- Email: [email protected]
The Hamiltonian structure of the expanding integrable model of the generalized AKNS hierarchy
Chaos, Solitons and Fractals 39 (2009) 271–276 www.elsevier.com/locate/chaos
The Hamiltonian structure of the expanding integrable model of the generalized AKNS hierarchy Binlu Feng *, Bo Han Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Accepted 2 January 2007
Abstract With the help of a new loop algebra g~, a simple deducing method for the generalized AKNS hierarchy (briefly called e from which the integrable couplings of the GAH, i.e. a GAH) is given. Then, g~ is extended to a larger loop algebra G kind of expanding integrable model, is obtained. It follows that its Hamiltonian structure is worked out by making use e 3 is devoted to getting the conserved denof the quadratic-form identity. Finally, a subalgebra ~a3 of the loop algebra A sities of the integrable couplings by the HAKS algorithm. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In 1984, Giachetti R. and Johnson R. [1] proposed the generalized AKNS isospectral problem as follows k þ w u ux ¼ U u;u ¼ ðu1 ; u2 ÞT ; U ¼ : v kw
ð1Þ
The infinite dimensional Lie algebraic structure of the integrable Hamiltonian system related to (1) were established in e1 [2]. The spectral matrix in (1) can be expressed by the following loop algebra A 8 n < e ðnÞ ¼ 1 0 kn ; e ðnÞ ¼ 0 k ; e ðnÞ ¼ 0 0 ; 1 2 3 ð2Þ 0 1 kn 0 0 0 : ½e1 ðmÞ; e2 ðnÞ ¼ 2e2 ðm þ nÞ; ½e1 ðmÞ; e3 ðnÞ ¼ 2e3 ðm þ nÞ; ½e2 ðmÞ; e3 ðnÞ ¼ e1 ðm þ nÞ; i.e. U ¼ e1 ð1Þ þ we1 ð0Þ þ ue2 ð0Þ þ ve3 ð0Þ. As we all know, by using the loop algebras with square matrix bases [3,4], we can produce the generalized AKNS hierarchy. However, we find that the following simpler loop algebra may easily lead to it as well. For 8a ¼ a1 e1 þ a2 e2 þ a3 e3 2 A1 ; b ¼ b1 e1 þ b2 e2 þ b3 e3 2 A1 , an operator [a, b] is defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞT ;
*
Corresponding author. E-mail address: [email protected] (B. Feng).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.139
ð3Þ
272
B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
then g ¼ spanfa ¼ ða1 ; a2 ; a3 ÞT g is a Lie algebra with (3), and isomorphic to R3. Assume that g~ ¼ g Cðk; k1 Þ, where Cðk; k1 Þ indicates a Laurent polynomial in k and 1k, a commutator in g~ is defined as ½aðmÞ; bðnÞ ¼ ½a; bkmþ1 ;
ð4Þ
then g~ is a loop algebra. In this paper, we want to use it to produce the generalized AKNS hierarchy. By enlarging it e the integrable couplings of the GAH and the Hamiltonian structure are obtained. Finally, to another loop algebra G, e 3 such that the conserved deisities of the intethe loop algebra ~a3 is enlarged into a subalgebra of the loop algebra A grable couplings of the GAH are produced by using the Herbman–Alberty–Koikawa–Sasaki (HAKS) algorithm. We e 3 can directly deduce many integrable couplings under the framework of the zero curvature find the loop algebra A equation.
2. The GAH and its Hamiltonian structure Consider an isospectral problem U ¼ ðk þ w; u; vÞT ; u ¼ ðu1 ; u2 ÞT : ð5Þ P P P T m m m Let V ¼ ða; b; cÞ ; a ¼ mP0 am k ; b ¼ mP0 bm k ; c ¼ mP>0 cm k , solving the stationary zero curvature equation ux ¼ U u;
V x ¼ ½U ; V
ð6Þ
8 amx ¼ ucm vbm ; > > > < 2b mþ1 ¼ bmx þ 2wbm 2uam ; > 2c ¼ cmx þ 2wcm 2vam ; mþ1 > > : a0 ¼ a ¼ const:; b0 ¼ c0 ¼ 0;
ð7Þ
gives
a1 ¼ 0;
b1 ¼ au;
c1 ¼ av:
ðnÞ
