Available online at www.sciencedirect.com

Chaos, Solitons and Fractals 37 (2008) 574–580

www.elsevier.com/locate/chaos

(2 + 1)-Dimensional Dirac hierarchy and its integrablecouplings as well as multi-component integrable system

Zhu Li a,*, Huanhe Dong b

a College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Chinab College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266510, PR China

Accepted 7 September 2006

Abstract

Under the frame of the (2 + 1)-dimensional zero curvature equation and Tu model, (2 + 1)-dimensional Dirac hier-archy is obtained. Again by use of the expanding loop algebra the integrable coupling system of the above hierarchy isgiven.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

In the past 20 years, the theory of integrable systems has undergone a quick development. Professor Tu proposed aefficient method for generating integrable system in Ref. [1]. Later, Professor Ma developed the method and called it Tumodel [2]. Following this method, a host of integrable systems associated with physics backgrounds were worked out,such as the results in Refs. [3–13]. With the development of soliton theory, integrable couplings are a new subject in thestudy of integrable systems, some integrable coupling systems of the integrable hierarchies were obtained in Refs. [14–16]. In recent years, in order to obtain multi-component integrable systems, professor Guo and Zhang constructed atype of multi-component loop algebra eGM , then some multi-component integrable systems were given in Refs. [17–19]. Recently, professor Zhou expressed the (2 + 1)-dimensional three-wave equation as a (2 + 1)-dimensional zero cur-vature equation and obtained the almost-periodic solutions for it in Ref. [20].

The following isospectral problem [21–23]:

0960-0doi:10

* CoE-m

ux ¼q �kþ r

kþ r �q

� �u ¼ Uu; kt ¼ 0; ð1Þ

admits the Dirac hierarchy as follows:

ut ¼q

r

� �t

¼ JLn bq

br

� �; J ¼

0 1

�1 0

� �; L ¼ �qo�1r qo�1q� o

o� ro�1r ro�1q

!: ð2Þ

779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved..1016/j.chaos.2006.09.033

rresponding author.ail address: lizhu1813@163.com (Z. Li).

Z. Li, H. Dong / Chaos, Solitons and Fractals 37 (2008) 574–580 575

In this paper, a (2 + 1)-dimensional Dirac hierarchy is presented by the (2 + 1)-dimensional zero curvature equation.Further, by using of the loop algebra in Ref. [21], the integrable coupling of the above system is produced.

2. A (2 + 1)-dimensional integrable system

Denoting ooz ¼ o

oy � oox ;

oow ¼ o

ot � oox, then zero curvature equation:

U w � V z þ ½U ; V � ¼ 0 ð3Þ

may be written as a (2 + 1)-dimensional form:

U t � V y þ ½U ; V � þ V x � U x ¼ 0; ð4Þ

which is regarded as the compatibility of the following Lax pairs:

uy ¼ ux þ Uu;

ut ¼ ux þ V u; kt ¼ 0:

�ð5Þ

We want to use (5) or (4) producing a hierarchy of soliton equations systematically, not a single equation. In what fol-lows, we consider the Dirac isospectral problem.

Taking the loop algebra in Ref. [1]:

hðnÞ ¼ 12kn 1 0

0 �1

� �; eðnÞ ¼ 1

2kn 0 1

1 0

� �; f ðnÞ ¼ 1

2kn 0 �1

1 0

� �;

½hðmÞ; eðnÞ� ¼ f ðmþ nÞ; ½hðmÞ; f ðnÞ� ¼ eðmþ nÞ; ½f ðmÞ; eðnÞ� ¼ hðmþ nÞ;degðhðnÞÞ ¼ degðeðnÞÞ ¼ degðf ðnÞÞ ¼ n:

8>>><>>>: ð6Þ

Considering an isospectral problem:

uy ¼ ux þ Uu; kt ¼ 0;

U ¼ f ð1Þ þ qhð0Þ þ reð0Þ:

�ð7Þ

Set

V ¼XmP0

ðamhð�mÞ þ bmeð�mÞ þ cmf ð�mÞÞ:

From the stationary zero curvature equation:

V y � V x ¼ ½U ; V �; ð8Þ

one arrives at the recursion relation as follows:

amy � amx ¼ bmþ1 � rcm;

bmy � bmx ¼ qcm � amþ1;

cmy � cmx ¼ qbm � ram;

c0 ¼ b ¼ const 6¼ 0; a0 ¼ b0 ¼ 0; c1 ¼ a ¼ const; a1 ¼ bq; b1 ¼ br:

8>>><>>>: ð9Þ

Note that

V ðnÞþ ¼ ðknV Þþ ¼Xn

m¼0

ðamhðn� mÞ þ bmeðn� mÞ þ cmf ðn� mÞÞ;

V ðnÞ� ¼ knV � V ðnÞþ :

Then (8) can be written as

�V ðnÞþy þ U ; V ðnÞþh i

þ V ðnÞþx ¼ V ðnÞ�y � U ; V ðnÞ�� �

� V ðnÞ�x : ð10Þ

A direct calculation reads that the terms on the left-hand side in (10) are of degree (deg) P 0, while the terms on theright-hand side in (10) are of degree (deg) 6 0. Therefore,

�V ðnÞþy þ U ; V ðnÞþh i

þ V ðnÞþx ¼ �bnþ1hð0Þ þ anþ1eð0Þ:

576 Z. Li, H. Dong / Chaos, Solitons and Fractals 37 (2008) 574–580

Taking V ðnÞ ¼ V ðnÞþ , thus the zero curvature equation:

U t � U x � V ðnÞy þ U ; V ðnÞ� �

þ V ðnÞx ¼ 0 ð11Þ

leads to the following (2 + 1)-dimensional Dirac hierarchy

qt � qx ¼ bnþ1;

rt � rx ¼ �anþ1;

�ð12Þ

that is,

ut � ux ¼q

r

� �t

�q

r

� �x

¼0 2

�2 0

� � anþ1

2

bnþ1

2

!¼ J

anþ1

2

bnþ1

2

!: ð13Þ

From (9), a recurrence operation L meets as follows:

anþ1

2

bnþ1

2

!¼

�q o�1y � o�1

x

� �r q o�1

y � o�1x

� �q� ðoy � oxÞ

ðoy � oxÞ � r o�1y � o�1

x

� �r r o�1

y � o�1x

� �q

0B@1CA an

2bn2

� �¼ L

an2bn2

� �:

Therefore, system (13) can be written as

ut � ux ¼q

r

� �t

�q

r

� �x

¼ JLnbq2

br2

!: ð14Þ

If we take ox ¼ o�1x ¼ 0, the system (14) reduces to the Dirac hierarchy. We call system (14) a generalized Dirac

hierarchy.

3. Integrable couplings of the system (13)

By using of the Lie algebra in Ref. [21]:

½e1; e2� ¼ e3; ½e1; e3� ¼ e2; ½e3; e2� ¼ e1; ½e1; e4� ¼ ½e3; e4� ¼ ½e5; e2� ¼ 12e5;

½e1; e5� ¼ ½e2; e4� ¼ ½e5; e3� ¼ 12e4; ½e4; e5� ¼ 0:

(ð15Þ

A new loop algebra eG is formed if we set:

eiðnÞ ¼ eikn; i ¼ 1; 2; 3; 4; 5;

½eiðmÞ; ejðnÞ� ¼ ½ei; ej�kmþn; 1 6 i; j 6 5;

degðeiðnÞÞ ¼ n; i ¼ 1; 2; 3; 4; 5:

8><>:

Considering an isospectral problem:

uy ¼ ux þ Uu; kt ¼ 0;

U ¼ e3ð1Þ þ u1e1ð0Þ þ u2e2ð0Þ þ u3e4ð0Þ þ u4e5ð0Þ:

�ð16Þ

Set

V ¼XmP0

ðame1ð�mÞ þ bme2ð�mÞ þ cme3ð�mÞ þ dme4ð�mÞ þ fme5ð�mÞÞ:

Solving the stationary zero curvature equation:

V y � V x ¼ ½U ; V �; ð17Þ

yields

amy � amx ¼ bmþ1 � u2cm;

bmy � bmx ¼ u1cm � amþ1;

cmy � cmx ¼ u1bm � u2am;

dmy � dmx ¼ 12ð�fmþ1 þ u1fm þ u2dm � u4am � u3bm þ u4cmÞ;

fmy � fmx ¼ 12ðdmþ1 þ u1dm � u2fm � u3am � u3cm þ u4bmÞ;

c0 ¼ b ¼ const 6¼ 0; a0 ¼ b0 ¼ d0 ¼ f0 ¼ 0; c1 ¼ a ¼ const;

a1 ¼ bu1; b1 ¼ bu2; d1 ¼ bu3; f 1 ¼ bu4:

8>>>>>>>>>>><>>>>>>>>>>>:ð18Þ

Z. Li, H. Dong / Chaos, Solitons and Fractals 37 (2008) 574–580 577

Note that

V ðnÞþ ¼ ðknV Þþ ¼Xn

m¼0

ðame1ðn� mÞ þ bme2ðn� mÞ þ cme3ðn� mÞ þ dme4ðn� mÞ þ fme5ðn� mÞÞ;

V ðnÞ� ¼ knV � V ðnÞþ :

A direct calculation reads that

�V ðnÞþy þ U ; V ðnÞþh i

þ V ðnÞþx ¼ �bnþ1e1ð0Þ þ anþ1e2ð0Þ þ1

2fnþ1e4ð0Þ �

1

2dnþ1e5ð0Þ:

Taking V ðnÞ ¼ V ðnÞþ , thus the zero curvature equation:

U t � U x � V ðnÞy þ U ; V ðnÞ� �

þ V ðnÞx ¼ 0

gives

u1t � u1x ¼ bnþ1;

u2t � u2x ¼ �anþ1;

u3t � u3x ¼ � 12fnþ1;

u4t � u4x ¼ 12dnþ1;

8>>><>>>: ð19Þ

that is,

ut � ux ¼

u1

u2

u3

u4

0BBB@1CCCA

t

�

u1

u2

u3

u4

0BBB@1CCCA

x

¼ J

12anþ1

12bnþ1

12dnþ1

12fnþ1

0BBB@1CCCA; ð20Þ

here

J ¼

0 2 0 0

�2 0 0 0

0 0 0 �1

0 0 1 0

0BBB@1CCCA: ð21Þ

From (18), a recurrence operation L meets as follows:

L ¼

�u1 o�1y � o�1

x

� �u2 A 0 0

B u2 o�1y � o�1

x

� �u1 0 0

C D �u1 2ðoy � oxÞ þ u2

E F �2ðoy � oxÞ þ u2 u1

0BBBBB@

1CCCCCA: ð22Þ

Here

A ¼ �ðoy � oxÞ þ u1 o�1y � o�1

x

� �u1;

B ¼ ðoy � oxÞ � u2 o�1y � o�1

x

� �u2;

C ¼ u3 � u3 o�1y � o�1

x

� �u2;

D ¼ �u4 þ u3 o�1y � o�1

x

� �u1;

E ¼ �u4 � u4 o�1y � o

�1x

� �u2;

F ¼ �u3 þ u4 o�1y � o�1

x

� �u1:

578 Z. Li, H. Dong / Chaos, Solitons and Fractals 37 (2008) 574–580

Therefore, system (20) can be written as

ut � ux ¼

u1

u2

u3

u4

0BBB@1CCCA

t

�

u1

u2

u3

u4

0BBB@1CCCA

x

¼ JLn

12bu1

12bu2

12bu3

12bu4

0BBB@1CCCA: ð23Þ

Taking u3 = u4 = 0, u1 = q, u2 = r in (23), then system (23) is reduces to (14). Therefore, we call (23) an extending(2 + 1)-dimensional integrable model of generalized Dirac hierarchy.

4. A corresponding multi-component integrable system

Denoting GM ¼ fa ¼ ðaijÞM�3 ¼ ða1; a2; a3Þg, where ai(i = 1,2,3) stand for the ith column of the matrix a, M is anarbitrary positive integer.

Definition 1. If a = (a1,a2, . . .,aM)T, b = (b1,b2, . . .,bM)T, are two vectors, then define their vector product a * b andvector quotient a

b as

a � b ¼ b � a ¼ ða1 � b1; a2 � b2; . . . ; aM � bM ÞT;

ab¼ a � 1

b¼ a1

b1

;a2

b2

; . . . ;aM

bM

� �T

:

Definition 2. Let a = (a1,a2,a3), b = (b1,b2,b3) 2 GM, a commutation operation [a,b] is defined as

½a; b� ¼ ða3 � b2 � a2 � b3; a1 � b3 � a3 � b1; a1 � b2 � a2 � b1Þ: ð24Þ

Definition 3. Set

eGM ¼ faknja 2 GM ; n ¼ 0;�1;�2; . . .g; ð25Þ

along with a communtation operation presented by

½akm; bkn� ¼ ½a; b�kmþn; 8a; b 2 GM : ð26Þ

In terms of Eqs. (24)–(26), we conclude that GM is a loop algebra.

