integrable decomposition of a hierarchy of soliton equations and integrable coupling system by...

Nonlinear Analysis 69 (2008) 3450–3461www.elsevier.com/locate/na

Integrable decomposition of a hierarchy of soliton equations andintegrable coupling system by semidirect sums of Lie algebras

Lin Luoa,b,∗, Engui Fanb

a Department of Mathematics, Xiaogan University, Xiaogan 432100, Hubei, Chinab School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received 5 September 2007; accepted 21 September 2007

Abstract

Staring from a new spectral problem, a hierarchy of the soliton equations is derived. It is shown that the associated hierarchiesare infinite-dimensional integrable Hamiltonian systems. By the procedure of nonlinearization of the Lax pairs, the integrabledecomposition of the whole soliton hierarchy is given. Further, we construct two integrable coupling systems for the hierarchy bythe conception of semidirect sums of Lie algebras.c© 2007 Elsevier Ltd. All rights reserved.

MSC: 02.30.Ik

Keywords: Soliton equations; Nonlinearization; Integrable couplings; Semidirect sums of Lie algebras

1. Introduction

A central and difficult topic in the study of integrable systems is to find Liouville integrable systems which possessphysical significance. Nonlinearization technique, which was put forward first by Cao [1,2], has proved to be apowerful tool for obtaining new finite-dimensional integrable Hamiltonian systems. In the past two decades, Cao’swork has developed greatly and many kinds of generalizations such as the method of Binary nonlinearization [3], theconstraint method of higher order symmetry [4], and the adjoint symmetry constraint method [5] have been made andapplied to a wide variety of soliton equations successfully [6–13], including continuous and discrete soliton equations.

Recently, Integrable coupling systems have been receiving growing attention by the conception of semidirectsums of Lie algebra. There exist plenty of examples of both continuous and discrete integrable couples belongingto such a class of integrable equations [15–24]. We know that an arbitrary Lie algebra has a semidirect sumstructure of a solvable Lie algebra and a semisimple Lie algebra. The semidirect sum decomposition of Lie algebraallows more classifications of integrable equations supplementing existing theories [25]. Moreover, the study ofintegrable couplings also generates many interesting mathematical structures such as Lax representation, infinitelymany symmetries and local Hamiltonian structures in higher dimensions and hereditary recursion operators of higherorder [18,23,24].

∗ Corresponding author at: Department of Mathematics, Xiaogan University, Xiaogan 432100, Hubei, China.E-mail address: [email protected] (L. Luo).

0362-546X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2007.09.032

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L. Luo, E. Fan / Nonlinear Analysis 69 (2008) 3450–3461 3451

In this paper, we consider the nonlinearization of a generalized Kaup–Newell (KN) spectral problem and itsintegrable couplings by semidirect sums of Lie algebras. This paper is organized as follows: In Section 2, we willderive a new soliton hierarchy from a spectral problem. It is shown that the hierarchy is Liouville integrable, and itsbi-Hamiltonian structure is established by trace identity. In Section 3, Under the Bargmann symmetry constraints, theLax pairs of the hierarchy are nonlinearized into finite-dimensional completely integrable Hamiltonian systems. InSection 4, we would like to discuss the integrable couplings of the hierarchy by the conception of semidirect sums ofLie algebras. Some conclusions are given in the final Section.

2. The soliton hierarchy and Hamiltonian structure

We introduce a new generalized KN spectral problem

ϕx = Uϕ =

(λ2

+ αq λqλr −λ2

− αq

)ϕ, (2.1)

where q = q(x, t) and r = r(x, t) are two potentials, λ ∈ C is a spectral parameter and α is an arbitrary constant. Todeduce the new soliton hierarchy, we first solve the adjoint representation of Eq. (2.1),

Vx = [U, V ] = U V − V U, V =

(a bc −a

), (2.2)

which is equivalent to

ax = λqc − λrb,

bx = 2[(λ2+ αq)b − λqa],

cx = 2[−(λ2+ αq)c + λra].

Substituting

a =

∑j≥0

a jλ−2 j , b =

∑j≥0

b jλ−2 j−1, c =

∑j≥0

c jλ−2 j−1, (2.3)

into the above equations gives the following recursive formulae:

a j x = qc j − rb j ,

b j x = 2(b j+1 + αqb j − qa j+1),

c j x = 2(−c j+1 − αqc j + ra j+1).

