð1Þ
f2
f2 f2 f4 f3
Dc ; D
d¼
Dd ; D
ð49Þ
ð1Þ f3 ð1Þ f2 ð1Þ f1
ð1Þ
ð1Þ
f4
ð1Þ ð1Þ ð1Þ
ð50Þ
T ð1Þ ð1Þ ð1Þ ð1Þ and Da, Db, Dc, Dd, are produced from D by replacing its 1st, 2nd, 3rd, 4th column with k1 f1 ; k1 f2 ; k1 f3 ; k1 f4 , respectively. Under the condition u1 = 0, u2 = u, b = d = 0, Darboux transformation (47), (48) is reduced to the Darboux transformation of the mKdV Eq. (20) as follows
¼ u 2c; u
ð51Þ
10 1 0 1 0 /1 kþa 0 c 0 /1 B/ C B B C ka 0 c C CB /2 C B 2 C B 0 CB C; B C¼B @ /2 A @ c 0 ka 0 A@ /2 A 1 0 c 0 kþa /1 /
ð52Þ
with
a¼
2 ð1Þ ð1Þ 2 k1 f2 f1 ð1Þ 2
f2
ð1Þ 2
c¼
;
ð1Þ ð1Þ 2k1 f2 f1 ð1Þ 2
f2
þ f1
ð1Þ 2
;
ð53Þ
þ f1
ðjÞ ðjÞ where f1 ; f2 is solution for the Lax pair of the mKdV Eq. (20)
/1;x ¼ k/1 þ u/2 ; /2;x ¼ u/1 k/2 ; 1 1 /1;t ¼ bkð2k2 þ u2 Þ/1 þ bð4k2 u þ 2u3 þ 2kux þ uxx Þ/2 ; 2 4 1 1 2 2 /2;t ¼ bkð2k þ u Þ/2 bð4k2 u þ 2u3 2kux þ uxx Þ/1 ; 2 4 ðjÞ
ð54Þ
ðjÞ
with k ¼ kj ; /1 ¼ f1 ; /2 ¼ f2 ; ðj ¼ 1; 2; 3; . . .Þ. From (51) and (53), we obtain a solution of the mKdV Eq. (20) as follows
u½0 ¼ u þ
ð1Þ ð1Þ 4k1 f2 f1 ð1Þ 2
f2
ð1Þ 2
:
ð55Þ
þ f1
With the Darboux transformation (51), (52), after taking the solution u[0] in (55) as the seed solution, we can obtain another solution for the mKdV Eq. (20) as follows:
u½1 ¼ u½0 2c2 ;
ð56Þ
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
0
1
1 0 1 f ð2Þ k þ a 0 c 0 1 2 C B B ð2Þ C B ð2Þ C B C B B f2½1 C B 0 0 c C k2 a CB f2 C C¼B B C C; B B ð2Þ C @ c ð2Þ C 0 k2 a 0 AB B f2½1 C @ f2 A A @ ð2Þ 0 c 0 k2 þ a ð2Þ f1 f1½1 ð2Þ
f1½1
0
ð57Þ
with
a2 ¼
2 ð2Þ ð2Þ 2 k2 f2½1 f1½1 ð2Þ 2
ð2Þ 2
c2 ¼
;
ð2Þ ð2Þ 2k2 f2½1 f1½1 ð2Þ 2
f2½1 þ f1½1
ð2Þ 2
ð58Þ
;
f2½1 þ f1½1
ð2Þ ð2Þ is solution of (54) with k = k2. where a, b are give by (53) and f1 ; f2 In what follows, we shall apply the solution formulas (55) and (56) to construct analytical positon, negaton, complexiton, and their interaction solutions for the mKdV Eq. (20). If choose a trivial solution u(x, t) = u (u is a constant) of the mKdV Eqs. (20), (54) are written as
/1;x ¼ k/1 þ u/2 ; /2;x ¼ u/1 k/2 ; 1 1 /1;t ¼ bkð2k2 þ u2 Þ/1 þ bð4k2 u þ 2u3 Þ/2 ; 2 4 1 1 /2;t ¼ bkð2k2 þ u2 Þ/2 bð4k2 u þ 2u3 Þ/1 : 2 4
ð59Þ
5.1. Positon solution 2 Eq. (59) has a solutions with positive parameter k ¼ kj ðj ¼ 1; 2; 3; . . .Þ:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4 u2 kj A sin nj ; ¼ cj1 cos nj @ 2 kj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 4 ucj1 cj2 u2 kj A cos nj þ cj2 sin nj ; ¼@ 2 kj 0
ðjÞ f1
ðjÞ f2
ucj2 þ cj1
ð60Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 4 where nj ¼ 12 u2 kj ð2x þ t 2kj þ u2 bÞ and cj1, cj2 (j = 1, 2, 3, . . .) are arbitrary constants. Substituting (60) into (55) with j = 1 and parameter selections
c11 ¼ c12 ¼
1 ; 4
k1 ¼
1 ; 2
b ¼ 1;
u ¼ 1;
one can obtain the analytical positon solution for the mKdV Eq. (20) as follows:
h pffiffiffiffiffiffi i 1 15 9t8 þ 2x 4 h pffiffiffiffiffiffi u¼ i : 8 þ 2 sin 14 15 9t8 þ 2x 7 2 sin
ð61Þ
(a)
(b) 3 2 1
4 3 u 2 1 0 1
0 5 1 0
t 2
5 5
0 x
3 5
4
2
Fig. 3. One-positon solution for the mKdV Eq. (20) by (61).
0
2
4
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
Fig. 3 shows the nonsingular one-positon structure by (61). Substituting (55), (57), (58) and (60) into (56) with j = 1, 2, positon–positon solution for the mKdV Eq. (20) can be constructed. Fig. 4 shows the nonsingular positon-positon structure (56) with (55), j = 1, 2 and parameter selections
c11 ¼ c12 ¼
1 ; 4
k1 ¼ 1;
k2 ¼
5 ; 8
1 c21 ¼ c22 ¼ ; 5
b ¼ 1;
u ¼ 2:
5.2. Negaton solution and positon–negaton solution 2 Eq. (59) has a solutions with negative parameter k ¼ kj ðj ¼ 1; 2; 3; . . .Þ: ðjÞ f1 ðjÞ f2
cj2 k2j cj2
pffiffiffiffiffiffiffiffiffi ffi 4 k u2
j expðfj Þ; ¼ cj1 expðfj Þ þ u ffi 2 pffiffiffiffiffiffiffiffiffi cj1 kj cj1 k4j u2 expðfj Þ; ¼ cj2 expðfj Þ þ u
ð62Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 where fj ¼ 12 kj u2 ð2x þ tð2kj þ u2 ÞbÞ and cj1, cj2(j = 1, 2, 3, . . .) are arbitrary constants. Substituting (62) with j = 1, c11 ¼ c12 ¼ 14 ; k1 ¼ 12 ; b ¼ 1; u ¼ 16 into (55), one-negaton solution for the mKdV Eq. (20) can be constructed as follows
pffiffi
pffiffi
pffiffi 3 exp 114325t þ 3 exp 35x 14 exp 8645 ð11t þ 144xÞ
pffiffi
pffiffi
pffiffi : u¼
6 3 exp 114325t þ 3 exp 35x 4 exp 8645 ð11t þ 144xÞ
ð63Þ
Fig. 5 shows the nonsingular one-negaton solution structure by (63). Substituting
pffiffiffiffiffiffiffiffiffi ffi c k2 c k4 u2 ð1Þ expðf1 Þ; f1 ¼ c11 expðf1 Þ þ 12 1 12u 1 p ffiffiffiffiffiffiffiffiffi ffi c k2 c k4 u2 ð1Þ expðf1 Þ; f2 ¼ c12 expðf1 Þ þ 11 1 11u 1 pffiffiffiffiffiffiffiffiffi4ffi uc22 þc21 u2 k2 ð2Þ sin n2 ; f1 ¼ c21 cos n2 k22 pffiffiffiffiffiffiffiffiffi4ffi uc21 c22 u2 k2 ð2Þ cos n2 þ c22 sin n2 ; f2 ¼ k22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 4 4 4 f1 ¼ 12 k1 u2 2x þ t 2k1 þ u2 b ; n2 ¼ 12 u2 k2 2x þ t 2k2 þ u2 b
ð64Þ
and (57), (58) into (56), we have positon–negaton solution of the mKdV Eq. (20), and their graphs are shown in Fig. 6. 5.3. Complexiton solution If choose a trivial solution u(x, t) = 0 of the mKdV Eqs. (20), (54) are written as
/1;x ¼ k/1 ;
/1;t ¼ bk3 /1 ;
/2;x ¼ k/2 ;
/2;t ¼ bk3 /2 :
ð65Þ
Eq. (65) has a solutions with complex parameter k = aj + ibj (j = 1, 2, 3, . . .):
(a)
(b) 3 2 1 4 0
10 u
2 5 0
0 4 2
2
0 x
2
1 t 2 3
4
4
4
2
0
2
4
Fig. 4. Positon–positon (56) with (57), (58), (60) and c11 ¼ c12 ¼ 14, k1 ¼ 1; k2 ¼ 58 ; c21 ¼ c22 ¼ 15 ; b ¼ 1; u ¼ 2:
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(a)
1.5 u 1 0.5 0
10 0 t 10 0
10
x
10
Fig. 5. One-negaton solution for the mKdV Eq. (20) by (63).
