ELLIPTIC SOLUTIONS PETVIASHVILI SYSTEMS EQUATION OF I. M. OF THE KADOMTSEV- AND INTEGRABLE PARTICLES UDC 517.9 Kr[chever The main purpose of this p a p e r is to c o n s t r u c t "action-angle" type v a r i a b l e s for a s y s t e m of p a r t i c l e s with a pairwise interaction potential, whose Hamiltonian has the f o r m n where ~ (x) is W e i e r s t r a s s ' S - f u n c t i o n (see [1]). The sign of the potential, c o r r e s p o n d i n g to an a t t r a c t i v e s y s t e m of p a r t i c l e s , is chosen to simplify f o r m u l a s used l a t e r on f r o m which we can exclude the t e r m ] / - - L The change of this sign o c c u r s when we go o v e r to i m a g i n a r y t i m e . The methods we develop allow an i n t e g r a tion of the equations of motion of s y s t e m (1) in t e r m s of R i e m a n n ' s 0-function. It is known that the equations of motion of s y s t e m (1) have the L a x - t y p e r e p r e s e n t a t i o n (see [2]): L : [M, L], (2) where the matrices L and M depend on xi and Pi. It follows from this representation that the qualities J ~ = ~ t r L ~, k-~ I . . . . . n, are integrals of system (1). It is proved in [3] that they are independent and in involution. Then by Liouville's theorem, system ( i ) i s completely integrable. In [4] there was reported for the first time a remarkable connection between Hamiltonian systems of particles on a line and the dynamics of the poles of special solutions of nonlinear equations that can be integrated by a method of an inverse problem. For the Korteweg-de Vries equation it was shown that the dynamics of the N poles of an elliptic solution u,(x, t) = 2 ~ b~ (x--x~ (t)) is equivalent to restricting the equations of motion of a i=I system with the Hamiltonian J3 to fixed points of system (1), grad H = 0. It turns out that N has the form n(n+ 1)/2. A similar assertion was obtained for the Boussinesq equation. The restriction on the number of particles is connected with the need to consider equations of motion on stationary manifolds of Hamiltonian systems. The connection between Hamiltonian systems and elliptic solutions of nonlinear equations turns out to be a natural one in the case of two-dimensional systems. For example, in the case of the Kadomtsev- Petviashvili (K- P) equation: 3 1 o {ut + 7(6uq,,.--u.,.xx)} T '%y= ~ - " (3) " The elliptic solutions of this equation have the f o r m u = const ÷ 2 ~ e (x -- xl (y, t)). H e r e the dynam[cs of the poles with r e s p e c t to y and t is d e s c r i b e d by the c o m m u t i n g Hamiltonian flows c o r r e s p o n d i n g to H and J3, r e spectively. This a s s e r t i o n in the c a s e of a d e g e n e r a t e W e i e r s t r a s s F - f u n c t i o n - x -2 was obtained in [5] (see a l s o [6]). Subsequently, pole s y s t e m s c o r r e s p o n d i n g to different nonlinear equations w e r e investigated by many w o r k e r s (see the s u r v e y [7] and [8]). The connection between c o m p l e t e l y integrable s y s t e m s of p a r t i c l e s on a line and the pole solutions of nonlinear equations can be used in two d i r e c t i o n s . For all d e g e n e r a t e W e i e r s t r a s s functions c o r r e s p o n d i n g to one or two infinite periods (~ (z) goes into sinh-2(x) or x-2, r e s p e c t i v e l y ) , the i n t e g r a l s J k a r e known, and m o r e o v e r , G. M. Krzhizhanovskii Moscow E n e r g y Institute. T r a n s l a t e d f r o m Funktsional'nyi A n a l i z i Ego P r i l o zheniya, Vol. 14, No. 4, pp. 45-54, O c t o b e r - D e c e m b e r , 1980. Original a r t i c l e submitted M a r c h 11, 1980. 