Let 93~ (r, d, e), where r > 0, d > 0, 0 < e < ~r_,denotethe set of all convex open bounded polygons M (cB')
such that:
1) the v e r t i c e s of M belong to Z ~,
2) the coordinate origin 0 belongs to M,
3) the distance of 0 from the boundary of M is g r e a t e r than r ,
4) the length of each side of M is g r e a t e r than d,
5) the size of each angle of M is g r e a t e r than n - e.
THEOREM. Let A(~ o} be an invertible o p e r a t o r . Then ~[r (>0)~{d ('~0) ~te(~ ( O , n ) i ~ I C ( ~ O ) ~ M ( ~ g ~ ( r , d , e ) ) ,
the o p e r a t o r PM+>~APM is invertible as an o p e r a t o r in Hom ('PM~(T), PM+~tI-~(T)), and
'
I (PM+× APM)-* I • C,
where ~ = z(A), M +
x is the set of allpoints ofthe plane of the form m + ~t, where m ~ M.
COROLLARY.
W h e n the hypotheses of the theorem hold, it is always possible to choose a sequence of
polygons M. ~ ~ (r,d. e) such that the sequences P M n , PMn+]< of projections converge strongly to the iden£ity
o p e r a t o r , and hence the p r o j e c t i o n method using the s y s t e m of projections {PMa , PMn+×} is applicable to the
o p e r a t o r A.
CITED
LITERATURE
1o
2.
3.
4.
5.
6.
7.
I. Ts. Gokhberg, Izv. Mold. AkacL Nauk SSSR, 1_p_0(76), 39-50 (1960).
So G. MikhHn, Multidimensional Singular Integrals and Integral Equations [in Russian], Fizmaigiz,
Moscow (1962).
I. B. Simonenko, Izv. Akad Nauk SSSR, Ser. Mat., 29, 757-782 (1965).
I. C. Gohberg (I. Ts. Gokhberg), and I. A. Fel'dman, Convolution Equations and P r o j e c t i o n Methods for
T h e i r Solution, A m e r i c a n Math. Soc., P r o v i d e n c e , Rhode Island (1974).
A. V. Kozak, Dokl. Akad. Nauk SSSR, 212, No. 6, 1287-1289 (1973).
I. 13. Simonenko, Mat. Sb., 7~4, 298-313 (1967).
I. B. Simonenko, Mat. Issled., 3, No. 1, 108-122 (1968).
RATIONAL
SOLUTIONS
OF THE
EQUATION
AND INTEGRABLE
KADOMTSEV-PETVIASHVILI
SYSTEMS
OF N PARTICLES
ON
A LINE
I . M. K r i c h e v e r
UDC 517.9
Recently, interesting rational solutions have been found for the K o r t e w e g - d e Vries (KdV), K a d o m t s e v P e t v i a s h v i l i (KP), B~/rgers, and c e r t a i n other equations (cf. [1-4]). F o r the KdV equation it was shown in [1]
that the set of all rational solutions is a degenerate family of finite-zone solutions obtained by deformation of
the Lamd potentials n(n + 1) ~ {x) with r e s p e c t to t using the KdV equation and its higher analogs. The f i r s t
solution of this type was obtained in [5]. In [1], the connection was f i r s t r e m a r k e d between the dynamics of the
poles of these solutions and the motion of a Hamiltonian s y s t e m of p a r t i c l e s on a line.
In this note we set forth an algorithm for constructing c e r t a i n rational solutions of the Z a k h a r o v - S h a b a t
equations. In the p a r t i c u l a r c a s e of the K a d o m t s e v - P e t v i a s h v i l i equation, a m o r e complete description of all
the rational solutions is given. The method developed allows us to completely identify the motion of the poles
of the rational solutions of the KP equation with the Hamiltonian flows a r i s i n g in M o s e r ' s theory [6] of a s y s t e m
N
of N p a r t i c l e s on a line with Hamiltonian H -----~-.
3=I
i<j
G. M. Krzhizhanovskii P o w e r Institute, Moscow. T r a n s l a t e d from Funktsional,nyi Analiz i Ego P r i lozheniya, Vol. 12, No. 1, pp. 76-78, J a n u a r y - M a r c h , 1978. Original article submitted September 6, 1977.