Denoting V þ ¼ Rnm¼0 ðam ; bm ; cm ÞT knm ¼ kn V V ðnÞ , a direct calculation reads that ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ 2bnþ1 e2 ð0Þ 2cnþ1 e3 ð0Þ: ðnÞ
Taking V ðnÞ ¼ V þ þ ðan ; 0; 0ÞT , from the zero curvature equation ðnÞ ¼ 0; U t V ðnÞ x þ ½U ; V
ð8Þ
we obtain 10 1 1 0 1 0 o 1 1 0 0 2an 2an 0 w anx 2 0 0 CB C B C B C C B C B B ~ut ¼ @ u A ¼ @ bnx 2wbn A ¼ @ 0 ð9Þ 0 o 2w A@ cn A ¼ J @ cn A ¼ JLn1 @ av A; au v t cnx þ 2wcn bn bn 0 o þ 2w 0 0 1 0 2o1 uðo2 þ wÞ 2o1 vðo2 wÞ 1 v o A. When taking w = 0, the hierarchy (9) is reduced to the famous AKNS where L ¼ @ 2 þw vo v 2 u2 0 o2 þ w hierarchy. Hence, it is is known as the GAH. Obviously, application for the trace identity proposed by Tu Guizhang [3] can not directly worked out the Hamiltonian structure if we take U and V as the the forms in (5) and (6), respectively. For the sake, we rewrite (3) as 10 1 0 b1 0 a3 a2 CB C B ð10Þ ½a; b ¼ @ 2a2 2a1 0 A@ b2 A ¼: P ðaÞb: 0
2a3
0
2a1
b3
The isospectral Lax pair ux ¼ U u; ut ¼ V u; can be presented in a new form according to (10)
ð11Þ
B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
273
ux ¼ P ðU Þu; ut ¼ P ðV Þu;
ð12Þ
whose compatibility is P ðU t Þ P ðV x Þ þ ½P ðU Þ; P ðV Þ ¼ 0:
ð13Þ
We can prove that (13) is equivalent to U t V x þ ½U ; V ¼ 0;
ð14Þ
which is just the compatibility of (11). In what follows, we use the trace identity to produce the Hamiltonian structure of the GAH since P(U), P(V) presented in (12) are all square matrices. ¼ 8a; hV ; U u i ¼ 4c; hV ; U v i ¼ 4b; hV ; U k i ¼ 8a. Substituting them into the trace A direct calculation gives V ; oU ow identity yields 0 1 8a d o C c cB ð15Þ ð8aÞ ¼ k k @ 4c A: d~u ok 4b Comparing the coefficients of kn1 gives 1 0 2an d C B ð2anþ1 Þ ¼ ðn þ cÞ@ cn A: d~u bn
ð16Þ
Taking n = 0 leads to c = 0. Thus, we have 1 0 2an d 2anþ1 dH nþ1 B C ¼: ¼ @ cn A: n d~u d~u bn
ð17Þ
Hence, we obtain the Hamiltonian structure of (9) 1 0 1 0 w 0 dH nþ1 C B C B ~ut ¼ @ u A ¼ JLn1 @ av A ¼ J ; d~u v t au
ð18Þ
where H nþ1 ¼ 2a2nþ1 is the Hamiltonian function.
3. An expanding loop algebra and its application Some Lie algebras and expanding Lie algebras were given in Refs. [5–9], which are used to generating integrable soliton hierarchies, their Hamiltonian structure and soliton solutions. In particular, the classifications of the Lie algee which belongs bras were also presented in Refs. [10,11]. In this section, we expand the loop algebra g~ into a larger one G e an isospectral problem can be set up, whose compatibility leads to an to one type appearing in [5]. With the help of G, expanding integrable model of the GAH. First, we show a Lie algebra G. Let G ¼ spanfei g6i¼1 , and ei ði ¼ 1; 2; 3; 4; 5; 6Þ satisfy the following operational relations ½e1 ; e2 ¼ 2e2 ; ½e1 ; e3 ¼ 2e3 ; ½e2 ; e3 ¼ e1 ; ½e1 ; e5 ¼ 2e5 ; ½e1 ; e6 ¼ 2e6 ; ½e2 ; e4 ¼ 2e5 ; ½e2 ; e6 ¼ e4 ; ½e3 ; e4 ¼ 2e6 ; ½e3 ; e5 ¼ e4 ; ½e1 ; e4 ¼ ½e3 ; e6 ¼ ½e2 ; e5 ¼ ½e4 ; e5 ¼ ½e4 ; e6 ¼ ½e5 ; e6 ¼ 0; ½ei ; ei ¼ 0; 1 6 i 6 6; where ½a; b ¼ ab ba. For 8a ¼ R6i¼1 ai ; ei ; b ¼ R6i¼1 bi ei 2 G, it is easy to compute that ½a; b ¼ ða2 b3 a3 b2 Þe1 þ 2ða1 b2 a2 b1 Þe2 þ 2ða3 b1 a1 b3 Þe3 þ ða2 b6 a6 b2 þ a5 b3 a3 b5 Þe4 þ 2ða1 b5 a5 b1 þ a4 b2 a2 b4 Þe5 þ 2ða3 b4 a4 b3 þ a6 b1 a1 b6 Þe6 : 6
R6i¼1 ai ei
ð19Þ T
6
2 G, there exists a vector va ¼ ða1 ; a2 ; a3 ; a4 ; a5 ; a6 Þ 2 R , which satisfies Let d : G#R be a linear map, i.e. 8a ¼ d (a)=va, we can verify that G is isomorphic to R6, also can prove R6 is a Lie algebra equipped with the commutator e 6 is defined as (19). The corresponding loop algebra R