282 0016-2663/80/1404-0282507.50 © 1981 Plenum Publishing C o r p o r a t i o n there is a m e c h a n i s m of integrating the equations of motion. The coordinates xj(t) of p a r t i c l e s turn out to be the eigenvalues of the m a t r i x xj(0)Sij-L(0)t, where L(0) depends on the initial coordinates and impulses (see [9] and [10]). Substituting these coordinates into the formula u (x, y, t)-----2 ~ (x--x~ (y, t))-"- allows one to obtain i=1 e.g., rational solutions of the K - P equation. On the other hand, having a c o n s t r u c t i o n for rational solutions of the K - 1) equation, one can obtain independently the integration of the equations of a s y s t e m of p a r t i c l e s with the potential x-2. The main aim of [5] was p r e c i s e l y to achieve this second possibility; there we used the idea of "finite-zoned integration ' to find exact formulas for the rational solutions of the K - P equation. In c o n t r a s t to the rational and t r i g o n o m e t r i c c a s e s , for a s y s t e m (1) with a nondegenerate W e i e r s t r a s s F - f u n c t i o n neither a construction of "angle" type variables c o r r e s p o n d i n g to the involutive integrals Jk nor a more explicit integration of the equations of motion was known. An exception is the solutions of the K - dV equation that a r e the sum of the three elliptic functions found in [18] without any connection with the t h e o r y of integrable s y s t e m s on a line, u = 2F (x -- x, (t)) -~- 26~ (x -- x2 (t)) -!~ 2F (x -- x3 (t)). For two particles s y s t e m (1) was integrated in [11], where it was proved that the level manifold of the int e g r a l s J1, H is a two-dimensional Abelian manifold. 1. Linear a System Nonstationary of P a r t i c l e s SchrSdinger Equation and on a C i r c l e Methods of integrating the K - P equation (3) are based on the following commutation r e p r e s e n t a t i o n (see [12] and [13]): o _ M] 0, (4) 0~ L~--Tx.,--u(x,g,t), 0 '3 3 0 M~-g~s--yu~+w(x,y,t). It will be proved later that the Hamiltonian of a s y s t e m of type (1) a r e connected with the existence of solutions of a specific f o r m for linear o p e r a t o r s with elliptic potentials. F i r s t of all, we r e c a l l the basic definitions and p r o p e r t i e s of the c l a s s i c a l functions of W e i e r s t r a s s (see [1]). Let cot and co2 be a pair of periods. The sigma-function of W e i e r s t r a s s is the entire function defined by the product n • m, n~ o t where Wmn = m¢~l +n¢02. The remaining functions _ °mnJ can be defined by the relations ~' (--) _ ~, (6) In c o n t r a s t to the -functions, the (7_ and ~-functtons a r e not doubly periodic. Under t r a n s l a t i o n s of the periods they t r a n s f o r m as follows: (a÷(01)= ~(a)+~]l; lh(%--~l~o,=2ni; (* (a + ,oz)= - - (~(a) exp ~lt (a + "~-~z)l. (7) In a neighborhood of a = O, the W e i e r s t r a s s functions have the f o r m ,o" (z,) = ~ + 0 (,~); (~) = a-~ + 0 (a'~); F (a) = a-'-'+ 0 (~). (8) THEOREM 1. The equation (9) has a solution ~ of the f o r m n ~ -= y, ai (t, k, (*) • (x - - x~, ~) e ~'~+~2t, ,=, (10) 283 where: -- ¢ (x, a) ~--- z (x - - a) e~(~)x ' {Ii) if and only if the x i ( t ) s a t i s f y the equations of motion of the s y s t e m of p a r t i c l e s (1): ~i = 4 ~ be' (x~ - - x~.). (12) The choice of this r e p r e s e n t a t i o n for ~b is connected with the fact that the function @(x, a) is a solution of L a m 6 ' s equation [14] (~ -- 2~ (x) ) ¢ (x, a) = ~ (a) (l) (x, a). (13) F r o m t r a n s l a t i o n r e l a t i o n s (7) it e a s i l y follows that @(x, ~) is doubly periodic in cr, @(x, ~+Wl) = 4Kx, o~). As a function of x, 4Kx, ~) s a t i s f i e s the r e l a t i o n (1) (x ~- (oz, c¢) = (I) (x, a) exp [~ (a)o) z -- llta]. (7') A function ¢ of f o r m (10) has s i m p l e poles at the poles x = x i. By substituting it into (9) and equating to z e r o the coefficients of ( x - x i ) - 2 and ( x - x i ) - l , we obtain the following equations: a~i .-{-2kal + 2 ~, aj~P( x ~ - xj, a) -~-O, (14) .~i If we introduce the v e c t o r a = (a 1.... , a n) and the m a t r i c e s L u (a) = x ~ u ÷ 2 (i -- ~j)~P (xi -- x~, a), (15) Ti; (a) ~-- 6ij ( - - be(a) ÷ 2 ~ ~ (xi - - xs.)