Linear integral equations and multicomponent nonlinear integrable systems II

Physica A 160 (1989) 235-273 North-Holland, Amsterdam

LLNEAR INTEGRAL EQUATIONS AND MULTICOMPONENT NONLINEAR INTEGRABLE SYSTEMS II J. VAN DER LINDEN Instituut-Lorentz voor Theoretische Natuurkunde, Nieuwsteeg 18, 2311 SB Leiden, The Netherlands

H.W. CAPEL Instituut voor Theoretische Fy$ica, University of Amsterdam, Valckenierstraat 65, 1018 X E Amsterdam, The Netherlands

F.W. NIJHOFF 1 L . P . T . H . E . , Universit# Pierre et Marie Curie, 4 Place Jussieu, 75230 Paris Cedex 05, France

Received 13 April 1989

In a previous paper (I) an extension of the direct iinearization method was developed for obtaining solutions of multicomponent generalizations of integrable nonlinear partial differential equations (PDE's). The method is based on a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of the spectral parameter. In I the general framework has been presented, with as immediate application the direct linearization of multicomponent versions of the nonlinear Schr6dinger equation and the (complex) modified Korteweg-de Vries equation. In the present paper we treat a variety of examples of other multicomponent PDE's, and we also discuss Miura transformations and gauge equivalences. The examples include the direct linearization of multicomponent generalizations of the isotropic Heisenberg spin chain equation, the complex sine-Gordon equation, the Getmanov equation, the derivative nonlinear Schr6dinger equation and the massive Thirring model equations.

1. Introduction In the last few years many soliton bearing systems have been investigated on the basis of the direct linearizzdon (DL) method. In the DL method use is made of a linear integral equation containing one or more integrations over an arbitrary contour, with an arbitrary measure, in the complex k-plane. The t Present address: Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 136 76, USA.

0378-4371/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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J. van der Linden et al. I Multicomponent integrable systems H

solution of the integral equation depends, besides on k, on the coordinates of the soliton system as additional variables, in a way which is determined by a plane-wave factor 0ccumng in the equation. The important feature is that, if the measUre and cont0ur are such that this solution is unique, it can be shown to comply with all the properties of the wave function in the inverse-scattering t r a n s f o ~ (IST) formalism for the soliton system, with k as the spectral parameter. The potential, too, is obtained according to the D L method directly by integrating the wave function over k, using the same measure and contour as in the integral equation. This potential then satisfies the (nonlinear) partial differential equation (PDE), or difference equation, of the soliton system, independently of the choice of measure and contour. For a special choice it yields the soliton solutions. The D L method was introduced by Fokas and Ablowitz [1] for the linearization of the Korteweg-de Vries equation. The corresponding wave function could be solved from a linear integral equation containing only one integration over a contour C, with measure dA(k). In ref. [2] two other integral equations were introduced, namely an integral equation of type I and an integral equation of type II. The integral equation of type I contains a two-fold integration involving a contour C with measure dA(k), and the complex conjugate contour C* with measure dA*(k'), and it leads to the DL of the nonlinear Schr6dinger equation (NLS), the complex modified Korteweg-de Vries equation (CMKdV) and the complex sine-Gordon equation (CSG). With the integral equation of type II, which contains a two-fold integration involving only one contour C with measure dA(k), one obtains the D L of the modified Korteweg-de Vries equation (MKdV) and the sine-Gordon equation (SG). The integral equations of ref. [2] were formulated for wave vectors with infinitely many components: the wave functions O Ck0associated to factors k -~, i E T/, in the source term, are the components of the wave vector 6k" Furthermore, a g x 7/matrix potential • was defined with elements ~kc~.J) that are obtained by integrating •~c~)t,-J ~k ,~ , i. ] E 7/, over k. In this matrix structure the (0, 0) element plays a central role: the NLS and the CMKdV were obtained as closed expressions in terms of only ck(°" o), by reduction of the PDE derived for q~. But also the nonlinear PDE's for the (1, 0) and the (1, 1) elements of could be obtained, together with the Miura"t r a-n ~ -~ ' ~ivl' "•~~1 connecting t o l --m a u ": o n s - q, these elements to the (0, 0) element. In this way the DL of the equation of motion for the classical isotropic Heisenberg spin chain in the continuum limit (IHSC), and of the Getmanov equation, could be established in addition to the DL of the NLS, the CMKdV and the CSG. The corresponding treatment for the integral equation of type II does not lead to PDE's other than the MKdV and the SG. An integral equation of type I with an extra factor k in front of the

J. van der Linden et al. / Multicomponent integrable systems H

237

integrations was studied in ref. [3], yielding the DL of the derivative nonlinear Schr6dinger equation (DNLS), and of the equations of the massive Thirring model (MTM). Further applications of the DL method which are concerned with B~icklund transformations and the derivation of integrable lattice versions of the associated nonlinear PDE's can be found in refs. [4-6]. In a previous paper to be cited as I [7], we have shown how the DL method can be formulated for multicomponent (i.e. finite vector- and matrix-) generalizations of the PDE's and partial difference equations treated in refs. [1-6]. For a multicomponent generalization of the KdV and the KP (the counterpart of the KdV in 2 + 1 dimensions), see ref. [8]. Integrable multicomponent "N x N" matrix systems in 1 + 1 dimensions have been treated in ref. [9]. In paper I we have introduced the integral equation ~,[A, B] +

f

dA*(/')

C*

*

dA(l) A k t , B r / ( k -

PkPt'

l ' ) ( l ' - !) ~t[A

, B ] = PkCk ,

C

in which A kl' and Bkr 7/, i.e.

are

A,t, = E am,,,kml ' ' , m,n

f

(1.1)

finite linear combinations of products kml 'n, m, n

B k,' = E bm,nkml ' ' •

(1.2)

m,n

Because of the functional dependence o n Akl, and Bk/, , the wave vector in (1.1) is denoted by ~bk[A, B]. It has infinitely many components ¢b~i)[A, B], i ~ 7/, corresponding to the components Ctki) -- k -i of the vector c k in the source term of (1.1). The contour C and the measure dA(k) in (1.1) are arbitrary apart from the condition that the solution of the integral equation must be unique. Tb -h the plane-wave factor Pk in (1.1), dPk[A, B] is assumed to depend on t,,e coordinates x, t of the nonlinear PDE's. Besides ~bk[A, B], we also considered a wave vector #k[A, B] which is defined by grkiA,

BI = f

C*

(1.3)

dA*(l') Bk,, k -p, 1' (br[A' BI.

From (1.1) one can derive another relation between ~b,[A, B] and ~l*,[A, B] containing only one integration over C*, with the measure dA*(l'). Associated to the vectors tbk[A, B] and ~k[A, B] we have the Z x 7/matrix potentials (

• [A, B] = f dA(k) ~bk[A, B i t , , , C

*[A, B] = j dA(k) ~k[A, Blc k , c

(1.4a,b)

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J. van der Linden et al. I Multicomponent integrable systems H

in which the integrands are dyadic expressions. In the special c a s e A k l , Bkr = 1, eq. (1.1) reduces to the integral equation of type I treated in ref. [2]. S t ~ n g from (1,1) we have derived the constitutive relations, i.e. algebraic and differential relations, as well as some symmetry properties, in terms of the wave vectors and the potentials. From these constitutive relations we then obtained generalizations of the NLS and the CMKdV, in terms of finite submatrices of O[A, B]. In order to get these closed multicomponent PDE's, two special cases were considered: either that Akr = Bkr, or that A k r - AI*,k and Bkr = Brk (the "hermitian" case). We also investigated in I the B~icklund transformations and the lattice version associated to these finite-matrix PDE's. In addition to the investigation of (1.1), we derived the constitutive relations for an analogous generalization of the integral equation of type II with functions of the form (1.2). From such an integral equation the finite-matrix versions of the MKdV and the SG can be obtained. In the present paper we report some more results regarding multicomponent PDE's for which the DL is based on (1.1). In section 2 we obtain the DL of mu!ticomponent versions of the IHSC, the CSG and the Getmanov equation, and discuss the associated scattering problems, the MT's, and gauge equivalence. These equations are the finite-matrix generalizations of the PDE's in ref. [2] involving ~btl"°) and &tl.1). The special cases for Akl, and B k r considered in section 2, are again the case that A k r - " B k r , and the hermitian case, but only the latter case appears to be applicable throughout. In section 3 we investigate the case of "minimal departure from hermiticity": A kr = k A k i , , Bkr --- Bkl', or Akt, = Akl,l, Bkr = Bkl, (equivalent to Akr = Akl', Bkl' -" k B k i ' ) , or Akr = "~kr, Bkr =/~kr l', where now "~kt' and/~kt, are considered to be hermitian (the case A kr = Bkr could be considered for some PDE's too, but that will not be done in this paper). In this section we obtain the DL of a variety of multicomponent PDE's, for instance multicomponent versions of the DNLS and the MTM. Most of these equations are the finite-matrix generalizations of the PDE's i~ ref. [3], for which the DL was based on (1.1) with Akl, = k, B~r = 1.

2. Muiticomponent IHSC, CSG and Getmanov equation In this section we shall derive the finite-matrix generalizations of the IHSC and the related equationr, given in section 5 of ref. [2], and of the CSG and the Getmanov equation, cf. section 6 of ref. [2]. We also discuss the associated scattering problems, MT~s, and gauge equivalence. Before we do that, we shall give a summary of results of paper I [7] which will be used in the treatment.

J. van der Linden et al. I Multicomponent integrable systems H

239

2.1. Summary of results of I In the introduction we have already defined ¢}k[A, B], Ok[A, B], O[A, B] and ~ [ A , B]. For any linear combination Akr of products k"l", as given in (1.2), we define Z x 7/matrices An, p E T/, by

I

'~, JY. A- J Tp-z-y

/~-100-

A~ =

p>~O,

,

(2.1)

~: 1JP+I"A'J T - l - i

p~
j=o

A ~ ~ Om,,J". O. jr,-

(=A,),

(2.2)

m,vl

(d)0 ---/$j,,+,,

(J'r)o ~ 8,.j+,,

(O)q =---6/,06j,0 ,

(2.3)

and likewise for Bkr w e can define matrices Bp, with B = 8 1 expressed in terms of bin,.. With these definitions we have derived in I from the integral equation (1.1) algebraic and differential relations for the wave vectors and potentials. The algebraic relations are

kP~bk[A, BI +@[A, B]-Bp- q~k[B, AI = (jxp _ ~*[A, B]. Ap)-@k[A, B ] - F~P)[A, BI,

(2.4a)

kP$k[A, B I - (d "rp - qr[A, B]. A~). Ok[A, B] = ~*[A, B]- Bp • 4,k[B, A] -- G~P)[A, B],

(2.4b)

@[A, B]-(J p + Bp* . . [ B , A]) = (JVP - ~*[A, B]. Ap).~[A, BI -= F¢P)[A, BI,

(2.5a)

*

~P[A, B] .JP - (JXP - aP[A, BI-Ap)" aP[A, B] = ~*[A, B]. [tp . ~ [ B , A ] - 6(P)[A, B].

