The Sylvester Equation and Integrable Equations: The Ablowitz—Kaup—Newell—Segur System

Vol. 82 (2018)

REPORTS ON MATHEMATICAL PHYSICS

No. 2

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS: THE ABLOWITZ–KAUP–NEWELL–SEGUR SYSTEM S ONG -L IN Z HAO Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, P.R. China (e-mail: [email protected]) (Received February 15, 2018 — Revised July 10, 2018) In this paper, we seek connections between the Sylvester equation and the Ablowitz–Kaup– Newell–Segur (AKNS) system. By the Sylvester equation KM −MK = r s T , we introduce master function S (i,j ) = s T K j (I + M)−1 K i r. This function satisfies some recurrence relations. By imposing dispersion relations on r and s, we study the constructions of the AKNS system, where some AKNS type equations are investigated emphatically, including second-AKNS equation, second-modified AKNS (mAKNS) equation, third-AKNS equation, third-mAKNS equation and (−1)st-AKNS equation. The reductions of these equations to complex Korteweg–de Vries (KdV) equation, real and complex modified Korteweg–de Vries (mKdV) type equations, nonlinear Schr¨odinger (NLS) type equations and sine-Gordon (sG) equation are discussed. Keywords: AKNS system, Cauchy matrix approach, reductions, solutions.

1.

Introduction As one of the famous matrix equations in mathematics, the Sylvester equation XM − MY = Z

(1.1)

with known matrices X, Y , Z and unknown matrix M has been a popular topic and is still drawing more and more attention. This equation plays a central role in particular in systems and control theory, signal processing, filtering, model reduction, image restoration, and so on. The solvability of Eq. (1.1) was proved by Sylvester [1]. This result was subsequently extended to the operator case (X, Y , Z and M are operators) independently by Dalecki [2] and Rosenblum [3]. Rosenblum’s paper made the operator case widely known, and presented an explicit solution for the Sylvester equation (1.1). In the operator theory the solvability theorem for Eq. (1.1) is always known as Rosenblum Theorem. In [4] Bhatia and Rosenthal investigated lots of interesting and important theoretical results for the Sylvester equation (1.1), such as similarity, commutativity, hyperinvariant subspaces, spectral operators and differential equations. Some works revealed that there exist close connections between the Sylvester equation (1.1) and integrable systems. In the operator method (or trace method in scalar case) proposed by Aden and Carl [5] and developed by Schiebold and her [241]

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S.-L. ZHAO

collaborators [6, 7], suitable dispersion relations were imposed on M or Ŵ = I + M and solutions of nonlinear partial differential equations were expressed in the form of logarithmic derivative Ŵ −1 Ŵ x or its trace. This method relies on Eq. (1.1) with Z of rank one so as to get the needed trace property. In [8, 9] solutions of the Gel’fand–Levitan–Marchenko equation were expressed via a triplet (X, Y , Z) where matrix X and vectors Y and Z satisfy some Sylvester equations. In bidifferential calculus approach [10], integrable equations were derived by introducing bidifferential operators d and d¯ into graded algebras. Solutions of the obtained integrable equations can be parameterised in terms of some matrices which satisfy the Sylvester equation. Recently, Nijhoff and his collaborators [11] proposed a method, named Cauchy matrix approach, to investigate multisoliton solutions of the Adler–Bobenko–Suris (ABS) list [12]. In this method, a Cauchy-type matrix ρi cj M = (Mi,j )N×N , Mi,j = ki + kj was introduced, which satisfies the Sylvester equation KM + MK = r tc,

(1.2) T

where K = Diag(k1 , k2 , · · · , kN ); r = (ρ1 , ρ2 , · · · , ρN ) is a column vector with known plane wave factors     p − ki n q − ki m (0) ρi , ρi = p + ki q + ki

and tc = (c1 , c2 , · · · , cN ) is a constant row vector. By introducing scalar functions S (i,j ) = tc K j (I + M)−1 K i r

and S(a, b) = tc (bI + K)−1 (I + M)−1 (aI + K)−1 r

and discussing their dynamical properties, soliton solutions to lattice KdV type equations and the ABS list were derived. The Cauchy matrix approach is actually a by-product of the linearisation approach which was first proposed by Fokas and Ablowitz [13] and developed to discrete integrable systems by Nijhoff, Quispel et al., one can refer to [14–17]. Based on the Cauchy matrix approach, a generalized Cauchy matrix scheme proposed in [18] leads to more kinds of exact solutions for the ABS list. In addition, the Cauchy matrix approach was applied to elliptic integrable systems [19]. In [20] we used the generalized Cauchy matrix scheme to systematically investigate the links between the Sylvester equation (1.2) and some (1 + 1)-dimensional integrable systems, including the KdV, mKdV, Schwarzian KdV and sG equations. Besides, in [21] Zhao et al. discussed the connections between the following Sylvester equation LM − MK = r s T ,

(1.3)

and the Kadomtsev–Petviashvili system; in (1.3) L ∈ CN ×N , K ∈ CN ′ ×N ′ , M ∈ CN×N ′ , r ∈ CN×1 and s ∈ CN ′ ×1 . Here and in what follows, T represents transposition of matrix.

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . .

243

In this paper, we will focus on a Sylvester equation (2.2) and examine the links between this Sylvester equation and the AKNS system. The motivation for studying AKNS type equations can be found in the fact that their reductions lead to many physically meaningful systems, such as the KdV equation, mKdV type equations, NLS type equations and sG equation. By imposing some dispersion relations on r and s, we will consider the constructions of various of AKNS type equations, such as second-AKNS equation, second-mAKNS equation, third-AKNS equation, third-mAKNS equation and (−1)st-AKNS equation. Here and hereafter we use (-1)th-AKNS equation to represent the first negative order AKNS equation for short. We find that all these AKNS type equations can be generated by a matrix function S (i,j ) (see (2.8)). The paper is organized as follows. In Section 2, we start from the Sylvester equation (2.2) and see that S (i,j ) (defined as (2.8)) obeys some properties, such as recurrence relations and invariance. In Section 3, by imposing dispersion relations on r and s, some positive order AKNS type equations are obtained as closed forms. In Section 4, the negative order AKNS equations are discussed. In Section 5, we consider reductions of the resulting AKNS system. Section 6 is for conclusions. In addition, two appendices on solutions are given as a complement to the article. 2.

Sylvester equation and master function

2.1.