(2.4)

From these equations, we can successively deduce

a0 = 1, b0 = q, c0 = r, a1 = −12

qr,

b1 =12

qx − αq2−

12

q2r, c1 = −12

rx − αqr −12

qr2, (2.5)

a2 =14(qrx − rqx ) +

12

q2r2+ αq2r,(

c j+1b j+1

)= L1

(c jb j

), (2.6)

with

L1 =

−12∂ − αq − αr∂−1q2

−12

r∂−1q∂ αr∂−1qr −12

r∂−1r∂

−αq∂−1q2−

12

q∂−1q∂12∂ − αq + αq∂−1qr −

12

q∂−1r∂

. (2.7)

3452 L. Luo, E. Fan / Nonlinear Analysis 69 (2008) 3450–3461

Consider the auxiliary spectral problem associated with (2.1)

ϕtn = V (n)ϕ, (2.8)

where

V (n)= (λ2n+2V )+ +

(bn 00 −bn

)=

n∑j=0

(a jλ

2n−2 j+2 b jλ2n−2 j+1

c jλ2n−2 j+1

−a jλ2n−2 j+2

)+

(αbn 0

0 −αbn

). (2.9)

Then the compatibility conditions between (2.1) and (2.8) gives the zero curvature equation Ut −V (n)x +[U, V (n)

] = 0,that is,

utn , Kn(u) =

(bnx

2αanx + cnx

), u = (q, r)T, (2.10)

which is a hierarchy of new soliton equations associated with spectral problem (2.1).In the following we will establish the bi-Hamiltonian structure for the hierarchy (2.10), and show that they are

integrable in Liouville’s sense. In order to apply the trace identity [26], we need to rewrite (2.10) in another form. Weintroduce

G j ,

(2αa j + c j

b j

), (2.11)

then

(c j+1b j+1

)= L2G j =

−12∂ −

12

r∂−1q∂ −αr −12

r∂−1r∂

−12

q∂−1q∂12∂ − αq −

12

q∂−1r∂

G j , (2.12a)

and

G j+1 = LG j =

−12∂ −

12

r∂−1q∂ − α∂−1q∂ −αr −12

r∂−1r∂ − α∂−1r∂

−12

q∂−1q∂12∂ − αq −

12

q∂−1r∂

G j , (2.12b)

where L2L = L1L2.We take the Killing–Cartan form 〈A, B〉 as tr(AB), then direct calculation gives⟨

V,∂U

∂λ

⟩= 4λa + rb + qc,

⟨V,

∂U

∂q

⟩= 2αa + λc,

⟨V,

∂U

∂r

⟩= λb. (2.13)

By using trace identity, we have

δ

δu(4λa + rb + qc) = λ−γ ∂

∂λ(λγ (2αa + λc, λb)T), (2.14)

where u = (q, r)T.Substituting (2.3) into the above equation leads to

δ

δu(4an+1 + rbn + qcn) = (γ − 2n)Gn . (2.15)

To fix the γ , we let n = 0 in (2.15) and find γ = 0. Therefore we conclude that

Gn =δHn

δu, Hn = −

12n

(4an+1 + rbn + qcn), n ≥ 1 (2.16)

with H0 = 2αq + qr .


Combining (2.10) and (2.16) gives the desired Hamiltonian structure of the hierarchy (2.10).

utn = Kn(u) = JδHn

δu= K

δHn−1

δu, u = (q, r)T, (2.17)

where

J =

(0 ∂

∂ 0

), K = J L =

−12∂q∂−1q∂

12∂2

− α∂q −12∂q∂−1r∂

−12∂2

− αq∂ −12∂r∂−1q∂ −

12∂r∂−1r∂ − α∂r − αr∂

. (2.18)

Theorem 2.1. The Hamiltonian functions {Hm}∞

m=0 defined by (2.16) are conserved densities of the whole hierarchy(2.10). In other words, the hierarchy (2.10) is a completely integrable Hamiltonian system in the Liouville sense.