~ ÷ 2 (1 - - 6i;) @' (xl - - x~, a), k~=i \ / then Eqs. (14) can be written in the f o r m (L(a)+2k.l)a=O, (~+T) a=0. (16) For Eqs. (16) to be compatible, it is n e c e s s a r y and sufficient that the following r e l a t i o n holds: [L,~+ T] = 0 ~ L = I L , TI" (17) LEMMA 1. Equations (17) hold if and only if the xi(t) s a t i s f y (12). The a s s e r t i o n of the l e m m a follows f r o m a s t r a i g h t f o r w a r d substitution of the e x p r e s s i o n s for L and T into (17). In addition, we can use the a s s e r t i o n , well known in the t h e o r y of s y s t e m (1), that the c o m m u t a t i o n r e p r e s e n t a t i o n (17) is equivalent to (12) provided that @(x, ~) s a t i s f i e s the functional equation: [O'(x)q') (y) -- • (z)O'(y)](I)-~ (x + y) = [be (y) -- ~ (x)l (18) and the r e l a t i o n (I) (x)(1) (--x) = ~ (a) -- ~ (x). (19) To v e r i f y the f i r s t of t h e s e r e l a t i o n s , notice that the left-hand side of (18) is doubly periodic in both x and y (this follows f r o m (7')), and it has a s e c o n d - o r d e r pole at x = 0, y = 0. Hence, it is equal to the right-hand side. S i m i l a r l y , the left-hand side of (19) is periodic in x and ~, and has a s e c o n d - o r d e r pole at x = 0 and ~ = 0. Thus, we have found the solutions of the functional equations (18), (19), and so the c o m m u t a t i o n r e p r e s e n tations (17) for s y s t e m (1); t h e s e , in c o n t r a s t to all those used e a r l i e r , depend on an additional " s p e c t r a l par a m e t e r " a defined on an elliptic c u r v e F with periods w 1 and w 2. This additional p a r a m e t e r also allows us to p r o c e e d to the integration of (1) by using the methods of a l g e b r a i c g e o m e t r y . We c o n s i d e r a m a t r i x A (t, ~) satisfying the equation and n o r m a l i z e d by the condition A(0, ~) -= 1. 284 F r o m (17) it follows that L (t, a)A (t, ~) = A (t, a)L (0, a). (21) R (k, a) = det (2k -k L (t, a)) (22) Hence, the function is independent of t. M a t r i x L(t, ~), which has e s s e n t i a l s i n g u l a r i t i e s at ~ = 0, can be r e p r e s e n t e d in ~he f o r m L (t, a) = G (t, a) Z (t, a) G-~ (t, a), (23) w h e r e ~ does not have an e s s e n t i a l s i n g u l a r i t y , and G is a diagonal m a t r i x , GIj = 5ij exp (~ (a)x i). the c o e f f i c i e n t s r i ( a ) in the e x p r e s s i o n Consequently, (24) R (k, a) = ~ ri (a) k ~ a r e elliptic functions with poles at ~ = 0. The functions ri(~) a r e r e p r e s e n t a b l e as a l i n e a r c o m b i n a t i o n of a be-function and its d e r i v a t i v e s . The c o e f f i c i e n t s in this e x p a n s i o n a r e i n t e g r a l s of s y s t e m (1). E a c h s e t of fixed v a l u e s of t h e s e i n t e g r a l s defines an a l g e b r a i c c u r v e F n by the e q u a t i o n R(k, a ) = 0; F n is an n - s h e e t e d c o v e r i n g of the o r i g i n a l c u r v e F. E x a m p l e 1. Let n = 2, then R (k, a) = 4k 2 -]- 2k (21 q- ~2) + ~x2~ d- 4~ (xa - - xs) - - 4~ (a). In a n e i g h b o r h o o d of c~ -- 0 the s i n g u l a r p a r t of the m a t r i x ~ has the f o r m ~ i i = O(1), ~ij = - 2 ~ - 1 + O(1), i ~ j. The eigenvalue - 2 ~ -1 of this m a t r i x is ( n - D - f o l d d e g e n e r a t e ; in addition, t h e r e is a n o t h e r e i g e n v a l u e e q u a l t o 2 ( n - I)(~ -1. T h u s , in a n e i g h b o r h o o d of t~ = 0 the function R(k, oz) c a n be r e p r e s e n t e d in the f o r m R (k, a) = (k - - (n - - 1) a -I q- bn (a)) I ] (k -q- a -i q- bz (a)), (25) l~l w h e r e the b / ( a ) a r e r e g u l a r functions of a . Hence, function k defined on F n has s i m p l e poles on all the s h e e t s at points Pi lying o v e r a = 0. Its e x p a n s i o n s in t e r m s of the local p a r a m e t e r a on t h e s e s h e e t s a r e given by the f a c t o r s on the r i g h t - h a n d side of (25). It