The Sylvester equation

Let E (X) be the set of eigenvalues of a matrix X. In the matrix case, the solvability of Eq. (1.1) is described as follows. T THEOREM 1. If X and Y are matrices such that E (X) E (Y ) = ∅, then the equation XM − MY = Z has a unique solution M for every matrix Z. From (1.1) one can easily get the following result. PROPOSITION 1. For the matrix M defined by the Sylvester equation (1.1), the following relations hold: Xs M − MY s =

s−1 X

Xs−1−l ZY l ,

l=0

X−s M − MY −s = −

−s X

s = 1, 2, . . . ,

X−s−1−l ZY l ,

l=−1

s = 1, 2, . . . ,

(2.1a) (2.1b)

where X0 and Y 0 represent the 2N-th order unit matrix. The starting point for our discussion is the following Sylvester equation KM − MK = r s T

(2.2)

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S.-L. ZHAO

depending on matrices K, M, r and s given by      K1 0 0 M1 r1 K= , M= , r= 0 K2 M2 0 0

 0 , r2

s=



0 s2

s1 0



(2.3)

with K i ∈ CNi ×Ni , M 1 ∈ CN1 ×N2 , M 2 ∈ CN2 ×N1 , r i , s i ∈ CNi ×1 , (i = 1, 2) and N1 + N2 = 2N. This corresponds to X = Y and Z being of rank 2 in (1.1). To make the Sylvester equation (2.2) more explicit, we substitute (2.3) into (2.2) and get the following system K 1 M 1 − M 1 K 2 = r 1 s T2 , K 2 M 2 − M 2 K 1 = r 2 s T1 .

(2.4a) (2.4b)

In terms of Theorem 1, we know that for arbitrary vectors r i and s i (i = 1, 2), the system (2.4) is solvable and has unique solution for M 1 and M 2 when T E (K 1 ) E (K 2 ) = ∅. In the remaining part of this section, we assume that K 1 and K 2 satisfy such condition. Besides, in order to avoid difficulties we also assume 1∈ / E (M 1 M 2 ) ∪ E (M 2 M 1 ) and 0 ∈ / E (K). The condition 1 ∈ / E (M 1 M 2 ) ∪ E (M 2 M 1 ) implies that matrix I + M is invertible and its inverse reads   (I N1 − M 1 M 2 )−1 −M 1 (I N2 − M 2 M 1 )−1 −1 (I + M) = . (2.5) −M 2 (I N1 − M 1 M 2 )−1 (I N2 − M 2 M 1 )−1 Here and hereafter we assume that I m indicates the m-th order unit matrix and I = I 2N is the 2N-th order unit matrix. Along with matrices K, M, r and s we also introduce two matrices     I N1 0 1 0 , a= . (2.6) A= 0 −I N2 0 −1 From the form of matrices K, M, r and s given in (2.3), one can easily recognize the following facts: rs T A = −Ars T , KA = AK, MA = −AM, s T A = −as T , Ar = ra, a 2 = I 2 . (2.7) 2.2.

The master function S (i,j )

Now we introduce a 2 × 2 matrix function S

(i,j )

T

j

−1

i

= s K (I + M) K r =



s1 s3

s2 s4



(2.8)

for i, j ∈ Z, which we identify as the master function. Based on (2.3) and (2.5), one knows that entries of S (i,j ) can be expressed as j

s1 = −s T2 K 2 M 2 (I N1 − M 1 M 2 )−1 K i1 r 1 ,

(2.9a)

s2 =

− M 2 M 1 )−1 K i2 r 2 ,

(2.9b)

− M 1M 2)

(2.9c)

s3 =

j s T2 K 2 (I N2 j s T1 K 1 (I N1

−1

K i1 r 1 ,

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . . j

s4 = −s T1 K 1 M 1 (I N2 − M 2 M 1 )−1 K i2 r 2 .

245 (2.9d)

Based on (2.1), a system of recurrence relations for the master function S (i,j ) can be derived [25]. PROPOSITION 2. For the master function S (i,j ) defined by (2.8) with K, M, r, s satisfying the Sylvester equation (2.2), we have the following relations S S

(i,j +s)

(i,j −s)

=S

(i+s,j )

=S

(i−s,j )

− +

s−1 X

S (s−1−l,j ) S (i,l) ,

s = 1, 2, . . . ,

l=0

−s X

S (−s−1−l,j ) S (i,l) ,

s = 1, 2, . . . .

l=−1

(2.10a) (2.10b)

Another property satisfied by the master function S (i,j ) is invariance. In fact, ¯ is similar suppose that under the transform matrix T = Diag(T 1 , T 2 ), the matrix K to K, i.e. ¯ = T KT −1 . K (2.11a) We denote

¯ = T MT −1 , M

r¯ = T r,

s¯ T = s T T −1 .

(2.11b)

Then one can easily that ¯ j (I + M) ¯ −1 K ¯ i r¯ , S (i,j ) = s T K j (I + M)−1 K i r = s¯ T K

(2.12)

(i,j )

which means that S is invariant under the similarity transformation (2.11). For relating the master function S (i,j ) to the AKNS system together, we need to impose dispersion relations on r and s. In the following parts, we suppose that r, s and M are functions of (x, t1 , t2 , t3 , . . .) while K is a nontrivial constant matrix, where x is referred to as spatial variable and {tn }∞ n=1 are viewed as infinite time variables. 3.

The positive order AKNS system The dispersion relations of r and s are set as r x = AKr, s x = AK T s, s tn = −A(K T )n s, r tn = −AK n r,

n = 1, 2, 3, . . . ,

(3.1a) (3.1b)

where A is defined by (2.6). 3.1.

Evolutions of M

We now discuss the evolution of matrix M, i.e. the derivatives of M w.r.t. independent variables x and {tn }. For deriving the spatial evolution of M, we take x-derivative of the Sylvester equation (2.2) and get KM x − M x K = r x s T + r s Tx .

(3.2)

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S.-L. ZHAO

Substituting (3.1a) into (3.2) and using commutation relations (2.7) together with Theorem 1, we immediately arrive at M x = ras T .

(3.3)

A similar calculation yields the time evolutions of M, M tn = −

n−1 X

K l ras T K n−1−l ,

l=0

n = 1, 2, 3, . . . ,

(3.4)

where we have used equality (2.1a) and relation (2.7). 3.2.

Evolution of S (i,j )

From the evolution relations (3.3) and (3.4) there can be obtained evolution relations of S (i,j ) . We begin by introducing an auxiliary matrix function u(i) = (I + M)−1 K i r

(3.5)

S (i,j ) = s T K j u(i) .