Proof. It is easy to see that J L = L∗ J , where L∗ denotes the conjugate of L . Then we have

{Hm, Hl} =

(δHm

δu, J

Hl

δu

)= (Lm G0, J Ll G0) = (Lm G0, L∗ J Ll−1G0)

= (Lm+1G0, J Ll−1G0) = {Hm+1, Hl−1}.

Repeating the above argument gives

{Hm, Hl} = {Hl , Hm} = {Hm+l , H0}. (2.19)

On the other hand, we have

{Hm, Hl} = (Lm G0, J Ll G0) = (J ∗Lm G0, Ll G0) = −{Hl , Hm}. (2.20)

Combining (2.19) with (2.20) gives

{Hm, Hl} = 0,

which implies that {Hm}∞

m=0 are in involution. Furthermore, we have(∫Hmdx

)tl

=

(δHm

δu, utl

)=

(δHm

δu, J

δHl

δu

)= {Hm, Hl} = 0.

This implies that {Hm}∞

m=0 are conserved densities. So this proof is completed. �

Example. When n = 1, the soliton hierarchy (2.10) is reduced to a new generalized derivative nonlinear Schrodingerequation [4,14]

qt1 =12

qxx − 2αqqx −12(q2r)x ,

rt1 = −12

rxx − 2α(qr)x −12(qr2)x ,

(2.21)

which are Liouville integrable and possess the bi-Hamiltonian structure

utn = JδH1

δu= K

δH0

δu, u = (q, r)T,

with H0 = qr, H1 = −14 qrx +

14 qxr −

12 q2r2

− αq2r .

Remark. As α = 0, the spectral problem (2.1) reduces to the Kaup–Newell spectral problem [14]

ϕx =

(λ2 λqλr −λ2

),


and the soliton hierarchy (2.10) leads to the well-known Kaup–Newell hierarchy

utn = Kn(u) = JδHn

δu= K

δHn−1

δu, u = (q, r)TT .

where

J =

(0 ∂

∂ 0

), K = J L =

−12∂q∂−1q∂

12∂2

−12∂q∂−1r∂

−12∂2

−12∂r∂−1q∂ −

12∂r∂−1r∂

(2.22)

and H0 = qr, Hn, (n ≥ 1) satisfies Eq. (2.16).

3. Finite-dimensional integrable system

We consider N copies of the spectral problem (2.1) with N distinct nonzero parameters λ j , 1 ≤ j ≤ N . Then thefunctional gradient λ j with respect to u = (q, r)T is

δλ j

δu=

(δλ j

δq,δλ j

δr

)T

= (2αϕ1 jϕ2 j + λ jϕ22 j , −λ jϕ

21 j )

T. (3.1)

Making use of Eq. (2.1), direct verification indicates that

Lδλ j

δu= λ2

jδλ j

δu. (3.2)

Consider the Bargmann constraint

G0 =

N∑j=1

δλ j

δu, (3.3)

that is,

q = −〈∧ϕ1, ϕ1〉, r = −2 + 2α〈ϕ1, ϕ2〉 + 〈∧ϕ2, ϕ2〉, (3.4)

where ϕ1 = (ϕ11, . . . , ϕ1N )T, ϕ2 = (ϕ21, . . . , ϕ2N )T, ∧ = diag(λ1, . . . , λN ), and 〈., .〉 denotes the standard innerproduct in RN . Under the constraint (3.4), Eq. (2.1) are nonlinearized into a finite-dimensional Hamiltonian system(FDHS).

ϕ1x = ∧2 ϕ1 − α〈∧ϕ1, ϕ1〉ϕ1 − 〈∧ϕ1, ϕ1〉 ∧ ϕ2

=∂ H

∂ϕ2,

ϕ2x = − ∧2 ϕ2 + α〈∧ϕ1, ϕ1〉ϕ2 + (−2 + 2α〈ϕ1, ϕ2〉 + 〈∧ϕ2, ϕ2〉) ∧ ϕ1

= −∂ H

∂ϕ1, (3.5)

whose Hamiltonian function H is

H = 〈∧2 ϕ1, ϕ2〉 − α〈∧ϕ1, ϕ1〉〈ϕ1, ϕ2〉 −

12〈∧ϕ1, ϕ1〉〈∧ϕ2, ϕ2〉 + 〈∧ϕ1, ϕ1〉.

