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A new loop algebra and a multi-component integrable system similar to the TC hierarchy
Chaos, Solitons and Fractals 24 (2005) 627–630 www.elsevier.com/locate/chaos
A new loop algebra and a multi-component integrable system similar to the TC hierarchy Yuqin Yao School of Information Science and Engineering, Shandong University of Science and Technology, Taian 271019, PR China Accepted 14 September 2004 Communicated by Prof. Y. Aizawa
Abstract e with 3M dimensions is constructed by use of some properties of exterior algebra, which is A new loop algebra X devoted to establishing a new isospectral problem. As its application, a multi-component integrable system similar to the TC hierarchy is obtained, whose reduction case presents a multi-component KdV equation. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Searching for new integrable hierarchies of soliton equations has been an important and interesting topic in soliton theory. Guizhang Tu once proposed a simple and straightforward method for generating integrable Hamiltonian hierarchies of soliton equations in Ref. [1]. By making use of Tu scheme, many integrable systems, such as AKNS hierarchy, KdV hierarchy, Schrodinger systems and so on, have been obtained [1–5]. As far as the multi-component integrable hierarchies are concerned, there have been developed in Refs. [6,7]. Recently a Lie algebra with 3M dimensions was constructed in Ref. [8] and a simple method for producing multi-component integrable hierarchies of soliton equations was proposed. In this paper, another loop algebra is presented by use of exterior algebra. It follows that an isospectral problem is established and the multi-component integrable system similar to the TC hierarchy is worked out by using Tu scheme. As its reduction case, the multi-component KdV equation is obtained.
2. A type of new loop algebra Definition 1 [8]. If a = (a1, a2, . . . , aM)T, b = (b1, b2, . . . , bM)T are two column vectors, their vector product a * b is defined as a b ¼ b a ¼ ða1 b1 ; a2 b2 ; . . . ; aM bM ÞT ; where ai, bi(i = 1, 2, . . ., M) are real or complex numbers.
E-mail address: [email protected] 0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.066
ð1Þ
628
Y. Yao / Chaos, Solitons and Fractals 24 (2005) 627–630
Introducing the diagonal matrix ~a ¼ diagða1 ; a2 ; . . . ; aM Þ, then (1) can be written as a b ¼ ~ab:
ð2Þ
