I L NUOV0

CIMFNTO

Voz. 45 A., N. 2

21 Maggio 1978

Evolution Equations Associated to the Triangular-Matrix Schriidinger Problem Solvable by the Inverse Spectral Transform (*). M. ]~RUSCHI, D. LEVI

"rod O. RAG~NISC0

I s t i t u t o d i F i s i c a d e l l ' U n i v e r s i t & - 00185 R o m a , I t a l i a I s t i t u t o N a z i o n a l e d i F i s i c a N u c l e a t e - S e z i o n e eli R o m a

(rieevuto il 13 Febbraio 1978)

In this paper we eon~ider the evolution equations which are assoei~ted to the Sehr6dinger matrix problem for triangular matrices, a very simple subeb~ss of the non-Hermitian m~triees. These evolution equations, though linear in the simple case considered in the present paper, are ilot all solvable by means of stan&u'd techniques, thus making it worthwhile (~xploit,ing the ~pplie~bility of the inw'.rse spectral f,r~nsform to obtain pnrticular solutions. Tim 2 x 2 c~se is investigated thoroughly. Summary.

1. -

Introduction.

R e c e n t l y , t h r o u g h tile applie}~tion of t h e s o - c a l l e d (~g e n e r a l i z e d W r o n s k i a n t e c h n i q u e ,> to t h e o n e - d i m e n s i o n a l m a t r i x S e h r 6 d i n g e r e q u a t i o n , CALOGERO a n d I)EGASPERIS (~) h a x e b e e n a b l e t o o b t a i n :~ l a r g e class of nonline,~r e v o l u t i o n eqm~tions ( N E E s ) w h i c h ,~re s o l v a b l e b y t h e i n v e r s e s p e c t r a l t r ' m s f o r m (IST). T h e y r e a d (**) (1.1)

E~(x, y, t) = 2/~0(L, t) W~Ax, y, t) + ~.,(_L, t)[~.. W(x, y, t)] --

--/L@, t) (~,L + 7(L, t) UX E ( x , y, t) , (*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) F . CALOG:ERO "}Aid .~.. DEGASI'ERIS: ~¥'ltOt;O C i m e ~ t o , 3 9 B , 1 (1977). (**) Equation (1.1) is obtained from ref. ('), by using, instead of the w~ri~ble Q ( x , y , t), co

W(x, y, t) = ~dx' Q(x', y, t). 225

225

~I.

BRUSCII[.

D.

LEVI

~%nd

o.

RAGNISCO

where W ( x , y , t) an(l its deriw~tives are N ' < N matrices, {I, a,~ (n = 1, ..., N ~ - l)} form a complete set of matrices of r a n k N, and the operators L and ~re defined b y co

(LF)(x)

÷

F(x)} + x

(1.1a) co

x

Clearly the class of N E E s described b y (1.1) contains m a n y different subclasses according to the r a n k of the matrices involved, to tile L- and t-dependence asstoned for the quantities flo, c~n, fin, 3( a n d to the particular properties of the matrices in question. So far, people h a v e c o n c e n t r a t e d on matrices of r a n k '2 (2) a n d 4 (3), always assuming ~/'~ 0, rio, c~,, and fl~ c o n s t a n t (neither depending on L nor on t), and restricting to the class of H e r m i t i a n matrices. I n the present paper, we shall instead focus on a n o n - H e r m i t i a n class, t h a t of triangular matrices. As is well known, the triangular matrices f o r m an Mgebra, so t h a t their space is closed under the t i m e evolution described b y eq. (1.1). As a first task, we report some general properties of eq. (1.1) in this space, t h a t hold for a n y r a n k of matrices and for a n y dependence of rio, a,~, fl~ and ~" upon L and t. N e x t , as in the previous papers, we concentrate on the simplest case, i.e. y = 0, rio, ~,~ and fl,, constant a n d write down explicitly the corresponding evolution equations, which turn out to be linear, but, however, not trivial at all, since they involve variable coefficients. Finally, we derive some p a r t i c u l a r solutions of such equations, n a m e l y those corresponding to b o u n d states of the associated m a t r i x Schr6dinger equation: these solutions are obtained in a ~cry simple w a y via the I S T technique, which we show to be applicable in the t r i a n g u l a r - m a t r i x space; a deeper investigation of the oneb o u n d - s t a t e and the two-degenerate-bound-state solutions t h a t obtain for triangular matrices of r a n k 2 will allow us to show some interesting similarities with the soliton solution of the corresponding H e r m i t i a n case. The general properties of s y s t e m (1.1) in the t r i ' m g u l a r - m ~ t r i x space and the explicit f o r m of the equations corresponding to y = 0, fio, a~, fl~ = c o n s t are the subject of sect. 2. I n sect. 3 we show the applicability of the I S T technique for n o n - H e r m i t i a n matrices, focusing on b o u n d states, while in sect. 4 ve investigate the b e h a v i o u r of the one-bound-state a n d the two-degenerateb o u n d - s t a t e solutions for '2 × 2 matrices.