(3.6)

for i ∈ Z. Then S (i,j ) can be written as

The x-derivative of (3.5) together with (3.1a), (3.3) and (2.7) lead to i+1 (I + M)u(i) ra − ras T u(i) , x =K

(3.7)

which under (3.6) further implies (i+1) u(i) a − u(0) aS (i,0) . x =u

(3.8)

Eq. (3.8) is a linear recursion relation between the objects u(i) with the objects S (i,j ) acting as coefficients. By analogous analysis, we derive the time evolutions of u(i) n−1 X (i) (i+n) utn = −u a+ u(n−1−l) aS (i,l) , n = 1, 2, 3, . . . . (3.9) l=0

With (3.8) and (3.9) in hand, the evolutions of master function S (i,j ) can be obtained immediately. In fact, multiplying the relations (3.8) and (3.9), from the left by s T K j and taking the evolutions of s (3.1) together with the connection (3.6) between u(i) and S (i,j ) , we have ) S (i,j = −aS (i,j +1) + S (i+1,j ) a − S (0,j ) aS (i,0) , x (i,j )

S tn

= aS (i,j +n) − S (i+n,j ) a +

n−1 X l=0

S (n−1−l,j ) aS (i,l) ,

(3.10a) n = 1, 2, 3, . . . .

(3.10b)

Higher-order derivatives of S (i,j ) w.r.t x can be derived from (3.10a) by iterating ) ) calculation. Here we just list S (i,j and S (i,j xx xxx , which read

247

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . . ) (i,j +2) S (i,j + S (i+2,j ) − 2aS (i+1,j +1) a + 2aS (0,j +1) aS (i,0) xx = S

− 2S (0,j ) aS (i+1,0) a − S (1,j ) S (i,0) + S (0,j ) S (i,1) + 2S (0,j ) aS (0,0) aS (i,0) , (3.10c)

) (i,j +3) S (i,j + S (i+3,j ) a + 3S (i+1,j +2) a − 3aS (i+2,j +1) − 3S (0,j +2) aS (i,0) xxx = − aS

− 3S (0,j ) aS (i+2,0) + 6aS (0,j +1) aS (i+1,0) a − 3S (1,j ) S (i+1,0) a − 3aS (0,j +1) S (i,1)

− S (2,j ) aS (i,0) + 2S (1,j ) aS (i,1) − S (0,j ) aS (i,2) + 3S (0,j ) S (i+1,1) a

+ 3aS (1,j +1) S (i,0) + 6S (0,j ) aS (0,0) aS (i+1,0) a − 6aS (0,j +1) aS (0,0) aS (i,0) + 3S (0,j ) aS (1,0) S (i,0) − 3S (0,j ) S (0,1) aS (i,0) + 3S (1,j ) S (0,0) aS (i,0) − 3S (0,j ) aS (0,0) S (i,1) − 6S (0,j ) aS (0,0) aS (0,0) aS (i,0) .

(3.10d)

Up to now, we have obtained various derivatives of S (i,j ) w.r.t. independent variables. All the relations in (3.10) can be viewed as semi-discrete equations when the parameters i and j are recognized as discrete independent variables. For n = 2 and n = 3, we consider the constructions of some AKNS type equations in the next two subsections, where second-AKNS, second-mAKNS, third-AKNS and thirdmAKNS equations are involved. For the sake of the resulting closed-form equations, we introduce the following variables       µ1 µ2 v1 v2 u1 u2 (1,0) (−1,0) (0,0) , , S = , v=S − I2 = u=S = µ3 µ4 v3 v4 u3 u4 (3.11) where ui , vi and µi (i = 1, . . . , 4) are scalar functions. For convenience, we denote {ti = t}. It is worth noting that (2.10b) with s = 1 and i = j = 0 gives vw = −I 2 , wherew = I 2 + S (0,−1) .

(3.12)

It is not necessary to discuss the equation related to w. 3.3.

The AKNS system with n = 2

When n = 2, (3.10b) reduces to (i,j )

St

= aS (i,j +2) − S (i+2,j ) a + S (1,j ) aS (i,0) + S (0,j ) aS (i,1) .

(3.13)

In the following, second-AKNS and second-mAKNS equations are studied by means of relations (3.10a), (3.10c) and (3.13). 3.3.1.

The second-AKNS equation

To proceed, we take i = j = 0 in (3.10). In this case, the evolution relations (3.10c) and (3.13) directly give rise to uxx = S (0,2) + S (2,0) − 2aS (1,1) a + 2aS (0,1) au − 2uaS (1,0) a − S (1,0) u + uS (0,1) + 2uauau,

(3.14a)

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S.-L. ZHAO

ut = aS (0,2) − S (2,0) a + S (1,0) au + uaS (0,1) .

(3.14b)

Substituting the equalities (identity (2.10a) with s = 1, and respectively, i = 0, j = 1; i = 1, j = 0; i = j = 0) S (0,2) = S (1,1) − S (0,1) u,

S (2,0) = S (1,1) + uS (1,0) , S

(0,1)

=S

(1,0)

2

−u

(3.15a) (3.15b) (3.15c)

into (3.14) and by direct calculation we find −2aut + uxx = 2(ua − au)((ua − au)u + aS (1,0) − S (1,0) a).

(3.16)

Taking (3.11) into (3.16) and expanding, we get a system consisting of u2 and u3 : −2u2t + u2xx = −8u22 u3 , 2u3t + u3xx =

−8u2 u23 ,

(3.17a) (3.17b)

which is nothing but the second-AKNS equation, whose solution is given by u2 = s T2 (I N2 − M 2 M 1 )−1 r 2 ,

u3 = s T1 (I N1 − M 1 M 2 )−1 r 1 .

(3.18a) (3.18b)

3.3.2. The second-mAKNS equation

To derive the second-mAKNS equation, we consider the variable v defined by (3.11). Making use of (3.15c) and the following equalities (identity (2.10a) with i = −1, j = 0 and with s = 1, respectively, s = 2) S (−1,1) = −uv,

S

(−1,2)

= −S = −S

(3.19a)

(1,0)

(0,1)

v − uS

(−1,1)

(3.19b) (3.19c)

v,

we arrive at v x = (au − ua)v,

(3.20a)

v t = (S (1,0) a − aS (1,0) )v − (ua − au)uv,

(3.20b)

v xx = 2(−S

(3.20c)

(1,0)

+ aS

(1,0)

2

a − au a + uaua)v.

By straightforward computation, we get −2av t + v xx = 2(au − ua)2 v,

(3.21)

− 2v1t + v1xx = 8u2 u3 v1 , − 2v2t + v2xx = 8u2 u3 v2 ,

(3.22a) (3.22b)

and the entries are

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . .

2v3t + v3xx = 8u2 u3 v3 , 2v4t + v4xx = 8u2 u3 v4 .

249 (3.22c) (3.22d)

To derive the second-mAKNS equation from (3.22), we need two additional variables q = v1 + v2 ,

r = v3 + v4 .

(3.23)

Adding (3.22a) to (3.22b) and (3.22c) to (3.22d), we get −2qt + qxx = 8u2 u3 q, 2rt + rxx = 8u2 u3 r.