Definition 2. Set a = (a1, a2, . . ., aM)T, A = (0, . . . , 0, ai, 0, . . ., 0)M·N, we define their product as a A ¼ A a ¼ ð0; . . . ; 0; a ai ; 0; . . . ; 0Þ; where ai = (ai1, ai2, . . ., aiM)T, aij(j = 1, 2, . . ., M) are real or complex numbers, which have no relations with independent variables x, t. Definition 3. If x1 ¼ ðx11 ; 0; 0Þ, x2 ¼ ð0; x22 ; 0Þ, x3 ¼ ð0; 0; x33 Þ are linear independent real or complex M · 3 matrix vectors, which have no relations with x and t. X denotes a linear space expanded by {x1, x2, x3}. i.e. X ¼ fwa jwa ¼ a1 x1 þ a2 x2 þ a3 x3 g
ð3Þ
ai (i = 1, 2, 3) are the same as those in Definition 2, xii ¼ ðxii1 ; xii2 ; . . . ; xiiM ÞT , xiij are real or complex numbers. Define a commutation operation [wa, wb] as ½wa ; wb wa ^ wb ¼ ða1 b2 a2 b1 Þ x1 ^ x2 þ ða1 b3 a3 b1 Þ x1 ^ x3 þ ða2 b3 a3 b2 Þ x2 ^ x3 ;
ð4Þ
x1 ^ x2 ¼ x2 ^ x1 ¼ x3 ;
ð5Þ
x1 ^ x3 ¼ x3 ^ x1 ¼ x2 ;
ða x1 þ b x2 Þ ^ x3 ¼ a x2 b x1 ;
x2 ^ x3 ¼ x3 ^ x2 ¼ x1 ;
ða x2 þ b x3 Þ ^ x1 ¼ a x3 b x2 ;
ða x1 þ b x3 Þ ^ x2
¼ a x3 þ b x1 ;
ð6Þ
where we call the symbol ^ a kind of exterior product, wa, wb 2 X. A direct verification indicates that X along with (4)– (6) constitutes a Lie algebra. Specially, setting x1 ¼ ðI M ; 0; 0Þ;
x2 ¼ ð0; I M ; 0Þ;
x3 ¼ ð0; 0; I M Þ
then (4) becomes the formula (4) in Ref. [8]. Therefore, the Lie algebra (3) with (4)–(6) is the generalized form of that presented in Ref. [8]. Definition 4. Set x1 ðnÞ ¼ x1 k2nþ1 x1 k2nþ1 ; x1 ðmÞ ^ x2 ðnÞ ¼ x3 ðm þ n þ 1Þ;
x2 ðnÞ ¼ x2 k2nþ1 ;
x3 ðnÞ ¼ x3 k2n
x1 ðmÞ ^ x3 ðnÞ ¼ x2 ðm þ nÞ;
x2 ðmÞ ^ x3 ðnÞ ¼ x1 ðm þ nÞ:
ð7Þ
e. then {x1(n), x2(n), x3(n)} is a loop algebra, denote it as X Consider the following isospectral problem:
ux ¼ U ^ u; e; ut ¼ V ^ u; U; V ; u 2 X
ð8Þ
whose compatibility gives rise to uxt ¼ U t ^ u þ U ^ ðV ^ uÞ ¼ utx ¼ V x ^ u þ V ^ ðU ^ uÞ; U t ^ u V x ^ u þ ðU ^ V Þ ^ u ¼ 0; ðU t V x þ U ^ V Þ ^ u ¼ 0:
ð9Þ
Sine u is arbitrary, the formula (9) reduces to U t V x þ ½U ; V ¼ 0:
ð10Þ
Hence, the compatibility of the Lax pair (8) leads to the zero curvature equation (10). This implies that the evolution equations derived from (8) are Lax integrable.
Y. Yao / Chaos, Solitons and Fractals 24 (2005) 627–630
629
3. The multi-component integrable system similar to the TC hierarchy Consider the following isospectral problem ux ¼ U ^ u;
U ¼ I M x1 ð0Þ þ q x1 ð1Þ þ r x2 ð1Þ;
kt ¼ 0;
ð11Þ
where M
IM
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ ¼ ð0; 0; . . . ; 0ÞT ;
Set V ¼
X iP0
q ¼ ðq1 ; q2 ; . . . ; qM ÞT ;
r ¼ ðr1 ; r2 ; . . . ; rM ÞT :
ðað0; iÞ x1 ðiÞ þ bð0; iÞ x2 ðiÞ þ cð0; iÞ x3 ðiÞÞ; ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
where að0; iÞ ¼ ðai1 ; ai2 ; . . . ; aiM ÞT , bð0; iÞ ¼ ðbi1 ; bi2 ; . . . ; biM ÞT , cð0; iÞ ¼ ðci1 ; ci2 ; . . . ; ciM ÞT . Solving the stationary zero curvature equation V x ¼ ½U ; V
ð12Þ
