I L NUOVO CIMENTO

VOL. 75 B, N. 2

11 Giugno 1983

A Simple Approach to the Hamiltonian Structure of Soliton Equations. III. - A New Hierarchy. M. BOITI ~ n d G. Z. T u (*) Dipartimento di _Fisica dell' Universith - .Lecce, Italia Istituto -Yazionale di ~'isica ~Yzecleare - Sezione di Bari, Italia

(ricevuto il 12 Gennaio 1983)

Summary. - - A simple method for studying the H a m i l t o n i a n structure of the soliton equations is applied to a new hierarchy of p a r t i a l differential equations related to a Z h a k a r o v - S h a b a t - l i k e spectral problem. These soliton equations are shown to be integrable H a m i l t o n i a n systems and can be described as motions on a symplectie K/~bler manifold.

PACS. 02.30. - Function theory, analysis.

l. - Introduction.

T h e s o l i t o n e q u a t i o n s (1) h a v e a t l e a s t t w o i n t e r r e l a t e d p r o p e r t i e s of f u n d a m e n t a l i n t e r e s t : i) t h e y c a n b e s o l v e d b y t h e use of t h e s p e c t r a l t r a n s f o r m (ST) d e f i n e d f o r t h e a s s o c i a t e d s p e c t r a l p r o b l e m (2); ii) t h e y c a n b e g r o u p e d i n t o h i e r a r c h i e s w i t h a c a n o n i c a l s t r u c t u r e (3.5). (*) On leave of absence (up to A p r i l 1983) from Computing Centre of Chinese Academy of Sciences, Beijing, China. (1) See, for example, R. K. BULLOUGH and P. J. CAUI)REY: Solitons in Topics in Current Physics (Berlin, 1980). (3) See, for example, F. CALOGERO and A. DEGASPI~RIS: Spectral Trans]orm and Solitons (Amsterdam, 1982). (3) F. MAGRI: A geometrical approach to the nonlinear solvable equations, in Lecture Notes in Physics, No. 120, edited b y M. BoITI, F. P~MPINELLI and G. SOLIANI (Berlin, 1980). (a) A . S . FOKAS and B. FUCHSST]~INER: .Lett. Nuovo Cimento, 28, 299 (1980); Physica D, 4, 47 (1981). (5) A. S. FOKAS and R. L. ANDERSON: J. Math. Phys., 23, 1066 (1982). 145

146

~i. BO~TI and G. z. :ru

I n this paper we are m a i n l y interested in the second p r o p e r t y of soliton equations. I n general, the soliton equations of a hierarchy related to a spectral p r o b l e m are H a m i l t o n i a n systems and their Hamiltonians are constructed from their infinite set of conserved quantities, which are involutive with respect to the Poisson bracket. Moreover, it has been shown t h a t the soliton equations m a y be identified with the group of motions of a special geometric structure called a symplectie K/~hler manifold (~). I n this scheme the squared-eigenfunetion operator JL of the related spectral problem (or more precisely its adjoint) is at the same time a ~ i j e n h u i s operator (8) (or h e r e d i t a r y s y m m e t r y (s)) and a strong s y m m e t r y (4,5) (or recursion operator (7)) for the soliton equations considered. I n some previous papers (s) one of the authors (GZT) proposed a m e t h o d for showing t h a t the soliton equations are endowed with this canonical geometric structure. This method, in contrast with other methods t h a t use the properties of the spectral d a t a (9), uses directly the associated spectral problem. The procedure has been successfully applied to the KdV, A K N S and K a u p Newell hierarchies, to a new hierarchy related to t h e SchrSdinger-like spectral p r o b l e m - - y ~ ~ (qo 9 q1~-1_ ~)y = 0 (s) and to its generalization --y~= -~ ~- ( ~ q t 2 - z - - X ) y = 0

with N @ 1 independent potentials q~ (10).

o

I n this p a p e r we consider the 2 • (1.1)

spectral problem

T== UT

with (1.2)

U(x, t; 2) = --i2a~-~- P ( x , t) -F i2-1Q(x, t) ,

where P is an off-diagonal 2 X 2 m a t r i x and Q is a free 2 x 2 matrix. F o r t h e sake of simplicity, we reduce in the spectral problem (1.2) the six scalar independent potentials to three independent scalar potentials u, v and s b y choosing (1.3)

(~) (~) 88, (s) (9)