(2) F. CALOGERO and A. DEGAS~'E~IS: Lett. Nuovo Cimento, 16, 425 (1976). (3) 5[. BRuscIII, D. LEvi ~lld 0. ]~AGNISCO: -~UOVO Cimento, 43 B, 251 (1978).

227

E V O L U T I O N EQUA'PI()-N$ ASSOCIATED ETC.

2. - Nonlinear evolution equations for triangular matrices.

Tile most i m p o r t a n t consequence of triangularity is t h a t the evolution equations derived from system (1.1) for the diagonal elements are decoupled from the remaining ones. This implies t h a t these elements satisfy just the evolution equations one obtains starting from the one-channel Schr6dinger equation (4), t h a t is the wave equation, the Korteweg-deVries equation, the modified Korteweg-deVries equation, etc. On the other hand, for any L- and t-dependence of the coefficients y, flo, c~,,, fl,, the off-diagonal elements W i; ( i < j ) satisfy linear equations coupled to those elements W ~ such t h a t i < l < k < j. We will see in detail these properties in the simplest case, y - 0, rio, cG, fl,, constant. To write down explicitly the evolution equations in this ease, we found convenient to use tile base of N ~ matrices a~j, which have all elements equal to zero except for a 1 at the (ij)-position, and satisfy the following relation:

(~.1)

a~.j (Y1,,l - -

(~jl: c;it •

In fact, if we set W

('2.2)

--

W i~ a ~ ,

~n (~n ~ A ij (~ij ,

eq. (1.1) gives rise to _y2 scMar equations, t h a t read

(2.3)

iJ

A it tV~/--

il

l~

7it BlJ

@ ~7~il ~,~.Ik~kj

Bil

lj

uyilBlk ]~rkxd-

WixIBlk Wkd _~ Bil Wlk w k ] ,

where tile indices i, j, k, 1 run from 1 to N with tile restriction i < l < k < j and sums over repeated indices are understood. I t turns out t h a t eqs. (2.3) are linear; the evolution equations for the diagonal terms are

(2.4)

(no sum over i),

whence (2.5)

W . ( x , t) - W . ( x + 21~.t, o),

the evolution equations for the off-diagonal elements can, in principle, be (4) F. CALOGEI~O: 2gUOVO

Cimento,

31B, 229 (1976).

228

M. BRUSCHI,

D. LEVI and o. RAGNISCO

solved step b y step, starting from those concerning W i'i+l, t h a t involve just bilinear terms of the form W~'~Wi'i~l and Wi,~W~,~+1, then solving those for W ~'*+2 t h a t involve, besides the unknown variable, just W ~'~, W ~'~+1, W i+~'", W ~+~'~+2,W ~+%~42 and their derivatives. We have to notice t h a t all the n - k evolution equations (2.3) for the variables W ~,~+k (i --~ 1, 2, ..., n - - k) have the same structure. We must emphasize t h a t the features reported above are a general p r o p e r t y of eq. (1.1) for triangular matrices, while the explicit form of the evolution equation given b y eq. (2.3) depends on the particular case considered.