(a)
(b) 20
10
u
3 2 1 0 1 20
20
0
10 10
0 t 10
0 x
20
20
20
30
20
10
0
10
20
Fig. 6. Positon–negaton (56) with (57), (58), (64) and c11 ¼ c21 ¼ 1; c12 ¼ 12 ; c22 ¼ 5; k1 ¼ 45 ; k2 ¼ 13 ; u ¼ 12 ; b ¼ 1.
ðjÞ
ðjÞ
f1 ¼ expðfj Þðcos nj þ i sin nj Þ;
f2 ¼ expðfj Þðcos nj i sin nj Þ;
ð66Þ
where fj ¼ xaj þ t a3j bj 3aj b3j ; nj ¼ xbj þ t 3a2j b2j b4j and aj, bj (j = 1, 2, 3, . . .) are arbitrary constants. 1 Substituting (66) with j ¼ 1; a1 ¼ 1; b1 ¼ 12 ; b ¼ 20 , into (55), one-complexiton solution for the mKdV Eq. (20) can be constructed as follows
u¼
ð4 þ 2iÞ exp 40t þ 2x
: 1 þ expð20t þ 4xÞ cos 11t þ x þ i 1 þ exp 20t þ 4x sin 11t þx 80 80
ð67Þ
Fig. 7 shows the one-complexiton structure by (67).
(a)
(b) 40
20 15 u 10 5 0 4
20 0
0
t 20
2
20
0 x
2 4
40
4
2
0
Fig. 7. The module of one-complexiton solution for the mKdV Eq. (20) by (67).
2
4
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
(b)
(a) 60 40 20 15 10 u 5 0
50 25 0 t
4 2
0 20 40
25 0 x
60 2
50 4
4
2
0
2
4
1 Fig. 8. The module of complexiton–complexiton solution (56) with (55), (57), (58), (66) and a1 ¼ a2 ¼ 1; b1 ¼ 12 ; b2 ¼ 13 ; b ¼ 20 .
Substituting (55), (57), (58) into (56)and using (66) with j = 1, 2, complexiton–complexiton solution for the mKdV Eq. (20) can be constructed. Fig. 8 shows the complexiton–complexiton structure (56) with (66), j = 1, 2 and parameter selections
1 2
1 3
a1 ¼ a2 ¼ 1; b1 ¼ ; b2 ¼ ; b ¼
1 : 20
6. Conclusions In this paper, we have derived a new integrable generalized AKNS hierarchy (11) and its bi-Hamiltonian structure with the help of a 4 4 matrix spectral problem. We have demonstrated an approach to constructing a Darboux transformation for the generalized AKNS hierarchy (11) based on a gauge transformation between the corresponding 4 4 spectral problem with four potentials. The Darboux transformations for the coupled mKdV equation and mKdV equation are obtained through the reduction technique. As applications of Darboux transformations, some analytical positon, negaton, complexiton and their interaction solutions of the mKdV Eq. (20) are discussed. The analytical positon, negaton and complexiton solutions of the coupled mKdV Eq. (19) can be also obtained by its Darboux transformation (47), (48). How to construct the DTs for the coupled KdV Eq. (17) and the KdV Eq. (18) from DT (29), (30) in order to give out the corresponding positon, negaton and complexiton solutions is also an interesting and a significant topic. We shall consider it in the near future. Acknowledgments This work was supported by the High Education Science Research Program of China (Grant No 211034), the Natural Science Foundation of Inner Mongolia (Grant No 2009MS0108) and the High Education Science Research Program of Inner Mongolia Autonomous Region (NJ10045). References [1] Matveev VB, Salle MA. Darboux transformations and solitons. Berlin: Springer-Verlag; 1991. [2] Rogers C, Schief WK. Bäklund and Darboux transformations geometry and modern applications in soliton theory. Cambridge: Cambridge University Press; 2002. [3] Gu CH, Hu HS, Zhou ZX. Darboux transformations in integrable systems. In: Theory and their applications to geometry. Dortrecht: Springer-Verlag; 2005. [4] Neugebauer G, Meinel R. General N-soliton solution of the AKNS class on arbitrary background. Phys Lett A 1984;100:467–70. [5] Levi D, Neugebauer G, Meinel R. A new nonlinear Schrödinger equation, its hierarchy and N-soliton solutions. Phys Lett A 1984;102:1–6. [6] Li YS, Ma WX, Zhang JE. Darboux transformations of classical Boussinesq system and its new solutions. Phys Lett A 2000;275:60–6. [7] Zeng YB, Ma WX, Shao YJ. Two binary Darboux transformations for the KdV hierarchy with self-consistent sources. J Math Phys 2001;42:2113–27. [8] Geng XG, Ren HF, He GL. Darboux transformation for a generalized Hirota–Satsuma coupled Korteweg-de Vries equation. Phys Rev E 2009;79:056602. [9] Fan EG. Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation. J Phys A Math Gen 2000;33:6925–33. [10] Ling L, Liu QP. Darboux transformation for a two-component derivative nonlinear Schrödinger equation. J Phys A Theor 2010;43:434023. [11] Li HZ, Tian B, Li LL, Zhang HQ, Xu T. Darboux transformation and new solutions for the Whitham–Broer–Kaup equations. Phys Scr 2008;78:065001. [12] Zhaqilao, Sirendaoreji. A generalized coupled Korteweg-de Vries hierarchy, bi-Hamiltonian structure, and Darboux transformation. J Math Phys 2010;51:073501. [13] Xu SW, He JS, Wang LH. The Darboux transformation of the derivative nonlinear Schrödinger equation. J Phys A Math Theor 2011;44:305203. [14] Tu GZ. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30:330–8. [15] Sirendaoreji. On gauge equivalent constrained flow for AKNS hierarchy and Geng hierarchy. Adv Math 1999;28:437–46. [16] Guo FK, Zhang YF. The quadratic-form identity for constructing the Hamiltonian structure of integrable systems. J Phys A Math Gen 2005;38:8537–48. [17] Xia TC, Fan EG. The multicomponent generalized Kaup–Newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions. J Math Phys 2005;46:043510. [18] Qin ZY. A generalized Ablowitz–Ladik hierarchy, multi-Hamiltonian structure and Darboux transformation. J Math Phys 2008;49:063505.