P ( x , t) = u(x, t)al

B. FVCHSS~I~CV.R: Nonlinear Analysis T M A , Vol. 3, 849 (1979). P. J. OLVER: J. Math. Phys., 18, 1212 (1977); Math. Proc. Camb. Philos. Soc., 71 (1980). G.Z. Tu: 2Vuovo Cimento B, 73, 15 (1983) (I); Sci. Exploration (to appear) (II). B. G. KO~COPOLCHENKO:J. Phys. A, 14, 1237 (1981). (10) M. BOITI, C. LADDOMADA, F. PEM:PINELLIand G. Z. Tu: O~ a new hierarchy o] Hamilto~dan soliton equations, to appear in J. Math. Phys. (N.Y.).

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and

(1.4) The The systems The

Q(x, t) = s(x, t){~3 + iv(x, t){~ . aps are the 2 • Pauli matrices. soliton equations related to this spectral problem are Hamiltonian with involutive IIamiltonians and satisfy the Magri's scheme. first equation in the h i e r a r c h y is

(1.5)

'at ---- u~ -]- 2 v ,

(1.6)

vt

:--2us

(1.7)

st

---- - - 2 " a v

,

a n d it can be cast into the H a m i l t o n i a n form j 8~f

(1.8)

q'=

Tq-'

where q is the vector field

(1.9)

q=

J is the symplectic operator

(1.10)

J =

--

-- didx

--2"a

2u

didx]

J/~ is the H a m i l t o n i a n functional co

(1.11)

and

(1.12)

~

\~/~.!

is the variational derivative with respect to q.

,

148

~. BOITI and G. z. TU

2. - B a s i c n o t i o n s and o u t l i n e o f t h e m e t h o d .

The basic notions referred in this section a n d used t h r o u g h o u t the p a p e r can be f o u n d in a more extended and complete version in ref. (aa,11-ls). The q(x, t) defined in (1.9) is considered as a point in the linear space M of the vector-valued field functions regarded as functions of the space co-ordinate x only. q(x, t) is supposed to be defined on the whole real x-axis a n d to vanish (rapidly) as x--~ :~ c~. To each point q in the configuration space M one associates a t a n g e n t space T~ of smooth vector field on the real line

~(~:, t) = (~1(~, t), ~.~(x, t), ~(~:, t)) ~ (controvariant fields) satisfying the same b o u n d a r y conditions as q. The cotangent space T* is the dual of Tq via the continuous bilinear form co

(2.1)

fl~(x,t)~(x,t)dx,


geT

, fieT*.

--co

The elements of Tq are n a m e d covariant fields. We deal with functions G(q) -- G(q, qx, ...) defined in M and attaining values in M, T~, T*, or in a space of operators which are assumed to be differentiable according to the definition of Gateaux

(2.2)

a'(q)[~] = ~d ~(q + ,~) ~0 .

G'(q)[7] is also called the directional derivative of G at the point q in the direction a. A n operator ] : M - + 2"* is said to be a gradient operator (or potential operator) if it can be expressed as the variational derivative of a f u n c t i o n a l / E : M - + C: (2.3)

]-

3E

3q.

G. Z. Tu: J. Phys. A, 1S, 277 (1982). j. M. GEL'FAND and I. YA DORFMAN: tZunct. Anal. Appl., 13, 248 (1979). H. H. CHV,Z% Y. C. LEE and C. S. LIu: Physica Scripta, 20, 480 (1979). M. D. ARTHUR and K. M. CAS]~: J. Math. Phys., 23, 1771 (1982). M. BRUSCHI and 0. I~AGNISCO:The Hamiltonian structure o] the ~to~.Abelian Toda, hierarchy, preprint (Roma, 1982). (11) (12) (t3) (14) (15)

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I f one considers a point transformation ~ = ~(g) between the configuration space M a n d a new configuration space _M, the c o n t r a v a r i a n t fields g and the covariant fields fl transform according to the formulae

(2.4)

=

(2.5)

~'(q)[~] ~ ~(~(~))~,

= ~'(q)*@ =- ~*(~(q))ti,

where + means adjoint with respect to the bilinear form < , > and the operator ~ is defined b y (2.6)

~(~(q)) = ~: ( ~ ) ~ ,

is the operator of total differentiation with respect to x and (2.7)

Oj ~ ~q,.

with q.~ ~ ~Jq. The variational derivative operator 8/3~/ under the point transformation = ~(q) transforms as follows:

(2.s)

~q = ~+(~(q)) @

in agreement with the covariant character of a potential operator.