Consider an isospectral problem

u1

u2

..

.

uM

0BBBB@1CCCCA

y

¼

u1

u2

..

.

uM

0BBBB@1CCCCA

x

þ ðq; r; kIM Þ;

u1

u2

..

.

uM

0BBBB@1CCCCA

266664377775; ð27Þ

where IM ¼ ð1; 1; . . . ; 1ÞT1�M , q = (q1,q2, . . .,qM)T, r = (r1, r2, . . ., rM)T. Set

V ¼XmP0

ðam; bm; cmÞk�m;

where am = (am1,am2, . . .,amM)T, bm = (bm1,bm2, . . .,bmM)T, cm = (cm1,cm2, . . .,cmM)T.Solving the equation as follows:

PmP0

am1

am2

..

.

amM

0BBBB@1CCCCA

y

�

am1

am2

..

.

amM

0BBBB@1CCCCA

x

;

bm1

bm2

..

.

bmM

0BBBB@1CCCCA

y

�

bm1

bm2

..

.

bmM

0BBBB@1CCCCA

x

;

cm1

cm2

..

.

cmM

0BBBB@1CCCCA

y

�

cm1

cm2

..

.

cmM

0BBBB@1CCCCA

x

2666664

3777775k�m

¼P

mP0

½ðq; r; kIM Þ; ðam; bm; cmÞ�k�m

8>>>>>>>>><>>>>>>>>>:ð28Þ

yields

Z. Li, H. Dong / Chaos, Solitons and Fractals 37 (2008) 574–580 579

am1

am2

..

.

amM

0BBBB@1CCCCA

y

¼

bmþ1;1

bmþ1;2

..

.

bmþ1;M

0BBBB@1CCCCA�

r1

r2

..

.

rM

0BBBB@1CCCCA �

cm1

cm2

..

.

cmM

0BBBB@1CCCCAþ

am1

am2

..

.

amM

0BBBB@1CCCCA

x

;

bm1

bm2

..

.

bmM

0BBBB@1CCCCA

y

¼

q1

q2

..

.

qM

0BBBB@1CCCCA �

cm1

cm2

..

.

cmM

0BBBB@1CCCCA�

amþ1;1

amþ1;2

..

.

amþ1;M

0BBBB@1CCCCAþ

bm1

bm2

..

.

bmM

0BBBB@1CCCCA

x

;

cm1

cm2

..

.

cmM

0BBBB@1CCCCA

y

¼

q1

q2

..

.

qM

0BBBB@1CCCCA �

bm1

bm2

..

.

bmM

0BBBB@1CCCCA�

r1

r2

..

.

rM

0BBBB@1CCCCA �

am1

am2

..

.

amM

0BBBB@1CCCCAþ

cm1

cm2

..

.

cmM

0BBBB@1CCCCA

x

;

c01

c02

..

.

c0M

0BBBB@1CCCCA ¼

b1

b2

..

.

bM

0BBBB@1CCCCA;

a01

a02

..

.

a0M

0BBBB@1CCCCA ¼

b01

b02

..

.

b0M

0BBBB@1CCCCA ¼

0

0

..

.

0

0BBBB@1CCCCA;

c1 ¼

a1

a2

..

.

aM

0BBBB@1CCCCA; a1 ¼

b1

b2

..

.

bM

0BBBB@1CCCCA �

q1

q2

..

.

qM

0BBBB@1CCCCA; b1 ¼

b1

b2

..

.

bM

0BBBB@1CCCCA �

r1

r2

..

.

rM

0BBBB@1CCCCA:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð29Þ

Note that

V ðnÞþ ¼Xn

m¼0

ðam; bm; cmÞkn�m; V ðnÞ� ¼ knV � V ðnÞþ : ð30Þ

Then a direct calculation gives:

V ðnÞþy � V ðnÞþx � U ; V ðnÞþh i

¼ ðbnþ1;�anþ1; 0Þ: ð31Þ

Taking V ðnÞ ¼ V ðnÞþ , then the zero curvature equation U t � U x � V ðnÞy þ U ; V ðnÞ� �

þ V ðnÞx ¼ 0 admits the following (2 + 1)-dimensional multi-component integrable system:

q1

q2

..