follows f r o m (25) that one of the s h e e t s is i s o l a t e d ; f o r b r e v i t y , we call it the " u p p e r " s h e e t . LEMMA 2. The genus g of the s u r f a c e Fn is n. The c o v e r i n g elliptic c u r v e is such that 2 g - 2 = v, w h e r e r is the n u m b e r of b r a n c h points on the c o v e r i n g F n o v e r F. The b r a n c h points coincide with the z e r o s on F n of aR/ak. By d i f f e r e n t i a t i n g (25) with r e s p e c t to k and s u b s t i t u t i n g for k the c o r r e s p o n d i n g e x p a n s i o n s , we obtain that 0 R / 0 k has s i m p l e poles on all s h e e t s a p a r t f r o m the u p p e r one, on which the pole is of o r d e r n - 1. F o r any m e r o m o r p h i c function the n u m b e r of z e r o s is equal to the n u m b e r of p o l e s . Hence, v = 2 ( n - l ) , o r g = n. The J a c o b i manifold J ( F n) of F n is an n - d i m e n s i o n a l t o r u s . t o r u s a r e "angle" type v a r i a b l e s for s y s t e m (1). We show below that the c o o r d i n a t e s on this To e a c h point P of the c u r v e F n, i.e., to e a c h p a i r (k, a ) = P c o n n e c t e d by r e l a t i o n R(k, a ) , c o r r e s p o n d s a unique e i g e n v e c t o r a(0, P) = (el(0, P) ..... an(0, P)) of m a t r i x L(0, a ) , n o r m a l i z e d by the condition al(0, P ) - 1. All o t h e r c o o r d i n a t e s ei(0 , P) a r e m e r o m o r p h i c functions on the c u r v e s Fn outside the points Pj. The n u m b e r of poles of a(0, P ) i s n - 1 . To p r o v e this, we c o n s i d e r a m a t r i x F(c~) w h o s e c o l u m n s a r e v e c t o r s a{0, Pj(~)), w h e r e the P j ( a ) a r e the i n v e r s e i m a g e s of the point a . The function [det F(~)[ 2 does not depend on the e n u m e r a tion of the s h e e t s , i.e., it is p r o p e r l y defined a s a function of a . It is m e r o m o r p h i c , and has double poles at the i m a g e s of the poles of a(0, p). The z e r o s of this function coincide with the i m a g e s of the b r a n c h points of Dn. If N is the n u m b e r of poles of g(0, P), then 2N = v = 2 n - 2. In a n e i g h b o r h o o d of "infinitely distant" points Pj, ai(0, P) has the f o r m a~(O,P)=( - - ~ 1 -{- 0 (a) } ex], [~ (a) (x ° - x°)], i) t, ] :/~ ~. (26) 285 On the u p p e r sheet, j = n ai (0, P) = (1 --i- 0 (a)) exp {~ (a) (x° -- 2)]. (27) H e r e and in what follows x~ = xi(0) is the initial position of a p a r t i c l e . The f u n d a m e n t a l m a t r i x A(t, ~) of s y s t e m (20), A(0, c~) = 1, is an a n a l y t i c function of a outside ~ = 0. If Ai(t , ~) is the i - t b c o l u m n of A, then the c o l u m n - v e c t o r 7~ (28) act, P ) = ~ a~(O,P)A~(t,a) i=l is a solution of s y s t e m (20) that is e i g e n for L: (L (t, a) -:- 2k-t) a (t, P) = 0, P = (k, a). (29) To find the f o r m a(t, P) at the i n v e r s e i m a g e s of c~= 0, we turn f r o m a p a i r L, T s a t i s f y i n g (17) to a c a l i b r a tionally equivalent pair ~L, T = G-iOG ~ G-~TG, w h e r e ~ and G a r e as in (23). The following r e l a t i o n holds P (a)~ u -'- 7 (t, a) = c~-i'L (t, a) -I- 0 (1). Consequently, the e i g e n v a l u e s of the m a t r i x ~" have the following f o r m : (30) for j ¢ n p j (t, a) = --k~j + 0 (t) = --(a-'-' -:- 2bj (0)a -~ -!~ 0 (1)). H e r e the k J a : [ (~-1+ b j ( ~ ) ) a r e s h e e t we (31) the e x p a n s i o n s of the e i g e n v a l u e s of L on d i f f e r e n t s h e e t s F n. On the " u p p e r " p,~ (t, a) -= 2k~a -~ ~ ~-2 _~ 0 (1). (32) We denote by vj (a) =/lj (t, c~) + O(1) the s i n g u l a r p a r t s of the e i g e n v a l u e s of ~'. T h e y a r e independent of t, and so the solutions of the equation which a r e eigenfunctions of the m a t r i x ~, have the f o r m a (t, P) = a (0, P) (t + 0 (c¢)) exp (v~ (a) t). T h e v e c t o r s a and a ' a r e connected by the s i m p l e r e l a t i o n a (t, P) = G (t, a) a (t, P). (33) T h u s , we have the following r e s u l t . LEMMA 3. The c o o r d i n a t e s ai(t, P) of the v e c t o r - f u n c t i o n e(t, P) a r e m e r o m o r p h i c on the c u r v e F n outside the points pj: T h e i r poles T1 ..... Tn_l do not depend on t, In a n e i g h b o r h o o d of Pj, ai(t, P) has the f o r m a~ (t, P) --~ c~j (a) exp [~ (a) (x~ (t) - - x~) + ~) (a) t], (34) w h e r e the cij(a) a r e r e g u l a r in a n e i g h b o r h o o d of a = 0, and clj(0)=l, ]=1 . n; c~(0)=l, c~(0)= ~ ~>1 ]~n. (35) We r e t u r n a g a i n to the eigenfunction ix, t, P) = ~ ai (t, P) ¢ ( x - - zi, a) e~'X+~':t. i=l The function ~ ( x - x i, a) has e s s e n t i a l s i n g u l a r i t i e s on all the s h e e t s r n . It follows f r o m (31), (34), and (25) that ~(x, t, P) does not have e s s e n t i a l s i n g u l a r i t i e s at the points Pj, j ¢ n. F r o m (35) it follows that ¢ does not have a pole at this point. T H E O R E M 2. The eigenfunction ¢(x, t, P) of the n o n s t a t i o n a r y S c h r S d i n g e r equation {9) is defined on the n - s h e e t e d c o v e r i n g Fn of the o r i g i n a l elliptic c u r v e . ~(x, t, P) is m e r o m o r p h i c e v e r y w h e r e on Fn e x c e p t for 286 the one e s s e n t i a l l y singular point Pn" Its poles 71,.-., 7n-1 do not depend on t and x. In a neighborhood of Pn, ~(x, t, P) has the f o r m , t, p ) = + + (36) where X(~) = n(e-l+ bn(0) Thus, ¢(x, t, P) is a c l a s s i c a l t 3 a k e r - A k h i e z e r function (see [15-17]). It is defined uniquely by its poles Yl ..... ~n-1 and by the value x~. In fact, by shifting a r e f e r e n c e point we can move a pole f r o m a specified point and so a s s u m e that ¢ has n a r b i t r a r y poles 71,..., Yn" It was proved in [15] that a function ¢(x, t, P), withthe p r o p e r t i e s stated in T h e o r e m 2, is a solution of the n o n s t a t i o n a r y SchrSdinger equation. A g e n e r a l s c h e m e for c o n s t r u c t i n g e x p r e s s i o n s for B a k e r - A k h i e z e r type functions in t e r m s of R i e m a n n ' s 0-function is given in [16], where t h e r e is a l s o a f o r m u l a for the potential u(x, t) for the n o n s t a t i o n a r y SchriSdinger equation. A confrontation of r e s u l t s in [15] and [16], and those obtained above yields the following a s s e r t i o n . THEOREM 3. Coordinates xi(t) of the s y s t e m of p a r t i c l e s (1) a r e defined by the equation ?1 0 (~x + Vt A- W) = 0 = const × 1-I (~( x - - xi (t)). (37) To prove this, we note that the f o r m u l a for the potential for a n o n s t a t i o n a r y SchrSdinger equation has the for m 02 --~ u (x, t) ~ 2 0-~ In 0 (~x -1- Vt q- W) q- const. (38) As is c l e a r f r o m (9), the poles of u coincide with the z e r o s of 0 on the one hand, and with the xi(t) on the other. H e r e 0 is R i e m a n n ' s theta-function, U, V a r e constant v e c t o r s equal to the periods of Abelian d i f f e r e n tials of the second kind with s i n g u l a r i t i e s at Pn. A m o r e detailed definition of t h e m can be found in [16, 17]. Below we shall only d i s c u s s the c h a r a c t e r of our a n s w e r . Both the t h e t a - f u n c t i o n and ~, V" a r e defined uniquely by the c u r v e Fn, or equivalently, by the c h a r a c t e r i s t i c polynomial R(k, v~), which does not depend on t. Consequently, these p a r a m e t e r s depend only on the i n t e g r a l s of s y s t e m (1). Let 7i ..... Tn-i be the r o o t s of the equation det(2k. 1 + Lij) = 0, i, j > 1 (i.e., the roots of the lower r i g h t m i n o r of the c h a r a c t e r i s t i c m a t r i x ) , o n t h e c u r v e Fn. A b e l ' s t r a n s f o r m a t i o n m a p s the s y m m e t r i c power S n r n of the c u r v e (a n o n o r d e r e d collection of n points) into the Jacobi manifold J(Fn), a, : S n F n - - J ( F n ). The vector ~ in (37) is the i m a g e of the