(3.24a) (3.24b)

To relate the term u2 u3 to variables q and r, we consider Eq. (3.20a). The substitution of (3.11) into (3.20a) gives v1 v3 v4 v2 2u2 = x = x , −2u3 = x = x , (3.25) v4 v3 v1 v2 which implies qx = v1x + v2x = 2u2 (v3 + v4 ) = 2u2 r, rx = v3x + v4x = −2u3 (v1 + v2 ) = −2u3 q,

(3.26a) (3.26b)

and moreover u2 u3 = −

qx rx . 4qr

(3.27)

Therefore, the system (3.24) becomes qx rx , r qx rx = −2 . q

−2qt + qxx = −2 2rt + rxx

(3.28a) (3.28b)

This system is considered as second-mAKNS equation and for the first time appeared in [14]. This equation is related to the motion of the Heisenberg ferromagnet with uniaxial anisotropy. It is easy to find that q = v1 − v2 ,

r = v3 − v4

(3.29)

also satisfy the second-mAKNS equation (3.28). Thus, solution to Eq. (3.28) is given by T −1 −1 q = −s T2 M 2 (I N1 − M 1 M 2 )−1 K −1 1 r 1 ± s 2 (I N2 − M 2 M 1 ) K 2 r 2 − 1,

(3.30a)

r = s 1 (I N1 − M 1 M 2 )

(3.30b)

T

−1

K −1 1 r1

T

−1

∓ s 1 M 1 (I N2 − M 2 M 1 )

K −1 2 r2

∓ 1.

Naturally, the system (3.26) constitutes the Miura transformation between the secondAKNS equation (3.17) and the second-mAKNS equation (3.28).

250 3.4.

S.-L. ZHAO

The AKNS system with n = 3

In this subsection, we pay attention to the constructions of third-AKNS equation and third-mAKNS equation. When n = 3, (3.10b) yields (i,j )

St

= aS (i,j +3) − S (i+3,j ) a + S (2,j ) aS (i,0) + S (1,j ) aS (i,1) + S (0,j ) aS (i,2) .

(3.31)

3.4.1. The third-AKNS equation

Taking i = j = 0 in (3.10a), (3.10d) and (3.31), we get ux = − aS (0,1) + S (1,0) a − uau, ut = aS

(0,3)

−S

(3,0)

a + uaS

(0,2)

(3.32a) +S

(1,0)

aS

(0,1)

+S

(2,0)

au,

(3.32b)

uxxx = − aS (0,3) + S (3,0) a + 3S (1,2) a − 3aS (2,1) − 3S (0,2) au − 3uaS (2,0) 2

2

+ 6aS (0,1) aS (1,0) a − 3S (1,0) a − 3aS (0,1) − S (2,0) au + 2S (1,0) aS (0,1)

− uaS (0,2) + 3uS (1,1) a + 3aS (1,1) u + 6uauaS (1,0) a − 6aS (0,1) auau

+ 3uaS (1,0) u − 3uS (0,1) au + 3S (1,0) uau − 3uauS (0,1) − 6uauauau. (3.32c) Noting that the relations identity (2.10a), respectively, with s = 3; i = j = 0; s = 2; i = 1; j = 0; s = 1; i = j = 1 are equal to S (0,3) = S (3,0) − S (2,0) u − S (1,0) S (0,1) − uS (0,1) ,

(3.33a)

S (3,0) = S (1,2) + S (1,0) + uS (1,1) ,

(3.33b)

2

S (1,2) = S (2,1) − S (0,1) S (1,0)

(3.33c)

and (3.15), the direct substitution of (3.11) and using of Mathematica, gives 4u1t + u1xxx = −24(µ2 − u2 u4 )(µ3 − u1 u3 ) − 24u22 u23 , 4u2t + u2xxx = 48u2 u3 (µ2 − u2 u4 ), 4u3t + u3xxx = −48u2 u3 (µ3 − u1 u3 ),

4u4t + u4xxx = 24(µ3 − u1 u3 )(µ2 − u2 u4 ) + 24u22 u23 .

(3.34a) (3.34b) (3.34c) (3.34d)

From Eq. (3.32a) one knows that u1x = 2u2 u3 , u2x = −2(µ2 − u2 u4 ), u3x = 2(µ3 − u1 u3 ), u4x = −2u2 u3 .

(3.35a) (3.35b)

Therefore, the system (3.34) can be changed to 4u1t + u1xxx = 6u2x u3x − 6u21x ,

4u2t + u2xxx = −24u2 u3 u2x , 4u3t + u3xxx = −24u2 u3 u3x ,

4u4t + u4xxx = −6u2x u3x + 6u24x .

(3.36a) (3.36b) (3.36c) (3.36d)

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . .

251

Eqs. (3.36b) and (3.36c) compose the third-AKNS equation, whose solution is still expressed as (3.18). The discrete and semi-discrete versions for (4.11) were recently studied in [22]. 3.4.2.

The third-mAKNS equation

Similarly to the case of n = 2, when n = 3 there is also a system with q and r defined by (3.23) or (3.29) as two dependent variables. Set i = −1, j = 0 in (3.10d) and (3.31) and get v t = (S (2,0) a − aS (2,0) )v + (S (1,0) a − aS (1,0) )S (−1,1) + (ua − au)S (−1,2) , (3.37a)

v xxx = [aS (2,0) − S (2,0) a + 3(−S (0,2) a + aS (1,1) + uaS (1,0) + S (1,0) ua − uS (0,1) a) + (au − ua)S (−1,2) − 6(aS (0,1) + uau)aua]v + (aS (1,0) − 3aS (0,1) + 2S (1,0) a − 3uau)S (−1,1) ,

(3.37b)

where we have used the equality (identity (2.10a) with s = 3 and i = −1, j = 0) S (−1,3) = −S (2,0) v − S (1,0) S (−1,1) − uS (−1,2) .

(3.38)

From (3.19a) and (3.19c), through direct calculation we get 4v t + v xxx =3 − aS (2,0) + S (2,0) a − S (0,2) a + aS (1,1) + uau2

+ aS (1,0) u − 2S (1,0) au + S (1,0) ua + uaS (1,0) + auS (0,1)
− uaS (0,1) − uS (0,1) a + aS (0,1) u v − 6(uau + aS (0,1) )auav.

(3.39)

Furthermore, by using the identities (3.15) and noticing (3.35), we use Mathematica and get the following system: 4v1t + v1xxx = −12u3 u2x v1 − 24u22 u3 v3 ,

(3.40a)

4v2t + v2xxx = −12u3 u2x v2 − 24u22 u3 v4 ,

(3.40b)

4v3t + v3xxx = −12u2 u3x v3 +

(3.40c)

4v4t + v4xxx = −12u2 u3x v4 +

24u2 u23 v1 , 24u2 u23 v2 .

