1L NUOVO CI1KENTO VOL. 88 B, I~. 2 11 Agosto 1985

Nonlinear Evolution Equations Associated with the Chiral- Field Spectral Problem.

~V~. ~BI~USCHI alld O. ~AGSIISCO

Istituto 2gazionale di Fis iea 7V~eleare - Sezione di ~oma, I ta l ia Dipart imento di Fis iea dell' Universitd (( J~a Sa2ienza ~ - P. le A . More 2, 00185 Reran, I ta l ia

(ricevuto il 5 Dieembre 1984)

Summary. - - In this paper we derive and investigate the class of non- linear evolution equations (NEEs) associated with the linear problem ~ : ~A~. I t tarns out that many physically interesting NEEs pertain to this class: for instance, the chiral-field equation, the nonlinear Klein- Gordon equations, the Heisenberg and Papanicolan spin chain models, the modified Boussinesq equation, the Wadati-Konno-Ichikawa equa- tions, etc. We display also the Bgcklund transformations for such a class and exploit them to derive in a special case the one-solitou solu- tion.

PACS. 02.30. - Function theory, analysis.

1 . - I n t r o d u c t i o n .

lgecently (1), in a joint work with LEVI, the authors have derived the class

of nonlinear evolution equations (NEEs) associate4 with the linear problem

( i . i ) ~ = ~A~,

where A ( x , t) is an (arbitrary) n • matr ix. I n t h a t Iauper~ however, the der-

ivat ion seemed to lie on some unpleasunt b o u n d a r y conditions for the field A

(namely~ it was assumed to v~nish rapidly at sl)atial infinity), lVloreover, just

(1) IV[. BI~ISscI~I, D. L~vI and O. RAG~ClSCO: Phys . Left . A , 88, 379 (1982).

119

120 _~i. mCus(:IH and o. ICAGNISCO

a few interes t ing examples were gi'~eu, no a t i e m p t being m a d e to ident i fy in teres t ing subclasses in a sys temat ic way.

I n the p resen t paper , first of all we will show ~hat the recursion opera tor der ived in ref. (~) is ac tua l ly ~-alid for a larger class of fiehls, charac ter ized b y a cons tan t asy inpto t ic l imi t ; secondly we will derive I~articular s~]bclasses associated with different r anks and algebraic s t ruc tures of the ma t r i x field A ; finally, we will cons t ruc t the :Biicklund t rans format ions (BTs), e lucidat ing where and for wha t reasons they are or a~'e not consistent with the algebraic s t ruc tu re ~ss~rned for A.

I t is here worthwhile ment ion ing t h a t man.y authors C), ins tead of con- sidering the l inear p rob lem (1.1), ha'r focused the i r a t t en t ion on the p rob lem

(1.2) ~,~ + ~ _-: ).a~0,

where U(x~ t) is an n • n off-diagonal m a t r i x and ~ is a cons tan t diagonal m a t r i x of the saine dimension. Indeed , for p rob lem (1.2), the direct ' rod inYersc spectra l p rob lem is solved (a), the whole class of associated N E E s (and BTs) has been deri~-ed and the ~ a m i l t o n i a n s t ruc lure of this isospectra.l flows has been expl ic i t ly shown (9. La te ly , a sys temat ic gn'oup-theorctieal approach to its associated reduct ion p rob lem has been pur sued (~). On the other hand , it migh t be r e m a r k e d that~ whenever A is diagonalizable and its eigenvalues are x- independent , problein (1.1) can be obviously reduced to prob lem (1.2) th rough a gauge t r ans fo rma t ion ; titus lhe corresponding 2~'EEs are gauge equivalent . However , this ga.ugc equivalence does not imp ly a t MI t h a t p rob lem (1.1) does not deserve to be inves t igal~d b y i tself: indeed (( equivalent ,) equations m a y look ~rery different in different gaugcs and some gauges are appropr ia te for cer ta in reduct ions but fa ir ly unsui table for others, l~Ioreover, p rob lem (1.1) is ob'ciously r icher t h a n problem (1.2), the class of N E E s which are gauge equivalent to the N E E s associa ted with p rob lem (1.2) be ing s t r ic t ly included in the whole class of N E E s aris ing f rom isospectral deformat ions of p rob lem (1.1). I ndeed some in teres t ing examples of N E E s , for which the eigen- values of the field A cannot be chosen to be cons tan t , and thus not gauge re- ducible to isospectrM deformat ions of (1.2), will be expl ici t ly shown th roughou t

this paper .

{") A.C..NJ~W~LL: Proe. R. Soe. London, Set. A, 365, 283 (1979); V. S. (~ElCDJIKOV, and E. KH. KttlSTOV: Bulg. J. Phys., 7, 119 (1980) (in I~ussian); V.S.G.~m)JIKov ~{. I. IvANov and I).P. K~:LISI[: Tl~or. Math. Phys., 44, 342 (1980) (in l~ussian); B.G. Ko_~or~r.cn]~Ko: Phys. ~ett. A, 75, 447 (1980). (3) V. E. Z_axm~Rov, S. V. h{~_','axov, S. P. N o v ~ o v and L. P. Pn'AIm,'SKI: Solilor~ Theo~nj, Methods o] the Inverse I)roblem (Nauka, 5{oscow, 1980) (in R1Lusi~n); P. CA~-- I)m.~Y: Phys~ca D, 6, 51 (1982). (4) L.A. DIeK~-f : Commun. Math. Phys., 87, 505 (1983); B. G. KONOrELCnJ~XO: Lett. J, Ila$h. Phys., 6, 309 (1982). (5) A. FOIr and P .P . K~Tr,IS~: Commqzn. Math. Phys., 89, 427 (1983).

