The Riemann boundary value problem on closed Riemann surfaces and integrable systems

Physica 24D (1987) 1-53 North-Holland, Amsterdam

THE RIEMANN BOUNDARY VALUE PROBLEM ON CLOSED RIEMANN SURFACES AND INTEGRABLE SYSTEMS Yu.L. R O D I N Institute of Solid State Physics of the Academy of Sciences of the USSR, Chernogolovka, Moscow distr., 142432 USSR Received 5 March 1986

Review paper The Riemann problem has different forms. In several areas of physics (scattering theory, integrable systems and so on) it is necessary to build an analytic function or a matrix with prescribed poles and zeros on a Riemann surface. Another, similar, problem is to construct a section of a complex vector bundle over a Riemann surface (for example, anti-self-dual Yang-MiUs equation). As was established by A. Grothendieck and H. Rrhrl, this problem is equivalent to the Riemann boundary value problem: let M be a compact Riemann surface, F a contour on M, G(p) an n X n matrix on F; determine an analytic matrix F(p) in M \ F which is multiple of a given divisor* "t and satisfies the boundary condition on F

F+(p)=G(p)F_(p). This problem is well understood on the complex plane. It is related to singular integral operators, Wiener-Hopf equations and Banach algebras. In the case of a Riemann surface there are additional connections with the Riemann-Roch and Abel theorems, Jacobi varieties and Riemann theta-functions. These relations are described here. The Riemann problem is equivalent to the so-called 0-problem

Ou Recently this elliptic equation was used to study the inverse scattering problem. Finally, we consider some physical problems (finite-gap potentials, the Landau-Lifschitz equation, the Painlev6 problems) to illustrate the applications of the Riemann problem on a Riemann surface. Acquaintance with Riemann surface theory and algebraic topology is not assumed.

1. Riemann surfaces

1.1. Abelian differentials and integrals A c l o s e d ( c o m p a c t ) R i e m a n n surface is a c o m p a c t m a n i f o l d with a c o m p l e x structure. This m e a n s that o n the m a n i f o l d one fixes some finite coveting b y s i m p l y - c o n n e c t e d d o m a i n s , called coordinate patches; e v e r y p o i n t o f the m a n i f o l d belongs to at least one such d o m a i n . I n every c o o r d i n a t e p a t c h U we define a local coordinate function z u = z u ( p ) m a p p i n g U to the unit disk Izvl < 1 of the c o m p l e x plane. If the i n t e r s e c t i o n U A V of two c o o r d i n a t e p a t c h e s is n o t e m p t y , then o n e - t o - o n e c o n f o r m a l m a p p i n g s z v = z v ( z v ) , Z v = Z v ( Z v ) are defined in U N V. It is clear that this allows one to define a n a n a l y t i c function w = f ( p ) , p ~ M, as a function analytic with respect to a n y local c o o r d i n a t e of the p o i n t p. A c o m p a c t R i e m a n n surface can be t h o u g h t of as a sphere with g h a n d l e s (see, for e x a m p l e [3, 13]). T h e n u m b e r g is called the genus o f the surface. If g = 0 or 1, we o b t a i n a R i e m a n n sphere o r a torus, respectively. C u t t i n g the surface as in figs. 1, 2 we t u r n M i n t o a s i m p l y - c o n n e c t e d d o m a i n , a 4 g - s i d e d *See definitions below. 0 1 6 7 - 2 7 8 9 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

2

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

~z

1c.!

I Fig. 1.

polygon )~t with sides identified in pairs [13]. It is called the fundamentalpolygon. This canonical form will often be used below. The cycles K a , . . . , Kzg generate a (free) group called the one-dimensional homology group. Elements of this group can be thought of as combinations of the type E2gcjKj where the cj are real or complex numbers. Let an analytic function w = f ( p ) be defined on the compact Riemann surface M. It maps the surface M to a multisheeted surface over the w-plane. This surface is just one of many possible multi-sheeted realizations of the surface M. For example, the function w2 = R2g+l(z),

2g j=O

R2g+l(z) = I I (z - zj),

(1.1)

-

Fig. 2.

-4-

~~

Ytt L. Rodin / The Riemann problem on closed Riemann surfaces -4-

~o

(.j

20

3

/"

~,

----

0

-%

-..~÷

/ _ / j ' A e e z~

c)

Fig. 3.

defines a two-sheeted surface (fig. 3). This situation is common. It can be shown that every compact Riemann surface is the Riemann surface of some algebraic function P(z, w) = 0, where P is a polynomial with respect to z, w. Along with analytic functions on a Riemann surface, one can consider analytic differentials. In the fixed local coordinate z(p), a differential has the form

w(p)~f(z)dz, where

z=z(p),

p~U,

f(z) is an analytic function in the coordinate patch U, and the value ~0(p) is invariant when the

4

Y~ L. Rodin / The Riemann problem on closed Riemann surfaces

coordinate z(p) is replaced by the coordinate z*(p),

dz(p)

f(z*(p)) =f(z(p)) d z * ( p )

"

(1.2)

For the differential to it is natural to define the indefinite integral =

fPo~.

(1.3)

P0

It is locally independent of the path of integration, but the values ~2i= / o~, Kj

j = l . . . . . g,

are, generally speaking, different from zero. These values are called periods of the differential ~o (or of the integral ~2). If all periods are equal to zero, the integral ~2(p) is a single-valued function on M; otherwise, 12(p) is a multi-valued function determined up to terms of the type Ejcj~2j. The value I2(p) can be considered as a single-valued function on the simply-connected polygon M with finite discontinuities on the c u t s K j ( j = 1 . . . . . 2g). The residue of o~ at the point P0 is, as usual, Respo to = 21~r~--~fv~, where ~, is a small contour around Po. Triangulate the surface M. To this end it is sufficient to divide the polygon /~ into a finite number of triangles A k (k = 1 .... , n). As long as the sides of adjacent triangles have opposite orientations, n

zf

k=l

Ak

This entails the residue theorem Y'. Resp~ = 0.

(1.4)

p~M

Let f ( p ) be an analytic function with no essential singularities. From eq. (1.4) applied to the differential d l n f , it follows that f ( p ) takes on all values (taking into account the multiplicities) equally often. Therefore, the following theorem is valid:

Theorem 1.1 (Liouville). An analytic function holomorphic everywhere on M is a constant. Note that a harmonic function regular everywhere on M is a constant too. This follows from the fact that such a function has its maximum and minimum values on the boundary which is, however, empty in the case of a compact Riemann surface.

Yu.L Rodin/The Riemann problem on closed Riemann surfaces

5

A differential holomorphic everywhere on the surface M is called an Abelian differential of the first kind. The real dimension of the space of these differentials is 2g. One proves this as follows. If the dimension were > 2g, one could construct an Abelian differential of the first kind having pure imaginary periods. The real part of the corresponding integral would be a single valued harmonic function, hence equal to zero by the Liouville theorem. In order to show that the dimension of this space is equal to 2g one must construct a real harmonic differential with prescribed periods along the cycles of the homology basis ( K j}. This is done in Riemann surface courses (see, for example, [13]). The following theorem is valid:

Theorem 1.2 (Riemann). The complex dimension of the space of Abelian differentials of the first kind is equal to g. Consider the surface this surface are

=

eg_

Eg determined by

(z)dz

the function (1.1). The Abelian differentials of the first kind for

p = (z, + ) .

(1.5)

fR2 + (z) ' Here Pg_l(z) is a polynomial of degree g - 1. One should verify that 0~(p) is regular at the branch points and at infinity. At the point z = oo, introduce the local coordinate t = (¢~-)-1. We have

t2g+lP2g_2(t)dt oa(p) = t2g_2t3~4g+2(t )

/52s_ 2(t ) dt '

where R4g+2(0 ) 4: 0. At the point z = zj, introduce the local coordinate t = ~z - zj. T h e n R 2 g + l ( z ) q5(0) 4= 0 and

P2g_l(t)2tdt ~o(p) = t~/~(t) =

= (z - zj)qg(z) =

t 2r~(t),

2/32s_x(t) dt

It is very important to know the number of zeros of an Abelian differential of the first kind. It is clear that all such differentials have the same number of zeros. Clearly, the ratio of two differentials is a function f(p) having the same number zeros as poles (taking into account their multiplicities), by the residue theorem for the differential d l n f ( p ) . For the hyperelliptic surface (1.1) the answer is very simple: the differential (1.5) has 2g - 2 zeros, g - 1 on every sheet. One can show that there are 2g - 2 zeros for an arbitrary differential of the first kind on any compact Riemann surface of genus g. We shall prove that fact below by the Riemann-Roch theorem. Moreover, if some analytic differential has m poles, it has 2g - 2 + m zeros. The Riemann bilinear period relation is an important tool in the study of differentials. Let d f and d g be two Abelian differentials with a finite number of poles. If )Q is a fundamental polygon (fig. 1) then

f .fdg=2~ri~Resfdg, OM

M

f=fp~df.