3. - The IST technique for non-Hermitian matrix potentials. In the previous section we have derived a system of linear evolution equations, some of which (those for the diagonal elements) are immediately solvable (as t h e y reduce to the first-order wave equation), while for the others, which in general involve variable coefficients, the explicit solution for a r b i t r a r y initial data cannot be easily found. However, it is always possible to solve the Cauchy problem for some special initial data, taking advuntage of the fact t h a t these equations belong to the class of N E E s (1.11 solvable b y IST. As is well known (5,s), the IST technique allows us to reconstruct the potential from the spectral data through the following steps:

i) from the spectral data we construct the m a t r i x M(x)

(3.11

M(x) = ~

dk R(k) exp [ikx] -~ ~ Ca exp [--p~x], 5=1 --co

where the m a t r i x R(k) is the reflection coefficient, {P~}~=I,...,~ are the energies of the 1 bound states and the matrices Cj are related to the coefficients cj characterizing the asymptotic behaviour of the l bound-state vector eigenfunctions (3.1a)

y~¢(x) ~

c~ exp [-- p~x] ;

x---> co

ii) the function M(x) is the kernel of the Gel'fand-Levitan-Marchenko (GLM) equation co

(3.2 /

K(x, y) -~ M(x + y) + fdzK(x, z)M(z + y) : 0

(y•x);

x (5) Z.S. 2~GRANOVICHand V. A. 5[ARCHENKO: The Inverse Problem o] Scattering Theory (New York, N.Y., 1963). (6) M. WADATI and T. KA~IJo: Prog. Theor. Phys., 52, 397 (1974).

EVOLUTION

EQUkTIONS

ASSOCIATED

229

ETC.

iii) f r o m the solution of the GLS:[ equation we get (3.3)

(2(x) = - - W~ = - - 2 d K ( x ,

x),

where Q(x) is the potential of the m~trix Sehr6dinger equation

-T"÷QT

(3.4)

k~-T.

So f~r the I S T technique for the multich~nnel Schr6dinger equation h~s been ~pplied m a i n l y to H e r m i t i a n matrices, although some authors (6) h~ve derived t r e a t m e n t s which, in the p~rtieul~r e~se of potential vanishing at infinity faster t h a n exponentially (*), are expected to be ~pplicable also to the n o n - H e r m i t i ~ n case, provided the poles of R(k) be simple, so t h a t the matrices iC~ are just the residues of R(k) for k = / p j . CALOGERO ~nd D~(~ASPERIS (1) h~ve extended the applicability of the I S T technique, derived in ref. (6), to more general H e r m i t i a n potentials, for which the Bargm~.nn strip is finite. W e notice t h a t this is the e~se of the pure multisoliton solutions of eq. (:1.1) for H e r m i t i a n ma~trix potentials (**). As for the multisoliton potentials eq. (3.2) is immedi~ttely s o l w b l e (the kernel M(x -~ y) being of finite r~mk), we nre interested to extend the results of C~logero and Deg~speris (~) to n o n - H e r m i t i a n multisoliton potentials. To do so, let us consider the J o s t (m~trix) solution of eq. (3.4), defined ~s usual b y the bound~ry eonditions (3.5a) (3.5b)

lira (F(x, k) -- exp [ikx] I) = O,

x ÷ 1 co

f l _ % (F(,,,, 1~)- A(~) e~p [i~'x] - B(I~) e x p [ - i~x]) = O,

to ensure the existence of a b o u n d state, it is sufficient theft det A(k)[k=i~ = 0, which implies tlmt one of the eigenvalues be zero for a certain value of k in the u p p e r imagim~ry ~xis (if the zero eigenwilue is r-fold degenerate, this a m o u n t s to considering ~ system of r different solitons with the same amplitude). I n this case we can find a, vector v ~ 0 such t h a t A(k)Ik_~,v -- O. Thus, for ~ b o u n d state, f o r m u l a (3.5b) g'ives (3.5c)

l+'(x, ipj)v

~ x

B~ exp [p~x] v ,

-#--- oo

(*) In this case the Bergmann strip (7) extends to the whole complex k-plane. (7) R . G . NEWTOn-: Scattering Theory o] Waves and Particles (New York, N. Y., 1966). (**) In this case, and provided the bound state is nondegenerate, C~--cjc~, cj being tile vector defining the asymptotic beh~viour of tile nornmlized j-th bound-state eigcnfunction.

2~0

M. B R U S C I I I , D. L E V I ~ i l d

o.