NONLIN:EA/~ E'VOLLITION ~QUATiIONS ASSOCIATE/) W I T H THIE CHIRAL 3gTC. ] 2 1

2. - The reeursion operator.

Let us COltsider the following l inear problem:

(2a) ~ ( x , t; 4) = ~A(x, t)~(x, t; ~) ,

where ~v and A are n Xn matrices, and the (( potent ia l )) mat r ix A obeys the following boundary condit ions:

(2.2) l im A ( x , t) ~- A ,

Z being a constant (diagonalizable) ma t r ix (*). We look for evolution equations for the field A arising us compat ibi l i ty conditions between the l inear problem (2.1) and

(2.3) ~ ( x , t; ;0 = - - (M~)(x, t; ~),

where M is a suitable operator cons t ruc ted in terms of the (( potent ia l )) A. To find the class of admissible M operators, it is convenient to in t roduce the ~ollowing l inear operator L :

(2.4) L ~ - l S ~ - - A ~

in such a way t ha t the compat ibi l i ty condit ion between (2.1) and (2.3), provided t ha t the spectral pa ramete r ~ is t - independent , is equivalent to the Lax equa- t ions:

(2.5) /3~ = [33, M ] .

Of course eq. (2.5) implies t ha t

(2.6a) [35, M ] ~ Y,

where V is a k-independent pure mult ipl icat ion operator such tha t

(26b) lira V ( x , t) = O . Izl-->r

Let us assume t h a t there exists an operator M which fulfills conditions (2.6). Then we are enabled to const ruct a whole class of such operators , th rough the following ansatz:

(2.7) M ' = / ~ - I M Jr ~ .

(*) Troughout this paper ~ superimposed bar denotes the asymptotic limit (]x[ -->co).

122 ~ . BRITSCEI ~ n d o . I~AGNISCO

Indeed, by requiring tha t [Z, M'] ---- V ' , V ' being ,~gain ~ ~-independent n • ~ matr ix , taking into account (2.4) and (2.7), we obtain

( 2 . s ~ ) F ( x , t) = --fdx' V(x', t) - - c o

and

(2.8b) V ' = ~WV,

the reeursion operator ~ being defined as

- - cO

On the other h~nd, conditions (2.6) are cer~ainly s~tisfied b y

(2.9a) Mm ~ C,

C being an x-independent ma t r i x such t ha t

[_~, O] = 0 , (2.9b)

and by

(2.9e) Me2) = kS t ,

k being a constant scalar coefficient, yielding

(2.10a) [L, M (1)] : - - [.4, C] ~ V (1) ,

(2.10b) [L, M (~)] ~ k A , ~ V (~) .

Hence, by i terat ion, we obtain a whole class of nonlinear (< eYo]ution >) equations for A , reading

(2.11) ~(~e)A, = ~ ( ~ ) [ A , O],

and fl being ent ire functions of the i r a rguments . We notice tha t , up to a slight change in the notat ions, the class (2.11) is just the class in t roduced in ref. (1), and there in denoted as the chiral-field hierarchy. Indeed, by choosing fi(L~) = �89 and e ( ~ ) = 1 + �89 (1 being the ident i ty operator) , we get

(2.12a) A~ -k B~ ---- 0,

(2d2b) A , - B~ = [A, B] ,

~IONLIN:EAR EVOLUTION EQUATIONS ASSOCIATED WIT}I THE CttlI~AL ETC. 1 2 3

having defined B as

(2.13) B(x , t) = c - - j 'dx ' t) .

Equations (2.12) are easily recognized to yield the principal chiral-field equa- tion

( 2 , 1 4 ) (G iGx) t Jl~ ( ~ - l ~ t ) x ~ - 0

with the positions A ~-G-~G,, B = G-~G~. The time evolution of the eigen- function ~ corresponding to class (2.11) is clearly given by

the operator ~ being defined as

(2.15b ~ M = Z- 'M --fdx'EZ, MJ(x', t) -- fdx'EA, M](x', t). --co --co

3. - O n t h e s u b c l a s s c r 1 6 2 1.

Iu this section we investigate in some detail an interestiug equation that obtains from {2.11) by setting there a(Lf) ---- 1. This equation is just the first nontrivial one, i.e.

Although eq. (3.1) is, ia general; an integro-differential equation, it admits some special reductions (i.e. some special choices of A and C) yielding physically interesting purely differential equations. The first one obtains by considering the following cyclic str~ctm'e for A and C:

(3.2) A -----

0 as 0 "-.

"*., ..

0 "-... a-~_~ . ",.