set ~,l..... Yn-i, Pn (x~=0) under A b e l ' s t r a n s f o r m a t i o n . It t r a n s f o r m s l i n e a r l y in t, which a l s o p r o v e s that the c o o r d i n a t e s on the Jacobi manifold a r e angle-type v a r i a b l e s , 2. Elliptic Solutions of the Kadomtsev-Petviashvili Equation Without d i r e c t l y substituting elliptic solutions into the K - P equation, we can obtain in a s t r a i g h t f o r w a r d way an identification of the poles of these solutions with s y s t e m (1) f r o m the c o m m u t a t i o n r e p r e s e n t a t i o n (7). THEOREM 4. A function u(x, y, t) is an elliptic solution of the K - P equation if and only if n u (x, y, t) = c + 2 ~ ~ (x - - x j (y, t)) (39) j~l and the equations have a solution of the f o r m ~; = ~ ai (t, y, P) (1) (x --x~, a) e~'x+~'*u+s"q (40) ~1 COROLLARY 1. The d y n a m i c s of the poles of the xi(y, t) with r e s p e c t to y coincides with the d y n a m i c s of p a r t i c l e s of s y s t e m (1). 287 This a s s e r t i o n follows f r o m r e s u l t s in the preceding section. S i m i l a r l y to the equation O/Oy- L, the a v a i l a b i l i t y of solutions of f o r m (40) for the equation ~ / 0 y - M is equivalent to a c o m m u t a t i o n equation of the type {17), and coincides w~th the equations of a Hamiltonian flow c o r r e s p o n d i n g to the Hamiltonian J3. COROLLARY 2. The d y n a m i c s of the xi(Y, t) with r e s p e c t to t coincides with the third Hamiltonian flow of s y s t e m (1). COROLLARY 3. The elliptic solution ti(x, y, t) of the K - P equation (39) can be e x p r e s s e d in t e r m s of the 0-function of the c o v e r i n g Fn of the elliptic c u r v e F: u = const -}- 2 ~ ]n 0 (Ux + Vy + -Zt + W). (41) We use the connection between the poles of elliptic solutions of the K - P equation and s y s t e m s of type (1) in yet another way to prove that under obvious r e s t r i c t i o n s , these solutions have no singularities for r e a l x, y, t. We consider the equation 30~u 0[ ~ )] ~- oy---et ~ u.t + -.i (6uu.~.- u,.x.~ -----0, which differs f r o m (3) by a sign. It has the c o m m u t a t i o n r e p r e s e n t a t i o n I°i ~ - - - L , (42) ~ -- M ] -----0 and its elliptic solutions a r e connected with a Hamiltonian s y s t e m with the Hamiltonian n (43) H----- ~-~-~,Pi + k~j k~l This Hamiltonian differs f r o m (1) by the sign of the potential energy. Let wl, w 2 be the periods of the F - f u n c t i o n ; they a r e c o m p l e x conjugates. Then ~ (~) = ~-~). We cons i d e r r e a l solutions of the K - P equation of f o r m (39). They a r e d e t e r m i n e d by the initial coordinates xj(0, 0) and the initial i m p u l s e s Xjy(0, 0). Suppose that these data t o l e r a t e conjugation, i.e., n = 2 m and xj -- Xj+m, J = 1 ..... m. Then u(x, y, t) is r e a l for all r e a l x, y, t. COROLLARY. If xj(0, 0) does not lie on the r e a l axis, then the solution (39), (41) does not have a singularity for r e a l x and y. The existence of a singularity m e a n s that one of the p a r t i c l e s fails onto the r e a l a x i s , but then the conjugate p a r t i c l e m u s t collide with it. This c o n t r a d i c t s the law of c o n s e r v a t i o n of e n e r g y since the potential is r e pelling and s i n g u l a r . MATRIX SYSTEMS We b r i e f l y state conditions on c u r v e s so that the constructions [16] of "finite-zoned" solutions of the c o m mutation equations a a __ M ] 0, (44) w h e r e L and M a r e o p e r a t o r s with m a t r i x coefficients, lead to elliptic solutions. We follow the notation in [16]. By [16], e v e r y nonsingular a l g e b r a i c c u r v e F of genus g with l distinguished points Pl .... , Pl and fixed local p a r a m e t e r s zj (P) in t h e i r neighborhoods, and a l s o a s e t T1..... ,/g+/_lofpoints in g e n e r a l position, d e t e r m i n e s the solution of Eqs. (44). If the c u r v e F N is an