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B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
e 6 ¼ spanfei ðnÞg6i¼1 ; R
ei ðnÞ ¼ ei ðnÞ ¼ ei kn ;
i ¼ 1; 2; 3; 4; 5; 6; nþm
whose commutator is given by ½ei ðmÞ; ej ðnÞ ¼ ½ei ; ej k problem is given by
e 6 , an isospectral ; m; n 2 Z. From the above loop algebra R
wx ¼ U w; U ¼ ðk þ w; u; v; u1 ; u2 ; u3 ÞT :
ð20Þ
Take V ¼ ða; b; c; d; f ; hÞT , where a ¼ RmP0 am km ; . . ., solving the stationary zero curvature equation for V V x ¼ ½U ; V
ð21Þ
8 amx ¼ ucm vbm ; > > > > > 2bmþ1 ¼ bmx þ 2wbm 2uam ; > > > > > 2cmþ1 ¼ cmx þ 2wcm 2vam ; > > > < d ¼ uh vf þ u c u b ; mx m m 2 m 3 m > 2h mþ1 ¼ hmx þ 2whm 2vd m þ 2u1 cm 2u3 am ; > > > > > > 2f mþ1 ¼ fmx þ 2wfm 2ud m þ 2u1 bm 2u2 am ; > > > > a0 ¼ a; b0 ¼ c0 ¼ d 0 ¼ h0 ¼ f0 ¼ 0; > > : a1 ¼ 0; b1 ¼ au; c1 ¼ av; f 1 ¼ au2 ; h1 ¼ au3 ;
ð22Þ
yields
T nm
Denoting V ðnÞ ¼ Rnm¼0 ðam ; bm ; cm ; d m ; fm ; hm Þ k Ut V
ðnÞ x
þ ½U ; V
ðnÞ
d 1 ¼ 0; . . . T
þ ðan ; 0; 0; d n ; 0; 0Þ , the zero curvature equation
¼0
ð23Þ
admits the Lax integrable hierarchy 10 0 1 0 w anx 0 CB 0 BuC B b 2wb nx n CB B C B CB B C B CB 0 BvC B cnx þ 2wcn C ¼B CB ~ut ¼ B CB o Bu C B d 1 nx CB 2 B C B CB B C B @ u2 A @ fnx 2wfn 2u1 bn A@ 0 u3 t hnx þ 2whn þ 2u1 cn 0 1 0 2an þ 2d n B c þh C n C B n C B B bn þ fn C C ¼: JDn : B C B 2a n C B C B A @ cn bn
0
0
o2
0
0
0
0 0 0 o þ 2w
0 0 o 2w 0
0 o 2
0 0
0 0 o þ 2w 0 0 o 2w þ 2u1
0 o 2w
1
C C C C 0 C C 0 C C o þ 2w 2u1 A 0
ð24Þ
Obviously, J is a Hamiltonian operator. (24) is called the expanding integrable model of the GAH. We define a funce6 tional on R fa; bg ¼ aT Fb
ð25Þ
so that the Hamiltonian structure of (24) is given. Again, the commutator (19) can be presented as another expression form as follows [12] T ½a; b ¼ aT RðbÞ; 0 0 b3 B 2b2 2b1 B B 2b3 0 where RðbÞ ¼ B B 0 b6 B @ 2b5 2b4 2b6 0
b2 0 2b1 b5 0 2b4
0 0 0 0 2b2 2b3
0 0 0 b3 2b1 0
1
0 0 C C 0 C C and meets b2 C C 0 A 2b1
ð26Þ
B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
RðbÞF ¼ ðRðbÞF ÞT ; Solving the matrix 0 2 0 B0 0 B B B0 1 F ¼B B2 0 B B @0 0 0 1
FT ¼ F:
275
ð27Þ
Eq. (27) for F, we have 1 0 2 0 0 1 0 0 1C C C 0 0 1 0C C: 0 0 0 0C C C 1 0 0 0A 0 0 0 0
A direct calculation reads fV ; U w g ¼ 2a þ 2d; fV ; U u g ¼ c þ h; fV ; U v g ¼ b þ f ; fV ; U u1 g ¼ 2a; fV ; U u2 g ¼ c; fV ; U u3 g ¼ b; fV ; U k g ¼ 2a 2d. We can prove the following quadratic-form identity holds d o fV ; U k g ¼ kc kc fV ; U ui g: d~u ok
ð28Þ
Substituting the above results into (28) yields 1 0 2a þ 2d B cþh C C B C B C B b þ f d o c cB C: fV ; U k g ¼ k k C d~u ok B 2a C B C B A @ c b
ð29Þ
Comparing the coefficients of kn1 gives d ð2anþ1 2d nþ1 Þ ¼ ðn þ cÞDn ; d~u n , where let n = 0, we have c = 0. Thus, Dn ¼ dH d~u 0 1 1 0 o uðo þ 2wÞ o vðo 2wÞ B o o B2 þw 0 2 B Bu 0 w o2 B L¼B 2 B 0 0 0 B B 0 0 @ 0
0
0
0
ð30Þ nþ1 H n ¼ 2anþ1 þ2d . From (22), a recurrence operator L is presented as n 1 1 0 o ðu3 o 2vu1 2wu3 Þ o1 ðu2 o þ 2uu1 þ 2u2 wÞ C C u3 u1 0 C C u2 0 u1 C C; 1 1 C 0 o uðo þ 2wÞ o ðo 2wÞ C C o v þ w 0 A 2 o u 0 w2
and it satisfies Dnþ1 ¼ LDn . Hence, the hierarchy (24) can be written as 1 0 0 B aðu þ u Þ C 3 C B C B B aðu þ u2 Þ C n1 B C ¼ J dH n ; ~ut ¼ JL B C d~u 0 C B C B A @ av
ð31Þ
au which is just the Hamiltonian structure of the integrable couplings of the GAH. We can verify that JL ¼ L J . Therefore, the hierarchy (31) is Liouville integrable.