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332 Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambrige: Cambrige University Press; 1991. Wadati M, Konno K, Ichikawa YH. A generalization of inverse scattering method. J Phys Soc Jpn 1979;46:1965–6. Kaup DJ, Newell AC. An exact solution for derivative nonlinear Schrödinger equation. J Math Phys 1978;19:798. Qiao ZJ. A hierarchy of nonlinear evolution equations and finite-dimensional involutive systems. J Math Phys 1994;35:2971. Ma WX, Fuchssteiner B. The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy. Phys Lett A 1996;213:49–55. Zhang YF. A few expanding integrable models, Hamiltonian structures and constrained flows. Comm Theor Phys 2011;55:273–90. Cao CW, Wu YT, Geng XG. Relation between the Kadometsev–Petviashvili equation and the confocal involutive system. J Math Phys 1999;40:3948. Qiao ZJ. A new integrable equation with cuspons and W/M-shape-peaks solitons. J Math Phys 2007;48:082701. Li YS, Ma WX. Hamiltonian structures of binary higher-order constrained flows associated with two 3 3 spectral problems. Phys Lett A 2000;272:245–56. Ma WX, Chen M. Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras. J Phys A Math Gen 2006;39:10787–801. Lou SY, Tong B, Hu HC, Tang XY. Coupled KdV equations derived from two-layer fluids. J Phys A Math Gen 2006;39:513–27. Hu HC, Sang BW, Zhu HD. New nonsingular positon, negaton and complexiton solutions for a special coupled mKdV system. Phys Lett A 2010;374:1141–6. Ma WX. Complexiton solution to the Korteweg-de Vries equation. Phys Lett A 2002;301:35–44. Wu HX, Zeng YB, Fan TY. complexitons of the modified KdV equation by Darboux transformation. Appl Math Comput 2008;196:501–10.
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
A generalized AKNS hierarchy, bi-Hamiltonian structure, and Darboux transformation Zhaqilao College of Mathematics Science, Inner Mongolia Normal University, Huhhot 010022, People’s Republic of China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 July 2011 Accepted 9 October 2011 Available online 25 October 2011
A new generalized AKNS hierarchy is presented starting from a 4 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Darboux transformation Soliton solution Generalized AKNS hierarchy Bi-Hamiltonian structure
1. Introduction It is well known that the Darboux transformation (DT) method plays an important role in solving many nonlinear evolution equations [1–13] such as Korteweg-de Vries (KdV), nonlinear Schrödinger, sine–Gordon, and other nonlinear differential equations which have great application in physics. But the main difficulty of the DT method lies in finding an appropriate spectral problem for a given nonlinear evolution equation. Hence, it is interesting for us to search for as many new spectral problems and corresponding nonlinear evolution equations as possible in soliton theory. In 1989, Tu [14] proposed a so-called trace identity to generate integrable evolution equations form the spectral problem. According to this method, many integrable evolution equations, their soliton hierarchies, and Hamiltonian structures are successively obtained [14–18]. We note that most of these well-known soliton hierarchies are related to the 2 2 zero-trace matrix spectral problems [19–22]. It is more important to investigate their mathematical structures and integrable properties, such as Hamiltonian structures [23], constrained flows [24], Darboux transformations (DT) [3], soliton solutions [4–13], algebra–geometric solutions [25], etc. [26]. Some 3 3 and 4 4 matrix AKNS spectral problems have been studied in many papers [27,28], but they have not considered their Darboux transformations and explicit solutions. In this paper, we are interested in bi-Hamiltonian structure, Darboux transformation and multiple soliton solutions for a generalized AKNS hierarchy associated with the following 4 4 matrix spectral problem
0 /x ¼ U/;
u1
u2
Bu B 4 U¼B @ u3
k
u3 C C C; k u4 A u1 k
0
0 u2
0
1
k
0
which is introduced by a matrix Lie algebra [24]. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.10.010
ð1Þ
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The outline of the present paper is as follows. In Section 2, a new generalized AKNS hierarchy is derived. It is shown that the new hierarchy is integrable in Liouville’s sense and possesses bi-Hamiltonian structure. Most importantly, some important nonlinear equations are produced from the generalized AKNS hierarchy, which including the special coupled nonlinear equation, the coupled KdV equation [29], the KdV equation, the coupled mKdV equation [30] and the mKdV equation, etc. In Section 3, the Darboux transformation for the generalized AKNS hierarchy is constructed by considering the gauge transformation of its Lax pair. In Section 4, some multiple soliton solutions of the generalized AKNS hierarchy are derived by the application of the Darboux transformation. In Section 5, as a reduction, a Darboux transformation of the mKdV equation and its new positon [30], negaton and complexiton [31,32] solutions are also obtained. The conclusion is given in Section 6.
2. A generalized AKNS hierarchy and its bi-Hamiltonian structure To obtain the generalized AKNS hierarchy and its bi-Hamiltonian structure with the matrix spectral problem (1), we first consider the stationary zero-curvature equation
V x ½U; V ¼ 0;
V ¼ ðV ij Þ44 ;
ð½U; V ¼ UV VUÞ:
ð2Þ
Let
V 21 ¼ V 34 ¼ A;
V 24 ¼ V 31 ¼ B;
V 11 ¼ V 44 ¼ F;
V 22 ¼ V 33 ¼ F;
A¼
X
aj kj ;
B¼
jP0
X
bj kj ;
V 13 ¼ V 24 ¼ C;
V 12 ¼ V 43 ¼ D;
V 14 ¼ V 41 ¼ H; X
C¼
jP0
cj kj ;
D¼
jP0
ð3Þ
V 23 ¼ V 32 ¼ H;
X
dj kj :
ð4Þ
jP0
Substituting (3) and (4) into (2), we have