gives ax ð0; 1 þ iÞ ¼ r cð0; iÞ; bx ð0; 1 þ iÞ ¼ cð0; 1 þ iÞ þ q cð0; iÞ; cx ð0; iÞ ¼ bð0; 1 þ iÞ þ q bð0; iÞ r að0; iÞ; M
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ bð0; 0Þ ¼ cð0; 0Þ ¼ 0 ¼ ð0; 0; . . . ; 0ÞT ;
að0; 0Þ ¼ a ¼ ða1 ; a2 ; . . . ; aM Þ;
bð0; 1Þ ¼ a r;
ð13Þ
M
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ cð0; 1Þ ¼ a rx ; að0; 1Þ ¼ 0 ¼ ð0; 0; . . . ; 0ÞT ; cð0; 2Þ ¼ a rxxx a qx r 2a q rx ;
bð0; 2Þ ¼ a rxx a q r; a að0; 2Þ ¼ r r: 2
Denoting ðnÞ
Vþ ¼
n X ðað0; iÞ x1 ðn iÞ þ bð0; iÞ x2 ðn iÞ þ cð0; iÞ x3 ðn iÞÞ; i¼0 ðnÞ
2n V ðnÞ ¼ k V Vþ ;
then Eq. (12) can be written as ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V :
ð14Þ
It is easy to find that the power times of k on the left-hand side in Eq. (14) P1, while the power times of k on the right-hand side in Eq. (14) 60.Therefore, the power times of k on both sides in Eq. (14) are 1 and 0. Thus ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ ax ð0; n þ 1Þ x1 ð1Þ þ ðbx ð0; n þ 1Þ cð0; n þ 1ÞÞ x2 ð1Þ bð0; n þ 1Þ x3 ð0Þ: ðnÞ
Taking V ðnÞ ¼ V þ þ Dn , Dn = (q * r1) * b(0, n + 1) Æ x1(1) + b(0,n + 1) Æ x2(1), then ðnÞ ¼ ½ax ð0; n þ 1Þ ððq r1 Þ bð0; n þ 1ÞÞx x1 ð1Þ cð0; n þ 1Þ x2 ð1Þ; V ðnÞ x þ ½U ; V
the zero curvature equation ðnÞ ¼0 U t V ðnÞ x þ ½U ; V
determines the Lax integrable system að0; n þ 1Þ q o oðq r1 Þ ððq r1 Þ bð0; n þ 1ÞÞx ax ð0; n þ 1Þ ¼ ¼ bð0; n þ 1Þ r tn cð0; n þ 1Þ ðq r1 Þ o o ! 1 að0; n þ 1Þ að0; n þ 1Þ o o~q~r ¼ ¼J ; bð0; n þ 1Þ bð0; n þ 1Þ ~q~r1 o o where J is Hamilton operator, r1 is the reverse vector of the vector r, so is ~r1 .
ð15Þ
ð16Þ
630
Y. Yao / Chaos, Solitons and Fractals 24 (2005) 627–630
From (13), we find að0; n þ 1Þ ¼ bð0; n þ 1Þ
o1 q o o1 r o 1 r þoðq r Þ o q þo2 að0; nÞ ¼L : bð0; nÞ
!
að0; nÞ bð0; nÞ
¼
qo o1~ro o1 ~ 1 ~r þ o~ q~r o ~ q þ o2
Thus, the system (16) can be written as að0; n þ 1Þ 0 q ¼J ¼ JLn : bð0; n þ 1Þ ar r tn
!
að0; nÞ bð0; nÞ
ð17Þ
When M = 1, the system (17) is just the TC hierarchy, when M > 1, the system (17) is a format of the TC hierarchy. As a reduction case, taking q = r, n = 1, a = 1, the system (17) reduces to the multi-component KdV equation qt ¼ qxxx q qx :
Acknowledgment The author would like to thank Professor Yufeng Zhang for his guidance.
References [1] [2] [3] [4] [5] [6] [7] [8]
Tu Guizhang. J Math Phys 1989;30:330. Guo Fukui. Acta Math Phys Sin 1999;19:507. Zhang Yufeng, et al. Math Pract Theory 2003;33:109. Zhu Zuonong, et al. Phys Lett A 1997;235:227. Zhang Yufeng. Phys Lett A 2003;317:280. Tsuchida T, Wadati M. J Phys Soc Jpn 1999;69:2241. Tsuchida T, Wadati M. Phys Lett A 1999;257:53. Guo Fukui, Zhang Yufeng. J Math Phys 2003;44:5793.