%~176 ~a~ �9 . . . . . . . . . 0

C =

'0 . . . . . . . . e ~ c~ 0 0

0 """, "'0 "..%

(?n--1

124 ~. s s v s c l n and o. RA*~Sr

~Note t h a t , in o r d e r to fulf i l l co l l d i t i on ( 2 . 9 b ) , we m u s t ha.re c . _ k - a~ ~-k,

c,, : c (c b e i n g a n t~ rb i t r a ry co l l s t an t ) . B y s e t t i n g

(3.3) ak : g~ exp [~v~0] (k = 1, . . . , n),

i t is e a s i l y seelt t h a t eq. (3.1) t a k e s t h e fo rm

(3.4)

N o t i n g t, h a t , due to t h e s t r u c t u r e of eq. (3.1), t h e t i '~ees of a n y p o w e r of A

a r e co] l se rved q u a n t i t i e s , one c~n ~s smne t h e m to be x - i n d e p e n d e n t , so ~hat

d e t A = d e t A a n d l h u s ,~i ~k -=- 0. B y i n t r o d u c i n g now k

(3.5) ~ - - 0~+~--0~, 0~+~- 0~,

eq. (3.4) r e a d s

(3.6)

w h i c h is t h e w e l l - k n o w n n o n l i n e a r K l e i n - G o r d o n s y s t e m a l r e a d y d e r i v e d w i t h

a d i f f e ren t t e e h n i q u e in ref . (6). T h e r e one c a n a lso f ind a d e t a i l e d i n v e s t i g a t i o n

of s y s t e m (3.6). W e j u s t r e ca l l he re that~ in t h e cs.se n = 2, f r o m sysi~ml (3.6)

one c~n r e c o v e r t h e c e l e b r a t e d s i n e - G o r d o n or s i n h - G o r d o n e q u a t i o n s .

These two l a s t e q u a t i o n s c~n be a lso o b t M n e d w i t h a d i f f e r en t r e d u c t i o n

of eq. (3.1) for 2 • m~t r i ce s . I a d e e d , due to t h e t i m e - i n d e p e n d e n c e of t r A

~ud d e t A , we c~n ca s t A in t h e f o r m

(3.7)

with

(3.8) - - d e t . 4 := a : + q r - - k ( e o n s t ) .

lge,p~iring ~ : - ~ -~ 0 a n d Chris g - - k ~-, t h e m a t r i x C tM~es t h e f o r m

(e) A .V . MICm~ILOV: l ' h y s i c a D, 3, 73 (1981); A. F()~DY and J. GIn~o~s: C o m m u n .

M a t h . P h y s . , 77, 21 (1980).

N O N L I N E A I r EVOLIYTION EQUATIO:NS ASSOCIAT:~;1) W I T I I TII]~ C H I R A L ETC. 125

a n d eqs. (3.1) give r ise to t h e s y s t e m

( 3 . 1 o l ( i / - - q ? - i ~ ~ = - , t e q , ( (k r t q r ) � 8 9 4er

f o r m ~: ~ 1, t hese equa t ions a re cons i s t en t w i th t he pos i t ions

q -~- silt 0 exp [ i~] , r = q* = sin 0 exp [ - - iqq, (a.ua)

which y ieh i

0 ~ - - ( tg 0)~,cp, : - - 4e s in 0 , (3.115)

(J : tg : O) 0 ~ + ( tg 0 ) ~ , -{- 0 , ~ = 0 ,

whie, h i m m e d i a t C y reduces to t h e s iam-Gordon equa. t ion i[ ~ - - - - c o n s t (7). AnMogous ly , for k . . . . 1, we can se t

(3.12a) q = s inh 0 exp [~] , r - s inh 0 exp [ - - ep],

t h u s get.t,ing t h e s y s t e m

{ 0~t -}- ( tgh 0)9%~v~ ~ - - 4c s inh 0 ,

(3.12b) ( 1 - lgh-" 0 )0 ,~ , --: ( tgh 0)q~t ~ t - 0 ~ 9 ~ --~ 0 ,

which in t u r n reduces to t he s i n h - G o r d m t equal:ion in t i le subcase V : const .

4. - On the subclass ~ ( L P ) ~ _LP ~.

I n th is sec t ion we will i n v e s t i g a t e t he subc lass a ( s - - , 2 ~ of class (2.1]), n a m e l y we will (lea] wi th evo lu t ion equa t ions of the f o r m

[ / [ fl XAI ]] (4.1) c.~kA, " - A, A, A, .. . , , . . . . -oq~"[A, C~],

--co

where C., a re c o n s t a n t m a t r i c e s such t h a t [-4, C~] = 0. F o r m u l a (4.1) y ie lds evo lu t ion equa t ions in an i m p l M t f o r m : to m a k e t h e m expl ic i t , we h a v e to i n v e r t t.he r ecu r s ion o p e r a t o r ~ . ] n t he a p p e n d i x we show ]tow this can be

(7) Actually, eqs. (3.11b) "~re gaugcmquivMcnt to the Lund-Roggc system: this cquiv- Mence has been pointed mlt by A. Srr~: priva|,e communication.

126 -~ . B I ' , U S C H I 1:1,11(l 0 , l g A G N I S ( ] O

done in generai: unfor tunate ly , it t~rns out t ha t the explicit form of the in- verse c~a 1 of the recursion operator depends on the dimension n of the mat r ix field A and, moreovcr, it looks ra ther cumbersome for large n. Thus, in this

section, we will mainly restrict considerations to n ---= 2 a~ld n =- 3.

To s tar t with, let us assume tha t A is a 2 x2 matr ix :

(4.2) A = AoI -!- A .r ,

where I is the ident i ty mat r ix and a =- (oh, a~ ~ a.~) is the (~ vector )) whose com- ponents arc the usual Pauli matrices.

The first lmntriviM equation iu subclass (4.1) obtains with k----2 and reads

(4.3) [A, fdx' ([A, c~[A, xt]) ' = 0 . --r --r

As previously pointed out, the general inversion procedure is outlined in the

~ppendix: however, (~ cxempli grat,ia ,, we th ink worthwhile showing in some detail how it works in this simple case.