0016-2663/78/1201-0059 $07.50 © 1978 Plenum Publishing Corporation
59
1 . A s s u m e given a s e t of p o l y n o m i a l s Q(~) :
ri k t qt (k) ---- ~ j h i k t , hiVffiffit , complex n u m b e r s ~ts,
c~at, R (k) =
i=o
,
i=0
i =,0
S
and r e c t a n g u l a r m a t r i c e s A s = a i j , 1 _< i _< ls of r a ~ k ~,. ~,~, : N.
$
T h e r e exists a unique function ~',(x, y , t, k) of the f o r m
,
(,,~, t, ~,): (~ + q(::,q~,c~)t,k).)~(~(,)~÷,,(,,
such that the coefficients of its expansion at the points ~s of the f o r m
~p(~, y, t,/0
: ~
{;.~,,(=, ~, t) (/~-- ~,):;
~=~
N--1
s a t i s f y the s y s t e m of equations ~a~i~i., (~, y, t)= 0. T h e coefficients of the polynomial q (z, y, t, k ) : ~
~
b i (:¢,y,
t)k t
i=~
m o r e o v e r turn out to be rational functions of the a r g u m e n t s .
THEOREM 1. T h e r e exist unique ~ e ~ t o r s
~
~
~'~
L.~ = m
L,= --ax'''ui(x'Y't)""--~'vi(x'Y't)
~d~
i=0
4
(~-~)
~:~.
T~ ~m~t~
~ th~
o~rato~ ~
s~ch
t~t
,( L~ - - ~0) . ¢ =
i=0
~c~t~
~ o t y ~ o ~ , t ~ ~ th~ ~ t ~ o ~
~,
y, t ) ~
hence a r e ~ t i o n a i f u ~ c t i o ~ of t h e i r a r g u m e n t s .
c ~ ~ .
~
th~ ~
~
~ons~
~.
~ ~,~
~.-~. ~-~ ~
~o~.
T h e s y s t e m s of ~ u a t i o ~ for the coefficients of the operat¢r~ ~ and ~ which a r e ~ u i v a [ e ~ t tv the [a~t
~uali~ are caIIedthe Za~arov-Sha~t
~uatio~.
2. Let Q ~ ) = 0, ~ ) = k 2, [s : 1. The function ~(x, t, k) d e f i e d by these ~
s a t i s f i e s the n o n s ~ t i o n a ~ S c h r ~ e r
~uation
and the polynomial ~ )
( # _ d~ _ u ( x , Zt)~¢(=,t.k)=O.
-~ d::
/
(1)
N
T h e potential u(x, t) is equal to - 2 ~C~-zj(t))-2.- T h e function ~(x, t, k) can be w r i t t e n in the f o r m
2=1
N
~., ,, ~, = (~ ÷ ~ o , . , ~ (~_., (t~-,:
~::÷~fz| ~
(~)
w h e r e the aj(t, k) depend rationally o~ the v a r i a b l e k.
THEOREM 2. Equation ~ ) has a solution of the f o r m (2) if and only if the m a t r i c e s A and T with m a t r ~
2 (i -- 5j~)
2 (i -- 5~) ~
25i~
e~e~.~s ~,~:~
~ ~-~.
, ~:
(~_~: - ~ ~
~.~V ~ ~ . . o . ~
- ~, ~ : 0.
T h i s m a t r i x r e p r e s e n t a t i o n for the equations of motion of the Hamiltonian s y s t e m of N p a r t i c l e s on a line
iV
with H a m i l t o n i a n H = ~I- 2
~=~
p~ + E 2 (~--z~)-2 was found in [6] and used to p r o v e the c o m p l e t e integrabflity of the
~'<)
system.
T h e r e a r e thus 2N p a r a m e t e r s : the coefficients of the polynomial ql (k) and the points ~<s, 1 _< s _< N det e r m i n i n g the function ~,(x, t, k), which give the g e n e r i c solution of the equations of motion for a M o s e r s y s t e m
of p a r t i c l e s . T h i s r e s u l t allows us to find explicit f o r m u l a s for the solutions of the last equations. F o r m u l a s of
this type w e r e f i r s t obtained in [7]. The method developed m a k e s it easy to obtain analogous f o r m u l a s for all
the h i g h e r H a m f l t o n i a n flows (cf. the f o r m u l a s of the following p a r a g r a p h ) .