N - s h e e t e d c o v e r i n g of the elliptic c u r v e F, i.e., it is given by the equation N--1 k N ~- ~ ri (a) ki, (45) where the ri(~) a r e elliptic functions with a single pole at the point ~ = 0. We suppose that F N has no b r a n c h e s o v e r ~= 0. This m e a n s that the function k on FN has N s i m p l e p o l e s Pl ..... PN (the i n v e r s e i m a g e s of the point a = 0). We denote by vj the r e s i d u e of k on the j - t h sheet, i.e., k vj~ -1 =0{1) in a neighborhood of Pj. 288 A s s e r t i o n . We a s s u m e that vj = 1 if j > l. Then if we take the local p a r a m e t e r s zj(~) in the neighberhc~ca.~ of Pi ..... Pl to be zj(a) = ( k j ( ~ ) - ~(~))-1 then the c o r r e s p o n d i n g solutions of (44) a r e elliptic. It follows e a s i l y f r o m this a s s e r t i o n that to e v e r y N - s h e e t e d c o v e r i n g of an elliptic c u r v e t h e r e c o r r e sponds elliptic solutions for s y s t e m s with ( N - 1) x ( N - 1) m a t r i x coefficients. The p r o o f of the a s s e r t i o n follows f r o m the fact that the function ~ (P) = exp [k (P) (oi -- ~ (a) (~ + ~ a ] , i = 1, 2, is p r o p e r l y defined as a function of P. Outside the points P, .... , P l it is holomorphic, and in nei~hoo~hoods of these points it has the f o r m ¢~ (P) = (1 + 0 (~)) exp It follows f r o m the definition of B a k e r - A k h i e z e r (~--L)*=(~--'ll),.p=O, (aT~ (~) ~z). type functions that for ¢(x, y, t, P) satisfying the equations we have (x + o~,, y, t, p) = ~ (x, y, t, P) qh (P). Hence, the coefficients of the o p e r a t o r s L and M a r e m e r o m o r p h i c and periodic with periods co1 and o:~, ,:.e., they a r e elliptic functions. In conclusion, we c u r v e $ of genus n, the but leading to solutions new integrable s y s t e m s mention that it would be i n t e r e s t i n g to d i s c o v e r how to obtain, b y u s i n g c o v e r i n g s over a solutions of nonlinear equations e x p r e s s e d in t e r m s of a 0-flmction of high dimension, with the group of periods of ~. It is possible that these solutions a r e connected with of p a r t i c l e s . LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. CITED H. B a t e m a n and A. Erdelyi, Higher T r a n s c e n d e n t a l Functions. Elliptic and Automorphic Functions, L a m 6 and Mathieu Functions [Russian translation], Nauka, Moscow {1967). F. Calogero, "Exactly solvable o n e - d i m e n s i o n a l m a n y - b o d y s y s t e m s , " Lett. Nuovo Cimento, 13, 411-415 (1975). A . M . P e r e l o m o v , " C o m p l e t e l y integrable c l a s s i c a l s y s t e m s connected with s e m i s i m p l e Lie a l g e b r a s , " Lett. Math. Phys., 1, 531-540 {1977). H. Airault, H. McK~an, and J. Moser, "Rational and elliptic solutions of the KdV equation and r e l a t e d m a n y - b o d y p r o b l e m , " Commun. P u r e Appl. Math., 30, 95-125 {1977). I . M . K r i c h e v e r , "Rational solutions of the Kadomtse--v- Petviashvili equation and integrable s y s t e m s of N p a r t i c l e s on a line," Funkts. Anal. Prilozhen., 12, No. 1, 76-78 (1978). D . V . Choodnovsky and G. V. Choodnovsky, "Pole expansions of nonlinear p a r t i a l differential equations," Nuovo Cimento, 40B, 339-350 (1977). F. Calogero, " I n t e g r a b l e m a n y - b o d y p r o b l e m , " P r e p r i n t , Univ. di R o m a , No. 89 (1978). K . M . Case, "The N-soliton solutions of the B e n d g a m i n e - O n o equation," P r o c . Nat. Acad. Sci. USA, 75, 3562-3563 (1978). M . A . Olshanetsky and Ao M. P e r e l o m o v , "Explicit solutions of s o m e c o m p l e t e l y integrable s y s t e m s , " Lett. Nuovo Cimento, 17, 97-133 (1976). M . A . Olshanetsky and N. V. Rogov, "Bound s t a t e s in c o m p l e t e l y integrable s y s t e m s with two types of p a r t i c l e s , " Ann. Inst. H. Poincare, ~ 169-177 (1978). D . V . Choodnovsky, " M e r o m o r p h i c solutions of