Assuming tha t A has simple eigenvalues, the mill space of A will be the

line~r hull of 17 and A, a.nd thus

(4.4a) Oo- - : . + + , , ( = ,

which implies

(4.45) Po( - - ~ , t) - - O,

(4.4v) p~(-- c~, t) =-- Co.

Taking into account formula (2~.5b) we ha~,-e

(~Ad)

(4.4e)

2p,~ -i- (:p: t r A)~ --- 0,

P0., t r A -]-p~ t r A ~ t- ~Pa (tr A2)~ = 0 ,

which, together with (4.4b), (4.4c), completely determine Po a.nd Pl:

p ~ = col~i , po = �89 , (4.4I)

where

(4.4g) ] = (4no :- a b - ~ ,

%, a r being the opposite of coefficients of the characterist ic equation for A.

:NO~NLLNEA:R :EVOLUTION EQUATIONS ASSOCIATZD ~ V I T I I THE, CI I lRAL :ETC. 127

~Now, differentiating eq. (4.4a) and using (A.3a) and (A.3b) with . n - - 2 , we get

--do

w ] l e l l e e

(4.5b) A t - - c2[A, A] + co(PE A, A~]).I] -~ po.I -t- (~A):o.

At this stage, oaly the usymptotic values of ~0, P~ axe determined, by tuking il~to ~ceount (4.5a):

(4.6) Po(- - c~ , t) = O, p~(- - cm, t) = e , .

To determine 15o and /~ completely, we have to make flu'ther assumptions on A. We will consider here 3 cases:

i) ao and a~ (and thus /, see formula (,l.4e)) ~re constant , i.e.

(4.7a)

(4.7b)

We h~Ye to require

tr At = tr A A , = O.

~o ,a l I ~ l . (2ao + a, ~) = O,

which, due 1o the asymptot ic belh~Yiour (4.6), imply

~o = O, .~1 : : el.

Consequently, eq. (4.5b) reams

(4.8) A, = coiXA, A.~] + c lA, -t- c2[A, A] ,

or, in terms of representation (4.2),

(4.9a) Ao.~ - 0 ,

(4.9b) Ae -: 2ieo]2AAA~ + elA~ -- 2io2A/~A.

128 ~. B~aTJSCHI and O. I~AGNISCO

By set t ing

(4.9e) eo = (2i]2) -1 , el = O, B = - - 2ie 2 A ,

eq. (4.9b) red,ices to the well-known isotropic I-Ieiseaberg spin chain equat ion i~ the ex te rna l constant magnet ic field B (s).

ii) Assume now

(4.10) Ao ---- 0 , A.~ = i , A~ = 1 .

~r i th in these assumptions, it is easil$ seen tha t the coefficients T0 and fil are given by

( 4 . ] : [ ) ~o = o,

(4 . ] .2 ) 91 = c~ - 2Co(t~J~)/].

1goring tha t , in this case, ] = (2AJ -~, eq. (4.5b) yields

(4.13a) A~---- e~A~. + �89

By introducing the field q = A~ -~, eq. (4.13a) becomes

(4.13b) qt = ez q~ - - �89 co q2 q~/~l .

We point out tha t , since det A is not x- independent , eqs. (4.13) have no counter- pa r t in the class of NEEs re la ted to the l inear problem (1.2). This equat ion has been derived in a different context by BLv~A~ and Kwu~i (~).

iii) Assume finally

(4.14) A o = O , A a = I , �9 A ~ - - i A 2 = q ~ A l + i A 2 = r .

In this case, f rom the diagonal p a r t of eq. (4.5b), with the asympto t ic con- ditions (4.6) (i.e. lira q = l i r a r = 0), we get

b P - > ~ I~1-~

(4.15) /~o = 0, ~ = e2 - - eo(qr~ -- q~r)/da~,

(s) L.A. TAKTADhAN: Phys. Lett. A , 64, 235 (1977); see also F.W. NIJttOFF,J. VAN D~a LIND~g, G .R .W. QuISP~L sad H.W. CAP, L: Phys. Lett. A, 89, 23 (1982), and references therein quoted. (9) G. BLUMA~ and S. K~M]~I: J. Math. Phys. (N. Y.) , 21, 1019 (1980).

NONLINEAR EVOL'UTION :EQIJATIONS ASSOCIATED VTITI{ TIlE CIILRAL ETC. 129

while the off-diagonal p a r t y i e lds

(4.16a)

(4.16b)

q , : = 2 c ~ q - - c l q , ~co(ql(1-,'-qr)~),,,

+', - - - 2 e , r - - c ~ r : - - ~ c o ( r ~ ( ] t- q~)9, , .

B y se t t ing ci - c.~ := 0, Co = --2i, q - r*, eqs. (4.16) reduce to the single scalar equat ion

(4.17) iq< t (q/(1 -[ Iq'~)-9,,~ : O,

which was first in t roduced by WADATI aud col laborators (1o), while S~m~IZU a n d VV.tDA~t~ (n) solved the corresponding spectral problem. I t is also possible to show explici t ly tha t , in this pa r t i cu la r reduct ion, the inverse of the reeursion opera tor coincides wi th the corresponding opera tor in t roduced b y BOlTI and col laborators ('~) to cons t ruc t the hiera.rehy of NEI'3s re la ted to the Wada t i -Kormo - l eh ikawa ( W K I ) sI)ectral p roblem. For a deeper inves t igat ion of the nice proper t ies of this h ie ra rchy we can thus refer to Boit i ' s paper .

