2
(2)
(3)
I & (x) I < -b" '
g~ (r=) - - ~I, (:~) = Ik (:~).
i - - ek
L e t us e s t i m a t e the i n t e g r a l of ~Pk(X). To do so, note that ~ (E~) >~----V-- 't<~<ks" T h e r e f o r e ,
k~
~
k -- ~t - -kS k
f ~ (x) dF ~
I (p~(x) d~t~ ks. k~
t ~"l---~k
X
i= k~--k~@l Ek
Hence
I g~ (x) d~t = k31 ?~ (x) dl~~ I -- e~
k
X
X
S e t M~ = {x: gk (x) =~ 0}.
k3--1
M~C ; 2 0 T-jGkc'(
k5
k~--1
.U
E~)U( U
/0
U
T
-j ( ~ k \
~)')
i ,,El-'
c~
c~
We have ~ (M~)~<@ + -k8
~ = - 3~ .
Since ~ li(Mk) c o n v e r g e s , by the B o r e l - C a n t e l l i l e m m a g(x)= ~, gk(x) c o n -
verges #-almost everywhere.
But by v i r t u e of {2), ](~)= ~, f~(x) c o n v e r g e s u n i f o r m l y , and f(x) is continuous.
k ~2
~
k~2
S u m m i n g o v e r all k >- 9 in (3), we obtain g(Tx) - g ( x ) = f(x) a l m o s t e v e r y w h e r e .
In addition, g(x) >- 0 and
__oo,
s i n c e ~k -~ 0. This p r o v e s the t h e o r e m .
LITERATURE
CITED
D. V. Anosov, Izv. Akad. Nauk SSSR, S e t . M a t e m . , 37__, 1259-1274 (1973).
P. R. H a l m o s , L e c t u r e s on E r g o d i c T h e o r y , C h e l s e a Publ. Co., New Y o r k (1956).
lo
2.
ALGEBRAIC
CURVES
DIFFERENTIAL
I.
M.
AND
COMMUTING
MATRICIAL
OPERATORS
Krichever
In [1] we have p r e s e n t e d the a l g e b r a i c - g e o m e t r i c c o n s t r u c t i o n of the e x a c t solutions of the Z a k h a r o v Shabat equations which a r e conditionally p e r i o d i c functions of t h e i r a r g u m e n t s . By the Z a k h a r o v - S h a b a t
equations [2] we m e a n nonlinear d i f f e r e n t i a l equations which can be r e p r e s e n t e d in the f o r m
0
0
The condition of the c o m m u t a t i v i t y of two o p e r a t o r s is equivalent with the p r e s e n c e of a " s u f f i c i e n t l y
l a r g e " (here we do not define this c o n c e p t m o r e exactly) c o l l e c t i o n of functions, s i m u l t a n e o u s l y c o n v e r t e d by
t h e m into z e r o . In [1] we have c o n s i d e r e d o p e r a t o r s with s c a l a r coefficients and we have p r o v e d that f o r t h e m
sufficient c o l l e c t i o n s a r e the functions @(x, y, t, P), w h e r e P is a point of a nonsingular c o m p l e x c u r v e given
by its analytic p r o p e r t i e s on ~ and h a v i n g an e s s e n t i a l s i n g u l a r i t y of a s p e c i f i c f o r m at s o m e fixed point. In
the p r e s e n t note we c o n s i d e r functions which have e s s e n t i a l s i n g u l a r i t i e s in l points. This b r i n g s us to o p e r G. M. K r z h i z h a n o v s k i i E n e r g e t i c Institute. T r a n s l a t e d f r o m F u n k t s i o n a l ' n y i Analiz i Ego P r i l o z h e n i y a ,
Vol. 10, No. 2, pp. 75-76, A p r i l - J u n e , 1976. Original a r t i c l e s u b m i t t e d D e c e m b e r 22, 1975.
This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, N.Y. 10011. N o part I
]of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
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[
144
I
ators whose coefficients are l × i matrices.
Without examining physically interesting examples
(see [2]), we
shall show that the solutions of the corresponding
equations, constructed with respect to such functions, can
be explicitly written in terms of Riemann' s e functions.
i. Let ~ be a nonsingular complex curve of genus g with the distinguished
sider the functions ~(x, y, t, P), PE~, satisfying the conditions:
i. ~(x, y, t, P) is meromorphic
x, y, t, is nonspeoial and has degree
2. In the neighborhood
points Pt .....
Pl-
We
con-
on 8~ outside the points Pj, the divisor D of its poles does not depend
g + l-l.
on
of Pj it has the form
exp (kix ~- Qj ~kj)y -~ Ri (kj) t)- ( ~ -~ ~ ~{ (x, y, f) ~).
H e r e zj = 1/kj is a l o c a l p a r a m e t e r in the n e i g h b o r h o o d of P~, Q~ (}) = c~}~ ÷ . . .
polynomials.
+ c~, ~ (}) = b~k~ ÷ - . . + b~ a r e
As m e n t i o n e d in [4], f o r l = 1 t h e s e conditions, t o g e t h e r with the n o r m a l i z a t i o n ~0 = 1, d e t e r m i n e uniquely
~'. S i m i l a r l y , f o r l > 0 the n o r m a l i z a t i o n }i0 = 6ij d e t e r m i n e s uniquely the functions el(X, y, t, P).
T H E O R E M 1. T h e r e e x i s t unique o p e r a t o r s
L~ =-
u= ( x , y , t )
¢=0
0
s u c h that Liq~= ~ - m ,
0
L~m = - ~
-
-
dx=
=
L==
v~(x,y,t)
~=0
m, w h e r e 4~ is a v e c t o r whose i - t h c o m p o n e n t is ~i(x, y, t, P).