(3.40d)

For the variables q and r defined by (3.23) or (3.29), one knows that relations (3.26) and (3.27) still hold. Thus from (3.40) we arrive at qx rx qxx rx 4qt + qxxx = 3 + 3 2 (qx r − qrx ), (3.41a) r qr qx rxx qx rx 4rt + rxxx = 3 − 3 2 (qx r − qrx ), (3.41b) q q r which is the third-mAKNS equation, whose solution is also expressed as (3.30). The system (3.26) constitutes the Miura transformation between the third-AKNS equation (3.36b), (3.36c) and the third-mAKNS equation (3.41). The discrete and semi-discrete versions for (3.41) were presented in recent paper [23].

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S.-L. ZHAO

In what follows, we study the links between the Sylvester equation (2.2) and the negative order AKNS system, in particular, focus on the (−1)st-AKNS equation. 4. The negative order AKNS system Consider dispersion relations r x = AKr, s x = AK T s, r tn = −AK n r, s tn = −A(K T )n s,

(4.1a) (4.1b)

with n = −1, −2, −3, . . ., where {tn }−∞ n=−1 are viewed as infinite time variables. For convenience, we denote t−1 = t. 4.1.

Evolution of S (i,j )

Analogously to the positive order case, we can derive evolutions along with time direction for various objects, such as n X K l ras T K n−1−l , (4.2a) M tn = − l=−1

(i+n) u(i) a− tn = −u (i,j )

S tn

n X

u(n−1−l) aS (i,l) ,

(4.2b)

l=−1

= aS (i,j +n) − S (i+n,j ) a −

n X

S (n−1−l,j ) aS (i,l) ,

(4.2c)

l=−1

where n = −1, −2, . . .. Specially, when n = −1, Eq. (4.2c) leads to (i,j )

St

= aS (i,j −1) − S (i−1,j ) a − S (−1,j ) aS (i,−1) .

(4.3)

Taking x-derivative on both sides of (4.3) and using (3.10a), we get (i,j )

S xt

= − 2S (i,j ) + a(S (i+1,j −1) + S (i−1,j +1) )a − aS (0,j −1) aS (i,0)

+ S (0,j ) aS (i−1,0) a + (aS (−1,j +1) a − S (0,j ) + S (0,j ) aS (−1,0) a)S (i,−1)

+ S (−1,j ) (S (i,0) − aS (i+1,−1) a + aS (0,−1) aS (i,0) ). 4.2.

(4.4)

The AKNS system with n = −1

Eqs. (4.3) and (4.4) with i = j = 0 lead to ut = − a − vaw,

uxt = aS

(−1,1)

aw − vaS

(4.5a) (1,−1)

a + uavaw + vawau,

(4.5b)

where w is defined by (3.12). For (2.10b) with s = 1 and i = 0, j = 1, respectively, i = 1, j = 0, one can easily get the equality u = S (−1,1) w = −vS (1,−1)

(4.6)

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . .

253

or moreover S (−1,1) = uw−1 = −uv,

(4.7a)

S (1,−1) = −v −1 u = wu,

(4.7b)

utx = (au − ua)(a + ut ) − (a + ut )(au − ua),

(4.8)

u1x = 2u2 u3 , u2tx = 2u2 (−2 + (u4 − u1 )t ), u3tx = 2u3 (−2 + (u4 − u1 )t ), u4x = −2u2 u3 .

(4.9a) (4.9b) (4.9c) (4.9d)

u4 − u1 = −4∂ −1 u2 u3 ,

(4.10)

u2xt = −4u2 − 8u2 ∂t ∂ −1 (u2 u3 ),

(4.11a)

where we have used (3.12). Substituting (4.7) into (4.5b) and using (4.5a), we get whose entries read

Subtracting (4.9a) from (4.9d), we have

∂ ∂x

with the condition ∂∂ −1 = where ∂ −1 stands for an inverse operator of ∂ = −1 ∂ ∂ = 1. Taking (4.10) into (4.9b) and (4.9c), we get the (−1)st-AKNS equation [24]

u3xt = −4u3 − 8u3 ∂t ∂ −1 (u2 u3 ),

(4.11b)

whose solution is also expressed by (3.18). The discrete and semi-discrete versions for (4.11) were recently studied in [25]. In conclusion, we have in these two sections identified the constructions of some AKNS type equations from the Sylvester equation (2.2). In principle, when n ≥ 3 or n ≤ −2, higher order AKNS type equations can also be deduced similarly, which are not discussed here. We note that all the resulting AKNS type equations can be generated by master function S (i,j ) with specific i and j . Besides the equation themselves, the Miura transformations are also derived. In the next section, we show how to derive the complex KdV equation, real and complex mKdV type equations, NLS type equations and sG equation from the resulting AKNS type equations by reduction. For this purpose, we take N1 = N2 = N . 5. 5.1.

Reductions Reductions to complex KdV and real mKdV equations

Taking n = 3 and K 2 = −K 1 , then from (2.4) and (3.1) we can specially choose r 1 = r 2 , s 1 = s 2 and M 2 = −M 1 . In this case, by (2.9) we know that u1 = −u4 and u2 = u3 . We define u = u1 + iu2 . Adding (3.36a) to (3.36b) and using (3.35a), we get the potential complex KdV equation 4ut + uxxx + 6u2x = 0,

(5.1)

254

S.-L. ZHAO

which gives rise to the complex KdV equation 4γt + γxxx + 6γ γx = 0

(5.2)

under transformation γ = 2ux . Noticing that u2 = u3 , then both (3.36b) and (3.36c) reduce to the real mKdV equation, i.e. 4u2t + u2xxx + 24u22 u2x = 0.

(5.3)

The solutions for (5.2) and (5.3) are, respectively, given by γ = 2(s T1 (−iI N + M 1 )−1 r 1 )x ,

(5.4a)

u2 = s T1 (I N + M 21 )−1 r 1 ,

(5.4b)

where M 1 , r 1 and s 1 satisfy the system K 1 M 1 + M 1 K 1 = r 1 s T1 , r 1x = K 1 r 1 , s 1x = K T1 s 1 ,

(5.5a) (5.5b)

r 1t = −K 31 r 1 , s 1t = −(K T1 )3 s 1 ,

(5.5c)

which is nothing more but the scheme used in [20] for the KdV system. From (3.35) it is easy to find the complex Miura transformation between the complex KdV (5.2) and the real mKdV (5.3) γ = 4u22 + 2iu2,x , which has been used to investigate solutions to the complex KdV in [26]. 5.2.