Le t us now emisider the fu r the r nont r iv ia l equat ion of class (.I.1), ob ta ined wi th k = 3; for s implici ty , we set cl - - c2 = 0. According to the procedure previously outliued, we can wri te it as a s t anda rd ~N-EE, name ly

(4.18) A , = c~[A, A] -- {coJ~[A, ( p [ A , A , ] ) , ] I ] + ~ o I + ~ A } , ,

When / has been defined ia (4.4g) ~nd the scalar funct ions Po a n d ~ are to be de te rmined according to the restrict.ions imposed orl the ma t r i x A. We consider

here jus t two cases:

i) up, a'l (and thus ]) are constant .

11l this case the funct ions i~o and 1~1 are given by

(4.19) ~ , - - - e o ] ~tr ([A, A , ] ) 2 , ~o --- - - �89 # ,

�9 rod eq. (4.19), using the I->auli representa t ion (1.2), becomes

(4.20a) Ao,t = 0 ,

(4.20b) A , = C o ] a [ ( A . A : : ) A - - ( A . A ) A : = - - 8]"-AIAAA:'L2]: - - 2 i c 3 A A A .

(lo) M. WADAa'I, K. KOXXO an4 Y. H. ICIIIKAWA: J . Phys . Soc. Jpn . , 46, 1965 (1979). (11) T. SILIMIZ1~ and 5[. WADATI: Prog. Thear. Phys . , 63, 808 (1980). (12) 3[. BOVI?I, 1~ ~. PE~(PIN~IJLI grid G.Z. Tu: Prog. Theor. l 'hys . , 69, 48 (1983).

9 - II Nuovo Uimento B.

131} M. BRESCItI a n d o. :RAGNISCO

Recal l ing t h a t al : 2A0, a o - - - A ' A - - A 2 o , we cau assume with no restrictioll

(4.21a) Ao : 0 ~ A . A : 1 ,

so tha t , by set t ing col 4 - - / c and in t roducing the cons tant field B =- - -2 ic~A, eq. (4.20) takes the form

(4.21b) At --- B A A + k[A(A . A ~ - - 2 !A~]") - - A ~ ] . ,

which was first in t roduced by P A r A ~ I O O ~ (~3).

ii) Assume now

Ao : 0 , A ~ - - i , A.~ : 1 .

I u this case, eq. {4.20) becomes

or, by in t roducing q ~-A~ -~,

C ~ e 8- (4.22b) q~ - - oq ( q ) ~ [ q l ,

which is jus t the second m e m b e r of ~ h ie ra rchy of ~o~fiinear scalar evolution ecluatiolls~ a l ready derived ia ref. ('), whose generic t e r m reads

where the opera tor _L is defined as

(4.23b) 35]-- q]~.

We shall not wri te dowlt here the equat ion ob ta ined in case iii) ( formulae

(4.14)) sine% as a l ready poin ted out, i t Js jus t the second m e m b e r in the W K I h ie ra rchy (1~).

We come now back to eq. (4.1) wi th k -= 2 and el ~ e2 ~ 0, which we shall inves t iga te for matr ices of higher dimension. As the general fo rm of the inverse of t h e rccttrsion opera tor is r a t he r cumbersome even for 3)<3 matr ices , to make it s impler we will impose fu r the r restr ic t ions on the m a t r i x field A.

(13) ,N ~. PA~'A_~ICOr~Atr: Phys. Lett. A, 73, 134 (1979).

NOXLIN] ' ;A~ EVOLV:r lON ] tQUk:l ' IONS ASSOCIATED I F I T I I TI:t~ ClIII~.AL }YI'C~ 131

Firs t of all, we ~ssume tha t t.he 3 x 3 mat r ix A fulfills the characterist ic equat ion

(4.24) A 3 ~- I .

In t.his case, ex~ploifing the procedm'e given in the ~ppendix~ one obtairts

(~.2a z ~ v = ,~ "{L~4 ~, v l - ~ tr f(EA'-; (A'),] V) d~,'-- --co

- - ] A -~ tr f([A~, A.] V)d,v'}.,

Choosing for simplicity A to V being, as usual, a generic var ia t ion of A. be diagonal, name ly

~. [ [2~ik]~

and setting in eq. (4.1) Co = 3A, we get

(4.27a) A , - - [A'-, A~]~. = [_45 A.,.~J -}- [A, A~].

On the other hand, by set t ing Co = 3-~ ~, we get

(4.27b) At = [A ~, (A~) . ] . = (A[A, A~:IA),~.

(k - - 0, 1, 2),

Eq!mtious (4.27) are, to our knowledge, new solvable ma t r ix NEEs , which obviously involve six irrde13endent scalar fields. A sys temat ic invest igat ion of the possible h r r the r reduct ion of such equations is beyond the scope of the present paper. IIero we exhibit just a reduct ion of eq. (4.27b) involving two independent scalar fields, which obtains b y assmning for A the following pe- eulia.r s t ruc ture : *

(4.28a) A ....

with

(o 0 0

a.a 0

(4,28b) a l a ~ a 3 = l .

Accordingly, from eq. (4.27b) we get the sys tem

(3.29) aj~ = [ a j ( a j f , , . /ai . i l - - a s_ l , z / a j_ , ) ] . ( ar 3 = a~) .