A new loop algebra and a multi-component integrable system similar to the TC hierarchy Yuqin Yao School of Information Science and Engineering, Shandong University of Science and Technology, Taian 271019, PR China Accepted 14 September 2004 Communicated by Prof. Y. Aizawa
Abstract e with 3M dimensions is constructed by use of some properties of exterior algebra, which is A new loop algebra X devoted to establishing a new isospectral problem. As its application, a multi-component integrable system similar to the TC hierarchy is obtained, whose reduction case presents a multi-component KdV equation. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Searching for new integrable hierarchies of soliton equations has been an important and interesting topic in soliton theory. Guizhang Tu once proposed a simple and straightforward method for generating integrable Hamiltonian hierarchies of soliton equations in Ref. [1]. By making use of Tu scheme, many integrable systems, such as AKNS hierarchy, KdV hierarchy, Schrodinger systems and so on, have been obtained [1–5]. As far as the multi-component integrable hierarchies are concerned, there have been developed in Refs. [6,7]. Recently a Lie algebra with 3M dimensions was constructed in Ref. [8] and a simple method for producing multi-component integrable hierarchies of soliton equations was proposed. In this paper, another loop algebra is presented by use of exterior algebra. It follows that an isospectral problem is established and the multi-component integrable system similar to the TC hierarchy is worked out by using Tu scheme. As its reduction case, the multi-component KdV equation is obtained.
2. A type of new loop algebra Definition 1 [8]. If a = (a1, a2, . . . , aM)T, b = (b1, b2, . . . , bM)T are two column vectors, their vector product a * b is defined as a b ¼ b a ¼ ða1 b1 ; a2 b2 ; . . . ; aM bM ÞT ; where ai, bi(i = 1, 2, . . ., M) are real or complex numbers.
E-mail address: [email protected] 0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.066
ð1Þ
628
Y. Yao / Chaos, Solitons and Fractals 24 (2005) 627–630
Introducing the diagonal matrix ~a ¼ diagða1 ; a2 ; . . . ; aM Þ, then (1) can be written as a b ¼ ~ab:
ð2Þ
Definition 2. Set a = (a1, a2, . . ., aM)T, A = (0, . . . , 0, ai, 0, . . ., 0)M·N, we define their product as a A ¼ A a ¼ ð0; . . . ; 0; a ai ; 0; . . . ; 0Þ; where ai = (ai1, ai2, . . ., aiM)T, aij(j = 1, 2, . . ., M) are real or complex numbers, which have no relations with independent variables x, t. Definition 3. If x1 ¼ ðx11 ; 0; 0Þ, x2 ¼ ð0; x22 ; 0Þ, x3 ¼ ð0; 0; x33 Þ are linear independent real or complex M · 3 matrix vectors, which have no relations with x and t. X denotes a linear space expanded by {x1, x2, x3}. i.e. X ¼ fwa jwa ¼ a1 x1 þ a2 x2 þ a3 x3 g