(2.5b)

Given a dispersion tok = ErA,U, r E Z, i.e. a plane-wave factor P, --- exp(ikx i¢okt), the differential relations are -iOx4ik[A B] = F~')[A, B],

--iOxq~k[A, B] = G~')IA, BI ,

(2.6a,b)

-i0x~[A, BI = F(')[A, BI,

-i0,,W[A, B] = G(')[a, B],

(2.7a,b)

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J. van der Linden et al. I Multicomponent integrable systems H

iO,,bk[A, B] = ~ A,F(k')[A, a ] , r

i0,@[A, B I = ~ A,F(r)[A, B l,

i0,$kIA, B] = ~ A,G~')[A, a ] , ' (2.8a,b) iO,~P[A, BI = ~ A,G(')tA, a ] . r (2.9a,b)

The expressions F~P)[A, B], G(kP)[A,B], F(P)[A, B] and (I(P)[A, B] occurring in the differential relations (2.6)-(2.9) have already been defined in (2.4), (2.5). In accordance to (1.4), we have

F(P)[A, BI = f dA(k) F(kP)[A, B]ck, C

(2.10)

G(P)[A, B] = f dA(k) G~P)[A, a ] c k • c

From (2.1)-(2.7) we derived in I the iterative scheme F(kP+~)[A, B] = -i0~F(kP)[A, BI -G(P)*[A, B]. A.

~bk[A, B]- @[A, B]. B*. G~P)[B,A], (2.11)

6(kP)[A, B] = i0;](~*[A, B]" B. F(kP)[B,A] - F'P)*[A, B]-B. ~bk[B, A]),

(2.12)

and the corresponding integrated equations applying (1.4) and (2.10). Evaluating F(2)[A, B] and F(3)[A, B] according to this iterative scheme, (2.9a) gives in the case that ¢0k = A2k2 + A3k3 i0,O[A, B] = (-A202 + iA3O3)@-tA, B] + (-2A2@[A, B] + 3iA30x~[A, B])

• B*-@*[B, A]. A-~[A, BI + 3iA3@IA, B]. B*- @*[B, A]. A.

Ox@[A,BI.

(2.13)

Moreover (2.11) may be used together with the definition (2.4b) of G~P)[A, B], to evaluate -iaxF~-l)[A, B]. In this way we obtain from (2.9a) in the case that tok = k-

OxOt~YP[A,B] =~[A, B]-*[A, B]-JT-e* "J'(D*[B, A]-A..[A, B] -@[A, B]. B*.@*[B, A]-jT.A.J.@[A, B]. (2.14)

J. van der Linden et al. i Multicomponent integrable systems H

241

Eqs. (2.13) and (2.14) are nonlinear 7/x 7/matrix PDE's, which are linearized by the integral equation (1.1) with tok as mentioned. The scattering problem associated to (2.13) and (2.14) is therefore given by (2.6) with (2.4), --iO~bk[A,

B] =

k r b k [ A , B ] + O[A, B]-a~q~k[B, A],

--iO,,qJk[B, A] = (]D*[B, A]- A. rbk[A, B ] .

For the potentials @[A, B] and ~P[A, relations O*[A, B] = O*[B*, A*],

B] we have derived in I the

~Ir[A, BI = -aP*[A*, B*].

(2.15a) (2.15b) symmetry

(2.16a,b)

Here A* and B* denote factors Atkr and B*kr in the integral equation (1.1) r , instead of A kl, and B k r , with A~l,- A r k and B k*~,- B r k (just like hermitian conjugation). 2.2. Finite matrices

The relations of subsection 2.1 have been formulated in terms of vectors with infinitely many components and 7/x 7/matrices. However, it follows from (2.2) and (2.3) that the coefficients am.,, and bm.,,, occuring in the finite sums (1.2), constitute the set of matrix elements of A and B that do not Jc~entically vanish. To be specific, let us consider the case that N~ No, e

Akl '= E

E

a-r;,,-~k -~'l'-~

a = l t~'=l Nil

(2.17)

N~,

8,,,=E E b

~ S~

13=1 /3'=1

Then it follows from (2.2) and (2.3) that (A)r.r;, = a-r;,,-r. - (a).. (A)o=0

for

and t

i ~ r.,

j ~ r ., ,

(a = 1 , . . . , N," a ' = 1 , . . . , N',,,), (2.18)

(B)s~sh, = b_sk _s, - ( b ) m 3, (B)o=O

for

and ?

i~s~,

j~s~,,

r

( ~ = I, . . . , N~; 3 ' = I, . . . , N ~ , ) .

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J. van tier Linden et al. I Multicomponent integrable systems H

Thus all contractions involving/b, and B reduce to finite summations, i.e. to contractions with the finite matrices a and b introduced in (2.18). As a consequence We can rewrite the relations of subsection 2.1 in terms of vectors ~ t h a finite number of components and finite matrices. To this end we define the vectors 4~(k")(a,b) and Otk")(a,b), and the matrices @(m'")(a,b) and ~(m")(a, b), m, n • 7, by

(,(")(a, b))~,

(~k[A, B]),.;,+., (,(")(b, a)),8,-- (Ok[B, Al)sk+,,

(~)(a, b)),,,, = (~k[A, B]),.;,+., (0(")(b, a)),~, = (Ok[B, Al)sk+,, ('¢'(m'")(a, b)),,,.a = (d~[A,

=

B])rg.+m.s~+n,

(ll~(m'n)(~, b)),~,... = ('~[A, Bl),.=.+m.,. +,,

(~D(m'n)(b, a))la, a = ((]D[B, (.('n'")(b.

A])s~.+m.r,+n,

=

+re.s,,+,.

(a = 1 , . . . ,N,~; a ' = 1 , . . . ,

I

N,~,; f l = 1 , . . . ,N~; f l ' = 1 , . . . ,

l

Na,). (2.19)

The finite matrices d~(a, b) and . ( a , b) of paper I are the ones with m = n = 0 in (2.19), and we will use this shorthand notation also here, • (°' °)(a, b) -- d~(a, b),

~(°'°)(a, b) - . ( a , b).

(2.20)

With (2.18)-(2.20), the eqs. (2.13) and (2.14) become

iOt(I)(m'n)(a, b) = ( - a 2 0 ~ + ia30~)d~("'")(a, b) + {-2a20(a, b) + • b*.

b)}

d)*(b, a)- a-@(°'")(a, b)

+ 3iA3@tm'°)(a, b)- b*" tb*(b, a). a . a~tbt°'")(a, b ) , (2.21)

a~#,@<"'")(a, b) = ¢,(~"°(a, b) -- (I)(m'l)(a, b)" b*" (b ( l'°)*(b, a). a-(]D (°' n)(a, b)

-d)("°)(a, b)-b*. (I) (°'])*(b, a)-a-d) (~'")(a, b), (2.22)

where the dispersions are tok = A2k2 -I- A3k3 and to, = k -1, respectively. From the scattering problem (2.15) we get

J. van der Linden et ai. I Multicomponent integrable systems H

243

-iO~(k')(a, b) = k~(k")(a, b) + (b("'°)(a, b)-b*, qJ(k°)(b, a),

(2.23a)

-iOn# (') (b, a) = ~(.,o). (b, a). a. # (k°) (a, b),

(2.23b)

and the symmetry relations (2.16) are rewritten as

(]D(m")*(a, b) = (]D("'m)+(b +, at),

lIF(m'n)(a,

b) =

--'I~ (n'm)t (a~

bt). (2.24a,b)

Eq. (2.21) for m = n = 0 is the multicomponent Hirota equation [10] already considered in I. It is a dosed PDE in terms of O(a, b) in two special cases for the choice of A k r and B k r . In the first case one has Akr = B k r , with the consequence that a = b and O*(b, a ) = tb*(a, b). In the second case one has A*kr = Akr, B+kr= Bkr, SO that one can apply (2.24a) for r e = n = 0 with a + = a , b + = b , yielding (]D*(b,a) =(I)+(a,b). For A3 = 0 , or A2=0, the multicomponent Hirota equation, (2.21) for m = n = 0 , becomes a multicomponent NLS, or CMKdV, respectively. Other treatments of multicomponent generalizations of the NLS can be found in refs. [11-16]: In the case that tok = k -1, the plane-wave factor p k - - e x p ( i k x - i k - l t ) is invariant under the interchange of x and - t , combined with the replacement of k by k -z. As a consequence one can derive in this case from (1.1)-(1.4) and (2.17)-(2.19) the relations

(l-n)/,.j ~(k")(a, b) = k- 1.4. '~'k-' ~", b) ,

~')(a, b) = -g-'q,~;~')(a, b),

~)(m,')(a, b) = ~('-m,'-')(a, b),

~(m,')(a, b) = -~('-m'~-")(a, b),

"

(2.25)

where the tilde denotes the interchange of x and - t . For the derivation of (2.25), note that ~k[A, B] satisfies the integral equation (1.1) with Pk = Pg-' instead of Pk"

2.3. PDE for ~(l"°)(a, b) in the case of dispersion tok = A2k2 + A3k3 In appendix A it is shown that

(1)*(b, a) • a. (]D(a, b) = (o~(I) (°'')* (b, a)) a. N(a, b). Ox(1)("°)(a, b), •

(2.26a)

• *(b, a)-a. O~(]D(a,b) = (O~(1)(°")*(b, a))-a-N(a, b). {02x(]D("°)(a, b) + (D ( "°)(a, b)- b*. (0 x([:)(°'')* (b, a)). a- N(a, b). 0x~( ~'°)(a, b) },

(2.26b)

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J. van der Linden et al. I Multicomponent integrable systems H

where we have used the abbreviation

N(a, b) = {1 (N;') - (1)("°)(a, b). b* • @(°a)*(b, a). a} -~

(2.27)

which, according to (2.24a), satisfies the symmetry relation (2.28)

a-N(a, b) = N*(a*, b*). a.