(1.6)

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Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

Since alV1 = K 1 + K 2 - K 1 - K 2 + . . . .

j:l

j:l

K2g, we obtain the relation

2S-1

2s 1

I~j---- K 2 j - 1 + K2j - g 2 j - 1 - K 2 j .

Therefore,

L ( f , g) = 2~riE R e s f d g . M

(1.7)

By substituting different Abelian differentials in eq. (1.7), we can calculate periods of these differentials. Consider the space of Abelian differentials of the first kind. Its complex dimension is equal to g (Theorem 1.2). It is convenient to use the basis dwj ( j = 1 .... , g) determined by the relations

fK2

dWk=SJ~'

j,k= l

g.

(1.8)

j-1

Substituting d r = dwj, dg = dw k into (1.7) we obtain the relations

fK dw*=fKdws' 2j

j,k=l,...,g.

(1.9)

2k

A meromorphic differential is called an Abelian differential of the second kind if all residues of its poles are equal to zero. We use the normalized Abelian differentials of the second kind dtq, z having a pole of the n + 1-st order at the point p = q. In a fixed local coordinate z ( p ) the differential has the form

dtq'z(P) =

_

ndz(p) [ z ( p ) - z(q)] n+l + regular terms.

(1.10)

The periods of dtq, z(p) are normalized to be

fK

dtq, z ( p ) = 0,

j = l . . . . . g.

(1.11)

2/-i

Letting d f = dtq,,, dg = dw: in (1.7),one can obtain the relation

fK2jdtq, z ( p ) = - 2 ~ r i

d"Wdz(q), :(q------~) ,

j = l . . . . . g.

(1.12)

Here the symbol dn/dz(q) ~ means, as usual, the nth derivative with respect to the local coordinate z ( p ) fixed in (1.10). The subscripts z and n -- 1 are often omitted. Abelian differentials of the third kind have a finite number of poles whose residues can be nonzero. The normalized Abelian differential of the third kind dtoq~q2(p) has first order poles at the points p = ql, q2

Yu.L. Rodin/ The Riemannproblemon closedRiemannsurfaces

7

with residues :t: 1, respectively. Its periods are normalized by

fK

dtoq~q2(P)=O ,

j = l . . . . . g.

(1.13)

2j-1

Letting d f = dwj, dg

=

dtoqlq2 and

fK d%,q2(p)=2~ri f q2dwj, 2j ql

df =

dtoqlq2 ,

dg = dtopxp2 in (1.7) we obtain the relations

j = l ..... g,

t°,lp2( q2 ) - tople2(qt) = toq,q2( P2)

-

(1.14)

toqlq2(P t)"

(1.15)

For the hyperelliptic surface (1.1) the Abelian differentials of the second kind with a pole at p are (see [13])

d/'l=

{

1 t (z) [¢R2g+t(a ) + 12(~R2g+t(a) ) - t R ' 2 g + l ( a ) ( z - a ) l d z ' ( z - -1a ) 2 + CR 2g-~ z ( p ) = a --/:oo,

-2k+t

dtp

=

(2k + 1)zg+kdz 2¢R2g+l(Z ) ,

z(p)=oo,

(1.16)

k=O,1 ....

The Abelian differential of the third kind with poles z(pt) = a t, z(p2) = a2, a I 4: 00, a 2 v~ oo is

1[

CR2g+l(Z ) + CR2g+l(a2)

dtSPl-~2= 2¢R 2;+ l(Z)

z - a2

~R2g+,I(Z ) "1-

CR2g+l(al)

z - a1

dz.

(1.17)

The normalized differentials dtp and dtoq~q2 c a n be obtained from (1.16), (1.17) by subtracting linear combinations of the differentials of the first kind, chosen to make appropriate periods vanish, cf. (1.11), (1.13). 1.2. Riemann-Roch theorem We now introduce some spaces of functions and differentials. Let Pi be points of the surface M and let a i be integers (i = 1,..., n). The symbol y = ~,~aip i is called a divisor, the number ~ 0 ~ i = deg y is called the degree of the divisor. The sum of the divisors 7 and 7' = ~,ia;Pi is the divisor 7 + 7' = ~,~(ai + a'i)Pi. The divisor 7 is said to be non-negative, 7 - 0, if all a~ >__0. The zeros and poles of a function f (or of a differential to) form the divisor ( f ) = ~,~fliPi (resp. (to)); here the fli > 0 are the orders of the zeros and fli < 0 are the orders of the poles. A function f ( p ) is said to be a multiple of a divisor 7 if ( f ) >__7. We denote by L ( 7 ) the space of analytic functions that are multiples of the divisor - 7 , ( f ) + V-> 0 and by H(V) the space of analytic differentials that are multiples of the divisor V, (to) >- 7.

Theorem 1.3 (Riemann-Roch). dim L (7) - dim H ( 7 ) = deg 7 - g + 1

(1.18)

8

YmL. Rodin/The Riernann problem on closed Riemann surfaces

(the dimensions are complex).

Sketch of proof. The Abelian differentials of the second kind that are multiples of the divisor - 7, 7 = E~p~ and have zero periods along the cycles K2j_ 1 ( j = 1 , . . . , g) have the form d t v ( q ) = c o + ~ cjdtp,(q).

(1.19)

j=l

The space of these differentials has dimension n + 1. The corresponding integrals tv(q) will be single-valued if

~lc, fK2dt,=O,= n

fK2,dt~=

j = l ..... g.

(1.20)

Therefore dim L ( y ) = n + 1 - r,

(1.21)

where

r

= rank

£:2jdte,(q) .

(1.22)

Because of (1.12) the adjoint system is written in the form g xj d w j ( p , ) = 0 , j=l dz(p)

i=1 .... ,n.

(1.23)

Every solution of this system corresponds to an Abelian differential of the first kind E j x j dwj which is equal to zero at the points p~, i.e. is multiple of y. The number of such differentials is equal to dim H(7). On the other hand, the number of solutions of the system (1.23) is equal to g - r. Therefore, dim H ( 7 ) = g - r.

(1.24)

Eqs. (1.21) and (1.24) entail eq. (1.18). F r o m the R i e m a n n - R o c h theorem it follows that if 7 is the divisor of an Abelian differential then deg 7 = 2g - 2. Indeed, let 70 be the divisor of some Abelian differential ¢o of the first kind. It is clear that dim L(7o) = g since the space L(7o) consists of the functions 1, ~ 1 / % , - . - , %-x/~°o, where o~o. . . . . % - 1 is a basis of the space of differentials. Further, let ~o and ~ be the Abelian differentials of the first kind having the same zeros, ( % ) = (to~) = 7o. Obviously, ~ o / ~ is a constant and hence dim H(7o) = 1. Then deg Yo = dim L(Yo) - dim H(Yo) + g - 1 = 2g - 2.

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

1.3.

9

Cauchy type integrals

On the Riemann sphere, i.e. on the compact Riemann surface of genus 0, the Cauchy type integral kernel ( t - z)-ldt is the Abelian differential of the third kind with residues -T-1 at the points t = oo, z with respect to t, and an analytic function with first order pole at the point z = t and zero z = oo with respect to z. On a Riemann surface M of non-zero genus, one has an analogous concept. Let 0

M*(p,q)dz(p)= ~ ( p ) [~%op(q)-~pop(qo)] dz(p),

(1.25)

where Po, qo are arbitrary points. As follows from (1.15) M * ( p , q ) d z ( p ) is an Abelian differential of the third kind with respect to p, with poles p = q0, q, having residues T-1. Moreover, M*(p, q) is an Abelian integral of the second kind with respect to q with a pole 1 M * ( p, q) = z ( p ) - z (q) + regular terms

and zero periods along K2j_ 1 ( j = 1. . . . . g), i.e.

M*( p, q) = - tp(q). By eqs. (1.12), (1.14) the periods of

2jdqM*(p,q)

)

(1.26)

M*(p, q) along

the cycles K2j ( j = 1. . . . . g) are

0( ~2d%op(q) /

dz(P)=0z--~- ~

dz(p)=2~ridwj(p),

j = l . . . . . g.

(1.27)

The Cauchy type integral 1 F(q) = ~-Cf fr~(p) M * (p,q)dz(p)

(1.28)

is analytic in M \ F and its limiting values to the right (left) of F are :~determined by the PlemeljSokhotsldi formulae F:~(q)=

1 T-l~(q) + -~-~ frep(p)M*(p,q)dz(p ), q~ r.

(1.29)

Here we suppose that F is a Lyapunov contour* and cp(p) satisfies the H61der condition (is H-continuous)

I~0(p)-~(P')I
0<#<1.

The contour F may or may not separate the surface M. The function

*A Lyapunov contour is a contour whose a tangent rotates H/51der-continuously.

F(q) is multi-valued

on M and its

10

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

periods along the cycles

K2j are equal to

Fj= frep(p)dwj(p),

j = 1 .... , g.