132 ~. Br~SCHI and o. ~RAGN]'SCO

By in t roducing the new fields

(4.30a)

obeying of course the res t r ic t ion

(4.30b)

s y s t e m (4.29) reads

3

~1,=0 ,

(4.3~)

:By t ak ing advan tage of condit ion (4.30b) to express ]3 in t e rms of /1 a n d ]~, sys tem (4.31.) gives rise to the so-called modified Boussinesq equat ions, first der ived by l~o~]),z and G~]3o~s (~4).

~ y re laxing now conditioll (4.24), bu t 13resettling for A the s t ruc tu re (4.28a) wi th

(4.32) a 3 = 1 , ~ = a2 --- 1

(so t h a t (4.28b) is fulfilled only asympt.ot ical ly) , eq. (4.1) with k = 2 aud Co - - u~2 yields

(4.33) { a ~ , - - ~ ( a d ( a x a ~ ) t ) : : ,

a .~ = - - < ~ ( a j ( < , ~ a . ) q : : .

Syste m (4.33) is, in some sense, the analog for 3 • 3 matr ices of sys t em (4.16) : $ indeed i t also admi t s (for ~ = i) the reduct ion (xx = a~ = q yielding

(4.34) q , - - i(q/iqI~)~,

which is a new plLrely differential scalar solvable :NEE (resembling eq. (4.17)). Now we consider a single case of matr ices of dimension 4, namely we will

a s sume for A the form

(4.35a) _4 = ) 0 a 2

0 0 ~

a4 0 0

Qa) •. FORDY and J. GIBBONS: J. Math. Phys. (.N.Y.), 22, 1170 (1981).

:NONLt.N]~AZr EVOLUTION )IQUATIONS ASS()CIATiED YVITII TII~E CI I IRAL ~ETC. 133

with

(4.35b) 4

1-[ a~ = 1 .

~gain , according to the procedm'e given in the appendix, the explici t expression of eq. (4.:t) (with Co = 2-4 a) c'~n be obtained:

(4.36) A ~ - - - - ( A [ A 2, A ~ A ) ~ ,

which, in terms of the fields ]j = aj,/a~, reads

(4.37) ]~i ~- [fJ(],+~ - - ]~-~) -~ (]J+l - - ]~'-~)~]~,

where ]~+4 - - ]j. Not ing tha t sys tem (4.37) is the na tura l extension of (4.31) to n = 4, we conjecture tha t analogous extensions exist for any n.

On the other hand, the analog of eq. (4.33) also exists; it reads

(4.3S) a2~ = o ~ { a 2 ( a ~ . / a , - - a a . ~ / a ~ ) / ( a ~ a , a 3 ) ~ } . ,

<~, . . . . . ~ ( ~ / ( ~ . . . ~ ) % ,

this sys tem obtains by set t ing a~ - - 1 and requiring tha t condition (4.35b) be fnl- filled only asymptot ical ly .

5 . - B f i c k l u n d t r ~ n a f o r m a t i o n s .

According to the scheme of derivat ion tirst outl ined in ref. (~5), the B~.ekhmd t ransformat ion (BTs) for the h-EEs exhibi ted ill sect. 3 and 4 can be obta ined s ta r t ing f rom the spectral problems

(5.1) LW-- 0 , ~ = 0 ,

where the l inear operator Z has been defined in formula (2.4) and the operator Z, is merely obta ined by replacing therein the field A with a new field •. The matri- ces A and X are required to have the same asympto t ic beha~iour (A).

l u order t h a t V and v~ be solutions of the l inear problems (5.1) (for the same ~zalue of the spectral pa rame te r A see (2.4)) , they have to be re la ted by the

(,5) M. B~vsc~ and O. RAGXISCO: ~ett. ~Vuovo Cimento, 29, 331 (1980).

1 ~ M. B R U S C H I ~ l ] d O. ICAG~'ISCO

Darboux t r~nsformMion:

(5.2) ~(x, t; X) = D ( x , t; ).)yJ(x, t; ) .) ,

where the ma t r ix D has to flfltill the compatibi l i ty condit ion

(5.3) ).-LD~ q- D A - - ~ D = O .

Let us ~ssumc now tha t there exists a ma t r i x D obeying the following equat ion:

(5 .4a) 2 -~D~ + / ) A - - _rid - - W ,

where W is ~ ).-independent matrix such that

(5.4b) lira W ( x , t) ~ 0 . I~l-*~

Then we can const ruct a whole class of such matr ices th rough the following unsatz:

(5.5) D ' = ). ~ D + H .

Indeed, by requir ing t h a t / ) ' s~tisfies again eq. (5.4a) with W replaced by some new ~-indcpendent m~tr ix W', we get

(5.6) H = - i d x ' W(x', t) --co

and

(5.7) x

W,(x, t) = t) f t) :~~ (A W)(x , t) . - - a ) - - v o

On the other hand, any constant ma t r ix G comnmting with .4 satisiies condi- t ions (5.4). Hence, by iteration~ we get a whole h ie ra rchy of W matrices, given by

(5.8) W ~ 7 ( A ) ( A G - - GA) ,

7 being an a rb i t ra ry entire funct ion of its argument . The corresponding class of D matr ices reads

(5.9) D ~ 7(S)G,

NONLINI,.'AI ~. EVOLUTION ~QUATIONS ASSOCIATED WITL[ THE CHIRAL ETC. 135

the opera tor S being defined as

(5.10) S D = ;.-'D § DA - - AD)d '=f(XD - D A ) ( x ' , t) dx'.

]n order t h a t these matr ices / ) be D a r b o u x matr ices , we have to require t h a t the corresponding matr ices W be ident ical ly zero (see (5.3), (5.4a)). This resul ts in a funct ional re la t ion be tween the fields A and A, reading

(5.11) y ( A ) ( A G - - G A ) - - O .