F ¼ @ 1 ðu1 A þ u2 B u3 C u4 DÞ; kA ¼ u3 H þ u4 F 12 Ax ;
H ¼ @ 1 ðu2 A þ u1 B u4 C u3 DÞ;
kB ¼ u4 H þ u3 F Bx ;
kC ¼ u1 H þ u2 F þ 12 C x ;
ð5Þ kD ¼ u2 H þ u1 F þ 12 Dx :
ð6Þ
Thus from Eqs. (5) and (6) we obtain the Lenard gradient sequence
kðA; B; C; DÞT ¼ LðA; B; C; DÞT ;
ð7Þ
where
0 B B L¼B @
12 @ þ L11
L12
L13
L14
L12
12 @ þ L11
L14
L13
L31
L32
L32
1 @ 2
L31
L33
1 C C C; A
L34 1 @ 2
L34
L33
with
L11 ¼ u3 @ 1 u2 þ u4 @ 1 u1 ; 1
1
L33 ¼ u1 @ 1 u4 þ u2 @ 1 u3 ;
L13 ¼ u3 @ u4 u4 @ u3 ; 1
1
L32 ¼ u1 @ u1 þ u2 @ u2 ;
1
1
L12 ¼ u3 @ 1 u1 þ u4 @ 1 u2 ;
L14 ¼ u3 @ u3 u4 @ u4 ; 1
1
L34 ¼ u1 @ u3 u2 @ u4 ;
L31 ¼ u1 @ 1 u2 þ u2 @ 1 u1 ; @@ 1 ¼ @ 1 @ ¼ 1:
Substituting (4) into (7) and comparing coefficients for the same power of k gives rise to the recursion relation
LGj ¼ Gjþ1 ;
Gj ¼ ðaj ; bj ; cj ; dj ÞT ;
j P 0:
We take the initial value
G0 ¼ ðbu4 ; bu3 ; bu2 ; bu1 ÞT ; then we work out Gj, which is uniquely determined by (8), the first few terms are as follows
0
12 bu4;x
B 1 bu C 3;x C B G1 ¼ B 1 2 C; @ 2 bu2;x A 1 bu1;x 2
0b
1 G2 ¼
1 2u1 u23 þ u24 þ u4;xx 2 C 2u2 u3 þ u24 þ u3;xx C C; 2u3 u21 þ u22 þ u2;xx C A 2 b 2 4u u u 2u u þ u þ u 1 2 3 4 1;xx 1 2 4
4u2 u3 u4 4 Bb B 4 4u1 u3 u4 B B b 4u u u @4 1 2 4
ð8Þ
Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
2321
0 G3 ¼
1 b ð6u2 ðu4 u3;x þ u3 u4;x Þ þ 6u1 ðu3 u3;x þ u4 u4;x Þ u4;xxx Þ 8 B b C B 8 ð6u1 ðu4 u3;x þ u3 u4;x Þ þ 6u2 ðu3 u3;x þ u4 u4;x Þ u3;xxx Þ C B C: B b ð6u ðu u þ u u Þ þ 6u ðu u þ u u Þ u C @ 8 2 4 1;x 3 2;x 1 3 1;x 4 2;x 2;xxx Þ A 8b ð6u1 ðu4 u1;x þ u3 u2;x Þ þ 6u2 ðu3 u1;x þ u4 u2;x Þ u1;xxx Þ
Assuming the auxiliary problem
V ðnÞ ¼ ðkn VÞþ ;
/tn ¼ V ðnÞ /;
n P 0;
ð9Þ
where the symbol + denotes the choice of non-negative power of k. Then the compatibility condition of (1) and (9) gives rise to the zero-curvature representation ðnÞ U tn V ðnÞ ¼ 0; x þ ½U; V
ð10Þ
which is equivalent to
1 u1 Bu C B 2C B C ¼ X n ¼ JGn ¼ KGn1 ; @ u3 A 0
u4
ð11Þ
tn
where
0
0
B 0 B J¼B @ 0
0
0 2
1
2 0C C C; 2 0 0 A
2
0 0
0
0
K ¼ JL and L is given by Eq. (7). When n = 2, we have the auxiliary problem
0 /t2 ¼ V ð2Þ /;
B B B V ð2Þ ¼ B B @
ð2Þ
ð2Þ
bk2 M 5
M1
ð2Þ
bk2 þ M 5
ð2Þ
ð2Þ
M6
M4 M3
ð2Þ
M 6
ð2Þ
M3
1
ð2Þ
ð2Þ
M6
ð2Þ
bk2 þ M5
ð2Þ
ð2Þ
M1
ð2Þ
M2
M 6
ð2Þ
M2
ð2Þ
M4
ð2Þ
bk2 M 5
C C C C; C A
ð12Þ
where
b ð2Þ M 1 ¼ bu1 k þ u1;x ; 2 b ð2Þ M 4 ¼ bu4 k u4;x ; 2
b b ð2Þ ð2Þ M 2 ¼ bu2 k þ u2;x ; M 3 ¼ bu3 k u3;x ; 2 2 b b ð2Þ ð2Þ M 5 ¼ ðu1 u4 þ u2 u3 Þ; M6 ¼ ðu1 u3 þ u2 u4 Þ: 2 2
Substituting (1) and (12) into (10), we have a new coupled integrable equation
8 u1;t2 > > > > u > 3;t 2 > > : u4;t2
¼ 2b 4u1 u2 u3 2u4 u21 þ u22 þ u1;xx ; ¼ 2b 4u1 u2 u4 2u3 u21 þ u22 þ u2;xx ; ¼ 2b 4u1 u3 u4 2u2 u23 þ u24 þ u3;xx ; ¼ 2b 4u2 u3 u4 2u1 u23 þ u24 þ u4;xx :
ð13Þ
When u2 = u3 = 0, b = 1, Eq. (13) is reduced to a special coupled nonlinear equation
1 u1;t2 ¼ u1;xx þ u4 u21 ; 2
u4;t2 ¼
1 u4;xx u1 u24 : 2
ð14Þ
When n = 3, we have the auxiliary problem
0 /t3 ¼ V ð3Þ /;
ð3Þ
V 11
B ð2Þ B M k þ M ð3Þ B 5 V ð3Þ ¼ B 4ð2Þ B M k þ M ð3Þ 6 @ 3 ð2Þ ð3Þ M6 k þ M4
ð2Þ
ð3Þ
M1 k þ M2 ð3Þ
ð2Þ
ð3Þ
M 6 k M 4
ð2Þ
ð3Þ
M3 k þ M6
M2 k þ M3 M6 k þ M4
V 11 ð2Þ
ð3Þ
V 11
ð2Þ
ð3Þ
M1 k þ M2
M6 k þ M4 M2 k þ M3
ð3Þ
ð2Þ
ð2Þ
ð3Þ
ð2Þ
ð3Þ
ð2Þ
ð3Þ
M4 k þ M5 ð3Þ
ð3Þ
V 11
1 C C C C; C A
ð15Þ
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
where ð3Þ
ð2Þ
ð3Þ
ð3Þ
V 11 ¼ bk3 M 5 k þ M 1 ; M 1 ¼ 4b ðu4 u1;x u3 u2;x þ u2 u3;x þ u1 u4;x Þ; ð3Þ ð3Þ M2 ¼ 4b 4u1 u2 u3 2u21 u4 2u22 u4 þ u1;xx ; M 3 ¼ 4b 2u21 u3 2u22 u3 4u1 u2 u4 þ u2;xx ; ð3Þ ð3Þ M4 ¼ 4b ðu3 u1;x þ u4 u2;x u1 u3;x u2 u4;x Þ; M5 ¼ 4b ð4u2 u3 u4 2u1 u23 þ u24 þ u4;xx Þ; ð3Þ
M6 ¼ 4b ð2u2 u23 4u1 u3 u4 2u2 u24 þ u3;xx Þ: Substituting (1) and (15) into (10), we have a new coupled integrable equation
8 u1;t3 > > > > u > 3;t3 > > : u4;t3
¼ 4b ð6u1 ðu4 u1;x þ u3 u2;x Þ þ 6u2 ðu3 u1;x þ u4 u2;x Þ u1;xxx Þ; ¼ 4b ð6u2 ðu4 u1;x þ u3 u2;x Þ þ 6u1 ðu3 u1;x þ u4 u2;x Þ u2;xxx Þ;
ð16Þ
¼ 4b ð6u1 ðu4 u3;x þ u3 u4;x Þ þ 6u2 ðu3 u3;x þ u4 u4;x Þ u3;xxx Þ; ¼ 4b ð6u2 ðu4 u3;x þ u3 u4;x Þ þ 6u1 ðu3 u3;x þ u4 u4;x Þ u4;xxx Þ:
(i) When u4 ¼ 1; u3 ¼ 12 ; b ¼ 4 and t3 = t, Eq. (16) is reduced to the coupled KdV equation [29]
u1;t ¼ 3u2 u1;x þ 3u1 u2;x 6u1 u1;x 6u2 u2;x u1;xxx ;
ð17Þ
u2;t ¼ 3u1 u1;x þ 3u2 u2;x 6u2 u1;x 6u1 u2;x u2;xxx : When u2 = u1 = u, Eq. (17) is reduced to the celebrated KdV equation [1–3]
ut þ 6uux þ uxxx ¼ 0:
ð18Þ
(ii) When u4 = u1, u3 = u2, and t3 = t, Eq. (16) is reduced to the coupled mKdV equation [30]
(
u1;t ¼ 4b 6u21 u1;x þ 6u22 u1;x þ 12u1 u2 u2;x þ u1;xxx ; u2;t ¼ 4b 6u21 u2;x þ 6u22 u2;x þ 12u1 u2 u1;x þ u2;xxx :
ð19Þ
When u1 = u, u2 = 0, Eq. (19) is reduced to the celebrated mKdV equation [3]
ut ¼
b ð6u2 ux þ uxxx Þ: 4
ð20Þ
To investigate the generalized Hamiltonian structure of the hierarchy (11), we take the Killing-Cartan form hA, Bi as tr(AB). Then by direct calculations, we conclude that
@U ¼ 4F; V; @k
@U ¼ 2A; V; @u1
@U V; ¼ 2B; @u2
@U V; ¼ 2C; @u3
@U V; ¼ 2D: @u4
ð21Þ
By using trace identity [14], we have
d d d d ; ; ; du1 du2 du3 du4
V;
@U @k
@ @U @U @U @U ¼ kc kc V; ; V; ; V; ; V; : @k @u1 @u2 @u3 @u4
ð22Þ
Substituting (4) and (21) into (22), we have
d d d d ; ; ; du1 du2 du3 du4
Z 4 ðu1 aj þ u2 bj u3 cj u4 dj Þdx ¼ ðc jÞð2aj ; 2bj ; 2cj ; 2dj Þ;
j P 0:
ð23Þ
To fix the constant c, we compare the coefficient of j = 0 in the above equation and get c = 0. Substituting c = 0 into (23), we arrive at
Z d d d d 2 Hj ¼ GTj ; Hj ¼ ðu1 aj þ u2 bj u3 cj u4 dj Þdx; ; ; ; du1 du2 du3 du4 j
j P 0:
ð24Þ
Therefore, we obtain the generalized bi-Hamiltonian structure of the hierarchy (11)
0
u1
0 dHn1 1
1
du1
Bu C B 2C B C @ u3 A u4
0 dHn 1 du
B dHn1 C B dH1n C C B B C B du C B du C ¼ K B dH 2 C ¼ J B dH2 C; B n1 C B nC @ du3 A @ du3 A tn
dHn1 du4
where K and J are given by (11).