(1.30)

Note that from eq. (1.25) follows that there exists a branch of condition

F(q) in the polygon M (fig. 2) satisfying the

F(qo) = O. For some applications, a single-valued Cauchy kernel is preferable (H. Behnke, K. Stein [a], S.Ya. Gusman, Yu.L. Rodin [a]). Let 8 = Zg=lP~ be a divisor with dim H(6) = 0. Define the Abelian differentials of the first kind d Z ) = Z](z(p)) d z ( p ) by the conditions

Z](z,(pi))=6ij,

i , j = l ..... g,

(1.31)

where the z~(p) are the fixed local coordinates of the points p,, used for the normalization of the differentials dtp, (1.10). Suppose g

M(p, q) = M*(p, q) + ~., tp,(q)Zf(p).

(1.32)

i=1

It is clear that the kernel M(p, q) possesses the same properties as M * ( p , q). But it is single-valued. Indeed, the periods of M ( p , q) along K2j are equal to

2~ri[dwj(P)- ~ dwj(p')dZi(P)]

j = l . . . . . g,

since the left sides of these equations are Abelian differentials of the first kind that are multiples of the divisor 6 (see eq. (1.31)). As long as dim H(6) = 0, any Abelian differential that is multiple of 6 is zero. The divisor 6 is called the characteristic divisor. The quantity M ( p , q) has also g poles at the points q = Pi ( i = 1 . . . . ,g). The behaviour of M(p, q) is illustrated by fig. 4. The solid diagonal line corresponds to the pole p = q. The vertical solid lines correspond to the poles q = p, (i = 1. . . . . g). The horizontal solid line corresponds to the pole p = q0 (the polar point of the kernel) with residue - 1. The horizontal dotted lines correspond to zeros at p = p~ (see (1.31), (1.32)). The vertical dotted line corresponds to zero at q = qo, as follows from the normalization tp,(q0) = 0. The Cauchy type integral 1

F( q ) = -ff~i frrp( p ) M( p, q ) dz( p )

(1.33)

is analytic in M\F, single-valued on M and has poles of the first order at the points q =pj (j = 1.... , g). The principal parts of these poles in the local coordinates used for the determination of the integrals tp(q) (see (1.10)) are 2~ri

(p)dZj(p),

j = l ..... g.

(1.34)

Y~ L Rodin / The Riemann problem on closed Riemann surfaces

11

m 2 --

7-

L_ Fig. 4.

The boundary values of F(q) satisfy eq. (1.29). Consider the Cauchy type integral

g'(q)dz(q)= ( 2 ~

fr~(p)M(q' p)dz(p)} d z ( q ) ,

whose density ~k(p)dz(p) is a differential form on F. xr,d z ( q ) is an analytic differential in multiple of the divisor 8 and has a pole at q = q0 with residue

1 2~ri

fr~(P)dz(p).

(1.35)

M\I-"

that is a

(1.36)

As long as the residue of the kernel at the pole q = p is equal to - 1, the Plemelj-Sokhotskii formulae have the form

if/ (p)M(q,p)dz(p).

f f % ( q ) = _+ ~k(q)+~-~-~

(1.37)

2. The Riemann boundary value problem (the scalar case)

Existence of solutions The statement of the problem. Let

2.1.

F be a contour on the surface M. We suppose that F is a Lyapunov contour consisting of a finite number of closed non-intersecting curves that separates the surface into the domains T + and T-. If the contour does not separate the surface (for example, if F is a cyclic section of the torus), the contour may be completed by the necessary number of curves such that the completed contour separates the surface. On these additional curves we assume G - 1. If the contour has self-intersec-

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

12

tions or angular points, this may be taken into account by known methods, just as with jumps of the boundary function G(p) [4, 12, 19]. Let z~ be a divisor, A = A + + A-, A-+~ T -+, and G(p) a HSlder continuous function different from zero everywhere on F.

The Riemann problem. Determine analytic functions ~b_+(q) in the domains T !, respectively, which are multiples of the divisors - A - , continuous up to the boundary and satisfying the boundary condition • ÷(p)=G(p)~_(p),

p~F,

(~)>-~.

(2.1)

The number c° = i n d r G = 2 - ~

fr dargG

(2.2)

is called the Cauchy index of the problem. By the argument principle

lfr

2~ri

dln~+

lfr d l n ~ _ = ~

21ri

lfr

d l n G = c o.

(2.3)

Here c 0 = deg(~) is the degree of the divisor (~) of an arbitrary solution ~ ( q ) of the problem. Therefore, if A = 0, a holomorphic solution has c 0 zeros. If A ~: 0, a solution has x = x o + deg A

(2.4)

zeros and deg LI poles. Simultaneously, we consider the dual problem 1

't'+(p)dz(p) = G-G---(~P_(p)dz(p),

(kv) > A .

(2.5)

Here ~/'_+d z ( p ) are analytic differentials in T -+ which are continuous up to the boundary and multiples of the divisors A+-. We start with the simplest boundary condition

cb+(p)-~_(p)=g(p)

onF.

(2.6)

The multi-valued solution of the problem is

F+_(q) = ~

(p)M*(p,q)dz(p).

The periods of the solution along K z j

are

f g(p)dwj(p)=O, j=l ..... g.

(2.7)

determined by (1.30). This solution is single-valued if

(2.8)

These conditions are necessary for solvability of the problem (2.6) (multiply (2.6) by dwj and integrate

YttL. Rodin/ The Riemannproblemon closedRiemannsurfaces

13

over F). Another form of the solution is

F±(q) = ~

(p)M(p,q)dz(p).

(2.9)

This function is single-valued, satisfies (2.6) and have poles at the points of the characteristic divisor 3 of the kernel (see section 1.3); the principal parts of 3 are determined by (1.34). The conditions that the function (2.9) be regular at 3 coincide with (2.8). In accordance with this scheme the problem (2.1) will be studied by different methods. First, we consider the problem

~+(p)=G(p)~

(p),

(2.10)

(~)+3-q0>0,

where 8 is the characteristic divisor and q0 is the polar point of the kernel. The dual problem is 1

~ t ' + ( p ) d z ( p ) - G(p) q ' _ ( p ) d z ( p ) ,

(g')+qo-3>_0.

(2.11)

Let ~b(q) be an analytic function in the domains T ±, a multiple of the divisor - 3 + q 0 , ~ + ( p ) - ~ - ( p ) = cp(p) on F. Then 1

ff~(q ) = -~--~f/p ( p ) M( p, q ) dz( p ).

and

(2.12)

Indeed, if ~ l ( q ) is another analytic function in T -+ such that ~ - ( p ) - ~ - ( p ) = ~ ( p ) , (~1) + 3 - q0 > 0, then ~ ( q ) - ~ l ( q ) is analytic on M and ( ~ - ~1) + 3 - q0 > 0. Because of dim H ( 3 ) = 0, dim L ( 8 ) = deg 3 - g + 1 = 1. Hence ~ ( q ) - ~ l ( q ) = const. Since ~/i(q0) - ~l(q0) -- 0, it follows that ~ l ( q ) ----~(q). Therefore, the solution of the problem (2.10) is represented in the form (2.12). By the PlemeljSokhotskii formulae (1.28) we get the singular integral equation

1 - G(q) cp(q) + 1 + G(q) f/p 2

2~ri

(p)M(p,q)dz(p)=O,

q~l"

(2.13)

of index K0 [4, 12]. This equation has l = l' + Xo solutions. Here l' is the number of solutions of the dual equation

1 + G(q) ~(q) + 1 - 2~ri G(q) f / ( p ) M ( q , p ) d z ( p ) = O . 2 It is clear that

(2.14)

q~(q)dz(q) is a differential on F; this can be seen from the fact that the scalar product

(cp, ~ ) = Re

f/~p d z ( p )

generating the adjoint operator has to be invariant.

14

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

Let

~l,(q)dz(q)=(~-~l f?(p)M(q,p)dz(p))dz(q).

(2.15)

Then, as follows from (2.14), (1.37), rP(q)dz(q) is a solution of the problem (2.11). Therefore, since l is the number of solutions of the problem (2.10) and eq. (2.13) and l' is equal to the number h of solutions of the problem (2.11), l' = h, we have l - h = x o.

(2.16)

The problems (2.1) and (2.5) are equivalent to the problems

~+(p)=G(p)~_(p),

(~)>qo-6,

~+(p)dz(p)=(~-l(p)~_(p)dz(p),

(~)>6-qo,

(2.17)

d(p) = c( pl~--=~H(p), where A+-(q) are analytic functions in T + whose zeros and poles are determined by the divisors A ±, and is an analytic function in T + with a pole at qo and zeros at 6 (for simplicity, we suppose that 6 - q0 ~ T+) • Since also i n d r G = x 0 + degA - g + 1, we have the relation

H(q)

l - h = x 0 + d e g A - g + 1 = x - g + 1.

(2.18)

N o t e that for the case G - 1 we obtain the Riemann-Roch theorem. From (2.18) it follows that the problem is solvable for K = ~0 + deg A > g. Note that l < K + 1 (see the R i e m a n n - R o c h theorem). 2.2.

The Riemann problem and the Jacobi inversionproblem

Let the c o n t o u r / " consist of m components, F = Y~j~=IFj, and let m

xj=ind~G,

~o = ]~ Kj.

