Journal of Shanghai University (English Edition), 2007, 11(3): 255–258

Digital Object Identifier(DOI): 10.1007/s 11741-007-0313-3

Equivalent transformation between the matrices for expandingintegrable model of the hierarchy of evolution equation

YAO Yu-qin (姚玉芹), CHEN Deng-yuan (陈登远)Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, P. R. China

Abstract A direct method for obtaining the expanding integrable models of the hierarchies of evolution equations wasproposed. By using the equivalent transformation between the matrices, a new isospectral problem was directly establishedaccording to the known isospectral problem, which can be used to obtain the expanding integrable model of the knownhierarchy.

Keywords isospectral problem, expanding integrable model, matrix.

2000 Mathematics Subject Classification 35Q51

1 Introduction

Searching new integrable hierarchies is an impor-tant topic in the field of soliton theory. Some inte-grable hierarchies with physical significance, such as theAKNS hierarchy, the BPT hierarchy, etc., have beenobtained[1−4]. As far as the expanding integrable modelis concerned, it is a rather new interesting problem. Therelated research was initialized by Ma, et al.[5,6] whilethey studied the relation between the Virasoro algebraand hereditary operators. The basic procedure to es-tablish the expanding integrable model of a known inte-grable system can be expressed as follows. Firstly, a newsubalgebra of loop algebra A1 or loop algebra A2 is con-structed. Secondly, a higher-dimensional loop algebra ispresented in terms of the new subalgebra. Finally, bydesigning a new isospectral problem and using the Tuscheme the expanding integrable model of a known hi-erarchy can be established[7−10]. It could be found thatall the higher-dimensional loop algebra was expandedfrom the subalgebra of 2×2 loop algebra. In this paper,we want to construct the expanding integrable modelof 3 × 3 matrix isospectral problem. It is obvious thatthe method presented in [7− 10] cannot be directly ap-plied. Therefore a direct method for establishing theexpanding integrable model is proposed in this paper.Firstly, a new isospectral problem is directly formulatedaccording to the known isospectral problem by use of theequivalent transformation between the matrices. Thenthe expanding integrable model of the integrable systemcan be worked out by making use of the Tu scheme.

2 The expanding integrable model ofintegrable system (2)

Consider the following isospectral problem[11]:

ϕx = Uϕ, U =

⎡⎢⎢⎣2λ − 2s

√2q 0

−√2λr 0

√2q

0 −√2λr −2λ + 2s

⎤⎥⎥⎦ . (1)

Let

V =

⎡⎢⎢⎣

−2a −√2λ−1b 0√

2c 0 −√2λ−1b

0√

2c 2a

⎤⎥⎥⎦

=∑m�0

⎡⎢⎢⎣−2am −√

2λ−1bm 0√2cm 0 −√

2λ−1bm

0√

2cm 2am

⎤⎥⎥⎦λ−m,

V(n)+ =

n∑m=0

⎡⎢⎢⎣−2am −√

2λ−1bm 0√2cm 0 −√

2λ−1bm

0√

2cm 2am

⎤⎥⎥⎦λn−m,

V(n)− = λnV − V

(n)+ .

Let V (n) = (λnV )++

⎡⎢⎢⎣

0√

2λ−1bn 0

−√2cn 0

√2λ−1bn

0 −√2cn 0

⎤⎥⎥⎦ ,

then the zero-curvature equation

Ut − V (n)x + [U , V (n)] = 0

Received May 20, 2005; Revised Feb.20, 2006Project supported by the National Natural Science Foundation of China (Grant Nos.10371070, 10547123)Corresponding author YAO Yu-qin, PhD Candidate, E-mail: yyqinw@126.com

256 Journal of Shanghai University

determines the Lax integrable system

utn =

⎡⎢⎢⎣q

r

s

⎤⎥⎥⎦

tn

= JLn

⎡⎢⎢⎣

r

q

1

⎤⎥⎥⎦ , (2)

where

J =

⎡⎢⎢⎣

0 2 0

−2 0 0

0 0 ∂

⎤⎥⎥⎦

is a Hamilton operator, and

L=

⎡⎢⎢⎢⎢⎢⎣

−12∂ + s +

12r∂−1q∂

12r∂−1r∂ r∂−1s∂

12q∂−1q∂

12∂ + s +

12q∂−1r∂ q∂−1s∂

12∂−1q∂

12∂−1r∂ ∂−1s∂

⎤⎥⎥⎥⎥⎥⎦

is a recurrence operator.Set a scalar transformation

⎡⎢⎢⎣

2λ − 2s√

2q 0

−√2λr 0

√2q

0 −√2λr −2λ + 2s

⎤⎥⎥⎦

→[−a1(λ − s) −a2q

−a3λr a1(λ − s)

],

where a1, a2, a3 are constants to be determined. Decom-pose the above two matrices as follows:⎡⎢⎢⎣

2λ − 2s√

2q 0

−√2λr 0

√2q

0 −√2λr −2λ + 2s

⎤⎥⎥⎦

= −λ

⎡⎢⎢⎣−2 0 0

0 0 0

0 0 2

⎤⎥⎥⎦ + s

⎡⎢⎢⎣−2 0 0

0 0 0

0 0 2

⎤⎥⎥⎦

− q

⎡⎢⎢⎣0 −√

2 0

0 0 −√2

0 0 0

⎤⎥⎥⎦ − rλ

⎡⎢⎢⎣

0 0 0√2 0 0

0√

2 0

⎤⎥⎥⎦ ,

[−a1(λ − s) −a2q

−a3λr a1(λ − s)

]

= −a1λ

[1 0

0 −1

]+ a1s

[1 0

0 −1

]

− a2q

[0 1

0 0

]− a3rλ

[0 0

1 0

],

and let

v1 =

⎡⎢⎢⎣−2 0 0

0 0 0

0 0 2

⎤⎥⎥⎦ , v2 =

⎡⎢⎢⎣

0 −√2 0

0 0 −√2

0 0 0

⎤⎥⎥⎦ ,

v3 =

⎡⎢⎢⎣

0 0 0√2 0 0

0√

2 0

⎤⎥⎥⎦ ,

w1 =

[1 0

0 −1

], w2 =

[0 1

0 0

], w3 =

[0 0

1 0

].