I~AGNISCO

where l~'(x, ipj)v provides the vector eigenstate of the associ~ted SehrSdinger equation. F(x, 1,:) can also bc represented in the f o r m co

(3.6)

F(x, k) = exp [ikx] I

+ f dy K(x, y)

exp [iky] ,

2g

when K(x, y) is the solution of the G LM equation (3.2). Now we spe(:ialize to the case R(k) = 0 and t r y to connect the matrices Cj with the :~syml)totic b e h a v i o u r of the corresponding J o s t solui, ions P(x, ipj). To establish this relation, we t a k e into account eqs. (3.1), (3.2), (3.6) and express K(x, y) in t e r m s of the J o s t functions l

(3.7)

K(x, y) = -- ~ F(x, ip~) Cj cxp [-- p~y] . j=l

I n s e r t i n g (3.7) into (3.6), written for k = ip,,, we obtain the following linear s y s t e m in the l u n k n o w n matrices F(x, ip~): l

(3.8)

/ , F(x, ip,~) = exp [-- p,, x / [ I -- ~ F(x, ipj) Cj exp [-- PA~J/(P,~ + p j)/• j=l

This system can be obviously solved for a n y l, thus Mlowing us to ~'et the relation between the quantities Cj and Bj (*) in the m o s t general ease. F o r the s:~ke of simplicity, however, we report herebelow just the result holding" for l = 1: (3.9)

]F(x, ip) I + ~ exp [-- 2px]

exp [-- px].

I f v is the vector annihilated b y A(k)I~=~ , applying eq. (3.9) to v, we get (3.10)

C 2':~p

F(x, ip)v + F(x, ip) 7 exp [ - - 2 p x ] v = exp [ - - p x ] v .

To recover eq. (3.50) we h a v e to set

(3.H)

C

2p exp [2p~] P

(Pv --= v),

whence it follows t h a t (3.12)

F(x, i p ) v

~

exp [p(x-- 2~)] v ,

(*) B~ = B(ipj) only for potentials whose Bargnlann strip extends ~o the whole complex /c-plane.

:EVOLUTION

I~QUATIONS

ASSOCIATED

231

ETC.

implying B = exp [-- 2p~] P .

(3.13)

We notice t h a t eqs. (3.10)-(3.13) cau be extended, with some relewmt restrictions, also to a r-fold degenerate case ( r < N , 37 being the dimension of the matrices considered). Two (liffeient cases can be distinguished. a) The matrix C is nonsing'ular (none of its eigenvalues is zero); this implies t h a t the asymptotic behaviom' of the J o s t function F(x, ip) is

F(x, ip) ~

(3.14)

x~

2pC -~ exi) [px]. co

I t follows t h a t B 2pC -~, A(@) 0. I n this ease, there exist clearly 2V independent bound-state vector solutions which build up the X × N m a t r i x F(x, ip). This is the case of complete degeneracy. b) The m a t r i x C is singular, t h a t is r ( r > l ) of its eigenvalues are zero. I t is easy to realize t h a t a necessary and sufficient condition for F(x, ip), given b y eq. (3.9), to have the a.symptotic behaviour (3.5b) is t h a t C be diagonalizable: in fact, if this is true, C can be written as

(3.15)

C:

i cjPj

(Pjv~

vj, I>~P~; bj,:P,.:)

]=r+l

(with no restriction we have set the first r eigenvalues to zero), and it turns 2/

out that B = 2p ~cT~Pj, while AP,

0 (j

r-~-1, ..., N), t h a t is A(k)l,.=."

i=r+l

is annihilated b y a n y linear combination of the eigenvectors of C. As a particul~r ease w e recover the single bound-state so]udon, w h e n r --~ 2Y -- 1 ; provided we can write C = c.,.P:v, it follows t h a t B : 2pc:.~Pv, A(k)l}=~ P ~ : 0, which means t h a t A(k)lk=~ is amlihilated just b y vectors proportional to the 5"-th eigenvector of C (*).

4. -

T h e '2 x 2 t r i a n g u l a r - m a t r i x

case.

Behaviour

of the associated

~ soliton)~

solutions.

We now turn to a deeper discussion of eqs. (2.3) for '2 × 2 triangular matrices. I n this ease we have (4.1a)

q)~t

v,%.x,

(*) It is easy to prove that couditions (3.15) are preserved by tlle evolution described by eq. (1.1) for triangular matrices.

232

(4.1b)

M. BRUSCttI, D. L E V I a I i d O. RAGNISCO

~t

~

v~y&~,

(4no)

where we h a v e set (4.2a)

W n = ~,

W ~ = YJ,

(4.2b)

B~

B~

(4.2c)

A I I - - A ~2 = ~,

= ~Vl,

= ~v~,

B a2 = fl,

W ~ = Z,

u=v~--v~,

w=

(v~+v~),

A ~2 : y .