Introduce the m a t r i x ® with m a t r i x e l e m e n t s
o,~ ffi (, + 2.:).~, + ~.~-' - - ~
60
~
•
i ~~~= I ,~::)~,'
.#:o ~:;.)-'
N
COROLLARY. If xj (t) a r e the coordinates of the p a r t i c l e s of a M o s e r s y s t e m , the equality I-[ (~z~(0)"L
det O holds,
.~=x
3. T h e Z a k h a r o v - S h a b a t equations in the c a s e of the o p e r a t o r s I~ = d2/dx ~- + u(x, y , t), I.~ = dS/dx s +
3/2ud/dx + w(x, y , t) imply that the function u{x, y, t) s a t i s f i e s the KP equation
3 o~. + 0
~+~=0.
THEOREM 3. T h e function u {x, y , t) is a solution of KP equations depending rationally on the v a r i a b l e x
and d e c r e a s i n g to z e r o as x ~ ~ , if and only if u(~. y , t ) = - - 2 ~, (x~x~(y,t)) -2 , and t h e r e exists a function
j~l
N
*(X, y, t, ]~)= (i .~- j~----Iaj (y, t, ]¢)(x--xj (y], t)) -1) e ~x+k=y+~r'',
satisfying the equation (~--L1),----(0~-- r~.,)~ =0 •
COROLLARY 1. T h e dynamics of the poles of the rational solutions of the K1~ equations with r e s p e c t to
the v a r i a b l e y coincides with the motion of the M o s e r s y s t e m , and is given with r e s p e c t to the v a r i a b l e t by the
[o
]
higher Hamiltonian F a = 1/~SpAa o r , what is the s a m e , by the m a t r ~ equation -~- -- r,, A = o, w h e r e T~ =
3~k
3 (t -- 5~)
~" --3/2AT, ~ ----~-~ (z~-- ~)~ (x~-- ~)~
COROLLARY 2. T h e g e n e r a l r a t i o n a l solutions of the Kt ~ equation which decay as x - - ~o a r e given by the
equation
u (x, y, t) ~ 2 d~ In det ~',
(3)
w h e r e ~si(X, y, t) = ®si(X, Y) + 3~s +it.
The solutions which a r e nonsingular for all r e a l x and y w e r e found in [4]. We r e m a r k that the decay c o n dition in the v a r i a b l e y is a consequence of the decay condition in the v a r i a b l e x.
COROLLARY
solutions of the KP
F 3 r e s t r i c t e d to the
2, and the solutions
3. The dynamics of the poles of the rational solutions of the KdV equation, i.e., the rational
equation not depending on the v a r i a b l e y, coincides with the Hamiltonian flow d e t e r m i n e d by
fixed points of the initial Hamiltonian flow. The n u m b e r of t h e s e poles is equal to n(n + 1) /
t h e m s e l v e s a r e given by Eq. (3) if we put
~'n-.~-s 02~4-1 ekx+h'~t [
~si ~ 0a.3T~_~--~Ok~l~-I 3'1(/') I~=0*
LITERATURE
1,
2.
3.
4.
5.
6.
7.
CITED
H. A i r a u l t , H. McKean, and J. M o s e r , Courant Institute P r e p r i n t (1976).
L. A. Bordag and V. B. Matveev, Leipzig Univ. P r e p r i n t (1977).
D. V. Choodnovsky and G. V. Choodaovsky, Marconi Inst. P r e p r i n t (1977).
L. A. Bordag, A. R. Its, V. B. Matveev, S. V. Manakov, and V. E. Z a k h a r o v , Leipzig Univ. P r e p r i n t
(1977).
B. A. Dubrovin and S. P. Novikov, Zh. Eksp. T e o r . F i z . , 6~7, No. 12, 2131-2143 (1974).
J. M o s e r , Adv. Math., 16, 354-360 (1975).
M. A. Ol,shanetskii a n d A . M. P e r e l o m o v , Funkts. Anal. P r i l o z h e n . , 1_.~0, No. 3 , 86-87 (1976).
61