dHn du4
ð25Þ
Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
2323
3. Darboux transformation In this section, we proceed to construct the Darboux transformation of the generalized AKNS hierarchy (11). We assume that / = (/1, /2, /3, /4)T is a solution of Lax pair (1) and (9). Now we introduce a gauge transformation for the Lax pair (1) and (9)
¼ T/; /
ð26Þ
¼ ð/ 1; / 2; / 3; / 4 ÞT and T is a Darboux matrix, from which Lax pair (1) and (9) are transformed into new matrix specwhere / as follows: tral problems of /
x ¼ U /; / U ¼ ðT x þ TUÞT 1 ; t ¼ V ðnÞ /; / V ðnÞ ¼ ðT t þ TUÞT 1 :
ð27Þ ð28Þ
With the help of the Darboux matrix method [12], we can obtain the following propositions. Proposition 1. The matrix U determined by the second expression of (27) has the same form as U:
0
k Bu B 4 U¼B 3 @u 0
1 0 3 C k 0 u C C 4 A 0 k u 2 u 1 k u 1 u
2 u
and the transformation formulae from original potentials u1, u2, u3 and u4 into new ones are given by
1 ¼ u1 2b; u
2 ¼ u2 2c; u
3 ¼ u3 þ 2f ; u
4 ¼ u4 þ 2s; u
ð29Þ
10 1 0 1 0 /1 kþa b c d /1 C B B/ B C kþh g f C CB /2 C B 2 C B s CB C; B C¼B @ /3 A @ f g kþh s A@ /3 A 4 /4 d c b kþa /
ð30Þ
with
a¼
Da ; D
ð1Þ f1 ð2Þ f D ¼ 1ð1Þ f4 ð2Þ f4
b¼
Db ; D
ð1Þ
f3
ð2Þ
f3
ð1Þ
f2
ð2Þ
f2
f2 f2 f3 f3
ð1Þ ð2Þ ð1Þ ð2Þ
c¼
Dc ; D
d¼
Dd ; D
s¼
Ds ; D
h¼
Dh ; D
g¼
Dg ; D
f ¼
Df ; D
ð1Þ f4 ð2Þ f4 ð1Þ f1 ð2Þ f1
ð31Þ
ð32Þ
T ð1Þ ð2Þ ð1Þ ð2Þ and Da, Db, Dc, Dd, are produced from D by replacing its1st, 2nd, 3rd, 4th column with k1 f1 ; k2 f1 ; k1 f4 ; k2 f4 , respec
T ð1Þ ð2Þ ð1Þ ð2Þ tively. Ds, Dh, Dg, Df are produced from D by replacing its1st, 2nd, 3rd, 4th column with k1 f2 ; k2 f2 ; k1 f3 ; k2 f3 , respec
T
T ð1Þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ tively. Where f1 ; f2 ; f3 ; f4 and f1 ; f2 ; f3 ; f4 are solutions of the Lax pair (1), (9) with k = k1 and k = k2, respectively. Proof. Let T1 = T⁄/detT, where it is a known formula of the inverse matrix,
ðT x þ TUÞT ¼ ðfsl ðkÞÞ44 ;
ð33Þ
⁄
⁄
where T stands for the adjugate matrix of T, and every element of the matrix T is an algebraic cofactors of the matrix T. A direct calculation shows that fsl(k) are polynomials of k. Resorting to Eqs. (31) and (32), it can be verified that all kj(j = 1, 2) are roots of fsl(k)(s, l = 1, 2, 3, 4). Hence, Eq. (33) can be written as
ðT x þ TUÞT ¼ ðdet TÞPðkÞ;
ð34Þ
where
0
k
B ð0Þ Bp B PðkÞ ¼ B 21 B pð0Þ @ 31 0
ð0Þ
ð0Þ
p12
p13
k
0
0
k
ð0Þ
p43
p42
ð0Þ
0
1
C ð0Þ p24 C C C; ð0Þ C p34 A k
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332 ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
with p12 ; p21 ; p13 ; p31 ; p24 ; p42 ; p34 and p43 are functions of (x, t) independent of k. Eq. (34) can be written as
T x þ TU ¼ PðkÞT:
ð35Þ
By comparing the coefficients of k in Eq. (35), we obtain ð0Þ
ð0Þ
p12 ¼ p43 ¼ u1 2b;
ð0Þ
ð0Þ
ð0Þ
p13 ¼ p42 ¼ u2 2c;
ð0Þ
p24 ¼ p31 ¼ u3 þ 2f ;
ð0Þ
ð0Þ
p21 ¼ p34 ¼ u4 þ 2s;
which implies U ¼ PðkÞ. The proof is completed. h Proposition 2. The matrix V ðnÞ determined by the second expression of (28) has the same form as V(n), in which original potentials 1 ; u 2 ; u 3 and u 4 according to the same transformation (29), (30). u1, u2, u3 and u4 are mapped into new potentials u For each integer n P 2, we can prove Proposition 2 similarly to the proof in Proposition 1. According to Propositions 1 and 2, the transformation (29), (30), maps the Lax pair (1) and (9) into another Lax pair (27) and (28) of the same type. Therefore, both of the Lax pair lead to the same system (11). We call the transformation u 1 ; u 2 ; u 3 ; u 4 Þ a Darboux transformation of the generalized AKNS hierarchy (11). We obtain the ð/; u1 ; u2 ; u3 ; u4 Þ ! ð/; following assertion: 1 ; u 2 ; u 3 ; u 4 Þ of the generalized AKNS Theorem 1. Every solution (u1, u2, u3, u4) of the system (11) is mapped into a new solution ðu hierarchy (11) under the Darboux transformation (29), (30).