Making use of (3.3), by a direct and tedious calculation, then a j , b j , c j , j ≥ 0 in (2.3) can become respectively

a j = 〈∧2 j ϕ1, ϕ2〉 j ≥ 1,

b j = −〈∧2 j+1 ϕ1, ϕ1〉 j ≥ 0,

c j = 〈∧2 j+1 ϕ2, ϕ2〉 j ≥ 1,

(3.6)

with a0 = 1, c0 = −2 + 2α〈ϕ1, ϕ2〉 + 〈∧ϕ2, ϕ2〉.


Now we define

V =

∞∑j=0

(a jλ

−2 j b jλ−2 j−1

c jλ−2 j−1

−a jλ−2 j

), (3.7)

and according to the Ref. [4], we have the following Lemma:

Lemma 3.1. The Bargmann system (3.5) enjoys the Lax equation

Vx = [U , V ].

Thus, from Lemma 3.1, it is easy to see that there is a natural set of integrals of motion generated by the relation

Fx , ( a2+ bc )x =

(12

trV 2)

x= 0.

Let F =∑

j≥0 Fmλ−2 j , then, we obtain the following expressions:

F0 = 1,

F1 = 2〈∧2 ϕ1, ϕ2〉 − 〈∧ϕ1, ϕ1〉(−2 + 2α〈ϕ1, ϕ2〉 + 〈∧ϕ2, ϕ2〉) = 2H, (3.8a)

Fm = 2〈∧2m ϕ1, ϕ2〉 − 〈∧

2m−1 ϕ1, ϕ1〉[−2 + 2α〈ϕ1, ϕ2〉 + 〈∧ϕ2, ϕ2〉]

+

m−1∑i=1

〈∧2i ϕ1, ϕ2〉〈∧

2m−2i ϕ1, ϕ2〉 −

m−2∑i=0

〈∧2i+1 ϕ1, ϕ1〉〈∧

2m−2i−1 ϕ2, ϕ2〉,

m ≥ 2, (3.8b)

which implies F j , j ≥ 0 are the integrals of motion of the system (3.5).Now, in the same way, we discuss the nonlinearization of the temporal part, under the Bargmann constraint (3.3),

auxiliary spectral problem (2.8) becomes(ϕ1ϕ2

)tm

= V (m)

(ϕ1ϕ2

), (3.9a)

with

V (m)= (λ2m+2V )+ +

(αbm 0

0 −αbm

), (3.9b)

where V , bm are defined as in (3.6) and (3.7).Through a simple calculation, we find that (3.9) can be rewritten as the following finite-dimensional Hamiltonian

system

ϕ1tm =∂ 1

2 Fm+1

∂ϕ2, ϕ2tm = −

∂ 12 Fm+1

∂ϕ1, m ≥ 0 (3.10)

where Fm+1, m ≥ 0 is defined as in (3.8).Similarly, we also see that Bargmann constraint system (3.10) has the Lax equation

Vtm = [V (m), V ]. (3.11)

Thus we know that Fm, m ≥ 0 in (3.8) are also the integrals of motion for (3.10).Next we show the integrability of FDHS (3.5) and (3.10). We consider the standard symplectic structure on R2N ,

the Poisson bracket for two smooth functions f and g in the symplectic space (R2N , dϕ1 ∧ dϕ2) is defined:

{ f, g} =

N∑j=1

(∂ f

∂ϕ1

∂g

∂ϕ2−

∂ f

∂ϕ2

∂g

∂ϕ1

),


which is skew-symmetric, bilinear, and satisfies the Jacobi identity. In particular, f and g are called in involution if{ f, g} = 0.

Lemma 3.2. F1, F2, . . . Fm are in involution with each other.

Proof. A direct calculation gives

{Fm, Fn} = −dFm

dtn.

From Lax presentation (3.11) we get

dFm

dtn= 0,

which implies

{Fm, Fn} = 0.

so the Lemma is proved.

Lemma 3.3. F1, F2, . . . Fm are functionally independent over some region of R2N .