Eq. (2.26a) is eq. (A.6) in the special case m = n = 0, and eq. (2.26b) follows from eq. (A.14) in the special case p = 0, q = 1, m = n = 0, taking into account eq. (A.16) for p = 1. Using (2.26), we obtain from (2.21) with m = 1, n = 0 the PDE iig,@(l"°)(a, b) = (-A202 + ia3asx)@("°)(a, b) + {-2A2(1)(x'°)(a, b) + 3iA3#x(I)(1"°)(a, b)}

• b*. (a ~(o.1 ). (b, a))-a. N(a, b)- o/I) (1,o)(a, b) + 3iA3(1)t"°)(a, b ) - b * . (Ox(1)t°'~)*(b, a)). a. N(a, b). {02(1)(~'°)(a, b) + (I)(l"°)(a, b). b*- (igx(1)(°';)* (b, a))- a- N(a, b). (gxq)( "°)(a, b)}. (2.29) In the hermitian case, i.e. with a = a*, b = b * (but not in the case that a = b and not hermitian), eq. (2.29) is a closed PDE for @(l"°)(a, b), because then @(°'~)*(b, a ) = @(~'°)*(a, b). This PDE is the finite-matrix generalization of eq. (5.11) of ref. [2], which in the special case A3=0 has been shown to be equivalent [17] to the IHSC. This equivalence can in general not be established, however, in a multicomponent version in relation to (2.29). Higherorder PDE's of the hierarchy attached to (2.29), involving higher powers of k in ¢ok, can be derived similarly, using eqs. (A.14) and (A.16) of appendix A also for other values of p, q than in (2.26). Associated with the PDE (2.29) for ~(~'°)(a, b) we have from (2.15) two scattering problems. The first one is - i # x ~ k(')(a, b) = k~b b(~)(a, *b) + k(1)<~'°)(a, b) •

" #(k°) (b, a) ,

(2.30a)

-i#~ #(k°) (b, a) = ((3xtI)(°")*(b, a)). a. N(a, b). (9~~ ~kl) (a, b),

(2.30b)

where (2.30b) is obtained from (2.15b) taking into account eq. (A.5) of appendix A. The second one is

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J. van der Linden et al. I Multicomponent integrable systems H

--iOx4)(kO)(a, b)= kq)(k°)(a, b) + (axt]D(°'l)(a, b)). b*-N*(b, a). {o~qJ(1)(b, a) - ikqS(kl)(b, a)}, (2.31a)

-iO x qJ(kl)( b, a) = (]D(1,o) * (b, a)" a" 4) (k°) (a, b),

(2.31b)

where (2.31a) is obtained from (2.15a) taking into account eq. (A.8) of appendix A. Obviously the two scattering problems are equivalent: the substitution

q)(kl)(a, b)'-> - qJk(1)*(a, b)

e ikx ,

"/'(°)*eL" q,(,,°)(b, a)->.,.k ,,,,,,

a)

e ikx

(2.32)

transforms (2.30) into (2.31). In the scalar case the scattering problem (2.30) or (2.31) has been given (as part of the Lax representation of the corresponding PDE in the case of dispersion tok = k -1) by Neveu and Papanicolau [18]. Eqs. (2.26a,b) may be regarded as determining the spatial part of a finite-matrix generalization of the MT mapping a solution of eq. (5.11) of ref. [2] to a solution of Hirota's equation, or in the special case A3 = 0, a solution of the IHSC to a solution of the NLS. To be more specific, let us consider the special case in which the matrices a and b are hermitian, and in which a and • (a, b) are invertible, i.e. the case e

N~, = N'~, = N~) = NI), ,

a = a+,

b = b ÷,

det(a. 4)) ~- 0 ,

(2.33)

in which ~ - - - ~ ( a , b). Then eqs. (2.26a,b) imply a relation of the type

¢)-'. ax(1) - ((]D+. a- (]D)-' .4)+. a. axtl) = H,[(I)'],

(2.34)

in which ~ = ~ ( a , b) and H~[~'] is a matrix that is explicitly given in terms of • ' - ~ ( l ' ° ) ( a , b). From (2.21) with m = n = 0, using also (2.34), one obtains in addition to (2.34) a relation to similar type. ~ - 1 . O,~= (~+- a . ~ ) - ' . ~ + . a . O,~ = H2[~' ] ,

(2.35)

with H2[tI~' ] explicitly given in terms of 4)'. As in H 1, only derivatives of @(l'°)(a, b) with respect to x occur in H E. From (2.34) and (2.35) one can now express @(a, b) as an ordered exponential of an integral in terms of ~(~'°)(a, b) as follows:

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J. van der Linden et al. I Multicomponent integrable systems H

+dl2H2t(~)'])}]'((]~[x=t=o) F

1

Sl

Sn- 1

dsl f ds 2 --. 0

o

f

dxsI,:,j

o

÷ dtI+__+,H2[~'I+=+(+j).,=,(+j)]}]" (~1+=,--o),

(2.36)

where 3"is the path ordering operator, r is an arbitrary path connecting the origin of the x - t plane with the point (x, t), and (d/l, all2) is an infinitesimal vector tangent to F; in the second expression for O, the path F is parametrized by s ranging from 0 to 1, i.e. F consists of the points (x(s), t(s)) with x(0) = t(0)= 0 and x ( 1 ) = x, t(1)= t. Of course, due to the ordered exponential that arises from non-commuting matrices, it is in general not possible to evaluate • more directly in terms of O'. Nevertheless, we shall call (2.34) and (2.35) the multicomponent MT mapping a solution of the PDE (2.29) in the case (2.33) to a solution of the PDE (2.21) for m = n = 0.

2.4. PDE for q~t~'~)(a, b) in the case of dispersion ,ok = A2k2 q- A3k3 In appendix A it is shown that ~(,,,,o),(b, a)- 8- ~(o,~)(a,b) = (o ~d+~'', )* (b, a))- a. N(a, b)- Ox~( ''~ )(a, b ) , lID(m'0)* (b, a). a-

(2.37a)

OxlD(0'l)(a, b)

= (#xd~("'l)*(b, a))-a. {0~(1"1)(a, b) - iN(a,2 b). (Oxd~(~")(a, b)).

0xRT(b *, a*)},

R(a, b) = {1 (N:,') - 4(#xd~(~'i)(a, b)) • b* -o,:d~ (11). ' (b, a) . a} ' /z ,

(2.37b) (2.38)

where N(a, b) is defined by (2.27), and R(a, b) is related to N(a, b) according to

N(a, b) = 2{1N;) + R(a, b)} -~ .

(2.39)

Eq. (2.37a) is eq. (A.6) in the special case n = 1, and eq. (2.37b) follows from eq. (A.14) in the special case p =0, q = 1, n = 1, taking into account eq. (A.26). Eq. (2.38) is the (positive root) solution for R(a, b) from eq. (A.23). With (2.38) and (2.39), the right-hand sides of (2.37a,b) are expressed in terms

J. van der Linden et al. / Multicomponent integrable systems H

247

of (1)(]")(a,b), and we will regard R(a,b) and N(a,b) in this sense as abbreviating notations. We mention also the relations

b*" (O,(]D('")*(b, a))-a-N(a, b). #~(I)("l)(a, b) = ~ {1 (N~) - RT(b+, a+)}, (2.40a)

(I)(]'°)(a, b). b*- o/(I)(°'~)*(b, a). a = - (a2tl)t"1)(a, b)). b*. a~tl)('")*(b, a)

.a

- ! 2 (/)xR(a, b)). {N(a, b)}-'

(2.40b)

Eq. (2.40a) follows from (2.37a) for m - 1, using (2.24a), (2.27) and (2.39), and eq. (2.40b) follows from (2.37b) for m = 1, using (2.39), (2.28), (2.24a), (2.40a) and (2.38). Using (2.37a,b) for m = 0 , (2.39) and (2.40a,b), we obtain from (2.21)with m = n = 1 the PDE

iot(]D(l'Z)(a, b) = iA30~(]D('")(a, b) - A2{(a2tD('")(a, b))-RT(b *, a+) - (#xR(a, b)). #Jl)('")(a, b)} -6i,I 3(02(I)('" )(a, b))- {b*- (0 xtl)( ~'' )* (b, a)). a. 02(I)(' ")(a, b)

+ ~RT(b *, a+) • 0xRT(b *, a+)} + 3iA3(OxR(a, b)) • {(Oxt]Dt"l)(a,

b)) . 0 ~ R T ( b+, a+ )

- R(a, b)-O2(]D(1")(a, b)}.

(2.41)

Note that, calculating the right-hand side of (2.41.) in the way as indicated, it is intended that the expression for (0/(I)(~'°)(a, b)). b* .(1)*(b, a)- a. (lD(°'~)(a, b) is found from the one for (1)(~'°)(a, b ) . b * • (I)*(b, a ) - a . o/(1)(°'~)(a, b) by hermitian conjugation. Both in the hermitian case and in the case that a = b, eq. (2.41) is a closed PDE for (I)(l'~)(a,b). In this way we obtain finite-matrix generalizations of eq. (5.20) of ref. [2]. The equivalence of (2.41) for ~3--'0 to a multicomponent version of the IHSC will be derived in subsection 2.6. Associated to the PDE (2.41) for (1)(~'~)(a,b) we have from (2.15) the scattering problem

--iO~k (' ')(a, b) = kq) k(Z)(a, b) + (ox(]D('")(a, b)). b*. N*(b, a) (2.42a)

• {Ox#(k')(b, a) - ik~tkI)(b, a ) ) ,

-iOx#kt')(b,

a) = (O~(1)(l")*(b, a)).a. N(a, b ) - a ~ ) ( a ,

b),

(2.42b)

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J. van der Linden e~ al. I Muiticomponent integrable systems H

where (2.42a,b) are obtained from (2.15a,b) taking into account eqs. (A.8) and (A,5), ly, 0f appendix A. •For the determination of an MT connecting the solution of (2.41) to those of (2.29), we can use eqs. (2.37a,b) for m = 1. The treatment is analogous to the one in subsection 2.3. Here we must take ~ - ~t°'~)(a, b) and ~ ' - ~(l'~)(a, b) in eqs. (2.33)-(2.36), and determine H~[~'] and H2[~' ] according to (2.37a,b) with (2.41). 2.5. P D E ' s in the case o f dispersion tok = k - :

From the eqs. (2.7a) and (2.9a) with explicit expressions for F(1)[A, B] and F(-:)[A, B] as given by eq. (2.5a), we derive in the case that tok = k - : the relation