nonlinear p a r t i a l differential equations and particle integrable s y s t e m s , " J . Math. Phys., 20, No. 12, 2416-2424 {1979). V . S . D r y u m a , "An analytic solution of the t w o - d i m e n s i o n a l K o r t e w e g - d e V r i e s equation," P i s ' m a Zh. Eksp. T e o r . Fiz., 19, No. 12, 219-225 {1973). V . E . Z a k h a r o v and A. B. Shabat, "A s c h e m e for the integration of nonlinear equations of m a t h e m a t i c a l physics by an i n v e r s e p r o b l e m of s c a t t e r i n g t h e o r y . I," Funkts. Anal. Prilozhen., 8, No. 3, 43-53 (1974). E. Kamke, Handbook on O r d i n a r y Differential Equations [in German], Chelsea Publ. I . M . K r i c h e v e r , "An a l g e b r a i c - g e o m e t r i c construction of the Z a k h a r o v - S h a b a t equations and t h e i r p e r i odic solutions," Dokl. Akad. Nauk SSSR, 227, No. 2, 291-294 (1976). I . M . K r i c h e v e r , "The integration of nonlinear equations by the methods of a l g e b r a i c g e o m e t r y , " Funkts. Anal. Pr[lozhen., 11, No. 1, 15-31 (1977). 289 17. 18. H. Baker, "Note on the foregoing paper, 'Commutative o r d i n a r y differential equations,' " P r o c . R, Soc, London, Ser. A, 118, 570-576 (1928). B . A . Dubrovin and s. p. Novikov, " P e r i o d i c and conditionally periodic analogues of many-soliton solutions of the K o r t e w e g - d e Vries equation," Zh. Eksp. T e o r . Fiz., 67, No. 12, 2131-2143 (1974). EFFECTIVE CONSTRUCTION HERMITIAN-POSITIVE SEVERAL OF NONDEGENERATE FUNCTIONS OF VARIABLES L. A. Sakhnovich UDC 517.57 1. Let S be some set of points of n-dimensional space R n and A ---- S -- S. A function @(x) is called Hermitlan-positive on A if for any choice of points x l, x 2.... , xNES and n u m b e r s ~l, ~2,.'., ~N one has N i, j ~ l Let n = 2. We consider the lattice S(N l, N2) consisting of points M(m, Z) with integral coordinates 0 ~ m ~ N1, 0 ~ l ~ No.. The set h (N~, N2) consists of the points M(m, l) for which ] m [ ~ Ni, [ l [ ~.~ N,,.. By ~(N~, N~) we denote the class of functions, Hermitian-positive on h (Nt, N.~). Calderon, Pepinsky [1], and Rudin [2] proved the following t h e o r e m . THEOREM. In o r d e r that any function of the class ~ (N~, N~) should admit an extension to a function of class ~ (co, co), it is n e c e s s a r y and sufficient that any nonnegative polynomial of the f o r m i(x, y)= ~ ~ a~.~x~yz 0~/¢-..~2N10~l~2N~ admit a r e p r e s e n t a t i o n r ](x, y)= .~ q~(x, y), (2) where qj(x, y) a r e r e a l polynomials. As Hilbert [3] proved, t h e r e exists a nonnegative polynomial in two variables of the sixth degree which cannot be r e p r e s e n t e d in the form (2). Consequently, one has the following: Assertion (see [1, 2]). T h e r e exist functions of class ~ (3,3) which cannot be extended to !p(co, co). However, there have not been until now concrete examples of such functions. In the p r e s e n t paper c l a s s e s of concrete functions of. 0(2, 2) and ~ ( l , i, i) I which cannot be extended, r e s p e c t i v e l y , to !p (2, 3) and (l, i, 2) a r e constructed. We note that functions of the class !p(i, 2) can always be extended to fp (co, co) (see [4-61i, 2. With each function @(m, l) f r o m ~ (Ni, N2) we a s s o c i a t e the Toeplitz m a t r i x . Fo (0, k) lO (-t, k) • O, k) • (0, k) ... • (,v,, ~) ] ... • (NI-- i, k) ] 0 < k ~ N~ (3) F r o m t h e m a t r i c e s C k we construct the block Teoplitz matrix: [-Co A(N~, c, ,Y2)~/c~* . . . . . co .. • ... ON, -[ .... . . c.,._~[. .. $ LCN ~ - CN~-I " ' ' Co (4) J A. S. popov Odessa E l e c t r i c a l Engineering Institute. T r a n s l a t e d f r o m FunktsionalTnyi Analiz i Ego P r i l o zheniya, Vol. 14, No. 4, pp. 55-60, O c t o b e r - D e c e m b e r , 1980. Original a r t i c l e submitted April 2, 1979. 290 0016-2663/80/1404-0290507.50 © 1981 Plenum Publishing Corporation

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