E q u a t i o n s (4.1a), (4.1b) can be i m m e d i a t e l y integrated, providing (4.3a)

of(x, t) : 9)(x q- vlt, 0),

(4.3b)

~p(x, t) ~ ~f(x q- v2t , 0),

which m u s t satisfy the same a s y m p t o t i c conditions as W in eq. (1.1), t h a t is ~(q- ~ , t) ~ y~(q- 0% t) ~-- 0,

~(-- 0¢, t) = c o n s t ,

9 ( - - oo, t) ~ c o n s t .

As for eq. (4.1c), we notice t h a t it is an hyperbolic linear p a r t i a l differential equation, for which the Cauchy p r o b l e m is obviously well posed. F u r t h e ; m o r e , it is clearly easily solvable in the p a r t i c u l a r case vl ~ v2 ~ w, where it reads (4.4)

Z~ = ~Z.~÷ wz~.~ + fl[(~ ÷ ~)~x + (9 -- ~ ) ( ~ - -

q~)~] +

Y(~-- ~)~-

Looking for a solitary-wave solution Z(x, t) = Z(x q- wt, 0), we get

z=;(~-9)-

(~+~o)'+~(~o-~)~ .

The a b o v e equation, holding for a n y value of the a r g u m e n t x @ wt, implies t h a t the initial condition for Z c a n n o t be chosen arbitrarily, being instead d e t e r m i n e d b y those imposed on ~0 a n d ~o. W e can also notice t h a t , if we set ~ = 0, the solitary-wave solution for Z s t a y s c o m p l e t e l y undetermined, b u t it exists only if ~o and q~ fulfil the relation fl[(¢ + ~), + 1 ( 9 _

~)~] + Y(~-- ~) = O.

I n t h e general a n d m o r e interesting case, when the two p r o p a g a t i o n velocities are different, we are not able to solve the Cauchy p r o b l e m associated

614

L V. F A L O M K I N , F. N I C H I T I U , M. G. 8 A P O Z H N I K O V , ETC.

in the P S A gets positive at E~---- (130 --140) MeV. Km~L~, ~nd E ~ c s o ~ (~o) h a v e supposed t h a t the change of sign of the S-wave is c o n n e c t e d with the presence of a (in pion-nueleus interaction. The S-wave phase beh~viour can h a r d l y be d e t e r m i n e d b y the size resonance, because in the A r g a n d diag r a m the partial amplitude of the S-wave shows a clockwise loop not vice versa, us it should be for a resonance in t h e scattering potential. I t is w o r t h n o t i n g t h a t in t h e P S A a n d in the E D P S A the inelasticity param e t e r of the P - w a v e is a b o u t 1, u p : t o E~ ~ ( 7 0 - - 8 0 ) M e V , a n d the pion absorption in the elastic c h a n n e l only occurs in t h e S-wave. This s t a t e m e n t disagrees with the results of the optic:al model, in which t h e a b s o r p t i o n m a i n l y takes place in t h e P - w a v e . l~evertheless, in t h e low-energy region ( E . (70--80) lV[eV) the optical model is less consistent t h a n in the A3a-resonance region; therefore, the results of the phase-shift analysis are more reliable.

The authors express their g r a t i t u d e t o T. AI~GELESCU, ]=~. ~/[ACH, G. I~II~A~ O for valuable discussion, and to T . A . S ~ P O Z ~ z o v A a n d G. B. P0~TEC0RVO for help in writing up the m a n u s c r i p t . (e0) M. ERICSON and M. KRELL: Phys. ~ett., 38 B, 359 (1972).

•

RIASSUNTO

]~ stata fatta un'analisi in fuse indipendente dall'energia (PSA) della diffusi0ne di pioni da 4He, nell'intervallo energetico (60--260)MeV. Tutte le soluzioni possibili, derivanti dall'ambiguit~ dell'analisi in fase, sono state analizzate. Particolare attenzione stata posta nella scelta delle soluzioni con significato fisico. I risultati sono confrontati con quelli di un'analisi in fuse dipendente dall'energia (EDPSA) e con le previsioni del modello ottico.

A H a ~ H 3 ~a3OBMX C~BHVOB B ynpyFoM pacce~HHH HHOHOB H a

4He.