Reductions to complex mKdV type equations and NLS type equations

Under transformations x → ix and t → it together with the constraint K 2 = K ∗1 (asterisk denotes the complex conjugate), we can choose r 2 = r ∗1 , s 2 = s ∗1 and M 2 = M ∗1 , which imply u3 = u∗2 and r = q ∗ . In the following, we list some complex mKdV type equations and NLS type equations together with their solution formulae. • complex mKdV equation:

4u2t − u2xxx + 24|u2 |2 u2,x = 0,

(5.6a)

u2 = s 1 (I N −

|M 1 |2 )−1 r ∗1 ;

(5.6b)

|qx |2 qxx qx∗ + 3 (qqx∗ − qx q ∗ ) = 0, q∗ |q|2 q ∗

(5.7a)

∗T

• modified complex mKdV equation: 4qt − qxxx + 3 ∗

∗

∗

T 2 −1 −1 ∗ q = −s T1 M ∗1 (I N − |M 1 |2 )−1 K −1 1 r 1 ± s 1 (I N − |M 1 | ) K 1 r 1 − 1.

• NLS equation:

2iu2t + u2xx + 8u2 |u2 |2 = 0,

(5.7b) (5.8a)

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . . ∗

u2 = s 1T (I N − |M 1 |2 )−1 r ∗1 ;

255 (5.8b)

• modified NLS equation: 2iqt − qxx − 2

|qx |2 = 0, q∗

(5.9a)

∗

∗

∗

T 2 −1 −1 ∗ q = −s T1 M ∗1 (I N − |M 1 |2 )−1 K −1 1 r 1 ± s 1 (I N − |M 1 | ) K 1 r 1 − 1.

(5.9b)

In the above solutions all the entities M 1 , r 1 and s 1 satisfy the system ∗

K 1 M 1 − M 1 K ∗1 = r 1 s T1 , r 1x = iK 1 r 1 , s 1x = iK T1 s 1 ,

(5.10a) (5.10b)

and for (5.6b) and (5.7b) these entities additionally satisfy r 1t = −iK 31 r 1 ,

s 1t = −i(K T1 )3 s 1 ,

(5.11)

while for (5.8b) and (5.9b) these entities additionally satisfy r 1t = −iK 21 r 1 , 5.3.

s 1t = −i(K T1 )2 s 1 .

(5.12)

Reduction to sG equation

Let K 2 = −K 1 . Similarly to the reduction to real mKdV equation, we know u2 = u3 and (−1)st-AKNS equation reduces to u2xt = −4u2 − 8u2 ∂t ∂ −1 (u22 ).

(5.13)

By transformation φ = arcsin u2t , (5.13) can be transformed to the usual sG equation φxt = −4 sin φ.

(5.14)

For detailed calculation one can see [25]. Thus sG equation (5.14) has solution φ = arcsin(s T1 (I N + M 21 )−1 r 1 )t ,

(5.15)

where M 1 , r 1 and s 1 satisfy the system K 1 M 1 + M 1 K 1 = r 1 s T1 , r 1x = K 1 r 1 , s 1x = K T1 s 1 , r 1t = −K −1 1 r 1,

s 1t = −(K T1 )−1 s 1 ,

(5.16a) (5.16b) (5.16c)

which is nothing more but the scheme used in [20] for the sG equation. 6.

Conclusions

In this contribution, we have revealed the connections between the Sylvester equation (2.2) and the AKNS system. The technique used in this paper is the so-called generalized Cauchy matrix approach, which always takes the Sylvester

256

S.-L. ZHAO

equation as the starting point. The Sylvester equation (2.2) and the master function (2.8) introduced in this paper are consistent with the discrete case (see [22, 23, 25]). By imposing dispersion evolutions on r and s, we derive evolutions (x-, t-derivatives) of S (i,j ) . For getting the AKNS system from these evolutions, we use a series of recurrence relations (see Proposition 2). These recurrence relations can be viewed as discrete equations of S (i,j ) with discrete independent variables i and j . It can be found that in the construction of AKNS system either in discrete case (see [22, 23, 25]) or in continuous case the recurrence relations of S (i,j ) are indispensable. While for the lattice KdV case, recurrence relations of corresponding master function do not play any role in the construction of the lattice KdV type equations [18]. By imposing constraints on matrices K 1 and K 2 , the resulting AKNS system including second-AKNS equation, second-mAKNS equation, third-AKNS equation, third-mAKNS equation and (−1)st-AKNS equation are reduced to the complex KdV equation, real and complex mKdV type equations, NLS type equations and sG equation. Since the pioneering work for the nonlocal NLS equation has been done by Ablowitz and Musslimani [27], there has been a considerable amount of interest in the study of nonlocal integrable equations. Very recently, based on the double Wronskian solutions for the AKNS hierarchy, Zhang et al. [28, 29] developed a reduction technique by imposing a constraint on the two basic vectors in double Wronskians so that two potential functions in AKNS hierarchy obey some nonlocal relations and therefore solutions to the nonlocal NLS equation, nonlocal mKdV equation and nonlocal sG equation (in nonpotential form) were derived. So how to discuss the nonlocal reductions of AKNS system along with the procedure given in the present paper is an interesting problem worth consideration. We hope that the results given in the present paper can be useful to study the integrable system, specially to the multi-component integrable system. Acknowledgments

This project is supported by the Natural Science Foundation of Zhejiang Province (Nos. LY17A010024, LY18A010033), the National Natural Science Foundation of China (Nos. 11301483, 11401529). A. Some notations and properties • Diagonal matrices

N Ŵ [N] D ({ki }1 ) = Diag(k1 , k2 , . . . , kN ), .

• Jordan block matrices



a 1   [N ] Ŵ J 1 (a) =  0  .. . 0

0 a 1 .. .

0 0 a .. .

··· ··· ··· .. .

0 0 0 .. .

0

0

···

1

 0 0  0  ..  .

a

N1 ×N1

(A.1)

.

(A.2)

257

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . .

• Lower triangular Toeplitz matrix1  a1 0  a2 a1   a3 [N] N a2 T ({aj }1 ) =   .. ..  . . aN aN−1 • Skew triangular Hankel matrix  b1  b2   [N] N H ({bj }1 ) =  b3  ..  . bN

0 0 a1 ··· aN−2

··· ··· ··· .. .

0 0 0 .. .

···

a2

··· ··· ··· .. .

bN −2 bN −1 bN .. .

bN −1 bN 0 .. .

···

0

0

 0 0  0  ..  . 

a1

.

(A.3)

N×N

 bN 0   0   ..  .  0

.

(A.4)

N×N

The lower triangular Toeplitz matrices and the skew triangular Toeplitz matrices defined above have the following properties. PROPOSITION 3. Let

T [N] = {T [N] ({ai }N 1 )}, T¯ [N] = {H [N] ({bi }N 1 )}.