• (:")(a, b) • b*. 0/l)('")*(b,

a) = (o,(I)("°)(a, b)) • b* • ~(°,')*(b, a). (2.43)

Inserting successively the hermitian conjugated version of (2.37a) for m = 0 , and the relation (2.43), in (2.22) for m = 1, n = 0, we obtain the PDE

0~0,(I)(~'°)(a, b) = {N(a, b)} -~ • (lD(:'°)(a, b) - (0,(I) ( :'°)(a, b))- b*-(I )(°': )* (b, a). a. N(a, b)- o/I) ( z'°)(a, b),

(2.44)

with N(a, b) given by (2.27). In the special case that a and b are hermitian, eq. (2.44) is a closed PDE for (I)(:'°)(a, b). This PDE is the finite-matrix generalization of the Getmanov equation, eq. (6.9) of ref. [2]. According to (2.22) for m = n = 1, with (2.27) and (2.39), we have the PDE

o~ot~('")(a,

b) = ½{(I)(:':)(a, b) • RT(b *, a*) + R(a, b). (1)(~")(a, b ) } , (2.45)

with R(a, b) given by (2.38). In the two special cases, hermitian a, b, and a = b , eq. (2.45) is a closed PDE for ~(~'~)(a,b). This PDE is then a finite-matrix generalization of eq. (6.10) of ref. [2], which equation has been shown to be equivalent to the CSG, eq. (6.6) of ref. [2] with OxOO ,. Note that, instead of ~( : 'l )(a, b) satisfying (2.45), one can just as well consider • (a, b) satisfying (cf. (2.25))

o~o,(1)(a, b) = ½[~(a, b)- {1 (N~) - 4b*-(o,(1)*(b, a))-a- o,(1)(a, b)} ,/2 + {1 (N;') - 4(0,(lD(a, b)). b* • 9,(1)*(b, a) • a}:/2 • (1)(a, b ) ] .

(2.46)

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J. van der Linden et al. I Multicomponent integrable systems H

Multicomponent generalizations of the Getmanov equation and the CSG have also been obtained by reduction of N-dimensional nonlinear or-models, see

refs. [19-22]. The scattering problems and the spatial parts of an MT which were treated in subsections 2.3 and 2.4 are independent of dispersion. Therefore eqs. (2.26), (2.30), (2.31), (2.37) for m = 1, and (2.42) can be used in the ease of dispersion tok = k -1 too. Moreover, in this ease the dispersion-dependent relations which are complementary to the scattering problems can be inferred from scattering problems through (2.25) and the interchange of x and - t . In this way (2.23) for n = 0 provides the linear relations complementray to (2.42), and eqs. (2.30) and (2.31) the relations complementary to (2.31) and (2.30), respectively. Note also that in the derivation of a relation of type (2.35) for the MT's, where H2[~I)' ] is dispersion dependent, we can use (2.43) in the case that O-=O(l'°)(a,b), and the relation which follows from (2.43) by (2.25) and x <-,-t, in the ease that ~ - O(a, b).

2.6. Spin equations and gauge equivalence Consider matrices gp, p EE7/, constructed from 4 blocks of 7/x 7/matrices as follows:

= (JTV-**[A, B]'Ap -'IDtA;B]-Bp 1 **[B, A].Ap

gP \

(2.47)

jTV _.~[B, A]. B : ] '

cf. eq. (2.1) for the definition of Ap and B v, with in particular =

g'

(jT

\

_

~,[A, B]. A

~*[B,A].A

-(1)[A,BI'B* ) jT_~[B,A].B,

J + xP*[A, B ] . j T . A . J

g-: =\

AI.J,.A.j

(10~ '

g°=l=

• [A,B].jT.B*.J J + qt[B,

).

0

1}' (2.48)

A].jT.B *-J

As a consequence of the recursion relation for Ap, following from its definition (2.1), Ap+q

= Ap. jTq + jp. Aq ,

(2.49)

and the same relation for Bp, one can prove using (2.5) that (2.47) satisfies gp+q

=gp .gq,

(2.50)

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J. van der Linden et al. ! Multicomponent integrable systems H

with in particular g~ "g-1 = g - t "g~ = I. Hence we conclude that

gp = gP

with g-- gl.

(2.51)

Introducing, in addition to (2.47) with (2.51), a vector ~:k and a projection matrix P by

= (,,,[A, al #k[B, A] ) '

(1 P=

0

O)(0 0 =l-

O) 0

1 ,

(2.52)

the relations (2.4), (2.6)-(2.9) yield in terms of gk, g and P k~k = gP" gk,

(2.53)

- - i O ~ k = g " P " ~k ,

(2.54)

- i 0 ~ g = g-[P, g],

(2.55)

iOt~k = X Arg r" P" ~ k ,

(2.56)

ia,g = ~ Arg r" [P, g ] ,

(2.57)

where [P,g] is the commutator P ' g - g ' P = P . g . ( I - P ) - ( I - P ) . g . P . Since (2.5) has been contained in (2.51), the relations (2.53)-(2.57) are an equivalent form of the set of constitutive relations (2.4)-(2.9). In this subsection we use (2.53)-(2.57) for the derivation of spin equations, defining a matrix $ by

S=g.(2P-I)-g-'.

(2.58)

Substitution of g --- g l, I and g - =g_~ from (2.48), and P from (2.52), in (2.58) gives, with the us~ of (2.5b), S=

1-20[A,B].B*.O*tB, 2iOxO*[B, A ] - j T . A . J

A].JT.A.J,

-2iOxOtA,B].jT

B* J )

-(1 - 20 [B, A]-A-O[A, B] (2.59)

Since ( 2 P - l ) 2= I, it follows from (2.58) that S2=l.

(2.60)

J. van der Linden et al. I Multicomponent integrable systems H

251

It also follows from (2.58), using (2.55), that

S = 2ig -1 • (O~g). g -1 + 2 P - I ,

(2.61)

and subsequently, applying again (2.55), that

[g, S] = 2 i ( a ~ g ) . g - ' .

(2.62)

Differentiating (2.58), we get

a~S = [(a~g).g-', S],

(2.63)

o,S = [(o,g). g-', S].

(2.64)

With the help of (2.62) and (2.60), we obtain from (2.63) that

(oxS) • S

(2.65)

-i[g, S].

=

Now, using (2.57), (2.58) and (2.65), we can write (2.64) as

O,S = -½ ~ ~.,[g"-~. (O,,S). S, S].

(2.66)

r

Differentiation of (2.65) with respect to x yields, with the use of (2.62), (2.65) and (2.60),

g. OxS = iOx{(~S ) • S} + (O~S).g - -~[(OxS)- S .g, S]

= i{(02S)-S + 3(0~S)2}.

(2.67)

Then, using (2.67) and again (2.62) and (2.65), we have for p >i I

gP+'.oxS=igP. {(02S)-S+ ~(o~S) 2}

=i{o

(g"-oxs) - ½ Z g"-"-'-(OxS)'S'e "+' .o s .s q=O

+ 3 igP. (OxS)2.

(2.68)

The relation (2.68) enables one to calculate gP. OxS recursively for p >i 2 in terms of S, starting from (2.67). Hence the right-hand side of (2.66) can be expressed in terms of S for all r I> 2. This means that (2.68) is a direct iterative scheme for the hierarchy of PDE's for S. Note that eq. (2.68) is a nonlinear

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J. van der Linden et al. I Multicomponent integrable systems H

recursion relation, which indicates that this iterative scheme is not trivially related to the usUal recursion operator [23]. The result in the case of dispersion -

+

is

O,S= ½iA2[S, O~ZS]+ AzOx{O~S- ~(OxS)-S-O~S } ,

(2.69)

where also (2.60) has been used. According to (2.54) and (2.56) with (2.58), (2.53) and, for r >~2 the identity

gr-1 . S = S . g r - I

r-2 4" E p--O

gr-2-p • [g, S]-g p ,

(2.70)

we have, using also (2.65),

-iOx ~k=

~ k(S + I)- gk,

(2.71a)

r-2

iOt~k=½k~

r~2

A r { k ~ - ' ( S + l ) + i ~'. kPg ~-2-p • p =0

s}.

(2.71b)

By means of (2.67), (2.68) the fight-hand side of (2.71b) too can be expressed in terms of 8, and then (2.71) is a Lax representation for the hierarchy of PDE's for $. The result in the case of dispersion tok = A2k2 + A3k3 is iO,~k = ½kI(A 2 + A3k){k(S + I) + i(O~S)- S} - x

{a

s-

•

(2.72)

In the case of dispersion % = k -1 we have, according to (2.57) and (2.64),

o,S = -i[[g-;, P], S].

(2.73)

Using (2.61), (2.73) can be written as

a,S = ½[O~-'[S, P], S].

(2.74)

Similarly using (2.61), we obtain from (2.56) and (2.53) in this case iO,~:k = ( k - ' P + ½iO~-~[S,P]). ~:k"

(2.75)

Eqs. (2.71a) and (2.75) form a Lax representation of (2.74). In eqs. (2.69) and (2.74), the four 7/x 7/submatrices of S contain the matrix (ID[A, B], and also the matrices J and jT, as specified in (2.59). In relation to

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J. van der Linden et al. I Multicomponent integrable systems H

the finite matrices (1)(m'n)(a, b) and 1][F(m'n)(a, b) defined in subsection 2.2, we consider the finite matrices g(a, b) = g(l'°)(a, b) and S(a, b) = S(l'l)(a, b) which according to (2.48) and (2.59), respectively, are given by

g(a, b) = / 1(~') liar(1,°)* (a ' b).a \ ~(1,°)*(b, a). a [

R(a, b)

S(a, b) = ~2i0xd)t,,1),(b,a).a

-O(~'°)(a, b). b* 1 (Nk) W(l'°)(b, a). b* ) -

-2i(~x(I)(~'l)(a, b).b*)

-R*(b, a)

(2.76) '

"

(2.77)

where in (2.77) we have used that, according to (2.27) and (2.39),

R(a, b) =

1 (N~')

-

2(1)(l'°)(a, b)- b*-(]D (0'1)*(b, a)- a.

(2.78)

With R(a, b) as given in eq. (2.38) S(a, b) is expressed by (2.77) exclusively in terms of ~(l'~)(a, b). The matrix (2.77) has, according to (2.60), the property that (2.79)

{ S ( a , b ) } 2 = I(N~'+Nk ) .