4. The conserved densities of the integrable couplings (31) e 3: We first construct a subalgebra ~a3 of the loop algebra A
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B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
8 1 0 0 n k 0 0 0 0 > > > > n C B B > 0 0 C > B0 B 0 k > > C; e2 ðnÞ ¼ B > e1 ðnÞ ¼ B n > A @0 @ 0 0 k 0 > > > > n > > 0 0 0 k 0 < 1 0 0 n 0 0 k 0 0 0 > > B0 0 B 0 0 0 kn C > > C B B > > e4 ðnÞ ¼ B C; e5 ðnÞ ¼ B > > @0 0 @0 0 0 > 0 A > > > > > 0 0 0 0 0 0 > > : degðei ðnÞÞ ¼ n; i ¼ 1; 2; 3; 4; 5; 6:
kn 0 0 0 0 0 0 0
0
0
1
0
0
0
B kn 0 0C C B e3 ¼ B n C; @0 0 k A 0 0 0 0 n1 k 0 0 B0 0 0C C B C; e6 ðnÞ ¼ B @0 0 0A 0
0
0
0
0
1
0 0 0C C C; 0 0 0A 0 kn 0 1 0 0 kn 0 C C C; 0 0A 0
ð32Þ
0
a31 and an Abelian ideal ~ a32 , i.e. ~ a3 ¼ ~ a31 ] ~ a32 , where The loop algebra a~3 is a semi-direct sum of an ideal subalgebra ~ ~31 ¼ spanfe1 ðnÞ; e2 ðnÞ; e3 ðnÞg; ~a32 ¼ spanfe4 ðnÞ; e5 ðnÞ; e6 ðnÞg. Consider the isospectral problem a ux ¼ U u; U ¼ e1 ð1Þ þ ue2 ð0Þ þ ve3 ð0Þ þ we1 ð0Þ þ u1 e4 ð0Þ þ u2 e5 ð0Þ þ u3 e6 ð0Þ; 8 ðnÞ < ut ¼ V u; n P : V ðnÞ ¼ ðam e1 ðn mÞ þ bm e2 ðn mÞ þ cm e3 ðn mÞ þ d m e4 ðn mÞ þ fm e5 ðn mÞ þ hm e6 ðn mÞÞ;
ð33Þ ð34Þ
m¼0
then the zero curvature equation ðnÞ U t V ðnÞ ¼0 x þ ½U ; V
ð35Þ
also admits the integrable hierarchy (31). But, it is easy to find that the Hamiltonian structure (31) can not be produced by using the trace identity because some zero identities are generated. In order to get the conserved densities of the integrable couplings by Herbman–Alberty–Koikawa–Sasaki (HAKS) algorithm [13], we ought to use the spectral square e X Þ diagðh1 ; h2 ; h3 ; h4 Þ, where hi ¼ ð U e XÞ . matrices. Construct a matrix X ¼ ðX ij Þ with X ij ¼ uuji , and set H ¼ ð U D ii The notation AD represent the diagonal part of a matrix A. It is easy to verify that X x ¼ U X XH :
ð36Þ
For arbitrary constants c1 ; c2 ; c3 ; c4 , the quantity R h ¼ Ri ci hi is a Hconserved density of the hierarchy of Eq. (35) by e ¼ 1 hC; H idx, then d e employing the HAKS algorithm, i.e. if let H ¼ 0, where C ¼ diagðc1 ; c2 ; c3 ; c4 Þ; hA; Bi ¼ trðABÞ. 1 dt References [1] Giachetti R, Johnson R. Phys Lett A 1984;102:81. [2] Li Yishen, Cheng Yi, Zeng Yunbo. In: Chaohao Gu, Yishen Li, Guizhang Tu, editors. Research reports in physics, nonlinear physics. Berlin, Heideiberg: Springer; 1990. p. 47. [3] Tu Guizhang. J Math Phys 1989;30(2):330. [4] Ma Wenxiu. Chinese J Contemp Math 1992;13(1):79. [5] Zhang Yufeng, Guo Fukui. Commun Theor Phys 2006;46:812. [6] Zhang Yufeng, Zhang Hongqing. Acta Math Sin 2006;49(6):1287 [in Chinese]. [7] Guo Fukui, Zhang Yufeng. Commun Theor Phys 2006;45:799. [8] Zhang Yufeng et al. Commun Theor Phys 2006;45:411. [9] Guo Fukui, Zhang Yufeng. Commun Theor Phys 2006;46:577. [10] Ma Wen-Xiu et al. Phys Lett A 2006;351:125. [11] Zhang Yufeng, Fan Engui, et al. Phys Lett A 2006;357(6):454. [12] Guo Fukui, Zhang Yufeng. J Phys A 2005;38:8537. [13] Tu GuiZhang. Northeastern Math J 1990;6(1):26.