4. Soliton solutions of the generalized AKNS hierarchy In this section, we will obtain some multi-soliton solutions of the whole generalized AKNS hierarchy (11) by applying the Darboux transformation (29), (30). If choose a trivial solution u1 = 0, u2 = 0, u3 = 0 and u4 = 0 of the generalized AKNS hierarchy (11), then the Lax pair (1) and (9) are written as
/1;x ¼ k/1 ;
/2;x ¼ k/2 ;
/1;t ¼ bkn /1 ;
/3;x ¼ k/3 ;
/2;t ¼ bkn /2 ;
/4;x ¼ k/4 ;
/3;t ¼ bkn /3 ;
ð36Þ
/4;t ¼ bkn /4 :
Eq. (36) has a solutions with k = kj (j = 1, 2, 3, . . .):
1 C 1j exp kj x þ bknj t C B
C C B B C n B ðjÞ C B C 2j exp kj x bkj t C B f2 C B C C¼B B
C; B ðjÞ C B C n B f3 C B C exp kj x bkj t C A B @ C B 3j @ ðjÞ
A f4 n C 4j exp kj x þ bkj t 0
ðjÞ f1
1
0
ð37Þ
where C1j, C2j, C3j, C4j are arbitrary constants and kj(j = 1, 2, 3, . . .) are spectral parameters. According to (29)–(32) and (37) with j = 1, 2, we have a new explicit solution for the generalized AKNS hierarchy (11) as follows:
u1½0 ¼ u1 2b1 ; u2½0 ¼ u2 2c1 ; u3½0 ¼ u3 þ 2f 1 ; u4½0 ¼ u4 þ 2s1 ;
ð38Þ
where b1 = b, c1 = c, f1 = f, s1 = s, a1 = a, d1 = d, g1 = g and h1 = h. Substituting (31), (32) and (37) (with j = 1, 2) into (38) with the zero seed (u1 = u2 = u3 = u4 = 0) and constant parameter selections
C 11 ¼ 1;
C 21 ¼ 2;
C 31 ¼
8 ; 3
C 41 ¼ 5;
C 12 ¼ 3;
C 22 ¼ 3;
C 32 ¼ 2;
one can obtain a new solution for the generalized AKNS hierarchy (11) as follows:
1 C 42 ¼ ; 5
b ¼ 2;
ð39Þ
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
6 expðn1 þ n2 Þð1935 expðn1 Þ þ 952 expðn2 ÞÞðk1 k2 Þ ; 3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 Þ 6 expðn1 þ n2 Þð1215 expðn1 Þ þ 616 expðn2 ÞÞðk1 k2 Þ ¼ ; 3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 Þ 15ð275 expðn1 Þ 126 expðn2 ÞÞðk1 k2 Þ ¼ ; 2ð3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 ÞÞ 5ð1275 expðn1 Þ þ 602 expðn2 ÞÞðk1 k2 Þ ¼ ; 2ð3375 expð2n1 Þ þ 784 expð2n2 Þ þ 7710 expðn1 þ n2 ÞÞ
u1½0 ¼ u2½0 u3½0 u4½0
ð40Þ
where n1 ¼ 2 xk1 þ 2tkn1 ; n2 ¼ 2 xk2 þ 2tkn2 . Particularly, the solutions of the coupled integrable Eqs. (13) and (16) are obtained by (40) with n = 2 and n = 3 respectively. If different spectral parameters k1 and k2 are chosen, the solution determined by (40) may be of different properties. For example, when k1 = 1,k2 = 1, the functions (40) is a one-soliton solution of the generalized AKNS hierarchy (11) (see Fig. 1). After taking the Darboux transformation (29), (30) once again, we can obtain the explicit solution for the generalized AKNS hierarchy (11) as follows:
u1½1 ¼ u1½0 2b2 ;
u2½1 ¼ u2½0 2c2 ;
1 ð3Þ 0 f k3 þ a1 B 1½1 C B ð3Þ C B B f2½1 C B s1 C B B ð3Þ C ¼ B B f3½1 C @ f1 A @ d1 ð3Þ f4½1
u3½1 ¼ u3½0 þ 2f 2 ;
0
0
ð4Þ
c1
k3 þ h1
g1
g1
k3 þ h1
c1
b1
b1
c1
k4 þ h1
g1
g1
k4 þ h1
c1
b1
ð41Þ
1 0 1 f ð3Þ 1 B ð3Þ C C B f1 C CB f2 C CB ð3Þ C; C s1 AB @ f3 A ð3Þ k3 þ a1 f4
ð42Þ
1 0 1 f ð4Þ 1 B ð4Þ C C B f1 C CB f2 C CB ð4Þ C; C s1 AB @ f3 A ð4Þ k4 þ a1 f4
ð43Þ
d1
1
0 k4 þ a1 C B B ð4Þ C B B f2½1 C B s1 C B B ð4Þ C ¼ B B f3½1 C @ f1 A @ d1 ð4Þ f4½1 f1½1
b1
u4½1 ¼ u4½0 þ 2s2 ;
d1
with
u1 0
4
0.1
2
0.05
0 4
0 t
2 x
2 1 u2 0 0 1 2 4
0 t
2
0.05
0
0.1 0.05
x
2 4
0.05
0 2
0.1
4
0.1
0.5 u3 0 0 0.5
0
0.1 0.05
4 2 x
0 2 4
0.1
u4 0
0.5
0 t
1
0.05
4
0.1 0.05 0 t
2 x
0.05
0 2 4
Fig. 1. One soliton solution (40) with k1 = 1, k2 = 1, n = 2.
0.1
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
a2 ¼
D½1a 2 D½1
s2 ¼
D½1s 2 D½1
D½1
; ;
f ð3Þ 1½1 ð4Þ f1½1 ¼ ð3Þ f4½1 ð4Þ f4½1
b2 ¼ h2 ¼
D½1b
2
D½1 D½1h
2
D½1
f2½1
ð3Þ
f3½1
f2½1
ð4Þ
f3½1
f3½1
ð3Þ
f2½1
ð4Þ
f2½1
f3½1
; ;
c2 ¼
D½1c 2 D½1
g2 ¼
D½1g 2 D½1
; ;
d2 ¼ f2 ¼
D½1d
2
D½1 D½1f
2
D½1
; ð44Þ ;
ð3Þ f4½1 ð4Þ f4½1 ð3Þ f1½1 ð4Þ f1½1
ð3Þ ð4Þ ð3Þ ð4Þ
ð45Þ
and D½1a2 ; D½1b2 ; D½1c2 ; D½1d2 , are produced from D[1] by replacing its 1st, 2nd, 3rd, 4th column with
T ð3Þ ð4Þ ð3Þ ð4Þ k3 f1½1 ; k4 f1½1 ; k3 f4½1 ; k4 f4½1 , respectively. D½1s2 ; D½1h2 ; D½1g2 ; D½1f2 are produced from D[1] by replacing its 1st, 2nd,
T
T
T ð3Þ ð4Þ ð3Þ ð4Þ ð3Þ ð3Þ ð3Þ ð3Þ ð4Þ ð4Þ ð4Þ ð4Þ and f1 ; f2 ; f3 ; f4 3rd, 4th column with k3 f2½1 ; k4 f2½1 ; k3 f3½1 ; k4 f3½1 , respectively. Where f1 ; f2 ; f3 ; f4 are solutions of the Lax pair (1), (9) with k = k3 and k = k4, respectively. Substituting (40), (42)–(45) into (41) with the zero seed (u1 = u2 = u3 = u4 = 0) and constant parameter selections
C 11 ¼ 1;
C 21 ¼ 2;
C 31 ¼
1 C 42 ¼ ; 5
k1 ¼ 1;
C 43 ¼ 7;
C 14 ¼ 8;
8 ; 3
k2 ¼ 1;
C 41 ¼ 5; b ¼ 2;
C 24 ¼ 3;
C 12 ¼ 3;
C 13 ¼ 2;
C 34 ¼ 6;
C 22 ¼ 3;
C 32 ¼ 2;
8 C 33 ¼ ; 3 3 3 k3 ¼ ; k4 ¼ ; 4 4
ð46Þ
C 23 ¼ 1;
C 44 ¼ 9;
one can obtain two-soliton solution for the generalized AKNS hierarchy (11), and their graphs are shown in Fig. 2. Iterating the above method, we can obtain a series of multiple soliton solutions of the generalized AKNS hierarchy (11).
5. Reduction of the Darboux transformation and analytical solutions of the mKdV equation In this section, we will discuss a reduction of the Darboux transformation (29), (30), by which the Darboux transformations of the coupled mKdV Eq. (19) and the mKdV Eq. (20) are given. Most importantly, some new nonsingular positon, negaton solutions and complexiton solutions for the mKdV Eq. (20) are given out both analytically and graphically by means of the iterative Darboux transformation.
8 u1 1 6 4 2 0
0.1 0.05 0 t
4
2 x
0
2
2
x
0.05
0 2
0.1
4
0.05 0 t 0 x
0 t
4
0.1
2
0.1 0.05
0.05 4
0 u3 1 1 2 3 4
0 u2 1 1 2 3
0.05
2 4 6
0.1
0.1
0 1 u4 1 2 3 4 4
0.1 0.05 0 t
2 0 x
0.05 2 4
Fig. 2. Two-solution (41) with (42)–(46) and u1 = u2 = u3 = u4 = 0, n = 2.