(2.19)

j=l

Below, we suppose that A = 0 and K = K0. Fix a point qj on each curve Fj ( j = 1 . . . . . m) and refer the values l n G ( p ) on Fj to the points qj. The function l n G ( p ) has jumps at qj which are equal to 2~rigj. As long as the Cauchy type integral has logarithmic singularities at q j, the function

Co(q) =

exp 2 ~

frlnG(p)M*(p,q)dz(P)

(2.20)

has zeros of orders Kj at the points qj if Kj > 0, and poles if xj < 0. The function (2.20) satisfies the boundary condition

¢o+(P) = G(p)¢o_(p).

(2.21)

Yu.L. Rodin/The Riemann problem on closedRiemann surfaces

It is single-valued along the cycles

K2j_ 1 and

15

has multiplicative periods ~j along K2j,

/,

• y = exp Jr In G d wj,

g

j = 1

(2.22)

(see eq. (1.30)). If there exists a single valued holomorphic solution of the problem (2.21), it has the form

O(q)=Oo(q)ex p - ~KsCOqoq,(q)+ ~ O~qop,(q)+2rri CrWr(q) . s=l

(2.23)

k=l

Here q0 is an arbitrary point of M, Pk are as yet unknown points, and cr are integers. The Abelian integrals are determined by (1.8), (1.13). The conditions for single-valuedness of (2.23) along K2j are (see (1.14)) g

2~ri

~w,(p~)=2~ri~.Kswj(qs)-ln~j+2~ri~~CrL k=l

s=l

nj

where the

dwr+2rrinj,

j = l . . . . . g,

K2j

r=l

are integers. Denoting

1 ln~j, lj = ~ x,wj(qs ) - ~-~

j = 1 ..... g,

(2.24)

s=l

we obtain the system (see (1.9))

~ w j ( p k ) = l j + n , + Ec,. L dwj=lj+ E k=l

r=l

K2r

r=l

n,.

dwj+crf dw, 2r-t

CK2r

=lj+2eri

dwj,

]

g

K=

E ( n r g 2 r - I "~- crg2r). r=l

It can be written as a system of congruences,

w,(pk) - lj

(mod periods of wj),

j = 1 . . . . . g.

(2.25)

k=l

This system is known as the Jacobi inversionproblem (the problem of inversion of Abelian integrals). The problem is solvable for K _ g ([3, 13]), in accordance with the results of section 2.1. Several proofs of this fact are known. One proof was essentially presented in section 2.1. The other proof will be adduced below, in section 2.4. Note that for K = 0 we obtain the solvability conditions in the form lj -= 0

(mod periods of wj).

(2.26)

These conditions are equivalent to the famous Abel theorem, giving the conditions for the existence of an analytic function F(q) having prescribed poles and zeros determined by the divisor y=]~kakpk, deg y = ~kak = O.

16

Yu.L Rodin / The Riemann problem on closed Riemann surfaces

Indeed, draw the contour F such that "t e T-. We obtain the representation

F(q)=F+(q),

q e T +,

F ( q ) = y ( q ) F (q),

geT-, ~,(q)=VIp~k(q),

(2.27)

k

where F±(q) are holomorphic in T +- and Pk(q) is an analytic function in T- having a single zero of the first order at pk. The representations (2.27) yield the boundary relation

F + ( p ) = , f ( p ) F (p),

per.

The conditions (2.26) are

frlnV(p)dwi=-O

(mod periods of wj).

(2.28)

Calculating the integrals on the left side, we obtain the Abel theorem as expressed by the system n

~_, akwj(pk ) -- 0 (moO periods of wj),

j = 1 . . . . . g.

(2.29)

k=l

These conditions have a natural interpretation in terms of the Jacobi variety. Consider the integral lattice

Zg=(mjf K

dwi+njf

2j- 1

dwil, K2j

i , j = l ..... g,

(2.30)

]

where m j, nj are integers. The g-dimensional complex torus Jg = C g/z_ g is called the Jacobi variety of the surface M. The conditions (2.25) mean that the point (lj, j = 1. . . . . g } has to correspond to the zero point of Jg. 2.3. The Riemann theta-function and explicit formulae for solutions of the Riemann problem 2.3.1. The Jacobi inversion problem is Solved by the Riemann theta-function. We describe its properties briefly (see [2]). The theta-function of g variables is determined by the series

O(Wl,...,Wg) =

exp ~ri ~ mj= - ~

ak,m~mt+2~ri ~ mjwj ,

k,l=l

(2.31)

j=l

where the matrix [lak~LIis symmetric, akl = at~ and the matrix ILImaklll is positive definite. One can verify that the series (2.31) converges absolutely and that

O(w 1 + a 1. . . . . . Wg + ag,) = exp ( - ~ r i a s , - 2~riw, }O(w~ ..... Wg), O(w 1..... w s + l

. . . . . W g ) = O ( W 1 . . . . . Wg),

j,s=l

. . . . . g.

(2.32)

17

Yu.L R o d i n / T h e Riemann problem on closed Riemann surfaces

Let the arguments of (2.31) be the Abelian integrals of the first kind, wj =

ajk

-- f .

dw k,

j,k=l

wj(p),

and

. . . . . g.

(2.33)

2.,

We obtain a function on the surface M called the

Riemann them-function

O(Ws(p) ) ----O(Wl(p)..... Wg(p)). Instead of the arguments w~(p) we often use the arguments ws(p)- es, where the e s are arbitrary numbers. Because of (2.32), the function O(ws(p) - es) is single-valued along the cycles K2j_ 1 ( j = 1. . . . . g) and has multiplicative periods ~ ( p ) along K2j,

~(p)=expfKdlnO=exp(-~ria/j-2~iwj(p)+

27riei},

j = l . . . . ,g.

(2.34)

2j

The Riemann theta-function has g zeros on M. Indeed, in the fundamental polygon M (fig. 2) we have

deg(0) = ~-~

fodtdlnO(ws(p)-es)=

~

1 ~ [f __

"d,

j = 1 L K2/

dA2j_ 1 ln0

- fK2j ldA2j In0 ].

Here the symbol Zijf means the increment of f along the cycle Kj

Aj f=

fKjdf.

(2.35)

Here and below the values of multi-valued functions are computed on the left edge, 2). We obtain the relation

Kf,

of the cut Kj (fig.

g

deg(0) = E f,

(2.36)

dwj= g.

jffi 1 K2j-1

Denote the zeros of

O(ws(p) - es) by Pl . . . . .

ls(O )= y" ws(pj )=

p~ and calculate the sums

s(P)dlnO(Wk(p)--ek)

j=l

= ~

r=l

K2r-1

+ f z~Zr_l[ws(P)dlnO(wk(p)-ek)]l ,

s = l . . . . . g.

Vl( r

By (2.33) and (2.34) we obtain

A2rw~(p)=a~s, A2rdlnO(wk(p)-ek)=-2~idw~(p).

(2.37)

Yu.L. Rodin/The Riemann problem on closedRiemann surfaces

18

Furthermore,

A2r { w~(p)dlno(wk(p) - e~)} = [ w s ( p ) + a r s ] [ d l n O ( w k ( p ) - e k ) - 27ridwr(p)] - w s ( p ) d l n O ( w ~ ( p ) - e k ) =ar~dlnO(wk(p)--e~,). Therefore, the first integral on the right side of (2.37) is equal to Ors

27ri

fK 2,-1 dlnO(wk(p)--ek)+

=-ar~nr+ f. ~K

where the

n r are

f,,2.1w ' ( p ) d w r ( p )

w,(p)dwr(p)+ar~,

+ar,fr2,_1 d w r

s=l .... ,g,

2r-1

integers. Also,

A2r_lws(p)=Srs,

dA2r_llnO(wk(P)--ek)=O.

The second integral on the right side of (2.37) is equal to

~rs 2~ri

fr 2,

dlnO(w~(p)-e~)=

~r,(-- ar__z _ 2

wAP)+er).

Therefore, we conclude that

j=IEWs(pJ)= r=lE

Is(p)=

--arsnr'4-ars +

Zr-1ws dw r

-

-w,(p)+e,,

s = l . . . . . g.

Integrating one of the integrals on the right side by parts

fK

WrdWr=A2r-l(Wr)2-- fK

2r--1

wrdwr'

2r-1

we obtain that (2.38)

w r d w r = W r ( p ) "4-1" g2r - 1

We have g

g

E w,(p+)= ~-'(-ar~nr+ar')+

j~l

r=l

=--e s - ..ks

]~[ r~s

.tK

1

w~dwr+W'(P)+ 2

2r-1

(mod periods of ws),

s = 1..... g

ars

2

ws(p)+es

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

19

where i...g

k s = - ) - +1T - ass

E

fK~, 1ws dwr,

s = l . . . . . g.

(2.39)

r~s

Therefore, the divisor of zeros of the function O(wfip) - e,) is a solution of the inversion problem g

ws(pj ) =----e s - k s (mod periods of w,),

s = 1 , . . . , g.