Pe3mMe (*). - - I~pOBO~HTC~[aHa~n3 He aaBHc~u~nx OT aHeprnH ~a3OBBIXC~BHFOBynpy_

roro paccear[a~ nrmHOB Ha 4He B o6aacTH aueprru~ (60--260)MaB. Anaanai~py~oTc~t Bce BOaMOmUbte pemeunr[, BOaanrammae Ha-aa Heonpe~eneHHocTrI qbaaoBoro anaan3a. Oco6oe •HnMaHrte y~en~erca Bbx6opy dpnarI~ecroro pemeHrt~. BbI~mcnennbm ~aaoBr~Te c~Bnra cpaBnnaamTc~ c peayabTaraMr~ aHa~naa 3aBrm~mnx OT aaeprHrt qbaaOB~,IX c~Bnroa H c npe)~craaannaMri OnTH~ecro~ Mohenm

(*) llepeee3eno pec)amlue~t.

234

~f. BRUSCIII, D. LEVI

and

O. I~AGNISCO

the same; in this way we get (4.9a)

q~(x, to) = -- 2p{1 -- tgh [ p ( x - - ~dto))]},

(4.95)

v(~, to) = - 2v{1- tgh [p(x- ~(to))]},

(4.9c)

X(x, to) ------ 2pJV(to) sech [ p ( x - - Sdto))] sech [ p ( x - - $~(to))],

in which case the m a t r i x C reads (4.10)

4pJV(to) exp [p[G(to) ÷ G(to)]]

C(to) =- 2p exp [2p~dto)] 0

2p exp [2p~(to)]

The time evolution of the functions % ~v, Z is then given, in terms of the time evolution of ~ , ~2, JV', b y means of eq. (4.7), b y

(4.1]a)

~1(t) = ~l(to)--Vl(t--to),

(4.~1b)

G(t)

(4.1 ~c)

JV(t) = JV'(to) exp [pue(t - - to)] -~

= G(to) - - v ~ ( t - - to),

[

sinh [ P ( ( u - - u c ) / 2 ) ( t - - t o ) ] _ (1 + ~ ) e x p - - p p ((u -

uo)/2)

G + ~

(t--to)

•

~inh [p((~ + ~d/2)(t- to)] 1

p((u + ~,~)lO)

j,

where we have set G = G(to)--~l(to). Equations (4.9a), (4.9b)~ (4.11a), (4.115) show t h a t the functions qJ(x, t) and y~(x, t) are (, antikinks >~translating with velocity v~ and v2 and vanishing exponentially when x - + - ~ ~ . Equations (4.9c), (4.11c) show in t u r n t h a t Z(x, t) is a solitary wave vanishing exponentially as ix]-+ ~ and centred at _ 1 ( ~ ~_ G) (the centre of muss of the two degenerate ())~ its amplitude being modulated b y the time-dependent factor JV(t). I t is also interesting to report and study the equations giving the time evolution of the potential m a t r i x of the associated Schr6dinger equation (3.4) (4.12a)

Q~' = v~?~' ,

(4.12b)

Q~2 = v 2"5x ,o~

co Y0

(4ol 2C)

Qi2 :

(~Q12 _~_ ~2 ( Q 2 2

QII) . I d x ' Q i ~ ( x " t) -I- wQ~" + x

+ fl[(Qil +

co

Q22)x~_((222_Qi1)fdx,(Q~2_Qill(x,,t)]_~,(Q22_Q11), x

235

EVOLUTION :EQUATIONS ASSOCIATED ETC.

where t h e coefficients are defined as in eqs. (4.2b)~ (4.2c); eqs. (4.12a), (4.12b) ~re first-order w a v e e q u a t i o n s a n d eq. (4.12c), once solved (4.12a), (4.12b), is a lirlear integro-differential e q u a t i o n . T h e 2-fold d e g e n e r a t e b o u n d - s t a t e solutions r e a d (*)

{p[x-- ~l(t)]}

(4.13a)

Q11(x, t) :

(4.13b)

Q~2(x, t) = - 2p 2 sech 2 ( p [ x -

(~.13c)

Q~'(x, t) = - 8p~JV'(t) sinh { 2 p [ x - - ~(t)]}.