.

qM

0BBBB@1CCCCA

t�x

¼

bnþ1;1

bnþ1;2

..

.

bnþ1;M

0BBBB@1CCCCA;

r1

r2

..

.

rM

0BBBB@1CCCCA

t�x

¼ �

anþ1;1

anþ1;2

..

.

anþ1;M

0BBBB@1CCCCA:

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð32Þ

5. Conclusions

In this paper, (2 + 1)-dimensional Dirac hierarchy and its integrable couplings as well as multi-component integrablesystem are obtained. The system (13) is Lax integrable, but how to judge it Liouville integrable and how to generateLiouville integrable system are open problem to us. Furthermore, how do we look for Hamiltonian structure of the

580 Z. Li, H. Dong / Chaos, Solitons and Fractals 37 (2008) 574–580

obtained system and how do we work out the solutions of the above system? These problems are worthwhile research-ing in the future.

References

[1] Tu Guizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys1989;30(2):330–8.

[2] Ma Wenxiu. A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. Chin J Contemp Math1992;13(1):79–89.

[3] Tu Guizhang, Ma Wenxiu. An algebraic approach for extending Hamilton operators. J Partial Differen Equat 1992;3(2):53–6.[4] Ma Wenxiu. An approach for constructing nonisospectra hierarchies of evolution equations. J Phys A 1992;25:719–26.[5] Guo Fukui. A hierarchy of integrable Hamiltonian equations. Acta Math Appl Sin. 2000;23(2):181–7.[6] Engui Fan. Integrable systems of derivative nonlinear Schrodinger type and their multi-Hamiltonian structure. J Phys A

2001;34:513–9.[7] Hu Xingbiao. An approach to generate supper extensions of integrable system. J Phys A 1997;30:619–32.[8] Guo Fukui. Subalgebras of the loop algebra and integrable Hamiltonian hierarchies of equations. Acta Math Phys Sin

1999;19(5):507–12.[9] Zhang Yufeng. A general Boite–Pempinelli–Tu hierarchy and its bi-Hamiltonian structure. Phys Lett A 2003;317(3):280–6.

[10] Hu Xingbiao. A powerful approach to generate new integrable systems. J Phys A 1994;27:2497–514.[11] Zhang Yufeng et al. A subalgebra of loop algebra and its applications. Chin Phys 2004;13(2):132–8.[12] Fan Engui. A Liouville integrable Hamiltonian system associated with a generalized Kaup–Newell spectral problem. Phys A

2001;301:105–13.[13] Zhang Yufeng et al. An integrable coupling of the generalized AKNS hierarchy. Ann Differn Equat 2002;18(4):431–7.[14] Zhang Yufeng, Zhang Hongqing. Integrable coupling of TD hierarchy. J Math Phys 2002;43:466–73.[15] Guo Fukui, Zhang Yufeng. A unified expressing model of the AKNS hierarchy and the KN hierarchy, as well as its integrable

coupling system. Chaos, Solitons & Fractals 2004;19(5):1207–16.[16] Guo Fukui, Zhang Yufeng. A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling.

J Math Phys 2003;44(12):5793–803.[17] Zhang Yufeng. A generalized multi-component AKNS hierarchy. Phys Lett A 2004;327:438–41.[18] Zhang Yufeng, Zhang Yusen. A type of multi-component integrable hierarchy. Chin Phys 2004;13(8):1183–6.[19] Dong Huanhe, Zhang Ning. A multi-component matrix loop algebra and its application. Commun Theor Phys

2005;44(6):997–1001.[20] Zhou Zixiang. Finite dimensional Hamiltonians and almost-periodic solutions for (2 + 1)-dimensional three equations. J Phys Soc

of Jpn 2002;71(8):1857–63.[21] Sun Yepeng, Xu Xixiang. An integrable system of Dirac hierarchy and its integrable coupling. J Jiam Univ 2003;21(1):66–9.[22] Qiao Zhijun. Chin Quarter J Math 1997;12(1):1–3.[23] Zhu Li, Huanhe Dong. Integrable couplings of the multi-component Dirac hierarchy and its Hamiltonian structure. Chaos,

Solitons & Fractals 2008;36(2):290–5.