Then we have

(A.5a) (A.5b)

(i) AB = BA, ∀A, B ∈ T [N] ; T (ii) H = H , ∀H ∈ T¯ [N] ; T (iii) H A = (H A) = AT H , ∀A ∈ T [N] , ∀H ∈ T¯ [N] . Meanwhile, the following expressions need to be considered: Exponential functions: ρi = eξi ,

ξi = ki x − kin tn + ξi0

̺j = e ,

ηj = −lj x +

ηj

̟j = e ,

2 × 2N matrix:

1 More

+

ηj0

ζi = ki x − kin tn + ζi0

σi = eζi ,

ςj

ljn tn

with constants ξi0 ,

ςj = −lj x +

ljn tn

+

 r , r , . . . , r1,N1 r = 1,1 1,2 0, 0, · · · , 0  0, 0, · · · , 0 s= s1,1 , s1,2 , . . . , s1,N1

with constants

(A.6a)

ηj0 ,

with constants ζi0 , ςj0

0, 0, · · · , 0 r2,1 , r2,2 , . . . , r2,N2 s2,1 , s2,2 , . . . , s2,N2 0, 0, · · · , 0

properties of this kind of matrices can be found in [30, 31].

T

T

(A.6c) ςj0 ,

with constants

(A.6b)

(A.6d)

,

(A.6e)

,

(A.6f)

258

S.-L. ZHAO

2N × 2N matrix: [N1 ;N2 ] DD

G

N N ({ki }1 1 ; {lj }1 2 )

=

[N ;N ]T

N

N

1 2 −GDD;12 ({ki }1 1 ; {lj }1 2 )   1 [N1 ;N2 ] N N GDD;12 ({ki }1 1 ; {lj }1 2 ) = , ki − lj N1 ×N2

N ({ki }1 1 ; b)

[N ;N ]

N

G

=

[N ;N ]T

N

1 2 −GDJ;12 ({ki }1 1 ; b)

1 2 GDJ;12 ({kj }1 1 ; b) = (gi,j )N1 ×N2 ,

gi,j =

2N × 2N matrix: G

[N1 ;N2 ] JD

N (a; {lj }1 2 )

[N ;N ]

=

[N ;N ]T

N

1 2 −GJD;12 (a; {lj }1 2 )

2N × 2N matrix: G

(a; b) =

[N ;N ]

where Cij =

j! , i!(j − i)!

,

0

0

gi,j = Ci−1 i+j −2

, (A.6h)

N

!

!

,

1 2 GJD;12 (a; {lj }1 2 )

[N ;N ]

1 2 −GJJ;12 (a; b)

1 2 GJJ;12 (a; b) = (gi,j )N1 ×N2 ,

1 ki − b

1 2 GJJ;12 (a; b)

0 [N ;N ]T



0 j

 −1 i gi,j = − , a − lj

N

1 2 GJD;12 (a; {lj }1 2 ) = (gi,j )N1 ×N2 ,

[N1 ;N2 ] JJ

N

1 2 GDJ;12 ({ki }1 1 ; b)

[N ;N ]

0

!

(A.6g)

[N ;N ]

0

N

0

2N × 2N matrix: [N1 ;N2 ] DJ

N

[N ;N ]

1 2 GDD;12 ({ki }1 1 ; {lj }1 2 )

0

!

, (A.6i)

,

(−1)i+1 , (a − b)i+j −1

(A.6j)

j ≥ i.

B. Exact solutions To derive the exact solutions for the various of AKNS type equations, we just need to solve the following system KM − MK = r s T , r x = AKr, s x = AK T s, r tn = −AK n r, s tn = −A(K T )n s,

(B.1a) (B.1b) (B.1c)

for n ∈ Z, where the forms of K, M, r and s are given by (2.3). It is easy to find that the system (B.1) is formal invariant under similarity transformation (2.11). By virtue of the similarity invariance of S (i,j ) and formal invariance of (B.1), we now

259

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . .

consider the solutions to the following canonical equations M − M = r s T ,

r x = Ar,

(B.2a) T

s x = A s,

n

r tn = −A r,

(B.2b) T

n

s tn = −A( ) s,

(B.2c)

where  = Diag(Ŵ, 3) T is the canonical form of the matrix KT= Diag(K 1 , K 2 ). As for condition E (K 1 ) E (K 2 ) = ∅, hereafter we assume E (Ŵ) E (3) = ∅. Because both Ŵ and 3 are of form canonical structures, it is possible to give a complete classification for the solutions. The procedure for solving the Sylvester equation (B.2) can be found in [25]. The key point in the solving procedure is to factorize M into a triplet, i.e. M = F GH . Here we skip the detailed procedure and directly write out solutions. PROPOSITION 4. (i) When N

[N ]

we have [N ]

r=

N

r 1,D1 ({ki }1 1 ) 0 [N2 ] N 0 r 2,D ({lj }1 2 ) [N ]

s=

s

[N2 ] 2,D

N

0 s 1,D1 ({ki }1 1 ) N2 ({lj }1 ) 0

!

!

=



=



N

[N ]

3 = Ŵ D 2 ({lj }1 2 )

Ŵ = Ŵ D 1 ({ki }1 1 ),

(B.3)

0, 0, · · · , 0 r2,1 , r2,2 , . . . , r2,N2

r1,1 , r1,2 , . . . , r1,N1 0, 0, · · · , 0

T

,

(B.4a) T s2,1 , s2,2 , . . . , s2,N2 , 0, 0, · · · , 0

0, 0, · · · , 0 s1,1 , s1,2 , . . . , s1,N1

(B.4b)

r1,i = ρi , r2,j = ̺j , s1,i = σi , s2,j = ̟j ,

i = 1, 2, , . . . , N1 ; j = 1, 2, , . . . , N2 , (B.4c)

and M = F GH ,

(B.5a)

where N

[N ]

N

[N ]

F = Diag(Ŵ D 1 ({r1,i }1 1 ), Ŵ D 2 ({r2,j }1 2 )),

(B.5b)

({ki }1 1 ; {lj }1 2 ),

(B.5c)

[N ;N2 ]

G = GDD1

H = Diag(Ŵ

N

[N1 ] D

N

[N ] N N ({s1,i }1 1 ), Ŵ D 2 ({s2,j }1 2 )).