We can thus asser t t ha t the recm'sion opera tor A (defined in (5.7)) is jus t the opera tor which allows us to cons t r~c t a whole h ie ra rchy of BTs (5.11) for the N E E s derived in the previous sections (*). As an example , we will show in the following tha t , th rough the func t iona l re la t ion (5.11), we are able to con- s t ruc t ~ special solution of the t t e i senberg spin chain equat ion (eq. (4.9b), (4.9c)). To this a im, we notice t h a t the first nont r iv ia l BT per ta in ing t o class (5.11) reads

(5.i2)

G' and G" being two cons tan t mat r ices c o m m u t i n g with A. the new field B as

(5.13) B A)

By in t roducing

eq. (5.12) can be recas t in the fo rm

(5.14) B ~ ( B G " - - G ' ) -[- [A, B ( t " - - G'] = 0 .

I n eq. (5.1,1) A has to be regarded as a known solution, so t h a t , once solved for B, i t yields immedia te ly , t h rough (5.13), the new solution ~ .

I f we are in te res ted in the ~ one-soliton )~ solution, we have to set A = A, which is obviously a t r iv ia l solution of a n y equat ion of class (2.11), so t h a t

(*) Actually, to make this assertion more rigorous, one should also prove that, once the potentials A. and #I are related by (5.11), they also satisfy the same I~EE or, equivalently, that, if the wave functions ~v and v~ are related through the Darbotu~ transformation (5.9), their time evolution is generated by the same operator M (see (2.15) ). This can be checked case by ease but it is also straightforward to prove it, in general, if one admits that, at any time, the wave functions are uniquely determined by their asymptotic behaviour.

136 M. m~vseIrI trod o. ~r

eq. (5.14) takes the s impler fo rm

( 5 . 1 5 ) B ~ ( B - - G ) i- [T1 , B ] - - 0 ,

where "G - - G'~(G') - ~. On the other hand , for thc [[ciscnbcrg spin chain equa t ion (4.9), we have

AT-- A . a , - ~ = a ~

and ~hus

G := goI -[ g3a3,

so tha t , solving eq. (5.14), we get

A 1 -= (g] - - g~)-~2g 3 sech y [go sinh (g0.tllg3 -i- of) - -

- - g3 tgh y cosh (goY/g3 + ~v)],

(5.16a) A2 ~ ., - 1 �9 : (go--ga) 2~g 3 sech Y[go cosh (goY/g3 + q ~ ) -

- - g~ tgh y sinh (goY/g~ -[- ~)] ,

A , = (g~ - - a~)-12a~ seeh'- y + 1,

where

(5.16b) y(t) = ( g ~ - g ~ ) - ' 2 g o ( x - ~(t)) (!go/g~l < 1)

an4 the t ime evolut ion ()f the p a r a m c t e r s ~ and % which obtains by inser t ing (5.16) into eq. (4.9), is g iven by

Of course formulae (5.16) pro ' t ide ~lso the (( one-soli ton ~) solution for the ~'aI)a- nicolau equat ion (4.21b), since the solution (5.16a) depends j~st on the s t ruc tm 'e of the m a t r i x field A which is ttlc same in bo th cases.

Obv ious ly , the time-e~-olutiou of the p~ra.meters ~ and 9 is now different a n 4 can be ex-plicitly ob ta ined by plugging (5.16a) into (4.21b).

I t is worthwhile here m a k i n g some r e m a r k s abou t the general featm'cs of the class of BTs (5.11).

Due to the fo rm of the recursion opera tor A (5.7), a n y BT can be wr i t t en as

A B - - B ~ - O, (5.1s)

where

X

B - f W d x ' .

NONLINI'~AI{. E V O L U T I O N E Q U A T I O N S A S S O C I A T F D ) v I ' I ' l l TIIiE CI i l I~AL :ETC. 137

I n order t h a t (5.18) provide a nont r iv ia l :BT, we have clear ly to reqn_h'e B -/: 0, otherwise b y (~ invcrse ,) i tera t ion wc will be reduced to the pure s imi lar i ty t r ans fo rma t ion

(5.19) A ~ G A G -1 ,

G being ~s usual a cons tan t m a t r i x comnlut.ing with A. On the o ther hand, it is well known (~e) t ha t (5.18) admi t s nont r iv ia l solutions if and only if

(5.20) 2, ~ ]j (for some i , j ) ,

2~(t~) being the eigenvalues of A(~) . Hence the subclasses of N E E s re l a t ed to the spectral p rob lem (1.1) for which the eigenvalues of A are cons tants of the mot ion h~ve surely nont r iv ia l BTs.

Incidental ly , we notice t h a t this is jus t the sllbclass for which the l inear p rob lem (1.:1) is gauge equiva len t to (].2).

For the o ther N E E s the significance of (5.18) n m s t be inves t iga ted case by case. For ins tance, it is easy to see tha t , for the W K I sys tem (4.]6), eq. (5.]8) does not provide any nont r iv ia l BT. Indeed , the r equ i remen t (5.20), due to t.he s t ruc tu re of the m a t r i x A appropr ia t e to this case (see form~fla (4.14)), implies

(5.21) ~P ~ qr .

Howcver , as poin ted out in ref. (,o), condition (5.21) e i ther yields t r iv ia l new solutions or is not compat ib le wi th the t ime evolut ion of the WK1 hierarchy .