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
2327
Under the condition u3 = u2, u4 = u1, s = b, g = d, f = c, h = a, Darboux transformation (29), (30) is reduced to the Darboux transformation of the coupled mKdV Eq. (19) as follows
2 ¼ u2 2c; u
1 ¼ u1 2b; u
ð47Þ
10 1 0 1 0 /1 kþa b c d /1 B/ C B B C k a d c C CB /2 C B 2 C B b CB C; B C¼B @ /3 A @ c d k a b A@ /3 A 4 /4 d c b kþa /
ð48Þ
with
a¼
Da ; D
b¼
ð1Þ f1 ð1Þ f D ¼ 2ð1Þ f3 ð1Þ f4
Db ; D
c¼
ð1Þ
f3
ð1Þ
f4
ð1Þ
f1
ð1Þ
f2
f2 f2 f4 f3
Dc ; D
d¼
Dd ; D
ð49Þ
ð1Þ f3 ð1Þ f2 ð1Þ f1
ð1Þ
ð1Þ
f4
ð1Þ ð1Þ ð1Þ
ð50Þ
T ð1Þ ð1Þ ð1Þ ð1Þ and Da, Db, Dc, Dd, are produced from D by replacing its 1st, 2nd, 3rd, 4th column with k1 f1 ; k1 f2 ; k1 f3 ; k1 f4 , respectively. Under the condition u1 = 0, u2 = u, b = d = 0, Darboux transformation (47), (48) is reduced to the Darboux transformation of the mKdV Eq. (20) as follows
¼ u 2c; u
ð51Þ
10 1 0 1 0 /1 kþa 0 c 0 /1 B/ C B B C ka 0 c C CB /2 C B 2 C B 0 CB C; B C¼B @ /2 A @ c 0 ka 0 A@ /2 A 1 0 c 0 kþa /1 /
ð52Þ
with
a¼
2 ð1Þ ð1Þ 2 k1 f2 f1 ð1Þ 2
f2
ð1Þ 2
c¼
;
ð1Þ ð1Þ 2k1 f2 f1 ð1Þ 2
f2
þ f1
ð1Þ 2
;
ð53Þ
þ f1
ðjÞ ðjÞ where f1 ; f2 is solution for the Lax pair of the mKdV Eq. (20)
/1;x ¼ k/1 þ u/2 ; /2;x ¼ u/1 k/2 ; 1 1 /1;t ¼ bkð2k2 þ u2 Þ/1 þ bð4k2 u þ 2u3 þ 2kux þ uxx Þ/2 ; 2 4 1 1 2 2 /2;t ¼ bkð2k þ u Þ/2 bð4k2 u þ 2u3 2kux þ uxx Þ/1 ; 2 4 ðjÞ
ð54Þ
ðjÞ
with k ¼ kj ; /1 ¼ f1 ; /2 ¼ f2 ; ðj ¼ 1; 2; 3; . . .Þ. From (51) and (53), we obtain a solution of the mKdV Eq. (20) as follows
u½0 ¼ u þ
ð1Þ ð1Þ 4k1 f2 f1 ð1Þ 2
f2
ð1Þ 2
:
ð55Þ
þ f1
With the Darboux transformation (51), (52), after taking the solution u[0] in (55) as the seed solution, we can obtain another solution for the mKdV Eq. (20) as follows:
u½1 ¼ u½0 2c2 ;
ð56Þ
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0
1
1 0 1 f ð2Þ k þ a 0 c 0 1 2 C B B ð2Þ C B ð2Þ C B C B B f2½1 C B 0 0 c C k2 a CB f2 C C¼B B C C; B B ð2Þ C @ c ð2Þ C 0 k2 a 0 AB B f2½1 C @ f2 A A @ ð2Þ 0 c 0 k2 þ a ð2Þ f1 f1½1 ð2Þ
f1½1
0
ð57Þ
with
a2 ¼
2 ð2Þ ð2Þ 2 k2 f2½1 f1½1 ð2Þ 2
ð2Þ 2
c2 ¼
;
ð2Þ ð2Þ 2k2 f2½1 f1½1 ð2Þ 2
f2½1 þ f1½1
ð2Þ 2
ð58Þ
;
f2½1 þ f1½1
ð2Þ ð2Þ is solution of (54) with k = k2. where a, b are give by (53) and f1 ; f2 In what follows, we shall apply the solution formulas (55) and (56) to construct analytical positon, negaton, complexiton, and their interaction solutions for the mKdV Eq. (20). If choose a trivial solution u(x, t) = u (u is a constant) of the mKdV Eqs. (20), (54) are written as
/1;x ¼ k/1 þ u/2 ; /2;x ¼ u/1 k/2 ; 1 1 /1;t ¼ bkð2k2 þ u2 Þ/1 þ bð4k2 u þ 2u3 Þ/2 ; 2 4 1 1 /2;t ¼ bkð2k2 þ u2 Þ/2 bð4k2 u þ 2u3 Þ/1 : 2 4
ð59Þ
5.1. Positon solution 2 Eq. (59) has a solutions with positive parameter k ¼ kj ðj ¼ 1; 2; 3; . . .Þ:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4 u2 kj A sin nj ; ¼ cj1 cos nj @ 2 kj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 4 ucj1 cj2 u2 kj A cos nj þ cj2 sin nj ; ¼@ 2 kj 0
ðjÞ f1
ðjÞ f2
ucj2 þ cj1
ð60Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 4 where nj ¼ 12 u2 kj ð2x þ t 2kj þ u2 bÞ and cj1, cj2 (j = 1, 2, 3, . . .) are arbitrary constants. Substituting (60) into (55) with j = 1 and parameter selections
c11 ¼ c12 ¼
1 ; 4
k1 ¼
1 ; 2
b ¼ 1;
u ¼ 1;
one can obtain the analytical positon solution for the mKdV Eq. (20) as follows:
h pffiffiffiffiffiffi i 1 15 9t8 þ 2x 4 h pffiffiffiffiffiffi u¼ i : 8 þ 2 sin 14 15 9t8 þ 2x 7 2 sin
ð61Þ
(a)
(b) 3 2 1
4 3 u 2 1 0 1
0 5 1 0
t 2
5 5
0 x
3 5
4
2
Fig. 3. One-positon solution for the mKdV Eq. (20) by (61).
0
2
4
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Fig. 3 shows the nonsingular one-positon structure by (61). Substituting (55), (57), (58) and (60) into (56) with j = 1, 2, positon–positon solution for the mKdV Eq. (20) can be constructed. Fig. 4 shows the nonsingular positon-positon structure (56) with (55), j = 1, 2 and parameter selections
c11 ¼ c12 ¼
1 ; 4
k1 ¼ 1;
k2 ¼
5 ; 8
1 c21 ¼ c22 ¼ ; 5
b ¼ 1;
u ¼ 2:
5.2. Negaton solution and positon–negaton solution 2 Eq. (59) has a solutions with negative parameter k ¼ kj ðj ¼ 1; 2; 3; . . .Þ: ðjÞ f1 ðjÞ f2
cj2 k2j cj2
pffiffiffiffiffiffiffiffiffi ffi 4 k u2
j expðfj Þ; ¼ cj1 expðfj Þ þ u ffi 2 pffiffiffiffiffiffiffiffiffi cj1 kj cj1 k4j u2 expðfj Þ; ¼ cj2 expðfj Þ þ u
ð62Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 where fj ¼ 12 kj u2 ð2x þ tð2kj þ u2 ÞbÞ and cj1, cj2(j = 1, 2, 3, . . .) are arbitrary constants. Substituting (62) with j = 1, c11 ¼ c12 ¼ 14 ; k1 ¼ 12 ; b ¼ 1; u ¼ 16 into (55), one-negaton solution for the mKdV Eq. (20) can be constructed as follows
pffiffi
pffiffi
pffiffi 3 exp 114325t þ 3 exp 35x 14 exp 8645 ð11t þ 144xÞ
pffiffi
pffiffi
pffiffi : u¼
6 3 exp 114325t þ 3 exp 35x 4 exp 8645 ð11t þ 144xÞ
ð63Þ
Fig. 5 shows the nonsingular one-negaton solution structure by (63). Substituting
pffiffiffiffiffiffiffiffiffi ffi c k2 c k4 u2 ð1Þ expðf1 Þ; f1 ¼ c11 expðf1 Þ þ 12 1 12u 1 p ffiffiffiffiffiffiffiffiffi ffi c k2 c k4 u2 ð1Þ expðf1 Þ; f2 ¼ c12 expðf1 Þ þ 11 1 11u 1 pffiffiffiffiffiffiffiffiffi4ffi uc22 þc21 u2 k2 ð2Þ sin n2 ; f1 ¼ c21 cos n2 k22 pffiffiffiffiffiffiffiffiffi4ffi uc21 c22 u2 k2 ð2Þ cos n2 þ c22 sin n2 ; f2 ¼ k22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 4 4 4 f1 ¼ 12 k1 u2 2x þ t 2k1 þ u2 b ; n2 ¼ 12 u2 k2 2x þ t 2k2 þ u2 b
ð64Þ
and (57), (58) into (56), we have positon–negaton solution of the mKdV Eq. (20), and their graphs are shown in Fig. 6. 5.3. Complexiton solution If choose a trivial solution u(x, t) = 0 of the mKdV Eqs. (20), (54) are written as
/1;x ¼ k/1 ;
/1;t ¼ bk3 /1 ;
/2;x ¼ k/2 ;
/2;t ¼ bk3 /2 :
ð65Þ
Eq. (65) has a solutions with complex parameter k = aj + ibj (j = 1, 2, 3, . . .):
(a)
(b) 3 2 1 4 0
10 u
2 5 0
0 4 2
2
0 x
2
1 t 2 3
4
4
4
2
0
2
4
Fig. 4. Positon–positon (56) with (57), (58), (60) and c11 ¼ c12 ¼ 14, k1 ¼ 1; k2 ¼ 58 ; c21 ¼ c22 ¼ 15 ; b ¼ 1; u ¼ 2:
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
(a)
1.5 u 1 0.5 0
10 0 t 10 0
10
x
10
Fig. 5. One-negaton solution for the mKdV Eq. (20) by (63).