(B.5d)

(ii) When [N ]

N

Ŵ = Ŵ D 1 ({ki }1 1 ),

[N ]

3 = Ŵ J 2 (l1 ),

we have  [N1 ]   N r , r , . . . , r1,N1 r 1,D ({ki }1 1 ) 0 r= = 1,1 1,2 [N2 ] 0, 0, · · · , 0 0 r 2,J (l1 )

0, 0, · · · , 0 r2,1 , r2,2 , . . . , r2,N2

(B.6) T

, (B.7a)

260 s=

S.-L. ZHAO





N

[N ]

0 s 1,D1 ({ki }1 1 ) [N2 ] s 2,J (l1 ) 0 j −1

r1,i = ρi ,

r2,j =

∂l1 ̺1 (j − 1)!

=



0, 0, · · · , 0 s1,1 , s1,2 , . . . , s1,N1

N −j

s1,i = σi ,

,

s2,1 , s2,2 , . . . , s2,N2 0, 0, · · · , 0

s2,j =

∂l1 2

T

, (B.7b)

̟1

, (N2 − j )! i = 1, 2, . . . , N1 ; j = 1, 2, , . . . , N2 ,

and

M = F GH ,

where

(B.8a)

N

[N ]

(B.7c)

N

F = Diag(Ŵ D 1 ({r1,i }1 1 ), T [N2 ] ({r2,j }1 2 )),

(B.8b)

G

(B.8c)

H

N =G ({ki }1 1 ; l1 ), [N ] N N = Diag(Ŵ D 1 ({s1,i }1 1 ), H [N2 ] ({s2,j }1 2 )). [N1 ;N2 ] DJ

(B.8d) [N ]

Likewise, we can also obtain the solution for Eq. (B.2) when Ŵ = Ŵ J 1 (k1 ), 3 = [N ] N Ŵ D 2 ({lj }1 2 ).

(iii) When

[N ]

[N ]

Ŵ = Ŵ J 1 (k1 ), 3 = Ŵ J 2 (l1 ),  [N1 ]   T r 1,J (k1 ) 0 r1,1 , r1,2 , . . . , r1,N1 0, 0, · · · , 0 r= = , [N ] 0, 0, · · · , 0 r2,1 , r2,2 , . . . , r2,N2 0 r 2,J2 (l1 ) T    [N ] 0 s 1,J1 (k1 ) 0, 0, · · · , 0 s2,1 , s2,2 , . . . , s2,N2 , s = [N2 ] = s1,1 , s1,2 , . . . , s1,N1 0, 0, · · · , 0 s 2,J (l1 ) 0 r1,i =

ρ1 ∂ki−1 1 (i − 1)!

N −i

j −1

,

r2,j =

∂l1 ̺1 (j − 1)!

and

,

s1,i =

∂k11 σ1

N −j

,

(N1 − i)! i = 1, 2, , . . . , N1 ;

s2,j =

∂l1 2

N

, (N2 − j )! j = 1, 2, , . . . , N2 , (B.10c)

]

3 = Diag Ŵ

N

G=G

(B.10f)

N

(k1 ; l1 ),

N N Diag(H [N1 ] ({s1,j }1 1 ), H [N2 ] ({s2,j }1 2 )). [N12 ]

Ŵ = Diag Ŵ D 11 ({ki }1 11 ), Ŵ J [N21 ] D

(B.10d) (B.10e)

H = [N

(B.10b)

F = Diag(T [N1 ] ({r1,j }1 1 ), T [N2 ] ({r2,j }1 2 )), [N1 ;N2 ] JJ

(iv) When

(B.10a)

̟1

M = F GH ,

where

(B.9)

[N13 ]

(kN11 +1 ), Ŵ J

[N1s ]

(kN11 +2 ), . . . , Ŵ J

(B.10g)
(kN11 +(s−1) ) ,


[N ] [N ] [N ] N ({lj }1 21 ), Ŵ J 22 (lN21 +1 ), Ŵ J 23 (lN21 +2 ), . . . , Ŵ J 2s (lN21 +(s−1) ) ,

(B.11a) (B.11b)

261

THE SYLVESTER EQUATION AND INTEGRABLE EQUATIONS. . .

Ps

Nij = Ni (i = 1, 2), we have    [N ] [N ] N N r 1,D11 ({ki }1 11 ) s 1,D11 ({ki }1 11 )  [N ]    [N ]   r 12 (k  s 1,J12 (kN11 +1 )  N11 +1 )  1,J        .. ..     . 0 0 .         [N ] [N1s ]  r 1s (k   s 1,J (kN11 +(s−1) )  N11 +(s−1) )  1,J    r= s =  [N ] ,  [N ] N N 21 21 21 21  s 2,D ({lj } )   r 2,D ({lj }1 )  1         [N22 ] [N22 ]    s 2,J (lN21 +1 ) r 2,J (lN21 +1 )          . . . .     0 . . 0     [N2s ] [N2s ] r 2,J (lN21 +(s−1) ) s 2,J (lN21 +(s−1) ) (B.12) and M = F GH , (B.13a) where 

j =1

where with

F = Diag(F 1 , F 2 ),

H = Diag(H 1 , H 2 ),

(B.13b)

 [N ] N N +N12 N F 1 = Diag Ŵ D 11 ({r1,j }1 11 ), T [N12 ] ({r1,j }N11 ), . . . , T [N1s ] ({r1,j }P1s−1 11 +1

j =1 N1j +1

 [N ] N N +N22 N F 2 = Diag Ŵ D 21 ({r2,j }1 21 ), T [N22 ] ({r2,j }N21 ), . . . , T [N2s ] ({r2,j }P2s−1 21 +1

j =1 N2j +1

 [N ] N N +N12 N ), . . . , H [N1s ] ({s1,j }P1s−1 H 1 = Diag Ŵ D 11 ({s1,j }1 11 ), H [N12 ] ({s1,j }N11 11 +1

 ) ,

(B.13c)  ) , (B.13d)  ) ,

j =1 N1j +1

 [N ] N N +N22 N ), . . . , H [N2s ] ({s2,j }P2s−1 H 2 = Diag Ŵ D 21 ({s2,j }1 21 ), H [N22 ] ({s2,j }N21 21 +1

(B.13e)  ) ,

j =1 N2j +1

and G is of the form G= where with



 G1 , 0

0 −GT1

(i,j )

G1 = (G1 ;N

]

N

(B.13g)

)s×s

N

G(1,1) = GDD11 21 ({ki }1 11 ; {lj }1 21 ), 1 [N ;N ] N (1,j ) G1 = GDJ11 2j ({ki }1 11 ; lN21 +j −1 ), [N ;N ] N G(i,1) = GJD1i 21 (kN11 +i−1 ; {lj }1 21 ), 1 [N1i ;N2j ] (i,j ) G1 = GJJ (kN11 +i−1 ; lN21 +j −1 ), [N

(B.13f)

1 < j ≤ s, 1 < i ≤ s, 1 < i, j ≤ s.

(B.13h)

262

S.-L. ZHAO

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