According to (2.69) and (2.77), S(a, b) satisfies the following PDE in the case of dispersion (.ok --" A2k2 W Aak3:

a,S(a, b) = ½iA2[S(a, b), a S(a, b)] + X,a {a S(a, b) -

b)) S(a, b)-~iS(a, b)}.

(2.80)

•

Eq. (2.80) is a multicomponent version of the IHSC with extra contributions for A3 #=0 as given in eq. (5.22) of ref. [2]. In the special case N',,, = N~, = 1, S is a 2 x 2 matrix and can be decomposed into Pauli matrices: S = SI~rx + Syerr + Szer z. Then one recovers from (2.80) the IHSC with higher-order contributions [24], in terms of the spin vector S = (S x, Sy, Sz) satisfying S- S = 1. See refs. [25, 26] for SU(n) type generalizations of the IHSC. According to (2.74) and (2.77), the equation satisfied by S(a, b) in the case of dispersion % = k-1 is t

o,S(a, b) = -~[o;l[S(a, b), diag(1 (s;''), -l(Nb'))], S(a, b)].

(2.81)

Eq. (2.81) is a multicomponent version of an equation which caa be found in ref. [27]. It is clear from (2.77) with (2.38) that the spin equations (2.80) and (2.81) are equivalent to the PDE's (2.41) and (2.45), respectively, for O(t'l)(a, b).

J. van der Linden et al. I Multicomponent integrable systems H

254

Introducing the finite-dimensional vectors X(k~)(n,b) derived from ~k given in (2.52) by

X(k")(a, b) = ~ [")(a, b) e - ( i / 2 ) ( k x - ' ' k t )

,

~")(a, b) = \O~-)(b, a ) , (2.82)

and using (2.76) and (2.77), the eqs. (2.53) and (2.71a) yield kx(,~)(a, b) = g(a, b) . x~(a, b) ,

x,(a, b) - X(k°)(a, b),

-ia~x(kl)(a, b) = ½kS(a, b). X(k~)(a, b).

(2.83) (2.84)

Eq. (2.84) is the scattering problem associated to the multicomponent spin equations (2.80) and (2.81). For the complete Lax representations the equations (2.72) and (2.75), respectively, have to be used to provide the corresponding expressions for iatx(1)(a, b). Note that eq. (2.84) also follows from (2.42), using (2.40), (2.39) and (A.21) of appendix A. The vector Xk(a, b), occuring in (2.83), has been shown in I to satisfy the Lax representation of the multicomponent Hirota equation, i.e. (2.21) for m = n =0, in the case of dispersion % = ~[2k2 -[- ,~3k3. Hence eq. (2.83) is the multicomponent version of the gauge equivalence between (the hierarchies of) the IHSC and the NLS. For other treatments of generalized gauge equivalences, see refs. [28-30].

3. Multicomponent DNLS, MTM and other nonlinear PDE's

In this section we shall derive the multicomponent generalizations of the DNLS and the MTM, cf. ref. [3], and some other related equations. Before we do that, we shall consider what follows from the results of I, when the case of minimal departure from hermiticity is applied to them. 3.1. Minimal departure from hermiticity We refer to the case of minimal departure from hermiticity when the factors A k r and Bkr occurring in the integral equation (1.1) are given by

A k r = k~l~krl, ~2 ,

Bk r = k,'3~krl, ~4,

(3.1)

where (i)

akl'

and/~kr are finite sums as in (1.2), which may be expressed in the

255

J. van tier Linden et al. I Multicomponent integrable systems H

form (2.17), and thus lead to the definition of finite matrices, as in (2.18) with (2.2); moreover we assume that A r k = "4kl' and B r k /]kl', resulting in hermitian finite matrices; (ii) one of the integers vv, 3, = 1 , . . . , 4, is equal to one, and the remaining three are equal to zero. Due to the assumption of hermitian "4kr and/~kr, we get in the following the finite-matrix PDE's as closed equations. It may be checked that in some, but not all of the eases, the alternative assumption that ~'kl' = /~kt' leads also to closed equations (then use eq. (3.9a) below, instead of eq. (2.16a)). According to (3.1), the coefficients of A k i , , B k r , and A k r , Bkr in the expressions of the type (1.2) differ only by a shift in their indices: am,n

= am-vl,n-v2

(3.2)

bm.n = bm-,,3,,,-.4 "

'

For the matrices A, B and A, B, we have, due to (2.2) and (3.2), A = J~2"A'J T~' ,

B = J~4"B'J x~3 .

(3.3)

So in the relations of subsection 2.1 A's and Ws can be changed into A's and B's, at the price of the insertion of some extra J's or ,IT's. Note, however, that there is yet another change introduced by (3.1) with (i) and (ii). The coupling between [A, B] and [B, A] dependence which is present in the constitutive relations leads here to a coupling between types of minimal departure from hermiticity with differe~at values of 3, = 1 , . . . , 4. In order to derive closed PDE's, this coupling must be removed, and for this purpose use will be made of some general symmetry relations which have been derived in subsection 2.3 of paper I. We will review them here for the special case of minimal departure from hermiticity. The factors A kt' and B kr in (1.1) can be expressed symbolically, according to (3.1), as A = F . P , , B=/~; A=fi,.F, B=/~; A=A, B=F.B; A=A, B = / ~ . F, in the 4 cases 3, = 1, 2, 3, 4, and accordingly we have 4 types of wave functions and potentials: 3'= 1:

~kk [ F " A , /~ l ,

qJk [ F " A , B I ,

dOfF',71, B I ,

qt[ F . ,4 , B I ,

y=2:

~k[A.F,/~],

0k[,4"F,/~],

~[A-F,/~],

~[A-F,/~],

3,=3 •

~k[,4, F . / ~ ] ,

qJk[A,F'/~I,

O[A,F'/~I,

~[,~i,F'/~],

3,=4:

tbk[A,B'f],

qJk[,~,/~'F],

O[,,i,/~'F],

~[,4,/~'F]. (3.4)

The wave functions and potentials of type 1 and 2 on the one hand, and of type

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J. van tier Linden et al. I Multicomponent integrable systems H

3 and 4 on the other hand, can be combined 'y= 1,2:

3,=3,4:

to

~#k[F. A . G, BI ,

¢,AF. A. G, B] ,

¢,IF. A . G, ~] ,

~IF.A.G,B],

4,,[A, F. ~-G],

¢,AA, e. ~. GI ,

¢,[A, F. B" GI ,

~[AiF.~.G],

withe,,= k',

G k -- k l - V ,

(3.5)

Vm-~{I-- (--1)~}.

Then, from eqs. (2.29a,b), (2.33a,b), (2.36a,b) of I, which have been derived directly from the integral equation, we have the symmetry relations

jT,.

~k[F" ,~. G, B] = k'tbk[A, G . B .

jT". ~k [ F "fit.

G,/~lk~-'=

F],

~k [ A , G . B . F ] ,

(3.6a) (3.6b)

ckk[F- A. G,/~] = {1 + (-1)"~*[F- A- G,/~]. A}- *kk[G • A. F, B], (3.7a)

#,[F. A.G, ~1

=

{1 + (-1)"aP[F. A - G , B] ..g.*}. qtk[G'/l" F,/~], (3.7b)

~kk[/l,F./~. G]- (-1)"O[A, F-/~. G]-~*. ~k[G./~- F, AI =~AA, G.~,FI,

(3.8a)

@k[~i, F" B- G] + (-1)"@*[A, F" B- G]-B" ~kk[G"/~"F, A I

= +,[A, G'~-FI.

(3.8b)

Besides these relations, we have their integrated versions. For (3.6a,b) they are given by JT"" @[F" A - G , B] =@[A, G" B" F ] - j ~ ,

(3.9a)

jx,. ~ [ F . A. G,/~]. jt+-~ = ~ [ ~ , G./~. F],

(3.9b)

and for (3.7a,b) and (3.8a,b) they involve just a change of wave functions into potentials. In the following we will use an abbreviating notation for the potentials @IF-/]- G,/~] and q~[ F. ~]. G,/~ ] in (3.5)"

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J. van der Linden et al. / Multicomponent integrable systems H m

@[F. A,/]1---@,,

u

~[F'A,B]=~,

,

with F k = k .

(3.10)

~[A.F,~I=-~,,

~[.4"F,/]I--~,,

From eqs. (2.13) and (2.14) we have, with (2.16a) and (3.3), two coupled equations for ~1 and ~2: i 0 t ~ , = (-A202 + iA303) ~, + ( - 2 A 2 @ , + 3iA30.@,) • B * - @ t 2 - A . J T

.@,

+ 3iA3~" ~* "@t2" A" JT" Ox@~,

(3.11)

i0,@2 = (-A202 + iA30~)@2 + (-2A2@ 2 + 3iA30~@2). B * - ~ . J . A . ~ 2 + 3iA3(I)2 • B*-(I)] .J-~, • ax@2,

(3.12)

in the case of dispersion tok = A2k2 + A3k3, and OxOtOl =~i~ 1 _ti~ 1 . j T . ~ , • J • (]Dr2 • ~, • J T • (]D] - - l ~ I * B * * ( I ) t 2 ° J T • A ° ( ] D 1

Ox 0 t(1)2 = (I) 2 -- (I) 2

•

jT. B, . j

,

(3.~3)

. ( I ) ] - J - A • ill)2 - (1)2 • d* • (I)] • A - J . tI)2,

(3.14) in the case of dispersion tok = k- ]. Eqs. (3.11)-(3.14) follow directly considering y = 1, 2 in (3.5), and the same information is obtained considering 3' = 3, 4 in (3.5) and using (3.9a). Similarly the scattering problem (2.15) yields -iO.4,k[F" Li,/]] = ktlik[F. A , B] + ~ , . B * . qtk[B, F . A ] ,

(3.15a)

--iOx~k[/~, F. A] = @~- ~,. tiT. 4~k[F" A,/]1,

(3.15b)

--i0,¢ik[F" A,/]l = k{~k[F,

fi~,B] + ~ .B*.

Ok[B. F,

AI},

(3.16a)

--iO,,qPk[F" B, A I = ~*[F"/L AI ",~" 4,,,[,,i • F,/~l,

(3.16b)

-i0x~k[A • F,/~1 = kCk[,~ • F,/~l + ~2" ~*" ~bk[B-,~" F],

(3.17a)

-i0x~k[/~, ,4. F] = ~

(3.17b)

. d . A . #k[fi~" F,/31,

where in (3.16) also (3.6) has been applied. In the next subsection we shall present decoupled versions of (3.11) and (3.12), and of (3.13) and (3.14), for dispersions % = A2k2 + A3k3 and o k = k -1, respectively.