(2.40)

j=l

Note that for some values of the parameters e~ the function O(wfip) - es) can be identically zero. In this case one can verify that the functions

8O(ws(p)-es) 19Wj '

j = 1 ..... g

(2.41)

possess the same periods as O(ws(p) - e~). This means that every non-zero function (2.41) has g zeros on M, and these zeros are the solutions of the problem (2.40). If all first derivatives are zero, we study the derivatives of second order, and so on. There exists a minimal integer et such that one of derivatives

O,,(w,(p)-e,)=

8'~O(ws(p)-es) Ow~,...Ow~, '

al + ' ' " + a s = ° t

(2.42)

is not identically zero. Note that this situation corresponds to the case dim L ( y ) > 1, 3' = )'-Pj [2]. 2.3.2. We now obtain the formula for the solution of (2.1). Let f ( q ) be some function with the multiplicative periods £ f 2 j = e x p / dl

din f ,

j=l ..... g

K2)

along K2j , and 1 along K2j_ 1" This function satisfies the boundary conditions g

f +(P) = -~zjf-(P),

p ~ K = [3 K2j-1-

(2.43)

j=l

E.I. Zverovich [16] noted that the function O(ws(p) - es) is a solution of the Riemann problem (see (2.34)) g

e+(e)=

0_(e),

= U/%-1,

(2.44)

j=l

O(p) = exp { - ~'iajj - 2~riwj(p) + 2*riej },

P ~ K2j-1.

We use this observation to obtain explicit formulae for solutions of Riem .ar_anproblems. Below we assume

20

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

for simplicity that the cycles K2j_ 1 and the contour F have empty intersection.* Case ~ = ~o = g. Since the solution of (2.44) is the 0-function having g zeros, the index of this problem is equal to g. Consider the boundary value problem

r+(p)=G(p)r_(p),

p~F=

0~,

~=indrG=~ind~G=~xj=g.

j~l

j

(2.45)

j

Let the values l s (s = 1. . . . . g) be determined by eqs. (2.24) and

e s = l s + ks,

(2.46)

where the k s are Riemann constants (2.39). Assume that O(ws(p) - es) ~ O. Then the zeros of O(ws(p) es) coincide with the solution of the problem (2.44). Therefore, the boundary value problem

*+(p)=G(p)*_(p),

p~P=FUK,

G(p)=

G(p), p~F, p~K,

t~(p),

(2.47)

has the unique regular solution (2.48)

~( q) = F( q)O-X(ws( q) - es).

Fix arbitrary points qk ~/'k (k = 1,..., m), pj ~ K2j_ 1 ( j = 1. . . . . g), s o ~ M. We refer the branches of the function ln G(p) to these points. Then

*(q)=exp

~1

lnG(p)M*(p,q)dz(p)+

=

2, ~ ( - - ~ f - - w ' ( p ) + e t ) M * ( p ' q ) d z ( P )

- ~ icjrOsoqj(q)+ k~=lWSOp,(q) .

(2.49)

j=l

This function is holomorphic in M \ F , different from zero and satisfies condition (2.47) on F. Let us verify that the function

(2.50)

F(q) = * ( q ) O ( w , ( p ) - es)

is single-valued. We calculate its periods along the cycles K2j ( j = 1. . . . . g). As is obvious from figs. 1, 2, the periods of a function f(q) along the cycles K2j are equal to the jumps of the function f(q) on the cuts K2j_ 1 (cf. (2.35))

AzJf= fK d f = f _ ( q ) - f + ( q ) , 2j

*If F = K, G(p)=

G(p)f(p).

q ~ K 2 j _ 1,

j = l . . . . . g.

21

YmL. Rodin/The Riemann problem on closed Riemann surfaces

Consider the periods of the integral f(q) = ~

(2.51)

~(p)M*(p,q)dz(p) 2j-1

along K2s. We have (see (1.27)) f+(q)=:~(q)+~-~

f_(q)

1

= - ~(q)

_

+ ~

1

p)M*(p,q)dz(p),

q ~ K 2÷j - 1 ~

fx2j_r(p)[M*(p,q)+ 21riwS(p) ] dz(q),

fK df=-w(q)+ fK 2j

(2.52)

j = l ..... g.

ep(p)dwj(p),

2j-I

We now calculate the multiplicative periods of the function (2.50) F2j=exp~

dlnF,

j=l

. . . . . g.

2j

Set

8j(q)

=

{0, 1,

q ~ K 2 j _ :,

q~K2j-:,

j = l .....

g.

Taking into account (1.30), (2.52), (1.14), (2.34), we obtain F2j = exp

In G dwj + 2~ri ~ 1=1

-2 iz( - - ~a,,- w , ( q ) + e

-- 7all -- w , ( p ) + e,

t

)3,(q)-2qri

1=1

+ E

dwj(p)

2/-1

K~wj(q~)+2~ri E w j ( p k ) s=l

[-Triatt-

k=l

2rriw,(q) + 2~rie,]31(q)

/=1

--exp

{f; lnGdwj + 2~ri Y'.

l~ 1 K2t-1

-2~ri

xswj(q~) +

( - -~-au - w t ( p )

2ori

+ eI

) dwj(p)

I

s~l

Because of (1.8) F2j

= exp

lnG d w j - 27ri

- ej -

2~ri E 1=1

wtdw j K2t-l

20ri s=l

xswj(qs ) +

20ri ~ w j ( p , ) k=l

.

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

22

Taking into account (2.46), (2.24), (2.22), we find

F2j exp { frlnGdwj

2~ri(aJJ - kj) + 2~ri ~ Gwj(G) - frlnGdwj --

~

s=1

g

g

EfK

-2~ri

I~l

g

w,dwj- 2~ri E Gwj(qs) + 2~ri E wj(Pk) 2t-1

s~l

k=i

- k j - 2~ri Et=l 2,-1w t d w j + 2¢rik=lEw j ( p k )

=exp -2~ri

•

By (2.38) and (2.39)

a j j l a ~j j F2j=ex p - 2 ~ i ( --~--+

-~ +

1..~ k~,jfK 2k 1wjdw* - 2 ~ i ~'"~f ~ l #:j

-2qri(wj(pj) +

+ 2~ri ~

wj(pk )

k~l

=

exp

-2wi ~ r--bj

=exp

d(%w~) -

wtdw j

K2I-1

I

wj(p~)

2r-1

-2~ri E [ [ w , ( p r ) + l ] w j ( P r ) - W j ( p ~ ) w , ( p ~ ) - w j ( p , rq:j

)

]}

=1.

(2.53)

Therefore, the function (2.50) is the solution of the boundary problem. If O(w~(q) - G) is identically equal to zero, it is necessary to use some function 0~, as was pointed out above. We have

F±(q) = 8 ~ ( w s ( q ) - k s - ls)exp {2-~ fr lnG(p)M*(p' q)dz(p) g

+~ r=l

+

2

wr(p)-l~-k~ M*(p,q)dz(p)

2r 1

'9%q,(q)-

m

(2.54)

The case ~¢>g. Fix K - g arbitrary points s1,...,SrE T +, r
in T +.

(2.55)

YmL. Rodin/The Riernannproblemon closedRiemannsurfaces

23

We obtain the problem - (~P ) F -°t' P )" F°+(P) = G(P) Y 7---:"

(2.56)

of index g. Therefore, in this case the solution depends on an arbitrary divisor of degree x - g. The case 0 < K < g. Let the divisor ~/+ ~ T +, deg~/+ = g - x. Assume

F+(q) = Fy +° ((qq)) ,

r_(q)=V°_(q).

(2.57)

We obtain the problem

F°+(p) = G(p)r+(p)F°(p)

(2.58)

of index g. The functions F ° are determined by the formula (2.54). The formula (2.57) determines a holomorphic solution if the function (2.54) is multiple of the divisor y +.

2.4. D-bar problem Finally, we consider the elliptic Carleman-Bers-Vekua (CBV) system

-Ou=Au,

5 - Oz(p) = 2

-0--~-+i

,

z(p)=x+iy,

(2.59)

where A ( p ) d z ( p ) is an invariant differential form. Usually one supposes [17] that the function A belongs to the Banach space Ls(M ), A ~ Ls(M), s > 2. For simplicity we suppose that A ( p ) is differentiable and has a finite number of points and fines of discontinuities of the first kind. If uo(q) is a particular solution of eq. (2.59), then the general solution has the form

u( q) = ep( q)uo( q),

(2.60)

where cp(q) is analytic. The function

,%(z)=-

l ffl,l<1a(t)d°t i-c-i

(2.61)

(do t is an area element) is such that O~o/Of = a(z) in the unit disk. This means that any solution of eq. (2.59) is a solution of the integral equation

u(q) + l f f M A ( p ) u ( p ) M ( p , q ) d a ~ ( , ) = ~ ( q ) , where ~ ( q ) is analytic. Conversely, any solution of eq. (2.62) satisfies (2.59).