Proof. Direct computation leads to

∂ Fm

∂ϕ1

∣∣∣∣ϕ1=0

= 2 ∧2m ϕ2, m ≥ 1.

Therefore we obtain

det

∂ F1

∂ϕ11

∂ F2

∂ϕ11. . .

∂ FN

∂ϕ11∂ F1

∂ϕ12

∂ F2

∂ϕ12. . .

∂ FN

∂ϕ12. . . . . . . . . . . .∂ F1

∂ϕ1N

∂ F2

∂ϕ1N. . .

∂ FN

∂ϕ1N

∣∣∣∣∣∣∣∣∣∣∣∣∣ϕ1=0

= 2N λ21 . . . λ2

N ϕ21 . . . ϕ2N det

1 λ2

1 . . . λ2N−21

1 λ22 . . . λ2N−2

2. . . . . . . . . . . .

1 λ2N q . . . λ2N−2

N

= 2N λ21 . . . λ2

N ϕ21 . . . ϕ2N

∏i 6= j

(λ2i − λ2

j ).

This means that functions F1, F2, . . . FN can be functionally independent at least over some region of R2N .

From Lemmas 3.2 and 3.3, we immediately arrive at the following Theorem.

Theorem 3.4. Both the finite-dimensional Hamiltonian systems (3.5) and (3.10) are completely integrable systems inthe Liouville’s sense.

Hence, from the above procedure we obtain the following proposition.

Proposition 3.5. If (ϕ1, ϕ2) satisfies both Eqs. (3.5) and (3.10), then q = −〈∧ϕ1, ϕ1〉, r = −2 + 2α〈ϕ1, ϕ2〉 +

〈∧ϕ2, ϕ2〉 solve Eq. (2.10).

This implies that we have obtained an integrable decomposition of the whole soliton hierarchy (2.10).

4. Integrable couplings of the hierarchy (2.10)

4.1. Integrable couplings from two specific semidirect sums

In this section, we will construct integrable coupling systems through semidirect sums of Lie algebra associatedwith soliton equations (2.10).


Let us first consider the semidirect sums of Lie algebra of 3 × 3 matrices (see [15] for further details)

G ⊕ Gc, G =

{(A 00 0

)∣∣∣∣ A ∈ C[λ, λ−1] ⊗ M2×2

},

Gc =

{(0 B0 0

)∣∣∣∣ B ∈ C[λ, λ−1] ⊗ M2×1

},

where C[λ, λ−1] ⊗ Mm×n = span{λk A | k ∈ Z, A ∈ Mm×n}. In this case, Gc is an ideal Lie subalgebra of G ⊕ Gc,

we define the corresponding enlarged spatial spectral matrix as

U = U (u, λ) =

(U Uα

0 0

)∈ G ⊕ Gc, Ua = Ua(s, λ) =

(s1λs2

), (4.1)

where s1 and s2 are two new dependent variables and

s = (s1, s2)T, u = (uT, sT)T

= (q, r, s1, s2)T. (4.2)

Upon setting

V =

(V Va0 0

), Va = Va(u, λ) =

(fg

), (4.3)

where V is a solution to Vx = [U, V ], defined by (2.2), the corresponding enlarged stationary zero curvature equationVx = [U , V ] becomes

Vax = U Va − V Ua, (4.4)


fx = λ2 f + αq f + λqg − s1a − λs2b,

gx = λr f − λ2g − αqg − s1c + λs2a.(4.5)

This system determines a solution for f, g as follows:

f =

∑j≥0

f jλ−2 j , g =

∑j≥0

g jλ−2 j−1. (4.6)

Now we define the the enlarged temporal spectral matrix as

V (m)=

(V (m) V (m)

α

0 0

), V (m)

a = (λ2m+2Va)+, m ≥ 0, (4.7)

where the V (m) is defined as in (2.9).Then, based on (4.5), we can compute that

V (m)ax − (U V (m)

a − V (m)Ua) =

(s1bm

λ(gmx + αqgm + s1cm)

), m ≥ 0. (4.8)

Therefore, the mth enlarged zero curvature equation

Utm = V (m)x − [U , V (m)