The m a t r i c e s u~(x, y, t) a r e d e t e r m i n e d f r o m the s y s t e m s of equations
The e l e m e n t ~isJ of the m a t r i x ~ is equal to.the c o e f f i c i e n t of zjS of the expans ion of ~i(x, y, t, P) in the n e i g h b o r h o o d of Pj. The m a t r i x c 7 is equal to c~8~j. One can find s i m i l a r l y the m a t r i c e s vfl(x, y, t, P).
C O R O L L A R Y 1. The o p e r a t o r s L 1 and L 2 s a t i s f y Eq. (1).
If on the c u r v e ~ t h e r e exists a m e r o m o r p h i c function E(p), having poles at the points Pj (the r i n g of
t h e s e functions will be denoted by A (9~, P, . . . . . Pt)~, w h o s e L a u r e n t s e r i e s e x p a n s i o n at Pj h a s p r i n c i p a l p a r t
Qj(kj), then ~(x, y, t, P) c a n be r e p r e s e n t e d in the f o r m ~0(x, t, P)exp (E(P)y).
0
C O R O L L A R Y 2. Under the a s s u m p t i o n s m a d e , we have L~¢0= E¢o, L2¢0 ----~
Now, if t h e r e e x i s t s H ( P ) ~ A (~, P1. . . . .
¢0 and so [L2,L~] --
OLz
Ot
&), equivalent to Rj(kj) in Pj, then O(x, y, t, P) = O0(x, P) exp (E
(P)y + H(l~)t).
C O R O L L A R Y 3. The function ~0 s a t i s f i e s the equalities LtO 0 = EO 0 and L2O 0 = HO 0, while the o p e r a t o r s
s a t i s f y the equation [L 1, L 2] =0.
Thus, e a c h d i v i s o r of d e g r e e l + g - 1 gives a h o m o m o r p h i s m XD of the r i n g h (~, P~. . . . .
of l i n e a r d i f f e r e n t i a l o p e r a t o r s with l × l m a t r i x coefficients.
Pz) into the r i n g
R e m a r k . We note that the c o n s t r u c t e d solutions of the Eqs. (1), as well as XD, depend only on the c l a s s
of the d i v i s o r D s i n c e going o v e r to an equivalent d i v i s o r D' r e d u c e s to the m u l t i p l i c a t i o n of • by a c o n s t a n t
matrix.
T H E O R E M 2. If in the c o m m u t a t i v e r i n g A of l i n e a r d i f f e r e n t i a l o p e r a t o r s with m a t r i c i a l c o e f f i c i e n t s
t h e r e e x i s t two o p e r a t o r s with r e l a t i v e l y p r i m e o r d e r s and with n o n s i n g u l a r leading c o e f f i c i e n t s , then t h e r e
e x i s t a c u r v e ~, points Pi, • • -, Pl, d i v i s o r D s u c h that h D gives an i s o m o r p h i s m b e t w e e n A(gL P~. . . . . Pl) and
A.
P
2. As
a
local p a r a m e t e r zj in the n e i g h b o r h o o d of Pj we s e l e c t the function
.f c0~, w h e r e P0 is a fixed
Po
point, w2 is a n o r m a l i z e d d i f f e r e n t i a l of the s e c o n d kind with s e c o n d - o r d e r p o l e s
at the points Pl . . . .
, Pl.
We
145
denote by 2viU the v e c t o r of its b p e r i o d s (for all n e c e s s a r y information and m i s s i n g definitions we r e f e r to
[5]) and by 2~iV and 27riW the b periods of the differentials co(Q) and c0(R), equivalent in Pj with d(Qj (1/zj)) and
d(Rj(1/zj)), r e s p e c t i v e l y . We also introduce the v e c t o r s 2viukj, which a r e the b periods of the differentials
dzi
having a unique singularity at Pj of the f o r m k~ ~ .
We c o n s i d e r the functions Xisj (x, y, t), given by the f o r m u l a s
=~ ~
o~'~llj... 0 s)lsi
~.j
The s u m m a t i o n is taken with r e s p e c t to all the collections ~ , . . . ,
"
~s such that ~, k~k = s .
k=l
correspondbyAbel'ssubstitutiontothedivisors
p~ ~ .
. . - ~ p g _ l - ~ p~, i ~ i ~ l , w h e r e
The v e c t o r s Z i
~-{-g--1
D=
~,
Ps.
Explicit e x p r e s s i o n s f o r the m a t r i c e s ~s(X, y, t), in t e r m s of which the solutions of the Z a k h a r o v - S h a b a t
equations a r e e x p r e s s e d , a r e given by the following t h e o r e m .
THEOREM 3. We have the equality L = ~x~,, where the e l e m e n t s of the m a t r i c e s ~s a r e given by the
equality
eo
~o
In p a r t i c u l a r , for the K a d o m t s e v - Petviashvili equation given in [1] we obtain that its solution is given by
the f o r m u l a
•
,
• ~
~
u (x, y, t) ----c - - 2"~-~x ~l (~, y, t) = c -~- 2"~'~x~ln @(Ux -]- Vy -~ W~ -b g).
LITERATURE
1Q
2.
3.
4.
5.
146
CITED
I. M. K r i c h e v e r , Dold. Akad. Nauk SSSR, 227, No. 2 (1976).
V. E. Z a k h a r o v and A. B. Shabat, Funktsfonal'. Analiz i Ego P r i l o z h e n . , 8, No. 3, 35-43 (1974).
B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Usp. Matem. Nauk, 31, No. 1, 55-136 (1976).
N.
I. Akhiezer, Dokl. Akad. Nauk SSSR, 142, No. 2 , 2 6 3 - 2 6 6 (1961).
p
E. I. Zverovich, Usp. Matem. Nauk, 26_, No. 1, 68-113 (1971).