(2.62)

Yu.L. Rodin/TheRiemannproblemon closedRiemannsurfaces

24

Another approach to the problem uses the multi-valued kernel M*(p, q). The function

o~(q) = -

ff ( p ) u . (

(2.63)

p, q) doz(p)

is such that (2.64)

0t~ = A ( q ) and has periods = - 2i f f . A ( p ) w j ( p ) doz(p),

j = 1 ..... g

(2.65)

along the cycles K2j. Thus, a particular solution can be taken in the form

uo(q) = exp ~o(q).

(2.66)

We obtain the general solution in the form

u(q) = q0(q) exp ~ ( q ) .

(2.67)

Here q0(q) is an analytic multiplicative multi-valued function whose periods along K2j are equal to (exp toj)- 1. The analogy between the formulas (2.67) and (2.1) (in the case xj = 0, j = 1..... m) is striking. The equivalence of the Riemann boundary value problem and the CBV equation can be easily proved. Let m

• +(p)=G(p)~_(p),

p~F=

~]~, j~l

indrG=0,

j = l ..... m.

(2.68)

Continue the function G(p) into the domain T- and suppose

u+(q)=~+(q),

q ~ T +,

u_(q)=G(q)~_(q),

veT -

(2.69)

(see B.V. Bojarsky [a]),

Ou=Au,

A(q)=

0,

q ~ T +,

G_l(q)OG(q),

q~T-.

Conversely, represent a solution of (2.59) in the form

u(q) = ep+(q)expo~ +(q),

q ~ T +-,

o ~ + ( q ) = - l f f v A ( p ) M + - ( p , q ) d % p ~,

(2.70)

where M+(p, q) are the single-valued Cauchy kernels whose characteristic divisors 8 +- belong to the domains M\2F+-. We obtain the Riemann problem

u+(p)=exp[o~_(p)-~o+(p)]u_(p)

onr.

(2.71)

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

25

The Abel theorem for the system (2.59) is written in the form

E a k w j ( p k ) ----00i (mod periods of wj),

j = 1 . . . . . g.

(2.72)

t, The R i e m a n n - R o c h theorem has the form l - h = deg 7 - g + 1

(2.73)

where l = dim L(y) is the dimension of the space of solutions of (2.59) that are multiples of the divisor - 7 and h = dim H(y) is the dimension of the space of differentials v d z ( p ) that are multiples of the divisor ~, and satisfy the equation

8o + Ao = 0 .

.

.

.

.

(2.74)

It is clear that the coefficient A(q) shifts the point ( q . . . . . rg), 5 = E~wj(Pk)' on the Jacobi variety to the vector ( % . . . . . %), just as the coefficient G ( p ) generates the shift (l 1.... , lg), lj= ~

lnGdwj,

j = 1 . . . . . g.

(2.75)

Now we can prove the Jacobi theorem about solvability of the inversion problem

wj(pk ) ==-lj (mod periods of wj),

j = l . . . . . g,

(2.76)

k=l

for x > g. By eq. (1.14) this system can be rewritten in the form

"/(f'2jd°Jq°Pk~ 2~ri/j

(mod periods of wj).

(2.77)

k=l

Let

h(q) = exp ~ ¢Oqop,(q ).

(2.78)

k=l

For the solvability of the problem (2.77) it is necessary and sufficient that the function h(q) have the multiplicative periods 2~rilj along the cycles K2j ( j = 1 .... , g). Choose the coefficient A(p) of (2.59) so that

p)w'j(p)doz(p)=lj,

j = l . . . . . g.

(2.79)

Then the function h(q)exp { - 00(q)} is single-valued, is a multiple of the divisor 8 - q0, and satisfies the eq. (2.59). The existence of such a function follows from the Riemann-Roch theorem (2.73) for K >__g.

Yu.L. Rodin/The Riemannproblemon closedRiemannsurfaces

26 3. M a t r i x R i e m a n n problem

3.1. Complex vector bundles 3.1.1. Presheaves and bundles An open coveting J g { Uj, j ~ I } of a compact Riemann surface M is defined by some finite set 1 of indices. T o every index j ~ I corresponds a coordinate patch Uj. Every point of M has to belong to a domain of the covering. We assume that some subset I 0 c I corresponds to a subset Jg0 c Jr' such that Ujn Uk'~ O, j, k ~ I o , and that the closure of the set U j ~ I ~ coincides with the whole surface. One example of such subset is a triangulation of the surface. Let every domain U c M determine some group (or ring, or module) P(U) such that any pair of domains U c V determines a homomorphism ruv: P(U)--*P(V) and ruv=ruwrwv . In this case a presheaf of groups (rings, modules) is said to be defined over M. We shall consider the presheaf of infinitely differentiable functions A °, the presheaves A"' ¢ of differential forms of type (a, fl) (a,/3 - 0,1)*, A ° ' ° - A °, their subpresheaves I2"'~ formed by holomorphic functions and forms, and the presheaf I2" of non-vanishing holomorphic functions with multiplication as the group operation (all these presheaves are, in reality, sheaves, [3]). A 0-cochain with values in the presheaf P associated to the covering ..¢t'{ Uj, j ~ I } is a set of functions ( q~j}, j ~ I, defined in the domains Uj and taking values in P(Uj); a 1-cochain is a set of functions ( +kj }, j, k ~ I defined in nonempty intersections Uj n Uk and taking values in P(Uj n Uk). The groups of 0- and 1-cochains with values in P are denoted by y , k ( p , ..¢t') or ~ek(p) (k = 0,1). The coboundary 8{ q,j} of a 0-cochain { q,j} is the 1-cochain (~jk }, ~jk = ~kj- q~k. The coboundary (+jk } of a 1-cochain { ~kjk} is the 2-cochain (q~jkt } (it is defined in nonempty intersections Uj n Uk n UI)

QJjkl = ~jk AI- ~ k l - l~ljl" A cochain whose coboundary is equal to zero is called a cocycle. The groups of cocycles are denoted by Zk(P) (k = 0,1). The group Z°(P) has a very simple interpretation: if {~j} is a cocycle, q~i= ~kk in any intersection Uj n U~ and hence (qg(P)} form a continuous function on M. Such functions with values in P are called sections of the presheaf P over M. Therefore, the group Z°(P) coincides with the group F(P, M) of sections of P over M. We shall consider also the groups F(P, G) of sections of P over domains G c M. Evidently, elements of the group F(A ~'~, G) are differential forms in the domain G. It is easy to see that the coboundary of the coboundary of a 0-chain is zero: 6(6 { qJj}) = 0, or "6 2 = 0." So, the group B I ( p ) = ~.~0(p) of coboundaries is a subgroup of the group Z I ( P ) . This allows one to define the cohomology groups

HI(p)=zI(p)/BI(p),

H°(P)=Z°(P).

(3.1)

The sequence of presheaves and homomorphisms ~j

ej_l

aj+i

4 --"

"

(3.2)

*The differential form ~o= adz(p) is a form of type (1,0), ~ =a dz(p) is a form of type (0,1), ~o= adz/x d~ (dz ^ d ~ - 2i d x d y, x + i y = z) is a form of type (1,1). A form of type (0, 0) is a function f(p).

Yu.L. Rodin/ The Riemannproblem on closedRiernannsurfaces

27

is called exact if Im aj = Ker %+ 1"" In particular, the sequence

O ~ P' ~ p ~ p

(3.3)

" ~0

is called a short exact sequence if K e r a = 0 , sequence ~*

I m a = Kerfl, I m f l = P " .

a 2

0 ---) H ° ( P ') ---) al H ° ( P ) ~ H ° ( P '') "-) H I ( p ') ~ H i ( p )

In this case the cohomology

H I ( P '') ---) " .

(3.4)

is exact. The homomorphisms ai, fli are induced by a, ft. The homomorphism 6" is constructed in the following manner. Let { ~j) ~ H°(P"). In the domain Uj of the covering ~ the element ~bj^has the preimage fl-t(ffj) = ~j determined up to terms belonging to P'(Uj) c Kerfl. Define elements ~kj - ~k ~ P'(Uj n Uk) in the intersections Uj n Uk. These dements determine some cohomology class of the group H I ( P l) denoted by 8*{ Cj). The reader may see details in [3]. Let us now define the concept of a complex vector bundle B over M with fibre C ~ and structure group G. The bundle B is determined by a set of non-degenerate n × n matrices hjk(p ) ~ G defined and holomorphic in the intersections Uj n Uk, j, k ~ I, such that

hjk(p ) =hjt(p)htk(p ) i n U j n U~n u~. The matrices {hjk ) are called the transition matrices of B. Let the functions cj(p) be defined in the coordinate patches Uj, take the values in C ' , cj ~ C ", and transform with the local coordinates according to the rule

cj(p) = h j k ( p ) c k ( p )

inUjn

(3.5)

U k.

Then the points of B are pairs (p, cj(p)). For n = 1, {hjk} is a cocycle belonging to the group Z1(~2"). The bundles B and B' with the transition matrices { hkt } and (h~/}, respectively, are equivalent if there exists a set of non-degenerate matrices h j, analytic in Uj, such that

hjk=hfl(p)hj~(p)hk(p)

inUjnU

k.

3.1.2. The matrix Riemann problem on the plane Consider the matrix Riemann problem

fh+(t) = G(t)@_(t)

on F.