Then the coefficients of λ, s, q, r have the following cor-responding relation:

vi → aiwi, i = 1, 2, 3.

Through a simple calculation, we have

[w1, w2] =−2a2

a1a2w2, [w1, w3] =

2a3

a1a3w3,

[w2, w3] =a1

a2a3w1.

Solving the equations −2a2a1a2

= 2, 2a3a1a3

= −2, a1a2a3

= 1yields a1 = −1, a2 = a3 = i. Thus the isospectral prob-lem (1) can be turned into the following 2 × 2 matrixisospectral problem

ϕx = U1ϕ, U1 =

[λ − s −iq

−iλr −λ + s

]. (3)

From (3), we can also obtain the integrable system (2)by using the Tu scheme. Therefore, the isospectral prob-lems (1) and (3) are equivalent.

Expand (3) into a 3 × 3 matrix isospectral problemwhich possesses 5 potential functions:

Ψx = UΨ , λt = 0, Ψ = (Ψ1, Ψ2, Ψ3, Ψ4, Ψ5)T,

U =

⎡⎢⎢⎣λ − u3 −iu1 u4

−iλu2 −λ + u3 u5

0 0 0

⎤⎥⎥⎦ . (4)

Let

V =∑m�0

⎡⎢⎢⎣−am ibmλ−1 dmλ−1

icm am em

0 0 0

⎤⎥⎥⎦λ−m,

then solving the auxiliary linear equation

Vx = [U , V ] (5)

Vol. 11 No. 3 Jun. 2007 YAO Y Q, et al. : Equivalent transformation between the matrices for expanding ... 257

yields⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

amx = −u1cm + u2bm,

bmx = 2bm+1 − 2u1am+1 − 2u3bm,

cmx = −2cm+1 + 2u2am+1 + 2u3cm,

dmx = dm+1 − iu1em+1 − u3dm + u4am+1 − iu5bm,

emx = −em+1 − iu2dm + u3em − iu4cm − u5am,

a0 = 1, b0 = u1, c0 = u2, e0 = 0, d0 = −u4.

(6)

Let

V(n)+ =

n∑m=0

⎡⎢⎢⎣−am ibmλ−1 dmλ−1

icm am em

0 0 0

⎤⎥⎥⎦λn−m, V

(n)− = λnV − V

(n)+ .

Then (5) can be written as

−V(n)+x + [U , V

(n)+ ] = V

(n)−x − [U , V

(n)− ]. (7)

A direct calculation leads to

−V(n)+x + [U , V

(n)+ ] =

⎡⎢⎢⎣

0 −i(bnx + 2u3bn)λ−1 (−dnx − u3dn − iu5bn)λ−1

i(2u3cn − cnx) 0 en+1

0 0 0

⎤⎥⎥⎦ .

Set V (n) = V(n)+ +

⎡⎢⎢⎣

0 −ibnλ−1 −dnλ−1

−icn 0 0

0 0 0

⎤⎥⎥⎦, then the zero-curvature equation

Ut = V (n)x + [U , V (n)] = 0 (8)

gives

utn =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

u1

u2

u3

u4

u5

⎤⎥⎥⎥⎥⎥⎥⎥⎦

tn

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

2bn

−2cn

anx

dn

enx − u3en + u5an

⎤⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 2 0 0 0

−2 0 0 0 0

0 0 ∂ 0 0

0 0 0 0 1

0 0 u5 0 ∂ − u3

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎣

cn

bn

an

dn

en

⎤⎥⎥⎥⎥⎥⎥⎥⎦

= J(cn, bn, an, dn, en)T. (9)

From (6), we obtain the recursive operator

L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−12∂ + u3 +

12u2∂

−1u1∂12u2∂

−1u2∂ u2∂−1u1∂ 0 0

12u1∂

−1u1∂12∂ + u3 +

12u1∂

−1u2∂ u1∂−1u3∂ 0 0

12∂−1u1∂

12∂−1u2∂ ∂−1u3∂ 0 0

u1u4 − 12u4∂

−1u1∂ iu5 − 12u4∂

−1u2∂ −iu1u5 − u4∂−1u3∂ ∂ + u1u2 + u3 iu1u3 − iu1∂

−iu4 0 −u5 −iu2 u3 − ∂

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

258 Journal of Shanghai University

Therefore, (9) can be written as

utn =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

u1

u2

u3

u4

u5

⎤⎥⎥⎥⎥⎥⎥⎥⎦

tn

= JLn

⎡⎢⎢⎢⎢⎢⎢⎢⎣

u2

u1

1

−u4

0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

. (10)

Let u1 = q, u2 = r, u3 = s, u4 = u5 = 0, then thesystem (10) is reduced to the integrable system (2). Ac-cording to the definition of integrable coupling[5], thesystem (10) is the integrable coupling of the integrablesystem (2), i.e., (10) is a type of expanding integrablemodel of (2).

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