-

2p ~ seeh 2

,

$2(t)]},

• {cosh

[2:,(x- ~(t))] + cosh [p(x-- ~(t))]}-~,

where we h a v e set ~(t) = ~2(t) -- ~(t), a n d ~(t), ~(t), JV(t) evolve in time according to eqs. (4.11). Obviously, t h e functions Q l i ( x , t) a n d Q22(x, t) r e p r e s e n t s o l i t a r y w a v e s c e n t r e d at x = ~(t) a n d x = $2(t), p r o p a g a t i n g with v e l o c i t y v~ a n d v~. More i n t e r e s t i n g is t h e b e h a v i o u r of Q~(x, t). F r o m f o r m u l a (4.]3c) it follows clearly t h a t Q~2(x~ t) is an o d d f u n c t i o n of t h e v a r i a b l e x - - ~ ( t ) ; moreover~ it bears t w o e x t r e m a l points (one a b s o l u t e m a x i m u m a n d one absolute m i n i m u m ) l o c a t e d at t h e (real) roots of t h e e q u a t i o n cosh 2 [2p(x -- ~(t))] -- eosh [p~(t)] cosh [2p(x -- ~(t))] -- 2 ~- O, t h a t is, b y s e t t i n g y = x - - 2 ( t ) , cosh

a(t)-

(2py±) =

cosh [p~(t)],

~ o ~ _ U, - S ] / 2 . [~(t) -:- ~¢~(t)

D u e to t h e t i m e e v o l u t i o n of t h e f u n c t i o n e(t), the t w o e x t r e m a l points m o v e as t w o pointlike particles of u n i t a r y mass in a r e p u l s i v e - p o t e n t i a l field, d e p e n d i n g on their m u t u a l dist,qnce y = y + - y_---2y~_ a n d given b y V(y) = 2p2u 2 c_osh (py) ~-_2 (cosh (py) + 5) 2 '

(4.14)

a c c o r d i n g t o the f o r m u l a

(4.15)

y(t) = p in

~(t) + ~'-'(t) + s ~

+

~(t) + ~ ( t ) + s ~-1

(*) As seen in eqs. (4.5), (4.8), the single-bound-state solution is nothing new, as it corresponds to initially set either ~ or ~ equal to zero, thus obtaining for the remaining variables the usual one-soliton solution.

M. BRUSCIII, D. LEVI and o. RAGNISCO

236

The large-It I b e h a v i o u r of the function c~(t) implies t h a t

y(t),tl'Z~,ul[t-tol[l

+ a(~)].

So, the two particles are infinitely s e p a r a t e d in the r e m o t e past, t h e n t h e y a p p r o a c h each other as t i m e goes by, a t t a i n i n g their m i n i m a l distance at t = t = to-- to/u, where their relative velocity is zero; afterwards t h e y reverse their motion, a n d their distance increases again to @ ~ as t -~ @ ~ .

5. -

Conclusions.

We h a v e seen t h a t triangular matrices provide a v e r y simple subclass of n o n - H e r m i t i a n matrices to whieh s y s t e m (1.1) can be specified. Tile resulting evolution equations, even in the very special ease eonsidered t h r o u g h o u t this paper, are not all solvable b y m e a n s of s t a n d a r d techniques, thus m a k i n g it worthwhile exploiting the applicability of tile I S T to obtain particular solutions. As an e x a m p l e we h a v e r e p o r t e d a n d investigated, in the 2 × 2 m a t r i x ease, the solutions eorresponding to one b o u n d state and two degenerate b o u n d states in the associated Sehr6dinger equation. As for the t w o - d e g e n e r a t e - b o u n d - s t a t e solution, the diagonM elements of the m a t r i x potential evolve in t i m e as two solitons of constant speed, while tile off-diagonal one exhibits two b u m p s , whieh, in the reference f r a m e translating with the centre of mass of the two (~diagonal >> solitons, interact m u t u a l l y via an a s y m p t o t i c a l l y vanishing repulsive force. W e point out t h a t the free-soliton b e h a v i o u r of tile diagonal elements, as well as the i n t e r a c t i n g - b u m p b e h a v i o u r of the off-diagonal ones are a general p r o p e r t y of the c o m p l e t e l y - d e g e n e r a t e - b o u n d - s t a t e t r i a n g u l a r - m a t r i x solutions, w h a t e v e r be the r a n k of the matrices considered, as t h e y depend merely on the structure of eqs. (2.3). Of course, the n u m b e r of b u m p s a n d the particular f o r m of their interaetion does instead depend on the r a n k or, to be more precise, on the difference between the column and the row indices of the element considered. We h a v e also to notiee the strong similarity between the m o t i o n of our b u m p s a n d t h a t of the ones observed b y CALOGE]~O and DEGASPERIS when s t u d y i n g degenerate solutions of the b o o m e r o n equation (s), as well as t h a t of the poles of the H a m i l t o n i a n density associated to the two-soliton solution of the (scalar) sine-Gordon equation investigated b y BOWTELL et al. (9). These analogies suggest a possible direction of research, aiming at defining the proper H a m i l t o n i a n density also for the b o o m e r o n equation and for eq. (2.3)