An operator

J(q)'T*-+Tq is called symplectic if it is skew symmetric: <~, J ~ > = - < ~ , J ~ > ,

(2.9) and if the bracket

{~, fl, 7} ----<~, J'(q)[J(q)fl]Y> satisfies the J a c o b i identity

(2.1o)

{~, ~, r) + (~, r, ~) + {y, ~, ~) = 0.

A symplectic operator J(q) enables us to introduce the following Poisson bracket for two scalar operators F , G: M - + C:

which is a n t i s y m m e t r i c and satisfies the Jacobi identity.

150

M. BOtTI and G. z. TrY A n y equation t h a t can be written in the form

(2.12)

q~---- J - ~q

is called a ~ a m i l t o n i a n system. E q u i p p e d with all these notions t h a t generalize the usual concepts of t h e classical Hamiltonian mechanics to the infinite-dimensional systems, we expose a m e t h o d for investigating the H a m i l t o n i a n structure of the soliton equations. I n t h e following section we will show t h a t the soliton equations related to the spectral problem q u o t e d in the introduction are a linear combination with arbitrarily t-dependent coefficients of the h i e r a r c h y of equations (2.13)

qt ~- JZ~l(q)

(n : 0, 1, . . . ) ,

where J is a symplectic operator, L an integro-differential operator and ](q) a suitable vector field in M. I f we define the formal series co

(2.14)

~(q; ~)--~ ~A-"Z"](q),

the specific structure of t h e flows of eqs. (2.13) can be uniquely determined b y assuming t h a t t h e covariant operator F satisfies the operatorial i d e n t i t y

(2.15)

L F = A F - - ~l(q).

The soliton equations (2.13) are i t a m i l t o n i a n systems if L'](q) is a potential operator for a n y n or, equivalently, if 2 ~ is a potential operator. I t is well known that, in general, the itamiltonians of the soliton equations are related to the infinite set of polynomial conserved quantities and, moreover, t h a t these quantities can be obtained directly from the associated linear spectral problem (1~). According to a well-known general procedure, we introduce a conserved q u a n t i t y ~ t h a t can be e x p a n d e d into a formal series (le) co

(2.16)

~ = ~ ~--~.,

t h e coefficients of which, 5(f~, are recursively d e t e r m i n e d b y a Riccati e q u a t i o n obtained directly from t h e spectral equation T~ : U T . (1~) See, for instance, T. M. -A-LBERTY,T. KOIKAWX and R, SASAKI: Physica D, 5, 43

(1982).

A

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We show t h a t the spectral equation can also be used to derive a linear differential equation in the x variable for the variational derivative ~%P/~q. The only technical tool t h a t we need is the chain rule for the variational derivative r e p o r t e d in (2.8). We use this differential equation to compute 35 &~%f/~q and we get the equation

(2.a7)

35~- =

~

- i~](q).

This equation furnishes, in the same way as eq. (2.15) for ~ , a recursion relation for t h e coefficients ~ ' . / ~ q in the series &;gf/3q. I t results t h a t

(2.18)

~q -- O, &ff~"

(2.19)

~'~f2,,+1 3q -- iL./(q)

(n = O, 1, 2, ...).

Therefore, a t the same time, we are able to prove the t t a m i l t o n i a n character of the soliton equations, to derive b y a recursion relation the explicit form of their t t a m i l t o n i a n s and to show t h a t all of t h e m are local partial differential equations in spite of the integro-differential character of the operator 35. ~ o r e o v e r , since the operators J and 35 satisfy the so-called first coupling condition (8) (2.20)

JL

=

.L+J ,

the infinitely m a n y conserved quantities {ovg.} are in involution with respect to the Poisson bracket { , } defined in (2.11). I n fact, the coupling condition (2.20) induces on the Poisson bracket

(2.21)

{~'~'., ~ , . } = _ <35-/(~), JZ'nl(q)>

the following recursion formula:

(2.22) and, b y iteration ( n < m), one finds

(2.23) and t h e n (2.24)

{Jeff, 5/zm} ~ 0

on account of the skew s y m m e t r y of the Poisson bracket.

152

~. BOITI and G. Z. TU

3. - A differential e q u a t i o n for

3W/3q.