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1. van der Linden et al. I Multicomponent integrable systems H

3.2. Decoupled P D E ' s in the infinite-matrix structure 3.2.1• Infinite-matrix P D E f o r • 1 in the case tok

=

A2k2 + A3k3

In appendix B it is shown that •

-I"

--

0,2. ~ . j x . O, = - 1 0 i • A-

O,,O,,

O2*-,~i-JT" i9,O, = - O I ' A ' ( i 9 2 0 , -

(3.18a) iO, • ~ * ' O I "A- 0xO,).

(3.18b)

Eqs. (3.18a,b) are the integrated versions of eqs. (B.8) and (B.15), respectively. Inserting (3.18) in (3.11), we obtain the PDE

i0,0, = (-A2 ~2 + iA303)0, + (2iA20, + 3A3OxO,)" B*-0~. A •0,0, + 3A30 , • B * . 0 ~ . A . (0~0, - i 0 , . B*. 0 ~ . A .

0,0,).

(3.19)

The associated scattering problem is, according to (3.15) with (B.8), -iOx~k[F. ,4,/~] = k tkk [F . A, B] + 0 , . B * . qJk[B, F . 2]], -iOxq~k[/~, F - , 4 ] = - i O ~ - A .

O,~k[F. ,4,

B].

(3.20a) (3.20b)

3.2.2. Infinite-matrix P D E for • 1 • j r in the case tok -" A2k2 -t- A3k3

In appendix B it is shown that O , - B * . "a~*~= (0xO,) • JT- 13" • J . 0xO~, (0.O,)" B*. O~ = {020, - i ( O . ~ ) .

(3.21a)

JT. S * . J .

• JT-I$*. J . a / b ~ ,

(3.21b)

Eqs. (3.21a,b) are eqs. (B.10) and (B.16), respectively, with ,4 and /~ interchanged. Inserting (3.21) in (3.19), we obtain the PDE -

3

iOtO1 = (-A202 + IA3O.)O , + 2iA2(OxO~).d r + 3,>,3{(0~O,) • j+ • [1" • d • (0xO]) • A • 0xO1 + a*. j . A. _ 2i(OxO~) "jT. B , . d. (0xO~) "A. (0xO1) •

a*. J.

0xo,},

(3.22)

which reduces to a closed equation for O 1 .jr. The associated scattering

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J. van der Linden et al. i Multicomponent integrable systems H

problem is, according to (3.16) with (B.12) and (B.9), --iOx~k[ F . fi,, B] = k { ~k[ F . A , / ~ 1 - i(0~Ot).,iv, d * - , k [ F - / ~ , ,41},

(3.230 -ia~#,[F-/~, ~i] = i,i. (0~O*~). A-4~,[F" ,4,/~],

(3.23b)

where in (3.23b) also (3.9a) has been used. 3.2.3. Infinite-matrix P D E f o r • 2 in the case o k = A2k2 + Aak3

In appendix B it is shown that

jT. 0 1

=

_ i ~ 0 2 + Oz "O,. 02*" A" 02 .

(3.24)

Eq. (3.24) is the integrated version of eq. (B.17). Inserting in (3.12) the hermitian conjugated version of the relation (3.24), we obtain the PDE i0,O 2 = ( - A202 + iA3O3)O2 + (-2AEO z + 3iA3O~Oz)- B*-(i0~O2' + O2*. A. 0 2 • d * . O*) • A. O 2 + 3iAzOz.B*.(iO,,O*z + O*z. A. Oz • B* -O])- A- 0xO2.

(3.25)

The associated scattering problem is, according to (3.17) with (3.24), -i~gx~k[A. F,/3] = k~k[fi~ • F,/~] + • z • 13"- tkk[/~, fi," F],

(3.26a)

-i0,,q,k[/~, A" El = (iOxO*Z + 0*2" A ' O 2 - B * "02*)" A. 4,k[A" F,/~l. (3.26b) 3.2.4. Infinite-matrix P D E f o r • 1 .,iv in the case % = k -~

Applying (2.16a) and (3.9a) to (3.18a), we get J . 0 2 • B*. O] = iO 1 • j T . d*" J" 0xO ] .

(3.27)

Inserting in (3.13) the relation (3.18a), and the hermitian conjugated version of the relation (3.27), we obtain the PDE

0x0tO t = O t + iOt-,iT" B* "`i'O~',~," ~xO1 + i(0xO, ) .jT. B, . j .O*~. A . O , ,

(3.28)

which reduces to a closed equation for O 1 • ,l x. The associated scattering problem is given by (3.23).

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J. van der Linden et al. / Multicomponent integrable systems H

3.2.5. Infinite-matrix P D E for

~2 in

the case tok = k - I

In appendix B it is shown that

4~2,.jr. A.4J, = -i(0,#'2)" A'~2.

(3.29)

Eq. (3.29) follows from the integrated version of eq. (B.19), inserting the relation F(-~)[fi, • F,/~] =iO,~ 2 which is (2.9a) in the case of dispersion tok = k -~. Applying (2.16a) and (3.9a) to (3.29), we get

(I)2. JT. 0*. J-~]- J = i(~,~2). B * - ~ 2t .

(3.30)

Inserting in (3.14) the hermitian conjugated version of the relation (3.29), and the relation (3.30), we obtain the PDE (3.31)

c9x~,~2 = ~ 2 - i(#,~2) "B*" ~2t" A. ~2 - i~2" d*. ~ ] . A. ~,~2,

which is the same as the PDE (3.28) for @: .d x, apart from the replacement x ~ - t . The associated scattering problem is given by (3.26). 3.2.6. Coupled infinite-matrix P D E ' s for tb~ and J ' c I ~ 2 in the case ook

--k-:

In appendix B it is shown that

(ID]. A. (1)2 -- I~I~I"i• A

(3.32)

• l~I~l ,

Applying (2.16a) and (3.9a) to (3.32), we get j . 4)2. e,, .(i)2t . j T = '(I),. j T . B . " J • (I)].

(3.33)

Using (3.27), (3.32) and (3.33), one derives from (3.24) and (3.28) the pair of coupled PDE's icg,~, = J . ~2 - J" ~2" B* "~]" j T . A - ~ , .

(3.34a)

''~V' 1"*'* " A " ~ 1 .

(3.34b)

- - i J " O r ~ 2 = ~ ! -- d ° ~'~'2 -" B *

The derivation of (3.34b) from (3.24) is obvious. For the derivation of (3.34a), use (3.34b) with (3.27) inserted to eliminate the term ~ : on the right-hand side of (3.28), then integrate both sides over x and use (3.33). Eq. (3.34a) can be derived alternatively, without integration, differentiating in appendix B both sides of eq. (B.4b) with respect to t in the case of dispersion tok = k -~, and proceeding analogously to the derivation of eq. (B.17). Differentiating both

J. van der Linden et al. / Multicomponent integrable systems H

261

sides of eq. (3.24) with respect to t, and using (3.34a) and (3.29), eq. (3.31) follows again.

3.3. Finite matrices Finite matrices are introduced just as in (2.17)-(2.19), but with "4~t,, Bkt, as starting point, rather than A kt,, Bkt,. When r,, (a = 1 , . . . , N,,), r'~, (t~'= I r F 1 , . . . , N',,) and s8 (/3 = 1 , . . . , N 8), s8, (/3 = 1 , . . . , Ns,) are the indices specifying the r,ets of those matrix elements of ,3, and/~, respectively, which do not vanish identically, then matrices a and b are constructed according to

(A)ro ,,:,, ------(a),,~,,

(B),,,,,~,-- (b)~,~,,, P

(,~ = 1 , . . . , N,,;,~' = 1 , . . . , N;,; p = 1 , . . . , N,~; p' = 1 , . . . , N ~ , ) .

(3.35) From the infinite-matrix PDE's of subsection 3.2 we immediately obtain the finite-matrix PDE's in terms of (- ]~D ( m ' n ) ( a ,

(a'=

b))a.

8 ~

((]D3,)r~.+m.sl3+n

3,=1,2

...,N',~.;/3=l,...,Ns).

(3.36)

As before, we shall omit m and n when both are zero. For the meaning of 3', see (3.10). 3.3.1. P D E for (1)l(a, b) in the case tok = A2k2 + A3k3 From (3.19) we have with (3.35), (3.36) the PDE

ia,~l(a, b) = (-A2a ~ + iA3d3),tI),(a, b) + 2iA2,t]D~(a, b).b*- ,l)*~(a, b). a-axOl(a, b) +

3A3 {(0x(li)l (a, b)) • b*- ":P,-*(a, b) • a • axe, (a, b)

+ ,P,(a, b)-b*, o*,(a, b). a-ax*, (a, b)} -

3iA3{(]D, (a, b ) . b * . ,tl)*~(a, b)-a} 2 • ax,l),(a, b).

(3.37)

Eq. (3.37) is a multicomponent generalization of a PDE treated in refs. [31-33], cf. also eq. (8) of ref. [3]. The associated scattering problem follows ? from (3.20), and can be formulated in terms of an (N',, + Ns,)-dimensional

J. van der Linden et al. I Multicomponent integrable systems II

262

vector XJa, b) in the following way:

-ia,x,(a, b) $kl t&o)

k”2?pl(a,b) b* l

=

l

i k”*@:(a, b) a

x&9

w 9

-jkl(N~J+@:(a,b)~a4,(a,b)~b*

l

(3.38)

*

3.3.2. PDE for @yvl)(a,b) in the case ok = A,k* + A,k3 From (3.22) we have with (3.35), (3.36) the PDE

iat@~9’)(a,b) = (-h,az

+ ih,a~)@~“)(a, b)

+ 2iA2(d,9(,09’)(a, b)) b* . (ax@y-‘)‘(a, b)) . a 9a,@y9’)(a, b) l

+ 3h,{(az@(,O.‘)(a, b)) . b* (a$D~*‘)‘(a,

b)) a a,#oyl)(a,

l

l

+ (a,@yvl)(a, b)) b* (ax@y9*)‘(a, b)) a a#09’)(a, l

l

l

b)

l

l

b)}

- 6iA,{(d,@~~‘)(a, b)) mb* ax@:091J’(a,b) ma}* a,!D\“*‘)(a, b) . l

l

(3.39)

Eq. (3.39) is a multicomponent generalization of eq. (7) of ref. [3], which PDE for A, = 0 is the potential form of the DNLS. The DNLS has been treated before in refs. [34-381, and multicomponent generalizations of the DNLS can be found in refs. [39] and [15]. The scattering problem associated to eq. (3.39) follows from (3.23), and can be formulated similarly to (3.38) as