ð3Þ
ai (i = 1, 2, 3) are the same as those in Definition 2, xii ¼ ðxii1 ; xii2 ; . . . ; xiiM ÞT , xiij are real or complex numbers. Define a commutation operation [wa, wb] as ½wa ; wb wa ^ wb ¼ ða1 b2 a2 b1 Þ x1 ^ x2 þ ða1 b3 a3 b1 Þ x1 ^ x3 þ ða2 b3 a3 b2 Þ x2 ^ x3 ;
ð4Þ
x1 ^ x2 ¼ x2 ^ x1 ¼ x3 ;
ð5Þ
x1 ^ x3 ¼ x3 ^ x1 ¼ x2 ;
ða x1 þ b x2 Þ ^ x3 ¼ a x2 b x1 ;
x2 ^ x3 ¼ x3 ^ x2 ¼ x1 ;
ða x2 þ b x3 Þ ^ x1 ¼ a x3 b x2 ;
ða x1 þ b x3 Þ ^ x2
¼ a x3 þ b x1 ;
ð6Þ
where we call the symbol ^ a kind of exterior product, wa, wb 2 X. A direct verification indicates that X along with (4)– (6) constitutes a Lie algebra. Specially, setting x1 ¼ ðI M ; 0; 0Þ;
x2 ¼ ð0; I M ; 0Þ;
x3 ¼ ð0; 0; I M Þ
then (4) becomes the formula (4) in Ref. [8]. Therefore, the Lie algebra (3) with (4)–(6) is the generalized form of that presented in Ref. [8]. Definition 4. Set x1 ðnÞ ¼ x1 k2nþ1 x1 k2nþ1 ; x1 ðmÞ ^ x2 ðnÞ ¼ x3 ðm þ n þ 1Þ;
x2 ðnÞ ¼ x2 k2nþ1 ;
x3 ðnÞ ¼ x3 k2n
x1 ðmÞ ^ x3 ðnÞ ¼ x2 ðm þ nÞ;
x2 ðmÞ ^ x3 ðnÞ ¼ x1 ðm þ nÞ:
ð7Þ
e. then {x1(n), x2(n), x3(n)} is a loop algebra, denote it as X Consider the following isospectral problem:
ux ¼ U ^ u; e; ut ¼ V ^ u; U; V ; u 2 X
ð8Þ
whose compatibility gives rise to uxt ¼ U t ^ u þ U ^ ðV ^ uÞ ¼ utx ¼ V x ^ u þ V ^ ðU ^ uÞ; U t ^ u V x ^ u þ ðU ^ V Þ ^ u ¼ 0; ðU t V x þ U ^ V Þ ^ u ¼ 0:
ð9Þ
Sine u is arbitrary, the formula (9) reduces to U t V x þ ½U ; V ¼ 0:
ð10Þ
Hence, the compatibility of the Lax pair (8) leads to the zero curvature equation (10). This implies that the evolution equations derived from (8) are Lax integrable.
Y. Yao / Chaos, Solitons and Fractals 24 (2005) 627–630
629
3. The multi-component integrable system similar to the TC hierarchy Consider the following isospectral problem ux ¼ U ^ u;
U ¼ I M x1 ð0Þ þ q x1 ð1Þ þ r x2 ð1Þ;
kt ¼ 0;
ð11Þ
where M
IM
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ ¼ ð0; 0; . . . ; 0ÞT ;
Set V ¼
X iP0
q ¼ ðq1 ; q2 ; . . . ; qM ÞT ;
r ¼ ðr1 ; r2 ; . . . ; rM ÞT :
ðað0; iÞ x1 ðiÞ þ bð0; iÞ x2 ðiÞ þ cð0; iÞ x3 ðiÞÞ; ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
ð0Þ
where að0; iÞ ¼ ðai1 ; ai2 ; . . . ; aiM ÞT , bð0; iÞ ¼ ðbi1 ; bi2 ; . . . ; biM ÞT , cð0; iÞ ¼ ðci1 ; ci2 ; . . . ; ciM ÞT . Solving the stationary zero curvature equation V x ¼ ½U ; V
ð12Þ