(3.6)

Here G(t) is a non-degenerate n × n-matrix satisfying the Hrlder condition, ~ ± ( z ) are n-vectors or n × n-matrices analytic in the domains T -+ respectively. The functions ~+_(z) are supposed to be holomorphic or to have a finite number of poles. A vector solution having a pole of the order m at infinity is represented in the form

• ±(z) = 2~r-----~

(~) "r-z +P,,(z),

*Ker a is the kernel of a, Im a is the imageof a.

(3.7)

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

28

where P,,(z) is a polynomial of degree m with vector coefficients. Substituting (3.7) in (3.6) we obtain the system of singular integral equations, l+G(t) 2

q0(t) +

1- G(,) f: d~" )~--1(~') ~-z-~_t = [1 - G(t)] Pro(t)

(3.8)

of index = ind r det G. The number l of the solutions of (3.8) satisfies

l>r+(m+ l)n-l', where (m + 1)n is the number of free coefficients on the right side of (3.8) and l' is the number of solutions of the adjoint equation. Therefore, if m is sufficiently large, there are n vector solutions of (3.6) forming an n × n matrix F(z) such that det F(z) ~ O. Note that the matrix solution F(z) of the problem (3.6) satisfies the boundary condition det F+(t) = det G(t) det F_(t) and hence det F(z) has N + r zeros, where N is the order of the pole of det F(z) at infinity. Using appropriate linear combinations of the columns of F(z) with rational coefficients, it is possible to obtain a certain matrix solution X(z) called canonical and possessing the following two properties: The matrix X(z) is nondegenerate at any finite z and the order K of a zero* of det X(z) at infinity is equal to the sum of the orders K: of the columns Xj(z), j = 1..... n [12]. The integers Kj are called the partial indices and are independent of the canonical solution chosen. If the point z = 0 belongs to T ÷, the matrix G(t) can be factorized in the form

G(t) =H+(t)diag(t ~, ... tK")H_(t),

(3.9)

where H+_(z) are holomorphic nondegenerate matrices. The representation (3.9) is unique. If x = 0, we adduce the two following criteria:

Criterion 1. In order that the problem (3.6) have zero partial indices it is necessary and sufficient that the problem possess a holomorphic matrix solution. Indeed, such a solution generates the factorization (3.9) with K1 . . . . .

~n = O.

Criterion 2. (I.Ts. Gohberg, M.G. Krein [5]). In order that all partial indices of the matrix G(t) be zero it is sufficient that one of the matrices ½ [G(t) + G+(t)] and (1/2i) [G(t) - G+(t)] be definite* *. For further results in this area see, for example, I.M. Spitkovskii [a] and H. Bart, I.Ts. Gohberg and M.A. Kaashoek [a-e]. *A pole of order ]rl if r < 0. **G + (t) is the Hermitean conjugate to G(t); a matrix ]laijl] is called definite if the quadratic form ~.aij~i~j takes values of a single sign.

Yu. L Rodin / The Riemann problem on closed Riemann surfaces

29

3.1.3. The Riemann problem on a Riemann surface Let us now turn to the study of the problem

q~+(p)=G(p)~_(p),

p~F,

(3.10)

on a surface M of genus g > 0. Our assumptions about F and G(p) are the same as above. The problem can be reduced to a system of singular integral equations or a system of Fredholm equations via the Cauchy type integral. It can be shown that the problem has n independent solutions having a finite number of poles. To every Riemann problem (3.10) corresponds some vector bundle Br. It is constructed by the following method. Let qoj be nondegenerate holomorphic matrices in the domains Uj belonging to the covering ~t' { ~ , j ~ I }. If Uj N F is nonempty, the matrices qoj have to satisfy the boundary condition

~pj+(p)=G(p)~pj_(p), p~rC~ G..

(3.11)

The coboundary of the cochain { qoj} is { qojk},

epjk(q) =Cpfl(q)epk(q),

q ~ ~ A U~.

(3.12)

The matrices (qojk(q)} are holomorphic, nondegenerate and satisfy ¢pjkrpk/qVlj= 1. The vector bundle B e with fibre C n and structure group GL(n,C) is defined by the transition matrices {q~jk}. Choosing another cochain ( rp~}, we obtain a bundle equivalent to Be*. This relation was established by A. Grothendieck [a] for g - 0 and H. Rrhrl [a] for g > 0. It can be shown that for every complex vector bundle B with structure group GL(n, C) there exists a matrix G ( p ) defined on the given contour F such that the bundle B is equivalent to the bundle B e (Yu.L. Rodin [d]). The matrix solution X(q) of the problem (3.10) is called canonical if the order of det X+(q) is equal to the sum of the orders of the columns at every point q ~ M.

Theorem 3.1. (Yu.L. Rodin [d]). In order that the problem (3.10) have a canonical solution it is necessary and sufficient that the bundle B c be equivalent to a direct sum of line bundles. Indeed, let B c be decomposed into a direct sum of line bundles ¢Pjk----hf I diag(h~k

"-"

hjnk)hk

in~n

Uk.

(3.13)

Taking into account (3.12), we obtain the relations

hjepTa=diag(h~, k ... hj~k)hkg~ 1 inUjf) U~. The transition functions {h~k } of the corresponding line bundles can be represented in the form h~k = A~/A~k, where A~(q) are meromorphic functions in ~ [3, 8]. We obtain that d i a g ( A ) . . . . A n~-l"j}njepj-l=diag(A1...A~)

-1

hkq0k- 1 = x - X ( q ) .

(3.14)

The matrix X(q) is a canonical solution of the problem (3.10). *Two b u n d l e s are e q u i v a l e n t if there exists a h o l o m o r p h i c n o n - d e g e n e r a t e m a t r i x c o c h a i n { hj } such that qojk = hj-1 ~)khk. , l n }. b u n d l e B is a d i r e c t sum of line b u n d l e s if the t r a n s i t i o n matrices of B are diagonal, qojk -- - d i ag ( q0)k - • • ~0)k

The

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

30

Conversely, a canonical solution is represented in Uj in the form X ( q ) = X j ( q ) d i a g ( A ~ . . . A~),

q~ ~,

(3.15)

where the nondegenerate holomorphic matrices ~ ( q ) satisfy the boundary condition

Xj+(p)=G(p)Xj_(p),

p~UjNF

and the functions A~.(q) are meromorphic in the domains Uj. The nondegenerate matrices hj = X7 lcpj are holomorphic in Uj and ¢Pjk =

01)7 lfpk = hf 1X7 1Xkh k =

hf I

diag (A1/A~ ... A~/A~) h~.

(3.16)

This means that the bundle B a is equivalent to a direct sum of line bundles with transition functions ( A~(q)/A~(q)}. Q.E.D. Let X(q) be a fundamental solution of the problem (3.10) and Xl(q) ..... X,(q) be its columns, X = (X x.... X.), and set the divisor y = (det X). Let 7j = (Xj), j = 1..... n be the divisors of Xj*. Then 3,='/0 + ~ '/j,

'/0>0,

(3.17)

j=l

where 70 is the divisor determined by the linear dependence relation between columns. The general solution of the problem has the form

F(q) = ~ Pj(q)Xj(q),

(3.18)

j~l

where the Pj(q) are analytic functions that are multiples of the divisors -(./j + ./0). Singularities at the points of the divisor '/0 have to vanish. The general solution of the problem was described by Yu.L. Rodin

[d]. The problem

•P + ( p ) d z ( p ) = G * - l ( p ) ~ P _ ( p ) d z ( p )

(3.19)

for vector-differentials ' P ( p ) d z ( p ) is called adjoint to (3.10). Let / and h be the numbers of holomorphic solutions of (3.10) and (3.19), respectively. The index formula l - h = indr det G - ng + n

(3.20)

follows from the Riemann-Roch theorem for vector bundles. The formula (3.20) and the Riemann-Roch theorem can be obtained from the analysis of the general solution (Yu.L. Rodin [d], G.D. Merzlyakova and Yu.L. Rodin [a]). • A v e c t o r - c o l u m n h a s a zero o f o r d e r m if eoery c o m p o n e n t h a s a z e r o w h o s e o r d e r is > m , a n d a pole o f o r d e r s if at least o n e c o m p o n e n t h a s a p o l e o f o r d e r s.

Yu.L Rodin/The Riemannproblemon closedRiemannsurfaces

31

3.2. Approximate solutions of the matrix Riemann problem In this section we adduce one approximate method for solving a Riemann problem. Let*

G(p)=R~_l(p)[l + g ( p ) ] R _ ( p ) ,

p~F

(3.21)

where the matrices R ±(q) are analytic in the domains T +-, respectively, and posses s a finite number of poles and zeros, 1 is the unit matrix and the matrix g(p) = Ilgull, Igu(P)l -< go, i, j = 1 . . . . . n satisfies the H f l d e r condition. The constant go will be determined below. Consider the problem

~+(p)=G(p)~_(p).

(3.22)

Letting

F±(q)=R±(q)~±(q),

q ~ T ±,

(3.23)

we obtain the problem

F+(p)-F_(p)=g(p)F_(p).