-ia,xJa, b) $kl W&4

-ik”*a,

(oyl)(a,b) b* l

= l

- $ kl (q3.J

ik1’2ax@(o*1)t(a, b) a l

xl@9

b) 9

(3.40) (&[F

l

A9

m,# e-(i/2)(kx-wkr)

Xk@,

w

=

i

k”*(*JF

l

B,

A

9

I),+1

cf. ref. [34] for the scalar case. The PDE’s (3.37) and (3.39) are equivalent in the following sense: there exist multicomponent M’T’s (of tb.e type discussed in section 2.3) both ways

J. van der Linden et al. I Multicomponent integrable systems H

263

between them. To see this, let us consider the special case that N, = N',, = N a = N a, and that a , ( I ) l ( b , a ) and 0x(lD~°'l)(b,a) are invertible. Then eqs. (3.21a,b) lead to relations of the type (2.34) with t

(I) = (]DT(b, a),

~ , = 0 xtl)~O,l )T(b, a ) ,

(3.41a) H1 [(]D' ] = (]D'-1. ~gxt]D, _ i b * .(I)' + . a . rID',

and inversely (I) = o~(1)~°'l)r(b, a ) ,

~ ' = (IDa(b, a ) ,

(3.41b) H~[(1)'] = ~ ' -1 • 0 x ~ ' + ib* -(D' + . a . ( D ' ,

From these relations and the PDE's (3.37) and (3.39), one can also infer the complementary relations of the type (2.35). Hence, with (2.36), (1)it(b, a) can be expressed as path-ordered exponential of an integral in terms of 0xtI~]°'l)T(b, a), and inversely Oxtl~°'l)T(b, a) can be expressed in this way in terms of @iT(b, a). Extending the treatment of the multicomponent DNLS as given above in the case of minimal departure from hermiticity A k~, = kA kr, Bkt' = Bkl', to the case where A kt, and Bkt, are given by

Remark.

A kl'

-

"

Bkl, = ~ k r

Fk'4 k l ' '

with F k = fo + f~k ,

fo and f~ real,

(3.42)

one can show that the matrix ~(a, b) defined by ((1)(a, b)~,~ = ((1)[F- A, B])r;,,s~,

(3.43)

P

(1)[f- A,/~] = J dA(k) (f0 --f~Ox)tllk[F" "4, /~]Ck(f0 + f~k) -~ , c satisfies a multicomponent version of the so-called "generalized NLS" [40, 31], hamely the PDE i0,(1)(a, b) = - 0 ~ ( a ,

b)

-2(fo-iflOx)~(a,b).b*.(I)+(a,b).a.~(a,b)

,

(3.44)

when tog-- k 2. Then in the special cases f0 ~ 0, f~ = 0 and f0 = 0, f; % 0, the multicomponent NLS and DNLS, respectively, are recovered.

264

J. van tier Linden et al. / Multicomponeni integrable systems H

3.3.3. P D E f or ~2(a, b) in the case tok = A2k2 d- A3k3

From (3.25) we have with (3.35), (3.36) the PDE

i0,@2(a, b)

=

(-~2 a2 + i~3O3)tI~2(a,b) + {-2~2¢I~2(a , b) + 3i~3axcIi2(a, b)}-b*-

{iox@t2(a,b)

+ (]D2t(a, b). a. (]D2(a, b)-b*-(]D2t (a, b)}. a. (1)2(a, b) + 3iA3q~2(a, b). b*- {iox~2*(a, b)

+ q)t2(a, b). a. (]D2(a, b)-b*-(I)t2(a, b)} • a. Oxt]D2(a, b ) . (3.45) Eq. (3.45) is a multicomponent generalization of a PDE which in the special case A3 = 0 has been given in refs. [41] and [42]. The associated scattering problem follows from (3.26), and can be formulated similarly to (3.38) and (3.40) as --iqg~Xk(a, b) =

!kl(N;.)

(1)(a, b) • b* /

2

•

t

{,0~d)2(a, b) + d)~(a, b). a. d)2(a , b ) - b * . (1)2t(a, b)}

-a

- ½kl (st') ]

• xk(a, b),

(3.46) [ (4Jk[fi~" F,/~]),;. ] e_(i/2)(,x_,,,k,)

x+(a' b)= ~(*ktB, ,~-Fl)s~,}

It can be established that the PDE's (3.37) and (3.45) are equivalent (in the same sense as (3.37) and (3.39)). We shall not give the details here. 3.3.4. P D E for =1m(°'1)(a, b) or d)2(a , b) in the case tok

=k-1

From (3.28) we have with (3.35), (3.36) the PDE (3~a,(I)~ °' ~)(a, b) = (]D~°")(a, b)

+ i(I)~°" )(a, b)-b*-(]D~ °''>t(a, b). a-(3~(~°" )(a, b) • + ffo~d)<~°")(a, b)) • b* • (bc~°'~)t(a, b) • a • ...(o,1)(=, ...~ ~_, b) ,

(3.47) and from (3.31) we have the same PDE (with interchange of x and - t ) for (IDE(a, b) instead of (1)~°'~)(a, b). Eq. (3.47) is a multicomponent generalization of eq. (10) of ref. [3] (see also ref. [37]). The associated scattering problem is given by eq. (3.40), or by eq. (3.46) after the replacement x <-->-t.

J. van tier Linden et al. / Multicomponent integrable systems H

265

3.3.5. Coupled P D E ' s f o r (pl(a, b) and (p(21'°)(a, b) in the case tok = k -1

From (3.34) we have with (3.35), (3.36) the set of coupled PDE's io,(]D,(a,

b) =

(p(l'°)(|l,

b)-

{ 1 (N~) - b * .

(]D(21'°)l(a, b ) - a

• (pl(a,

b)},

(3.48)

-iOx(]D(2"°)(a, b) = {1 (N;'') -

(p(21'°)(a,

b ) . b * . (pl(a, b).a} • (pl(a, b).

Eq. (3.48) is a multicomponent generalization of the equations of the MTM, as given in eq. (11) of ref. [3] (see also refs. [43-45]; a multicomponent generalization of the "massless" Thirring model has been given in ref. [46]). The associated scattering problem is given by eq. (3.38), cf. ref. [44] for the scalar case. It can be established that the PDE (3.47) and the set of PDE's (3.48) are equivalent. To prove this, the corresponding proof in ref. [3] is generalized to the multicomponent case, which implies the use of the path-ordered exponena). tial of an integral in terms of either (piT(b, a) and (pET(b,a), or m(°,')'r(b, -~-1 With this we have completed a treatment of the DL of a variety of multicomponent PDE's, all on the basis of one and the same linear integral equation (1.1). Acknowledgement

We thank Dr. G.R.W. Quispel for his interest. Appendix A

In this appendix we give the derivation of eqs. (2.26), (2.37), (2.38)-(2.40). Also eqs. (2.30b), (2.31a) and (2.42) are found with the use of relations derived in this appendix. From the algebraic relation (2.5b) for p - 1 it follows that

{I (N'') .

. . .

,

b) • a*} • {I

"

"

u ) " a- * ' /

= 1 (N~') -~(~'°)*(a, b ) - b * . (p(°")(b, a),

(A.1)

or equivalently that a . N(a, b) = {1 (s,,) + a . w(o,l)*(a, b)} -1 • a - { 1 (N;,,) _ ~ ( 1 , o ) . ( a ' b ) . a } - ' ,

(A.2) where the matrix

N(a, a) has been defined in (2.27).

J. van tier Linden et al. I Multicomponent integrable systems H

266

From the differential relations (2.6a) and (2.7a) we have, with (2.4a) and (2.5a), respectively, -iox(k°)(a, b) = {1 (N:'') - ~(~'°)*(a, b). a}. ~k(k°)(a,b),

(A.3)

-iOxdp(m'a)(b, a )

(A.4)

=

(I)(S'°)(b, a). {1 (N') + a*- ~(°';)(a, b)}.

Using (A.3) and (A.4) with (A.2), we obtain the relation

a). a- (k ~O)(a, b)

~(m,O).(b,

(A.5)

= (O:dP(m")*(b, a)). a. N(a, b)-0x(k(k~)(a, b), and in integrated version

a)- a- (1)(°'")(a, b)

~(m,O).(b,

(Ox(~)(rn'l ) * (b, a))- a- N(a, b)- (gx(I)("")(a, b).

=

(A.6)

Eq. (A.5) is used in eqs. (2.30b) and (2.42b), and eq. (A.6) yield eqs. (2.26a) and (2.37a). Analogously to (A.3), we have from the differential relation (2.6b), with (2.4b), •

(l

-10x# k )(b, a)

- kl~t(1)(b, a)

= - {1 (Nh') --

~(l'°)(b, a). b* } • O(k°)(b, a),

(A.7)

so that, with (A.4) and (A.2),

(I)(m'°)(l, b)-b*. ~(k°)(b, a) = (0x(I)(m")(a, b))-b*" N*(b, a)- {0xO~')(b, a) - ik#(k')(b, a)}. Eq. (A.8) is used in eqs. (2.31a) and (2.42a). From the d i f f e r e n t i n l r o l n t i n n (9 it fallax~,c ~ , s , ~"/h) e,v] .

.

.

.

.

.

.

.

.

.

.

v .

~ 4 , ~ . m, v

s, t

J~

A . V A l V

VV *~7,

u, lth (~ 9~.r. d~h~ ~'] IV

11,11

9

(A.8)

that i, 11

- i0x~(L°)(a, b) = (lD(~'°)*(a, b)- b. (b(b, a).

l l ~ . W.,

(A.9)

Differentiating both sides of (A.3) p times with respect to x and using (A.9), one finds that the result can be written in the form _i@~+l(k(kl)(a,b ) = {l(S;,) - q , (, •o). (a, _ b ) . a } - o~.,.k .p,~(o).~a, b ) ,

(A.10)

J. van der Linden et al. / Multicomponent integrable systems H

267

where the action of the operator ~x is defined recursively by ~P-~(~)(a •P+l-•(•)(a,b)=a x

~"k

x

x ~P'k

~

b)

+ @(l'°)(B, b)" b*" (I)*(b, a)with ~x~k(kl)(a, b)--- ax~k(~)(a, b),

a . vx,-~p- 1'9"k '~ (0) ~q4,/'* b )

p = 1,2,3, . . . .