L e t us consider the 2 X2 spectral problem (3.1)

k~ = U T

with (3.2)

~=

kCt"

and

a n d let us suppose t h a t the entries of U, a, b and c depend linearly on the v e c t o r - v a l u e d potential q ~ (u, v, s) T in the following way:

(3.4)

= ;,q + ;,o,

where the 3 ;~ 3 nonsingular m a t r i x y and the vector Yo are rational functions of the spectral p a r a m e t e r ;t. The so-specified spectral problem includes as a special ease the spectral problem defined in the introduction b y eqs. (1.2)-(1.4). B y introducing the projective variable (16) (3.5)

z = T2/T1,

the spectral equation (3.1) is transformed into the Riccuti equation (3.6)

Z. :

c --2aZ

-- bZ ~ .

According to the A K N S (17) m e t h o d the t-dependence is fixed b y compelling 7 t to satisfy the auxiliary spectral equation (3.7)

~t :

V~

with V ~ V ( x , t; 2) a 2 • m a t r i x with fixed rational singularities in )t. The compatibility condition for the principal spectral equation (3.1) and

(17) M. J. ABLOWlTZ,D. J. KAUP, A. C. NEWELLand It. SEG17R:Stud. A p p l . M a t h . , 5 3 , 249 (1974).

A

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TtIE

IIAMILTONIA~"

STRUCTURE

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153

the auxiliary spectral equation (3.7) furnishes to so-called Lax-pair representation (3.8)

U~-- V=+ [U, V] = O.

I t is easy to show (see e.g. ref. (as)) that the quantity

H = a -}- bZ

(3.9)

is a conserved density whatever the spectral operator V satisfying the Lax representation m a y be. We are interested in computing the variational derivative of the corresponding conserved quantity -t-co

gf ~jdx H.

(3.10)

-co

It is convenient to introduce the auxiliary function

R = bZ

(3.11)

and to consider a, b and c as functionals of the three independent functions H, Z and R: (3.12)

a = H--R

(3.13)

b = R/Z,

(3.14)

c = Z~ ~ 2 H Z - - Z R .

Let us compute for this specific point transformation the operator $/" in (2.6). In the following equation the matrix elements of r are explicitly defined by using a self-evident notation and computed:

(3.15)

$/~

= /~(b)

--R/Z ~

$Pz(b) ~ ( b ) ] :

\~(c)

~(c)

~(c) l

2z

D+2~--R

According to the chain rule (2.8) for the variational derivative we get

(3.16)

11 -

[~/~H~ ( |8/SZ] =

I I N u o v o Cimento B .

1

0 --R/Z ~

2Z --D +

) [~/~a~

9

154

~. BOITI and G. z. TU B y applying this operational identity to ~%f, it results t h a t

(3.17)

_4 ~ 1 - - 2 Z C ,

(3.18)

B ~ Z--Z~C

(3.19)

C : 2(a-~ bZ)C--b,

,

where A -= ~.;,~/~a, B = ~%f/~b and C = ~J%f/~c. B y applying once more t h e chain rule (2.8) to t h e point t r a n s f o r m a t i o n (3.4), we obtain t h a t

~q--

9

F r o m (3.17), (3.19) and (3.6), it is easy to verify t h a t

(3.21)

(:):(i ?ot(i) --2a

C

0

2a]

which together with (3.20) yields (3.22)

~J/f ~ = ( y T M ? ~ - ' ) ~q ,

where M is the m a t r i x appearing ia (3.21). I t is worthwhile stressing t h a t the conserved q u a n t i t y ~ f and the differential equation satisfied b y ~Jt'~/~q are determined only b y the principal spectral problem without a n y reference to the specific form of the auxiliary spectral problem. Moreover, up to this point the form of the y - m a t r i x and the ?o vector h a v e not been specified.

4. - A n e w h i e r a r c h y o f s o l i t o n equations.

L e t us make the following specific choice for the elements a, b and c of the spectral operator U: (4.1)

a = - - i ~ -~ i ~ - l s ,

(4.2)

b = u ~- i ~ - l v ,

(4.3)

e = u--i~-Iv

A 8IMPL]~ A P P R O A C H TO T H E H A M I L T O N I A N

S T R U C T U R E ETC.

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155

a n d let us suppose t h a t t h e spectral o p e r a t o r V is a p o l y n o m i a l of odd order n = 2m q- 1 ( m ~ 0 ) in the spectral p a r a m e t e r 2:

(4.4)

V ( x , t; ~) = ~ V~(x, t) ~ - ~ J=0

with traceless m a t r i x coefficients

(4.5)

V~ = d ~ + 8 9

lf~..