],

leads to

stm =

(s1s2

)tm

, Sm(u, s) =

(s1bm

gmx + αqgm + s1cm − s2bm

), m ≥ 0 (4.9)


together with the mth soliton hierarchy in (2.10). Therefore, we obtain a hierarchy of coupling systems for thehierarchy (2.10)

utm =

(us

)tm

, Km(u) =

(Km(u)

Sm(u, s)

), m ≥ 0, (4.10)

where u = (uT, sT)T= (q, r, s1, s2)

T.Second, let us consider the semidirect sum of Lie algebra of 4 × 4 matrices as in [19,20]

G ⊕ Gc, G =

{(A 00 A

)∣∣∣∣ A ∈ C[λ, λ−1] ⊗ M2×2

},

Gc =

{(0 B0 0

)∣∣∣∣ B ∈ C[λ, λ−1] ⊗ M2×2

},

where C[λ, λ−1] ⊗ Mm×n = span{λk A | k ∈ Z, A ∈ Mm×n}. In this case, Gc is a ideal Lie subalgebra of G ⊕ Gc,

we define the corresponding enlarged spatial spectral matrix as

U = U (u, λ) =

(U Uα

0 U

)∈ G ⊕ Gc, Ua = Ua(s, λ) =

(s1 λs3λs2 −s1

), (4.11)

where s1, s2 and s2 are new dependent variables and

s = (s1, s2, s3)T, u = (uT, sT)T

= (q, r, s1, s2, s3)T.

If we set

V =

(V Vα

0 V

), Va = Va(u, λ) =

(f hg − f

), (4.12)

where V is a solution to Vx = [U, V ], defined by (2.2), then the corresponding enlarged stationary zero curvatureequation Vx = [U , V ] becomes

Vax = [U, Va] + [Ua, V ], (4.13)


fx = λqg + λs3c − λs2b − λrh,

hx = −2λq f + 2s1b − 2λs3a + 2λ2h + 2αqh,

gx = 2λr f + 2λs2a − 2s1c − 2λ2g − 2αqg.

(4.14)

This system can determine a solution for f, h, g as follows:

f =

∑j≥0

f jλ−2 j , h =

∑j≥0

h jλ−2 j−1, g =

∑j≥0

g jλ−2 j−1. (4.15)

Now we define the the enlarged temporal spectral matrix as

V (m)=

(V (m) V (m)

α

0 V (m)

), V (m)

a = (λ2m+2Va)+ + ∆m,a, m ≥ 0, (4.16)

where the V (m) is defined as in (2.9). Choose ∆m,a as

∆m,a =

(hm 00 −hm

), m ≥ 0.

Then, based on (4.14), we can compute that


V (m)ax − [U, V (m)

a ] − [Ua, V (m)]

= (λ(2m+2)Va)+x + ∆m,ax − [U, (λ2m+2Va)+] − [U,∆m,a] − [Ua, V (m)]

=

(hmx λ(hmx − 2s1bm + 2s3bm)

λ(gmx + 2αqgm + 2s1cm − 2s2bm − rhm) −hmx

). (4.17)

Therefore, the mth enlarged zero curvature equation

Utm = V (m)x − [U , V (m)

],

leads to

stm =

s1s3s2

tm

, Tm(u, s) =

hmxhmx − 2s1bm + 2s3bm

gmx + 2αqgm + 2s1cm − 2s2bm − rhm

, m ≥ 0

(4.18)

together with the mth soliton hierarchy (2.10), we obtain a hierarchy of coupling systems for whole hierarchy (2.10)

utm =

(us

)tm

, Km(u) =

(Km(u)

Tm(u, s)

), m ≥ 0, (4.19)

where u = (uT, sT)T= (q, r, s1, s2, s3)

T.

4.2. Illustrative examples

We now work out two examples as follows, one in each of the two above cases. Let us first compute an example ofthe hierarchy (4.10). It directly follows from (4.5) that

f j x = f j+1 + αq f j + qg j − s1a j − s2b j ,

g j x = r f j+1 − g j+1 − αqg j − s1c j + s2a j+1,

where j ≥ 0. Here we can obtain that

f0 = 0, g0 = s2,

f1 = s1, g1 = −s2x − αqs2 −12

qrs2,

then the vector-valued function S1(u, s) defined by (4.9) becomes

S1(u, s) =

s1

(12

qx − αq2−

12

q2r

),(

−s2x − αqs2 −12

qrs2

)x

+ αq

(−s2x − αqs2 −

12

qrs2

)+s1

(−

12

rx − αqr −12

qr2)

− s2

(12

qx − αq2−

12

q2r

) .