The Hamiltonian structure of the expanding integrable model of the generalized AKNS hierarchy Binlu Feng *, Bo Han Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Accepted 2 January 2007
Abstract With the help of a new loop algebra g~, a simple deducing method for the generalized AKNS hierarchy (briefly called e from which the integrable couplings of the GAH, i.e. a GAH) is given. Then, g~ is extended to a larger loop algebra G kind of expanding integrable model, is obtained. It follows that its Hamiltonian structure is worked out by making use e 3 is devoted to getting the conserved denof the quadratic-form identity. Finally, a subalgebra ~a3 of the loop algebra A sities of the integrable couplings by the HAKS algorithm. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In 1984, Giachetti R. and Johnson R. [1] proposed the generalized AKNS isospectral problem as follows k þ w u ux ¼ U u;u ¼ ðu1 ; u2 ÞT ; U ¼ : v kw
ð1Þ
The infinite dimensional Lie algebraic structure of the integrable Hamiltonian system related to (1) were established in e1 [2]. The spectral matrix in (1) can be expressed by the following loop algebra A 8 n < e ðnÞ ¼ 1 0 kn ; e ðnÞ ¼ 0 k ; e ðnÞ ¼ 0 0 ; 1 2 3 ð2Þ 0 1 kn 0 0 0 : ½e1 ðmÞ; e2 ðnÞ ¼ 2e2 ðm þ nÞ; ½e1 ðmÞ; e3 ðnÞ ¼ 2e3 ðm þ nÞ; ½e2 ðmÞ; e3 ðnÞ ¼ e1 ðm þ nÞ; i.e. U ¼ e1 ð1Þ þ we1 ð0Þ þ ue2 ð0Þ þ ve3 ð0Þ. As we all know, by using the loop algebras with square matrix bases [3,4], we can produce the generalized AKNS hierarchy. However, we find that the following simpler loop algebra may easily lead to it as well. For 8a ¼ a1 e1 þ a2 e2 þ a3 e3 2 A1 ; b ¼ b1 e1 þ b2 e2 þ b3 e3 2 A1 , an operator [a, b] is defined as ½a; b ¼ ða2 b3 a3 b2 ; 2ða1 b2 a2 b1 Þ; 2ða3 b1 a1 b3 ÞÞT ;
*
Corresponding author. E-mail address: [email protected] (B. Feng).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.139
ð3Þ
272
B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
then g ¼ spanfa ¼ ða1 ; a2 ; a3 ÞT g is a Lie algebra with (3), and isomorphic to R3. Assume that g~ ¼ g Cðk; k1 Þ, where Cðk; k1 Þ indicates a Laurent polynomial in k and 1k, a commutator in g~ is defined as ½aðmÞ; bðnÞ ¼ ½a; bkmþ1 ;
ð4Þ
then g~ is a loop algebra. In this paper, we want to use it to produce the generalized AKNS hierarchy. By enlarging it e the integrable couplings of the GAH and the Hamiltonian structure are obtained. Finally, to another loop algebra G, e 3 such that the conserved deisities of the intethe loop algebra ~a3 is enlarged into a subalgebra of the loop algebra A grable couplings of the GAH are produced by using the Herbman–Alberty–Koikawa–Sasaki (HAKS) algorithm. We e 3 can directly deduce many integrable couplings under the framework of the zero curvature find the loop algebra A equation.
2. The GAH and its Hamiltonian structure Consider an isospectral problem U ¼ ðk þ w; u; vÞT ; u ¼ ðu1 ; u2 ÞT : ð5Þ P P P T m m m Let V ¼ ða; b; cÞ ; a ¼ mP0 am k ; b ¼ mP0 bm k ; c ¼ mP>0 cm k , solving the stationary zero curvature equation ux ¼ U u;
V x ¼ ½U ; V
ð6Þ
8 amx ¼ ucm vbm ; > > > < 2b mþ1 ¼ bmx þ 2wbm 2uam ; > 2c ¼ cmx þ 2wcm 2vam ; mþ1 > > : a0 ¼ a ¼ const:; b0 ¼ c0 ¼ 0;
ð7Þ
gives
a1 ¼ 0;
b1 ¼ au;
c1 ¼ av:
ðnÞ
Denoting V þ ¼ Rnm¼0 ðam ; bm ; cm ÞT knm ¼ kn V V ðnÞ , a direct calculation reads that ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ 2bnþ1 e2 ð0Þ 2cnþ1 e3 ð0Þ: ðnÞ