B y e q u a t i n g to zero the coefficients of the powers of 2 in t h e L a x represent a t i o n (3.8), we get the following recursion relations for the dr, ej and fj's:

(4.6)

dOx =

0

(4.7)

dj~ :

--iuf~

(4.8)

eo

=0

(4.9)

el

= 2iudo ,

(4.1o)

ej+l :

(4.11)

1o

(4.12)

1~+1 = slj_~ § i ej. § 2ivd~_~

~ ivej_l

sej_l--lf~x-~

(j :

2iudi

1, . . . , n ) ,

(~ = 1, ..., n - - l )

,

= I1 : 0 , (j = 1 , . . . , n)

and the evolution equations (4.13)

~t :-- fn+l ,

(4.14)

~)t

~-

--

86n

(4.~5)

8 t

=

--

Wen ,

,

where we h a v e introduced for convenience ].+1. E q u a t i o n s (4.6) a n d (4.7) with a convenient choice of t h e constants of integration can be i n t e g r a t e d to (4.16)

d o = - - i

(4.17)

dr :

,

-- iI(ufj--

vej_,) ,

where (4.18) --vo

x

156

~. BOITI and G. z. TV

B y inserting (4.16) a n d (4.17) in the recursion relations for the e2s a n d t h e ]2s, it is easy to p r o v e b y induction t h a t (4.19)

e2k : 0

(k : 0, ... , m) ,

(4.20)

/2k+~ = 0

(k = 0, ..., m ) .

T h e recursioa relations for t h e ]j a n d er

can be cast into the f o r m

|,

(4.21)

\--ve~/

= | \ --2vu ]

\--re d

\--ver

w h e r e 25+ is the adjoint w i t h respect to the bilinear f o r m <, > of the integrodifferential operator -- ~ 2 ~- uI~--

(4.23)

25 =

2ulv+u~ ~ s

-

-

892 s + 2 u l s u

8

89 u -- Iu~-- 2Iv

2I~s

__ 1 2 v + 2 u l u v ~

'0

]. |

2Iuv

T h e first t e r m (4.21) in t h e recursion relation can be expressed as follows:

(4.24)

- - sel

= J25

,

\--re1/ where (4.25)

or =

--

- - 2

One easily verifies t h a t J is a symplectic o p e r a t o r according to t h e definition given in sect. 2. Since t h e two operators J a n d 25 satisfy the first coupling condition (4.26)

J25 ~ 25+J,

t h e evolution equations (4.13)-(4.15) can be w r i t t e n in the canonical f o r m a n n o u n c e d in sect. 2: (4.27)

qt = J L ' ~ ] ( u ) ,

A SIMPLE

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157

~here (4.28)

](u) :

.

I n addition to the first equation in the hierarchy for n ~ 1 which is given in the introduction we write explicitly also the soliton equation for n ~--2:

(4.29)

ut:

~u2u~--a-u~ ~ - - 8 9

2u~s + 2 u s ~ - ~ 2vs ~ v u 2,

(4.30) (4.31)

5. -

The

Hamiltonian

structure.

The explicit form of the infinitely m a n y polynomial conserved quantities for the soliton equations in the h i e r a r c h y (4.27) can be obtained b y solving the Riccati e q u a t i o n (3.6) with the following formal expansion in 2 for Z:

(5.1)

z = ~: A - - z . .

B y substituting (5.1) into (3.6) and b y using (4.1)-(4.3), we can determine the coefficients Z . b y the recursion formula

(5.2)

Zo -~ 0 ,

ZI = 89i u , n

n--1

k~0

k~O

(5.3) (n = 1, 2, ...) . Consequently the conserved density H ~ a q- bZ can be e x p a n d e d in powers of 2: (5.4)

• = -i~

+ ~f H . ~ - n~0

and t h e polynomial conserved densities H~ are given b $ t h e equations (5.5)

Ho ~ 0 ,

(5.6)

H . ~ i s ~ . l + u Z . -k- i v Z . _ l

(n=1,2,...).