(s) F. CALOGERO and A. DEGASPERIS: Lett. Nuovo Cimento, 19, 525 (1977). CT. BOWTELL and A. E. G. STUAnT: I)hys. Rev. D, 15, 3580 (1977).

(9)

237

EVOLUTION ]~IQUATIONS ASSO(!IAT]~I) ETC.

a n d a t e s t a b l i s h i n g t h e relatioH b e t w e e n its poles (if a n y ) a n d C a l o g e r o ' s a n d our bumps, respectively. On the other hand, the similarity between the two-solitou sine-Gordon s o l u t i o n s an(1 t h o s e p r o v i d e d b y CALOGERO a n d h e r e u b o v e , s t r o n g l y s u g g e s t s t h a t eq. ( 1 . l ) for m a t r i c e s of r a n k 2 m a y c ( m t a i n in s o m e w a y t h e s i n e - G o r d o n e q u a t i o n (see also ref. (6)), t h u s :fllowing us to ~'et its f f e n e r a l i z a t i o n s b y s t u d y i n g eq. (1.1) for m a t r i c e s of r ~ n k h i g h e r th:~n 2.

W e w ~ r m l y t h a n k Profs. F . enliglltenin~" d i s c u s s i o n s .

•

C.xLO(~Et~O a m l A. ])EGASPEI'~IS for m a n y

RIASSITNT()

In questo ~rticolo si studia~m le equazioni di evoluzimm ~l.ssoci~l,e ~ll'equ,~zionc matriciale di Schr6dinger Hel e,~so di matrici tri~ulgoh~ri, un~ ~.l~sse assai semplice di matrici non herrniti~um. T~fli equazioni di evoluzio~m, ~nclw se ]im~ri nel caso qui consider~to, non sono t u t t e risolubili con tecniche sta.n&u'd: di qui l'interesse di utilizzare la tccnica delb~ trasfm'm~t~ spcttra,h; ittversa, per trowu'e dellc soluzioni p~rticob~ri. Si es~unin~ in dett~glio il ca so di matri~i tria,~goh~ri 2× 2.

YpaBHeull~l 9BOYIIoUttH~ CBn3aHHh]e C npo~YteMo~

lllpe~uHrepa ~laa TpeyroYihUblX MaTpHU

n pemaeMbm c HOMOIE[blO o6paTnoro eneKTpaYzbU0FO npeo6pa3oBaHn~l.

Pe3mMe (*). - B 3TOil pa6oTe Mbl pacCMaTpHBaeM ypanHeHi4n 3BO.alOIIrlH, CB~13arIHble C n p o 6 - ~ e M o ~ l l l p e , ~ z H r e p a , a . ~ TpeyFOYlbHblX MaTpHLt, OqeHb HpocToFO nO~K~acca He3pMIdTOBblX MaTpnH. ~Tri ypaBHeHH~ 9BOJ'ItOIIHI4, XOTJ:I OHI4 ~BYI~ItOTC~ YlI,IHe~HblMI4 B paCCMOTpeHHOM B p a 6 o T e c.~y'-Iae, He MOryT 6bITb pellJeHbI C HOMOI/3blO CTaH~2apTHO~ TeXHHKVI. B CBJ/3H C 9THM HCrIOYlb3yeTcfl o6paTHOe cneKTpaYlbHOe rlpeo6pa30BaHHe )l.rl~

noY[y'~eHHfl qaCTHblX pemeH[~. I/Iccne,ayeTcn c.~yqah 2 × 2.

(*) Hepeee3eHo peOaKque(t.