Taking V ðnÞ ¼ V þ þ ðan ; 0; 0ÞT , from the zero curvature equation ðnÞ ¼ 0; U t V ðnÞ x þ ½U ; V
ð8Þ
we obtain 10 1 1 0 1 0 o 1 1 0 0 2an 2an 0 w anx 2 0 0 CB C B C B C C B C B B ~ut ¼ @ u A ¼ @ bnx 2wbn A ¼ @ 0 ð9Þ 0 o 2w A@ cn A ¼ J @ cn A ¼ JLn1 @ av A; au v t cnx þ 2wcn bn bn 0 o þ 2w 0 0 1 0 2o1 uðo2 þ wÞ 2o1 vðo2 wÞ 1 v o A. When taking w = 0, the hierarchy (9) is reduced to the famous AKNS where L ¼ @ 2 þw vo v 2 u2 0 o2 þ w hierarchy. Hence, it is is known as the GAH. Obviously, application for the trace identity proposed by Tu Guizhang [3] can not directly worked out the Hamiltonian structure if we take U and V as the the forms in (5) and (6), respectively. For the sake, we rewrite (3) as 10 1 0 b1 0 a3 a2 CB C B ð10Þ ½a; b ¼ @ 2a2 2a1 0 A@ b2 A ¼: P ðaÞb: 0
2a3
0
2a1
b3
The isospectral Lax pair ux ¼ U u; ut ¼ V u; can be presented in a new form according to (10)
ð11Þ
B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
273
ux ¼ P ðU Þu; ut ¼ P ðV Þu;
ð12Þ
whose compatibility is P ðU t Þ P ðV x Þ þ ½P ðU Þ; P ðV Þ ¼ 0:
ð13Þ
We can prove that (13) is equivalent to U t V x þ ½U ; V ¼ 0;
ð14Þ
which is just the compatibility of (11). In what follows, we use the trace identity to produce the Hamiltonian structure of the GAH since P(U), P(V) presented in (12) are all square matrices. ¼ 8a; hV ; U u i ¼ 4c; hV ; U v i ¼ 4b; hV ; U k i ¼ 8a. Substituting them into the trace A direct calculation gives V ; oU ow identity yields 0 1 8a d o C c cB ð15Þ ð8aÞ ¼ k k @ 4c A: d~u ok 4b Comparing the coefficients of kn1 gives 1 0 2an d C B ð2anþ1 Þ ¼ ðn þ cÞ@ cn A: d~u bn
ð16Þ
Taking n = 0 leads to c = 0. Thus, we have 1 0 2an d 2anþ1 dH nþ1 B C ¼: ¼ @ cn A: n d~u d~u bn
ð17Þ
Hence, we obtain the Hamiltonian structure of (9) 1 0 1 0 w 0 dH nþ1 C B C B ~ut ¼ @ u A ¼ JLn1 @ av A ¼ J ; d~u v t au
ð18Þ
where H nþ1 ¼ 2a2nþ1 is the Hamiltonian function.
3. An expanding loop algebra and its application Some Lie algebras and expanding Lie algebras were given in Refs. [5–9], which are used to generating integrable soliton hierarchies, their Hamiltonian structure and soliton solutions. In particular, the classifications of the Lie algee which belongs bras were also presented in Refs. [10,11]. In this section, we expand the loop algebra g~ into a larger one G e an isospectral problem can be set up, whose compatibility leads to an to one type appearing in [5]. With the help of G, expanding integrable model of the GAH. First, we show a Lie algebra G. Let G ¼ spanfei g6i¼1 , and ei ði ¼ 1; 2; 3; 4; 5; 6Þ satisfy the following operational relations ½e1 ; e2 ¼ 2e2 ; ½e1 ; e3 ¼ 2e3 ; ½e2 ; e3 ¼ e1 ; ½e1 ; e5 ¼ 2e5 ; ½e1 ; e6 ¼ 2e6 ; ½e2 ; e4 ¼ 2e5 ; ½e2 ; e6 ¼ e4 ; ½e3 ; e4 ¼ 2e6 ; ½e3 ; e5 ¼ e4 ; ½e1 ; e4 ¼ ½e3 ; e6 ¼ ½e2 ; e5 ¼ ½e4 ; e5 ¼ ½e4 ; e6 ¼ ½e5 ; e6 ¼ 0; ½ei ; ei ¼ 0; 1 6 i 6 6; where ½a; b ¼ ab ba. For 8a ¼ R6i¼1 ai ; ei ; b ¼ R6i¼1 bi ei 2 G, it is easy to compute that ½a; b ¼ ða2 b3 a3 b2 Þe1 þ 2ða1 b2 a2 b1 Þe2 þ 2ða3 b1 a1 b3 Þe3 þ ða2 b6 a6 b2 þ a5 b3 a3 b5 Þe4 þ 2ða1 b5 a5 b1 þ a4 b2 a2 b4 Þe5 þ 2ða3 b4 a4 b3 þ a6 b1 a1 b6 Þe6 : 6
R6i¼1 ai ei
ð19Þ T
6
2 G, there exists a vector va ¼ ða1 ; a2 ; a3 ; a4 ; a5 ; a6 Þ 2 R , which satisfies Let d : G#R be a linear map, i.e. 8a ¼ d (a)=va, we can verify that G is isomorphic to R6, also can prove R6 is a Lie algebra equipped with the commutator e 6 is defined as (19). The corresponding loop algebra R