(a)
(b) 20
10
u
3 2 1 0 1 20
20
0
10 10
0 t 10
0 x
20
20
20
30
20
10
0
10
20
Fig. 6. Positon–negaton (56) with (57), (58), (64) and c11 ¼ c21 ¼ 1; c12 ¼ 12 ; c22 ¼ 5; k1 ¼ 45 ; k2 ¼ 13 ; u ¼ 12 ; b ¼ 1.
ðjÞ
ðjÞ
f1 ¼ expðfj Þðcos nj þ i sin nj Þ;
f2 ¼ expðfj Þðcos nj i sin nj Þ;
ð66Þ
where fj ¼ xaj þ t a3j bj 3aj b3j ; nj ¼ xbj þ t 3a2j b2j b4j and aj, bj (j = 1, 2, 3, . . .) are arbitrary constants. 1 Substituting (66) with j ¼ 1; a1 ¼ 1; b1 ¼ 12 ; b ¼ 20 , into (55), one-complexiton solution for the mKdV Eq. (20) can be constructed as follows
u¼
ð4 þ 2iÞ exp 40t þ 2x
: 1 þ expð20t þ 4xÞ cos 11t þ x þ i 1 þ exp 20t þ 4x sin 11t þx 80 80
ð67Þ
Fig. 7 shows the one-complexiton structure by (67).
(a)
(b) 40
20 15 u 10 5 0 4
20 0
0
t 20
2
20
0 x
2 4
40
4
2
0
Fig. 7. The module of one-complexiton solution for the mKdV Eq. (20) by (67).
2
4
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Zhaqilao / Commun Nonlinear Sci Numer Simulat 17 (2012) 2319–2332
(b)
(a) 60 40 20 15 10 u 5 0
50 25 0 t
4 2
0 20 40
25 0 x
60 2
50 4
4
2
0
2
4
1 Fig. 8. The module of complexiton–complexiton solution (56) with (55), (57), (58), (66) and a1 ¼ a2 ¼ 1; b1 ¼ 12 ; b2 ¼ 13 ; b ¼ 20 .
Substituting (55), (57), (58) into (56)and using (66) with j = 1, 2, complexiton–complexiton solution for the mKdV Eq. (20) can be constructed. Fig. 8 shows the complexiton–complexiton structure (56) with (66), j = 1, 2 and parameter selections
1 2
1 3
a1 ¼ a2 ¼ 1; b1 ¼ ; b2 ¼ ; b ¼
1 : 20
6. Conclusions In this paper, we have derived a new integrable generalized AKNS hierarchy (11) and its bi-Hamiltonian structure with the help of a 4 4 matrix spectral problem. We have demonstrated an approach to constructing a Darboux transformation for the generalized AKNS hierarchy (11) based on a gauge transformation between the corresponding 4 4 spectral problem with four potentials. The Darboux transformations for the coupled mKdV equation and mKdV equation are obtained through the reduction technique. As applications of Darboux transformations, some analytical positon, negaton, complexiton and their interaction solutions of the mKdV Eq. (20) are discussed. The analytical positon, negaton and complexiton solutions of the coupled mKdV Eq. (19) can be also obtained by its Darboux transformation (47), (48). How to construct the DTs for the coupled KdV Eq. (17) and the KdV Eq. (18) from DT (29), (30) in order to give out the corresponding positon, negaton and complexiton solutions is also an interesting and a significant topic. We shall consider it in the near future. Acknowledgments This work was supported by the High Education Science Research Program of China (Grant No 211034), the Natural Science Foundation of Inner Mongolia (Grant No 2009MS0108) and the High Education Science Research Program of Inner Mongolia Autonomous Region (NJ10045). References [1] Matveev VB, Salle MA. Darboux transformations and solitons. Berlin: Springer-Verlag; 1991. [2] Rogers C, Schief WK. Bäklund and Darboux transformations geometry and modern applications in soliton theory. Cambridge: Cambridge University Press; 2002. [3] Gu CH, Hu HS, Zhou ZX. Darboux transformations in integrable systems. In: Theory and their applications to geometry. Dortrecht: Springer-Verlag; 2005. [4] Neugebauer G, Meinel R. General N-soliton solution of the AKNS class on arbitrary background. Phys Lett A 1984;100:467–70. [5] Levi D, Neugebauer G, Meinel R. A new nonlinear Schrödinger equation, its hierarchy and N-soliton solutions. Phys Lett A 1984;102:1–6. [6] Li YS, Ma WX, Zhang JE. Darboux transformations of classical Boussinesq system and its new solutions. Phys Lett A 2000;275:60–6. [7] Zeng YB, Ma WX, Shao YJ. Two binary Darboux transformations for the KdV hierarchy with self-consistent sources. J Math Phys 2001;42:2113–27. [8] Geng XG, Ren HF, He GL. Darboux transformation for a generalized Hirota–Satsuma coupled Korteweg-de Vries equation. Phys Rev E 2009;79:056602. [9] Fan EG. Darboux transformation and soliton-like solutions for the Gerdjikov-Ivanov equation. J Phys A Math Gen 2000;33:6925–33. [10] Ling L, Liu QP. Darboux transformation for a two-component derivative nonlinear Schrödinger equation. J Phys A Theor 2010;43:434023. [11] Li HZ, Tian B, Li LL, Zhang HQ, Xu T. Darboux transformation and new solutions for the Whitham–Broer–Kaup equations. Phys Scr 2008;78:065001. [12] Zhaqilao, Sirendaoreji. A generalized coupled Korteweg-de Vries hierarchy, bi-Hamiltonian structure, and Darboux transformation. J Math Phys 2010;51:073501. [13] Xu SW, He JS, Wang LH. The Darboux transformation of the derivative nonlinear Schrödinger equation. J Phys A Math Theor 2011;44:305203. [14] Tu GZ. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30:330–8. [15] Sirendaoreji. On gauge equivalent constrained flow for AKNS hierarchy and Geng hierarchy. Adv Math 1999;28:437–46. [16] Guo FK, Zhang YF. The quadratic-form identity for constructing the Hamiltonian structure of integrable systems. J Phys A Math Gen 2005;38:8537–48. [17] Xia TC, Fan EG. The multicomponent generalized Kaup–Newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions. J Math Phys 2005;46:043510. [18] Qin ZY. A generalized Ablowitz–Ladik hierarchy, multi-Hamiltonian structure and Darboux transformation. J Math Phys 2008;49:063505.
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