gives ax ð0; 1 þ iÞ ¼ r cð0; iÞ; bx ð0; 1 þ iÞ ¼ cð0; 1 þ iÞ þ q cð0; iÞ; cx ð0; iÞ ¼ bð0; 1 þ iÞ þ q bð0; iÞ r að0; iÞ; M
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ bð0; 0Þ ¼ cð0; 0Þ ¼ 0 ¼ ð0; 0; . . . ; 0ÞT ;
að0; 0Þ ¼ a ¼ ða1 ; a2 ; . . . ; aM Þ;
bð0; 1Þ ¼ a r;
ð13Þ
M
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ cð0; 1Þ ¼ a rx ; að0; 1Þ ¼ 0 ¼ ð0; 0; . . . ; 0ÞT ; cð0; 2Þ ¼ a rxxx a qx r 2a q rx ;
bð0; 2Þ ¼ a rxx a q r; a að0; 2Þ ¼ r r: 2
Denoting ðnÞ
Vþ ¼
n X ðað0; iÞ x1 ðn iÞ þ bð0; iÞ x2 ðn iÞ þ cð0; iÞ x3 ðn iÞÞ; i¼0 ðnÞ
2n V ðnÞ ¼ k V Vþ ;
then Eq. (12) can be written as ðnÞ
ðnÞ
ðnÞ V þx þ ½U ; V þ ¼ V ðnÞ x ½U ; V :
ð14Þ
It is easy to find that the power times of k on the left-hand side in Eq. (14) P1, while the power times of k on the right-hand side in Eq. (14) 60.Therefore, the power times of k on both sides in Eq. (14) are 1 and 0. Thus ðnÞ
ðnÞ
V þx þ ½U ; V þ ¼ ax ð0; n þ 1Þ x1 ð1Þ þ ðbx ð0; n þ 1Þ cð0; n þ 1ÞÞ x2 ð1Þ bð0; n þ 1Þ x3 ð0Þ: ðnÞ
Taking V ðnÞ ¼ V þ þ Dn , Dn = (q * r1) * b(0, n + 1) Æ x1(1) + b(0,n + 1) Æ x2(1), then ðnÞ ¼ ½ax ð0; n þ 1Þ ððq r1 Þ bð0; n þ 1ÞÞx x1 ð1Þ cð0; n þ 1Þ x2 ð1Þ; V ðnÞ x þ ½U ; V
the zero curvature equation ðnÞ ¼0 U t V ðnÞ x þ ½U ; V
determines the Lax integrable system að0; n þ 1Þ q o oðq r1 Þ ððq r1 Þ bð0; n þ 1ÞÞx ax ð0; n þ 1Þ ¼ ¼ bð0; n þ 1Þ r tn cð0; n þ 1Þ ðq r1 Þ o o ! 1 að0; n þ 1Þ að0; n þ 1Þ o o~q~r ¼ ¼J ; bð0; n þ 1Þ bð0; n þ 1Þ ~q~r1 o o where J is Hamilton operator, r1 is the reverse vector of the vector r, so is ~r1 .
ð15Þ
ð16Þ
630
Y. Yao / Chaos, Solitons and Fractals 24 (2005) 627–630
From (13), we find að0; n þ 1Þ ¼ bð0; n þ 1Þ
o1 q o o1 r o 1 r þoðq r Þ o q þo2 að0; nÞ ¼L : bð0; nÞ
!
að0; nÞ bð0; nÞ
¼
qo o1~ro o1 ~ 1 ~r þ o~ q~r o ~ q þ o2
Thus, the system (16) can be written as að0; n þ 1Þ 0 q ¼J ¼ JLn : bð0; n þ 1Þ ar r tn
!
að0; nÞ bð0; nÞ
ð17Þ
When M = 1, the system (17) is just the TC hierarchy, when M > 1, the system (17) is a format of the TC hierarchy. As a reduction case, taking q = r, n = 1, a = 1, the system (17) reduces to the multi-component KdV equation qt ¼ qxxx q qx :
Acknowledgment The author would like to thank Professor Yufeng Zhang for his guidance.
References [1] [2] [3] [4] [5] [6] [7] [8]
Tu Guizhang. J Math Phys 1989;30:330. Guo Fukui. Acta Math Phys Sin 1999;19:507. Zhang Yufeng, et al. Math Pract Theory 2003;33:109. Zhu Zuonong, et al. Phys Lett A 1997;235:227. Zhang Yufeng. Phys Lett A 2003;317:280. Tsuchida T, Wadati M. J Phys Soc Jpn 1999;69:2241. Tsuchida T, Wadati M. Phys Lett A 1999;257:53. Guo Fukui, Zhang Yufeng. J Math Phys 2003;44:5793.