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e 6 ¼ spanfei ðnÞg6i¼1 ; R
ei ðnÞ ¼ ei ðnÞ ¼ ei kn ;
i ¼ 1; 2; 3; 4; 5; 6; nþm
whose commutator is given by ½ei ðmÞ; ej ðnÞ ¼ ½ei ; ej k problem is given by
e 6 , an isospectral ; m; n 2 Z. From the above loop algebra R
wx ¼ U w; U ¼ ðk þ w; u; v; u1 ; u2 ; u3 ÞT :
ð20Þ
Take V ¼ ða; b; c; d; f ; hÞT , where a ¼ RmP0 am km ; . . ., solving the stationary zero curvature equation for V V x ¼ ½U ; V
ð21Þ
8 amx ¼ ucm vbm ; > > > > > 2bmþ1 ¼ bmx þ 2wbm 2uam ; > > > > > 2cmþ1 ¼ cmx þ 2wcm 2vam ; > > > < d ¼ uh vf þ u c u b ; mx m m 2 m 3 m > 2h mþ1 ¼ hmx þ 2whm 2vd m þ 2u1 cm 2u3 am ; > > > > > > 2f mþ1 ¼ fmx þ 2wfm 2ud m þ 2u1 bm 2u2 am ; > > > > a0 ¼ a; b0 ¼ c0 ¼ d 0 ¼ h0 ¼ f0 ¼ 0; > > : a1 ¼ 0; b1 ¼ au; c1 ¼ av; f 1 ¼ au2 ; h1 ¼ au3 ;
ð22Þ
yields
T nm
Denoting V ðnÞ ¼ Rnm¼0 ðam ; bm ; cm ; d m ; fm ; hm Þ k Ut V
ðnÞ x
þ ½U ; V
ðnÞ
d 1 ¼ 0; . . . T
þ ðan ; 0; 0; d n ; 0; 0Þ , the zero curvature equation
¼0
ð23Þ
admits the Lax integrable hierarchy 10 0 1 0 w anx 0 CB 0 BuC B b 2wb nx n CB B C B CB B C B CB 0 BvC B cnx þ 2wcn C ¼B CB ~ut ¼ B CB o Bu C B d 1 nx CB 2 B C B CB B C B @ u2 A @ fnx 2wfn 2u1 bn A@ 0 u3 t hnx þ 2whn þ 2u1 cn 0 1 0 2an þ 2d n B c þh C n C B n C B B bn þ fn C C ¼: JDn : B C B 2a n C B C B A @ cn bn
0
0
o2
0
0
0
0 0 0 o þ 2w
0 0 o 2w 0
0 o 2
0 0
0 0 o þ 2w 0 0 o 2w þ 2u1
0 o 2w
1
C C C C 0 C C 0 C C o þ 2w 2u1 A 0
ð24Þ
Obviously, J is a Hamiltonian operator. (24) is called the expanding integrable model of the GAH. We define a funce6 tional on R fa; bg ¼ aT Fb
ð25Þ
so that the Hamiltonian structure of (24) is given. Again, the commutator (19) can be presented as another expression form as follows [12] T ½a; b ¼ aT RðbÞ; 0 0 b3 B 2b2 2b1 B B 2b3 0 where RðbÞ ¼ B B 0 b6 B @ 2b5 2b4 2b6 0
b2 0 2b1 b5 0 2b4
0 0 0 0 2b2 2b3
0 0 0 b3 2b1 0
1
0 0 C C 0 C C and meets b2 C C 0 A 2b1
ð26Þ
B. Feng, B. Han / Chaos, Solitons and Fractals 39 (2009) 271–276
RðbÞF ¼ ðRðbÞF ÞT ; Solving the matrix 0 2 0 B0 0 B B B0 1 F ¼B B2 0 B B @0 0 0 1
FT ¼ F:
275
ð27Þ
Eq. (27) for F, we have 1 0 2 0 0 1 0 0 1C C C 0 0 1 0C C: 0 0 0 0C C C 1 0 0 0A 0 0 0 0
A direct calculation reads fV ; U w g ¼ 2a þ 2d; fV ; U u g ¼ c þ h; fV ; U v g ¼ b þ f ; fV ; U u1 g ¼ 2a; fV ; U u2 g ¼ c; fV ; U u3 g ¼ b; fV ; U k g ¼ 2a 2d. We can prove the following quadratic-form identity holds d o fV ; U k g ¼ kc kc fV ; U ui g: d~u ok
ð28Þ
Substituting the above results into (28) yields 1 0 2a þ 2d B cþh C C B C B C B b þ f d o c cB C: fV ; U k g ¼ k k C d~u ok B 2a C B C B A @ c b
ð29Þ
Comparing the coefficients of kn1 gives d ð2anþ1 2d nþ1 Þ ¼ ðn þ cÞDn ; d~u n , where let n = 0, we have c = 0. Thus, Dn ¼ dH d~u 0 1 1 0 o uðo þ 2wÞ o vðo 2wÞ B o o B2 þw 0 2 B Bu 0 w o2 B L¼B 2 B 0 0 0 B B 0 0 @ 0
0
0
0
ð30Þ nþ1 H n ¼ 2anþ1 þ2d . From (22), a recurrence operator L is presented as n 1 1 0 o ðu3 o 2vu1 2wu3 Þ o1 ðu2 o þ 2uu1 þ 2u2 wÞ C C u3 u1 0 C C u2 0 u1 C C; 1 1 C 0 o uðo þ 2wÞ o ðo 2wÞ C C o v þ w 0 A 2 o u 0 w2
and it satisfies Dnþ1 ¼ LDn . Hence, the hierarchy (24) can be written as 1 0 0 B aðu þ u Þ C 3 C B C B B aðu þ u2 Þ C n1 B C ¼ J dH n ; ~ut ¼ JL B C d~u 0 C B C B A @ av
ð31Þ
au which is just the Hamiltonian structure of the integrable couplings of the GAH. We can verify that JL ¼ L J . Therefore, the hierarchy (31) is Liouville integrable.