158

M. BOITI and ~. z. Tv

:From (3.22) with the specific choice of the m a t r i x y corresponding to eqs. (4.1)(4.3) we derive the following differential equation for the variational derivative ~gg/~q:

(5.7)

~

=

( o \

--2+~-~v

2u

:o:)

~q "

Since q with all its x-derivatives is supposed to vanish at x ~ • c~, eqs. (5.2), (5.3) and (5.5), (5.6) furnish the b o u n d a r y value of ~gf/Sq at x - - - - 4 - c ~ . Precisely it results t h a t

- -~q - + - -

i~

as x -+ • c<~.

B y applying the operator L to 8J~ff/Sq and b y using eqs. (5.7) and (5.8), we can easily obtain the equation L ~gf

(5.9)

"2 ~#f

t h a t we used in sect. 2 to prove t h a t the soliton equations in the hierarchy (4.27) are ]~amiltonian systems. E q u a t i o n s (2.19), together with the reeursion relation ( 5 . 2 ) - ( 5 . 4 ) a n d eqs. (5.5), (5.6) furnish the explicit form of the corresponding ]~amiltonians. Moreover, it results, in spite of the integro-differential character of the ~-operator, the soliton equations are local differential equations.

6.

-

The

geometrical

structure.

L e t us recall t h a t a linear operator 2q(q) mapping the tangent space T~ into itself is called a ~ijenhuis operator or a h e r e d i t a r y s y m m e t r y if

(6.1)

2V'(q) [N(q)a]fl - -

N(q)N'(q)[a]fl

is s y m m e t r i c with respect to ~ and fl (3,e). An operator 2V:Tr T~ is a strong s y m m e t r y (4-s) or a recursion operator (7) of an evolution equation qt = p(q) if it is invariant along the traject o r y of t h e vector field p(q). This is the case if and only if N satisfies the operational equation (6.2)

N'[s] - - s'_Y § N s ' = 0 .

A SIMFLE

APPROACH

TO THE

HAMILTONIAN

STRUCTURE

:ETC. - I I I

A h e r e d i t a r y s y m m e t r y and a symplectic operator are said led ,) (3) if a n d only if (6.3)

N J ----J N +

(6.4)

<~, _,v'[r

159

to

be (( coup-

,

- <~, 2r

+ <3, _,v'[j~]m> + + <3, 2r

- -
= o.

According to Magri general result (3) ( l s t t h e o r e m of Magri) the nonlinear evolution e q u a t i o n in the h i e r a r c h y (6.5)

q~ = J.L~J(q)

with J a s y m p l e e t i c operator a n d h r = Z + a h e r e d i t a r y s y m m e t r y satisfying the first coupling condition (6.3) are H a m i l t o n i a n s y s t e m s w i t h H a m i l t o n i a n s in involution if /(q) a n d .EJ(q) are p o t e n t i a l operators. I f in addition (2nd t h e o r e m of Magri) (3) j a n d N = L + satisfy t h e second coupling condition (6.4), ~V is a strong s y m m e t r y for all the equations in t h e hierarchy (6.5). B y direct, a l t h o u g h tedious, c o m p u t a t i o n s one can v e r i f y t h a t t h e opera t o r L + defined in sect. 4 is a h e r e d i t a r y s y m m e t r y a n d t h a t L + and J satisfy also t h e second coupling condition (6.4). Therefore, we recover the t t a m i l t o n i a n structure t h a t we f o u n d b y direct m e t h o d s a n d we get t h a t t h e soliton equations can be t h o u g h t as groups of m o t i o n of a special geometric s t r u c t u r e defined b y t h e coupled operators L + a n d J , which has b e e n called a s y m p l e c t i c Ki~hler manifold (3). The m a i n a d v a n t a g e of the direct m e t h o d we propose t u r n s out to be its simplicity a n d its constructive c h a r a c t e r since it allows us to c o m p u t e explicitly the p o l y n o m i a l conserved quantities a n d to relate t h e m to the H a m i l tonians of t h e soliton equations.

The a u t h o r s aknowledge m a n y useful discussions with Prof. F. PEIVIPINELLI a n d Dr. C. LADDOIV[ADA.

9

RIASSUNTO

Si applica un metodo semplice per lo studio della struttura hamiltoniana delle equazioni sotitoniche ad una nuova gerarchia di equazioni alle derivate parziali collegate ad un problema spettrale del tipo Zakharov-Shabat. Si mostra che queste equazioni solitoniche sono sistemi hamiltoniani integrabili e possono essere descritte come moti su una variet~ simplcttica di K~hler.

160

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