To end this section, we poin t out t h a t a fu r the r class of BTs can be der ived s t a r t i ng a.gain f rom (5 .4a) , but b y replacing ansa tz (5.5) b y the following one:

(5.22) D ' : ~ D I- ) .H ,

which yields

(5.23a) W ~: z t H - - H A ,

(5.23b) W"-- H~.

I~-o~ tha t , in order to have a nont r iv ia l sta.rting point for the i terat ion, we have to require t h a t the homogeneous equat ion A H - - H A =: 0 has a nonzero so- lution. Again, this occurs if and only if condit ion (5.20) is fulfilled. Thus the new class of BTs (5.23) surely exists for t h a t subclass of -NEEs for which the e igenvahms of A are cons tan ts of the mot ion.

(1~) Sop, for inst~mce, R. BELLSfAN: Introd,uct ion to M a t r i x A n a l y s i s (McGraw-IIill, New York, N.Y. , 1960).

138 M. B1~som and o. tlAGNISCO

As previously pointed out, for such NEEs also the first class of BTs is avail- able.

For the other NEEs re la ted to the l inear problem (1.1), the existence of a nontr iv ia l s ta r t ing point mus t be checked case by case. For instance, it does not exist for the W K I equation, jus t for the same reasons out l ined before. This new class of Bgcklun4 t ransformat ions would indeed deserve a deeper investigation, at least because they are in a pure ly differential form. This more detai led analysis is pos tponed to a future research.

6 . - C o n c l u s i o n s .

I n our opinion, the main achievement of this paper is to have shown t h a t a large number of physical ly in teres t ing solvable evolution equations arising in different contexts and so far der ived through different techniques are indeed re la ted to each other : t hey are, in fact , jus t different members of the same Class of evolution equations, namely tha t corresponding to isospectral deformations of the single l inear problem (1.1). Among them, we just recall here the chiral field equation, the nonlinear Klein-Gordon equations, the t Ie isenberg and Papanicolau ferromagnet ic models, the W K I equation~ the modified Bous- sinesq equations.

Moreover, we have derived new solvable simple evolution equations, which could be interes t ing bo th for thei r physical relevance and for their mathemat ica l propert ies . For a large subclass of such evolution equations we have also ex- h ib i ted two hierarchies of Bgekhmd transformat ions.

A P P E I ~ D I X

t t e re we show in some detai l how to obtain the inverse of the recarsion operator ~e. Le t us denote for implici ty by _C the commuta t ion operator

(A.1) ~ F = [A, F ]

and by D the different iat ion operator , so t h a t (see formula (2.8c))

(A.2)

thus the first s tep is to inver t C. This has been done in closed form in ref. (25) to which we refer for fur ther details. We use here just the result

(~.3) .D__~V= _V-~V'= CV'+ P(.A.),

NONLIN~AI'r EVOLVTION I'QVATIONS ASSOCIATED WITIt THE CHIIfAL ETC. 139

where P(A) is the generic element of the null space of C which (if A is ':~ regl~la,r mat r ix of dimension n) is a polynomial of degree n - - 1 in A, and _(~ is defined by (~4)

Tt--1

(A.4a) C_V-- ~ Cj~(A.'VA~--A~VAO. t > k = o

The n(n-- l ) /2 coefficients C~'s can be fotmd in terms of the invar iants os the mat r ix A by solacing the linear sys tem given by formnla (1i) of ref. (~').

Thus we get

(A.5) v-. ~ - a v ' = .og_v'+ 2)l,(A).

Formula (A.5) does not define uniquely (~O--1 the n coeflieients p~:(x) of the polynomial P(A) being at this stage still undetermined. However, f rom the structlzre of the reeursion opera,for c~a it is readily seen t h a t aU the elements V belonging to the domain of s are restr icted to the following submanifold:

( A . 6 a ) l i m V ( x ) = 0 ,

(A.6b) trACK= 0 (k =: O, ..., n - - l ) .

][enc% the n mlknown functions p~(x) can bc Ulfiquely determined by sol~'ing the linear first-order differential system tossing from (A.5), (A.6b) with the boundary conditions given by (A.6a).

The only possible exception to condition (A.6b) could arise when V=:: kA,, which (see eq. (2.10b)) is one of the s tar t ing points of the iteration. In this case, if one requires again condit ion (A.6b) to be fulfilled, i t comes out t h a t the eigenvalues of A are constants of the mot ion for eq. (2.11). I f not, to determine the coefficients of P(A) one has to impose suitable, s tructure restric- tions on the matr ix field A : this occurs, for instance, for cases it) and iii) considered in sect. 4.

(aT) 5I. B~vsoliI and O. I~,GNISCO: Lett. N~wvo Cimento, 38, 41 (1983).

@ R I A S S U N T O

In questo lavoro si deriva c stadia la elassc di cquazioni nonlineari di evoluzionc associate con il problema lineare V, = 2A~'. A questa classe appartengono moltc equa- zionc intorcssanti: per csempio l'equazione dcl campo ehirMe, le cquazioni di Klein- Gordon non lineari, i modelli di spin di Heisenberg e l'apanicolau, l'equa.zione di Bous- sinesq raodi_ficata, le cquazioni di Wadati-Konno-Iehikawa, ece. Per cluesta classe di cquazioni sono anehe mostrate le trasforrnate di B/~cklund, utilizza~o per dcrivare, in unoaso partieolare, la soluzione ad un solitone.

Pe3roMe He HO~q'yqeHO.