4. The conserved densities of the integrable couplings (31) e 3: We first construct a subalgebra ~a3 of the loop algebra A
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8 1 0 0 n k 0 0 0 0 > > > > n C B B > 0 0 C > B0 B 0 k > > C; e2 ðnÞ ¼ B > e1 ðnÞ ¼ B n > A @0 @ 0 0 k 0 > > > > n > > 0 0 0 k 0 < 1 0 0 n 0 0 k 0 0 0 > > B0 0 B 0 0 0 kn C > > C B B > > e4 ðnÞ ¼ B C; e5 ðnÞ ¼ B > > @0 0 @0 0 0 > 0 A > > > > > 0 0 0 0 0 0 > > : degðei ðnÞÞ ¼ n; i ¼ 1; 2; 3; 4; 5; 6:
kn 0 0 0 0 0 0 0
0
0
1
0
0
0
B kn 0 0C C B e3 ¼ B n C; @0 0 k A 0 0 0 0 n1 k 0 0 B0 0 0C C B C; e6 ðnÞ ¼ B @0 0 0A 0
0
0
0
0
1
0 0 0C C C; 0 0 0A 0 kn 0 1 0 0 kn 0 C C C; 0 0A 0
ð32Þ
0
a31 and an Abelian ideal ~ a32 , i.e. ~ a3 ¼ ~ a31 ] ~ a32 , where The loop algebra a~3 is a semi-direct sum of an ideal subalgebra ~ ~31 ¼ spanfe1 ðnÞ; e2 ðnÞ; e3 ðnÞg; ~a32 ¼ spanfe4 ðnÞ; e5 ðnÞ; e6 ðnÞg. Consider the isospectral problem a ux ¼ U u; U ¼ e1 ð1Þ þ ue2 ð0Þ þ ve3 ð0Þ þ we1 ð0Þ þ u1 e4 ð0Þ þ u2 e5 ð0Þ þ u3 e6 ð0Þ; 8 ðnÞ < ut ¼ V u; n P : V ðnÞ ¼ ðam e1 ðn mÞ þ bm e2 ðn mÞ þ cm e3 ðn mÞ þ d m e4 ðn mÞ þ fm e5 ðn mÞ þ hm e6 ðn mÞÞ;
ð33Þ ð34Þ
m¼0
then the zero curvature equation ðnÞ U t V ðnÞ ¼0 x þ ½U ; V
ð35Þ
also admits the integrable hierarchy (31). But, it is easy to find that the Hamiltonian structure (31) can not be produced by using the trace identity because some zero identities are generated. In order to get the conserved densities of the integrable couplings by Herbman–Alberty–Koikawa–Sasaki (HAKS) algorithm [13], we ought to use the spectral square e X Þ diagðh1 ; h2 ; h3 ; h4 Þ, where hi ¼ ð U e XÞ . matrices. Construct a matrix X ¼ ðX ij Þ with X ij ¼ uuji , and set H ¼ ð U D ii The notation AD represent the diagonal part of a matrix A. It is easy to verify that X x ¼ U X XH :
ð36Þ
For arbitrary constants c1 ; c2 ; c3 ; c4 , the quantity R h ¼ Ri ci hi is a Hconserved density of the hierarchy of Eq. (35) by e ¼ 1 hC; H idx, then d e employing the HAKS algorithm, i.e. if let H ¼ 0, where C ¼ diagðc1 ; c2 ; c3 ; c4 Þ; hA; Bi ¼ trðABÞ. 1 dt References [1] Giachetti R, Johnson R. Phys Lett A 1984;102:81. [2] Li Yishen, Cheng Yi, Zeng Yunbo. In: Chaohao Gu, Yishen Li, Guizhang Tu, editors. Research reports in physics, nonlinear physics. Berlin, Heideiberg: Springer; 1990. p. 47. [3] Tu Guizhang. J Math Phys 1989;30(2):330. [4] Ma Wenxiu. Chinese J Contemp Math 1992;13(1):79. [5] Zhang Yufeng, Guo Fukui. Commun Theor Phys 2006;46:812. [6] Zhang Yufeng, Zhang Hongqing. Acta Math Sin 2006;49(6):1287 [in Chinese]. [7] Guo Fukui, Zhang Yufeng. Commun Theor Phys 2006;45:799. [8] Zhang Yufeng et al. Commun Theor Phys 2006;45:411. [9] Guo Fukui, Zhang Yufeng. Commun Theor Phys 2006;46:577. [10] Ma Wen-Xiu et al. Phys Lett A 2006;351:125. [11] Zhang Yufeng, Fan Engui, et al. Phys Lett A 2006;357(6):454. [12] Guo Fukui, Zhang Yufeng. J Phys A 2005;38:8537. [13] Tu GuiZhang. Northeastern Math J 1990;6(1):26.