Therefore, the integrable coupling of the generalized derivative nonlinear Schrodinger equation (2.21), defined by(4.10), reads as

qt1 =12


rt1 = −12


s1t1 = s1

(12

qx − αq2−

12

q2r

),


s2t1 =

(−s2x − αqs2 −

12

qrs2

)x

+ αq

(−s2x − αqs2 −

12

qrs2

)+ s1

(−

12

rx − αqr −12

qr2)

− s2

(12

qx − αq2−

12

q2r

). (4.20)

Second, let us compute an example of the hierarchy (4.19). we take the initial set of functions as follows:

f0 = 0, h0 = s3, g0 = s2.

Obviously from (4.14) and (4.15), we have

f j x = qg j + s3c j − s2b j − rh j ,

h j x = −2q f j+1 + 2s1b j − 2s3a j+1 + 2h j+1 + 2αqh j ,

g j x = 2r f j+1 + 2s2a j+1 − 2s1c j − 2g j+1 − 2αqg j ,

where j ≥ 0. It then follows that

f1 = −12(qs2 + rs3), h1 =

12(s3x − q2s2 − 2qrs3 − 2s1q − 2αqs3),

g1 =12(−s2x − 2qrs2 − r2s3 − 2s2r − 2αqs2),

then the vector-valued function T1(u, s) defined by (4.18) becomes

T1(u, s) =

12(s3x − q2s2 − 2qrs3 − 2qs1 − 2αqs3)x ,

12(−s2x − 2qrs2 − r2s3 − 2s1r − 2αqs2)x + 2αq(−s2x − 2qrs2 − r2s3 − 2s1r − 2αqs2)

−12

r(s3x − q2s2 − 2qrs3 − 2s1q − 2αqs3) + 2s1

(−

12

rx − αqr −12

qr2)

−2s2

(12

qx − αq2−

12

q2r

),

12(s3x − q2s2 − 2qrs3 − 2qs1 − 2αqs3)x + (2s3 − 2s1)

(12

qx − αq2−

12

q2r

)

.

Therefore, the integrable coupling of the generalized derivative nonlinear Schrodinger equation (2.21), defined by(4.19), reads as

qt1 =12


rt1 = −12


s1t1 =12(s3x − q2s2 − 2qrs3 − 2qs1 − 2αqs3)x ,

s2t1 =12(−s2x − 2qrs2 − r2s3 − 2s1r − 2αqs2)x + 2αq(−s2x − 2qrs2 − r2s3 − 2s1r − 2αqs2)

−12

r(s3x − q2s2 − 2qrs3 − 2s1q − 2αqs3) + 2s1

(−

12

rx − αqr −12

qr2)

− 2s2

(12

qx − αq2−

12

q2r

),

s3t1 =12(s3x − q2s2 − 2qrs3 − 2qs1 − 2αqs3)x + (2s3 − 2s1)

(12

qx − αq2−

12

q2r

).

5. Conclusions

In this paper, we have derived a hierarchy of soliton equations from a new generalized KN spectral problem, andobtained their integrable decompositions by nonlinearization method. Making use of the conception of semidirect


sums of Lie algebras, two specific integrable coupling systems are constructed. Thus, this kind of method providesa good source of the matrix spectral problem, and the study of integrable couplings using semidirect sums of Liealgebras will enhance our understanding of the classification of integrable systems.

Obviously, there are other questions worth further investigating for the resultant integrable coupling systems, suchas their Hamiltonian structure, nonlinearization, algebraic structure and so on. These investigations will be conductedin the future.

Acknowledgments

The authors would like to express their sincere thanks to Professor Ma Wen-Xiu for his enthusiastic guidanceduring his visit to Fudan University. This work was supported by grants from National Key Basic Research Project ofChina (2004CB318000) and the Research Project of Hubei Provincial Department of Education (D20082602).

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integrable decomposition of a hierarchy of soliton equations and integrable coupling system by...

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