(A.11)

Together with (A.11) we have its conjugated integrated version, cf. (2.24), i~x + P + l(]D("'l)*(b, a) -- {-i~xP+l(]D(l'm)(a *, b+)} + = (ap~(m'°)*(b, a)){1 (N~) + a . q,(°,')*(a, b)}.

(A.12) From (A.2), (A.11) and (A.12) one obtains the generalizations of (A.5), (A.6), p (re,o)*(b, a))- a • aq'~(°)t~ (ax(l) ,,x-,-k ,--, b) = (~ ~ p +14)(m,1)* (b, a))- a. N(a, b)- ~ q +1 d~(~1)(a, b)

(A.13)

and ~q (o,n)(a,b) (a~(m,°)*(b,a)).a.ox~

--(~+ P+l{lD(m'l)*(b a))-a-N(a,b)- ~q+l~(l'")(a, b) X

~

(A.14)

Application of (A.13) in the right-hand side of (A.11) gives ~xp+l ~(1)(a, b) = Ox~xP~(1)(a, b) +

(1)(l'°)(a, b) • b* • (axq)(°'l)*(b,

a)) • a • N ( a , b) " ~Pxg~k(1)( a , b ) ,

(A.15)

and consequently b) =

b)

+ q~(l'°)(a, b) . b* . (axq~(°'g)*(b, . . a)) a N(a, b). @xq J p (1,O)(a ' b).

(A.16) From eq. (A.16), ~P+l@t1'°)(a,b) can be evaluated in terms of @(l'°)(a,b) and @(°'l)*(b, a) by iteration for p = 1,2,3, . . . . Eq. (A.14) for p =0, q = 1 and eq. (A.16) for p = 1 yield eq. (2.26b). Application of (A.14) in the right-hand side of (A.15) gives

268

J. van tier Linden et al. I Multicomponent integrable systems H

~+1~(")(a, b) = #x °~p'~'(1):~'-x-~k , . , b) + (0xtI)(Z'~)(a, b ) ) - b * . N*(b, a). (9 2d)~1'l)*(b, a)) • a- N(a, b). *Z" o~xp.c.(~)(a, b) ~U" k

(A. 17)

and consequeL~tly

~xp+l (]D(1,1) (a, b) = ax~xPt]D(l'l)(a, b) + (#x(]D(l'l)(a, [3)) • b* • N*(b, a) . (~xti)2 (1,~). (b, a)) • a. N(a, b). ~:d)tl'])(a, b),

(A.18)

In order to be able to evaluate ~ + l ~ ( a ' l ) ( a , b) in terms of ~(1'1)(a, b) and • (1"1)*(b, a) from eq. (A.18) by iteration for p = 2, 3, 4 , . . . , it is necessary to have expressions for N(a,b) and ~2d~(l'l)(a,b) in these terms. We will proceed to derive such expressions now. From the algebraic relation (2.5a) for p = - 1 it follows that

(]D(°'l)(a, b). {1 (N#) - b*- ~It(l'°)(b, a)} = {1 (N;') + ~(°'l)*(a, b).a} • (]D(l'°)(a, b),

(A.19)

Applying successively the integrated version of (A.3), (A.19), and (A.4), one proves that

(]D(m'°)(a, b). b*. (D(°'l)*(b, a)-a-0xt]D(l'n)(a, b) = ((~x@('~")(a, b))- b*. (1~( 1,o), (b, a). a. (]D(°'")(a, b),

(A.20)

so that we find for (2.27) the property

(OxtI)(l'~)(a, b))-NT(b *, a*) = N(a, b)-axtl)("~)(a, b).

(A.21)

Now, according to (A.6) for m = n = 1, (2.27), (A.21) and (2.28), we have

1 (Nh') - {N*(b, a)}-' = (0.(I)(l")*(b, a))- a. (d~d)(~")(a, b))-b*. N*(b, a). (A.22) Expressing N(a, b) in terms of R(a, b) according to (2.39), it follows from (A.22) that {R(a, b)}2 = ltN~) _ 4(a:¢~t],l)(a, b)). b*- ax~t~'~)*(b, a ) - a , and solving R(a, b) from (A.23), we get eq. (2.38).

(A.23)

J. van der Linden et al. I Multicomponent integrable systems H

Furthermore, let us write

R'r(b +,a*),

269

using (2.39), (2.27) and (2.24a) as

RT(b *, a t) = 1 (N~) - 2b*- (]D(t'°)*(b, a). a. (I)(°'~)(a, b).

(A.24)

Differentiating both sides of (A.24) with respect to x, and applying (A.14), we obtain that b * - ( ~ ~(]D('")*(b, a ) ) - a +N(a, b). oflD('a)(a, b) = - { ½o,,RV(b *, a*) + b*(bx(lD("~)*(b, a)). a. N(a, b)- ~ 2(I)('")(a, b))}. (A.25) Then multiplication of both sides of (A.18) for p = 1 from the left by {N(a, b)} -~ yields, with use of (A.21), (2.28), (A.25) and (A.22),

~z('")(a, b) = {N(a, b ) } - ' . 0~z(1)(i't)(a,b) - ½(O,(I)(t't)(a, b))-O+RX(b ', a*). x

(A.26)

Eq. (A.14) for p =0, q = 1 and eq. (A.26) yield eq. (2.37b).

Appendix B In this appendix we give the derivation of eqs. (3.18), (3.21), (3.24), (3.27), (3.29) and (3.32). Also eqs. (3.20b) and (3.23) are found with the use of relations derived in this appendix. From the integrated version of the symmetry relation (3.7b) it follows that (1 -t- ~ 2 " h * ) . (1 - ~ , - A*) = 1 ,

(B.la)

(1 - 1111-A*). (1.4- "tIJ'2 • A*) = 1.

(B.lb)

Since, according to (2.16b), we have q~2 = -alrl,

(B.2)

the relations (B.la,b) imply that (1 - ~ , . A,*)+. A*. (1 - ~ , . A*) = A*,

(B.3a)

(1 + ~ 2 . A*)+. ~,*. (1 + ~2-A*) = A * .

(B.3b)

J. van der Linden et al. I Multicomponent integrable systems H

270

The symmetry relation (3.7a) gives m

m

~

m

~k[F" A, B] = (1 - llifl • Jr)"f][lk[A ° F~ n ] , • F, BI = (1 + , , .

A).

.4,

(B.4a) (B.4b)

From the differential relation (2.6a) with (2.4a) we get, using (3.3), --iCgx~k[F-/],/~] = (1 -- ~

,

-

• A)-

jT • 4}k[F. A,/}1,

(B.5)

and from the differential relation (2.7a) with (2.5a) we get, using (3.9b) and (B.2),

-iOx~[F./~, A] = ~[F-/}, A]-(1 - al[rx.A*)* "d.

(B.6)

Now, from (B.4a), the integrated and hermitian conjugated version of (B.4a), (B.5), (B.6) and the hermitian conjugated version of (B.6), we obtain on application of (B.3a) the relations

• ~. A. ~k[F. A, B] = ~*2"A. ~k[A. F, B],

(B.7)

- i ~ . A. OxC~k[F. A, B] = ~*2 "A.,I T. ~k[F. .4, B] ,

(B.8)

i(Cgx@*[F./},~il)-J r. A-@k[F. A, B] = @*[F./}, :] ] • A- 4}k[/] • F,/}1,

(B.9)

(O,,qo[F. B, .4]).JT.A*.J.O,,~*[F. ~, .~] = ~ [ F . / } , / ] ] - A * . ~ * [ F . / ~ , A].

(B.10)

Eq. (B.7) yields eq. (3.32), eq. (B.8) is used in eqs. (3.18a) and (3.20b), eq. (B.9) is used in eq. (3.23b), and eq. (B.10) yields eq. (3.21a). The symmetry relation (3.7b) gives a" w,/ c :r" ~ . . ,i ,,~

/~l

L.,.j

:

tt

\.

-- 'llt

1:1

. A*'~..I. r~

]

r ,,i.

'gJ,k[Z-a

~

I

1~]

~ Ul

to.li)

and applying again (B.3a), it follows from (B.6) and (B.11) that

-i(Ox~[F" B, A]).J T. A*. qtk[F. A,/~] = ~ [ F . / } , A]. A*- ~,[,4 • F,/}]. Eq. (B.12) is used in eq. (3.23a).

(B.12)

J. van der Linden et al. / Multicomponent integrable systems H

271

Analogously to the procedure in appendix A, one can generate more relations of the type (B.7)-(B.10) by differentiating both sides of their composing relations with respect to x. Differentiating ~1, we may then use according to (2.7b) with (2.5b) •

*

l O x ' ~ 1 --'(I~ 1

"a*

(B.13)

• (I~2t ,

or, applying the integrated version of (B.9) with ,4 and/~ interchanged and (3.9a) to the right-hand side of (B.13), -i#xqt , = id. (0x~*[F-/~, A]). B . ~ [ F - / ~ , fi,].

(B.14)

In this way we derive in addition to (B.8), using (B.13), and to (B.10), using (B.14), the relations

- i ~ . A'(O2tkR[F • .4,

BI -

i ~ .B* . ~ . A'Ox~k[F" A,/~l) (B.15)

= ~2' • A. JT • Ox4~k[t'/i,/~],

{02~[F./~,/]]_ i(0x~[F. ~, ,~1). jT. A* .a.(Oxaa*[F./~,/]1) o13 .,gx@[F. ~, fi=]} .jT. A.. a. Ox@,[F./~, .~1 = (OxdP[F. B, ft.]). A*. ~*[F-/~, ,41.

(B.16)

Eq. (B.15) is used in eq. (3.18b), and eq. (B.16) yields eq. (3.21b). Furthermore, differentiating both sides of (B.4a) with respect to x, using (B.13), and then multiplying both sides from the left by 1 + ~2 "A, we obtain according to (B.5), (B.lb) and the integrated version of (B.4b) jT • ~kk[F" A, B] = -iOxekk[fi~" F, B] + ~2 . B * . ~ . A .

6k[fi~" F, B]. (B.17)

Eq. (B.17) is used in eq. (3.24) Finally, we have according to (2.5a), using (3.3),

Ft-')[A • F,/~] = (l! + ~2 "A).J "~2,

(B.18)

and obtain from (B.4b) and the hermitian conjugated version of (B.18), on application of (B.3b), the relation

272

J. van der Linden et al. / Multicomponent integrable systems H

(F:C-'[,i- F,

• F,

=

4,,[F-

(B.19)

Eq. (B.19) is used in eq. (3.29).

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273