(3.24)

The problem (3.24) will be studied in the Hardy classes H2(T +-) defined below. Let d a ( p ) be some real differential on the contour F. The space L~(F) (the index n might be omitted) is the Hilbert space of vector functions F ( p ) = ( F 1.... , F n) with the scalar product

(F,F') = ~

j~l

f:.(p)F/(p)da(p).

(3.25)

Let 8 be a divisor such that deg 8 = g and dim 8 = 1. For simplicity we suppose that 8 ~ T-. We fix also an arbitrary point qo ~ T-, q0 ~ 8. The Hardy classes H2(T ±) (more exactly, H2(T +-,8, qo, da, n) are the classes of vector functions that are analytic in T ±, respectively, multiples of - 8 + qo in T - and possess boundary values on F belonging to the space L~(F). We use two properties of these classes [19]: a) If the vector functions F±(q) ~ H2(T ±), and if

F + ( p ) = F (p),

pEF,

then F±(q) form a single analytic vector function on M (that is a multiple of qo - 8). b) Vectors in H2(T ±) are represented by Cauchy integrals; the Plemelj-Sokhotskii formulae (1.29) are valid for the Cauchy type integrals with densities of class L2(F ). We search for a solution of the problem (3.24) in the form of a Cauchy type integral,

F ± ( q ) = l + - ~1- ~ f ~ ( p ) M ( p , q ) d z ( p )

(3.26)

*Let G(p) = H(p) + h(p), where H(p) = R71(p)R_(p). Then G(p) has the form (3.21), where g(p) = R+(p)h(p)RZl(p). If the elements of the matrix h(p) are small, the elements of g(p) are small too.

32

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

with the characteristic divisor 3 and the polar point qo- We obtain the singular integral equation

[1

ep(q)=g(q)-~-ep(q)+-~i

1::

(p)M(p,q)dz(p)

]

+g(q),

q~F,

(3.27)

in the space L2(F ). The operator

=

-

(p)M(p,q)dz(p), q r,

+

(3.28)

is a projection operator from L2(/' ) onto the space Im A of boundary values of vector functions of class H2(T- ). The kernel kerA of the mapping A is the space of boundary values of vector functions of class

H2(r+). Let cp(q) ~ L2(F ). Assuming 1

F(q)=~-~fr~(p )

M

(p,q)dz(p),

we obtain that

~p(p)=F+(p)-F_(p),

per,

F+_(q)~H2(T+-).

(3.29)

Therefore, L 2 ( F ) = KerA * Im A. The representation (3.29) is unique since KerA n Im A = 0. Indeed, if f(q) ~ KerA n Im A, then f(q) is analytic on M and is a multiple of the divisor q0 - 3. Since dim3 = 1, f(q) = const and hence f(q) = O. As follows from the well-known theorem of M. Riesz [19], the operator A is bounded in L2(F ), IIAI[L~tr) < oo. However, generally speaking, the operator A is not an orthogonal projection, and hence [[A[IL2(r) _> 1. The norm of the operator gA is _
- - , go < IIAIIL2(r) the norm

(3.30)

Ilghll < 1, and hence the iterations

[1

1::

%,(q)=g(q)-~%,-1(q)+~'-~

m-a(p)M(p,q)dz(p)

]

+g(q)

(3.31)

converge in the space Z2(y). If M is a hyper-elliptic surface (1.1) with real roots z t . . . . . z, and F is the set of c u t s ( Z 0 Z l ) , . . . , (Z2n , 00) (fig. 3), the Abelian differentials of the first kind are defined by eq. (1.5). Let P1(~1, - ) "'" P,(~,, - ) be n points belonging to T-. As follows from (1.5), there is no Abelian differential of the first kind that is multiple of the divisor 6 = E~pj, since any Abelian differential of the first kind has n - 1 zeros on every sheet. This means that there exists an Abelian differential da(q) of the third kind with poles at the points (40, - ) , ((o, + ) and zeros pj ( j = 1.... , n) that is real on F. Using the kernel M(p, q) with characteristic

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

33

divisor 8=Z~pj and pole at q 0 ( ~ 0 , - ) , we conclude that in this case K e r A _ L I m A . Indeed, let O(q) ~ I m A and qJ(q) ~ KerA. Introduce the involution of M by the rule q(~, + ) ~ ~(~, - ) . Then ~k(q) ~ H 2 ( T - ) and

£rp(p)¢(p)da(p)

=0

since the differential ep(q)+(gt)da(q ) is holomorphic in the domain T-. Therefore, A is an orthogonal IIAII = 1, and one can assume go < 1. If the genus of the surface M is equal to zero, the iteration process generates a holomorphic solution since 8 = 0. This means that all partial indices of the problem (3.24) are equal to zero. In the general case we obtain a solution that is a multiple of the divisor - 8 .

projector,

3.3. D-bar problem Here we consider the equation

Ou =Au,

(3.32)

where A is an n x n-matrix and u(q) is a vector-column or an n x n-matrix. If Uo(q) is a matrix satisfying eq. (3.32), and cp(q) is an analytic vector or matrix, then

u( q) = uo( q)ep( q)

(3.33)

is also a solution. Therefore, the problem is reduced to finding a particular, non-degenerate, matrix solution. We start with the case of the unit disk Izl < 1. The eq. (3.32) is equivalent to the integral equation

dot = O ( z ) , u(z) + l f~tl
(3.34)

where O(z) is an analytic matrix. Let u 1. . . . . u ' be linear independent vector solutions of the homogeneous equation Ku = 0. Every such solution is analytic in the domain Izl > 1, and u J ( o o ) = 0. If u(z) is a matrix solution of the equation Ku = 0, then ( det u) = Sp A . det u,

(3.35)

det u(z) is analytic in [z I > 1 and det u(oo) = 0. Hence it appears that det u(z) - O. Therefore, the number s of vector solutions of the equation Ku = 0 is less than n. Let vI..... v" be vector solutions of the equations Ko j = Pg, where Pg(z) are vectors whose components are polynomials of degree m; the matrix v(v 1..... v ' ) is formed by the columns v j ( j = 1 . . . . . n) and det v(z) ~ O. The function det v(z) has a pole at infinity of order m o = nm, and it has m 0 zeros in the z-plane. Let z 0 be such a zero, det V(Zo) = 0. Then there are constants cj such that C 1 U I ( Z o ) "J¢" " ' "

"~"cnon(zo) "~- O.

34

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

Let cj 4: 0. Set

z l z o (clvl(z)+ ... +cnvn(z))

u J ( z ) ~-

(3.36)

and substitute the column v~(z) for vJ(z) in the matrix v(z). The determinant det vl(z) of the new matrix is equal to det v(z)/(z - Zo). Therefore, the function det vl(s) has no more than m 0 - 1 zeros. Repeating this procedure m 0 times we obtain a matrix solution iT(z) such that det tT(z) is different from zero for Izl < oo. Return to the case of an arbitrary dosed Riemann surface M. Let J / ( Uj, j ~ I ) be an open covering of M. As was shown above, in every coordinate patch Uj there exists a non-degenerate matrix solution vj(q). If u(q) is a solution of (3.32) on M, we have

u(q) =vj(q)q~j(q)=v~(q)~p~(q),

q~ Ujcq Uk.

This means that (9~j} is a section of the vector bundle BA determined by the transition matrices (H. RiShrl [al)

hjk( q ) = vf t( q) vk( q).

(3.37)

If T is a finitely-connected domain in M, then any vector bundle possesses a holomorphic non-degenerate matrix section over T [3] and hence there exists a non-degenerate matrix solution Vo(q) of (3.32) (A. Turakulov [a]). Let the surface M be separated by the contour F into the domains T ±, and let vf(q) be non-degenerate solutions in T ±, respectively. We obtain the representations

u(q) =Vo~(q)~p ±(q), q~ T ±

(3.38)

for every solution of eq. (3.32). We obtain the Riemann problem

ep+(p)=G(p)~p_(p),

G(p)=[vg(p)]-lvo(P),

p~Y.

(3.39)

The bundle B c is equivalent to the bundle BA determined by (3.37). Therefore, the Riemann boundary value problem and the D-bar problem (3.32) are equivalent. Here we adduce an approximate method suitable for eqs. (3.32) which are almost scalar. Consider the operator

pu=

7rl ffra(p)u(p)M(p,q)doz(p),

(3.40)

where M ( p , q) is the Cauchy kernel with characteristic divisor 8 ~ M - T. The estimate

IIP:llc(r) < MpllallL,(r)llullc
P>2

is valid. Here Ul

IlulIc
u=

i

Un

(3.41)

Yu.L Rodin / The Riemann problem on closed Riemann surfaces

35

This inequality is an elementary consequence of the corresponding scalar estimate for a simply-connected domain [17]. Let

A(q) = a(q). 1 + Ao(q), where 1 is the unit matrix and the norm

(3.42)

IIPoullc(r)

of the operator

Pou= _ l ffrao(P)u(p)M(p,q)do~(v)

(3.43)

is less than one, IlPollc(r ) < 1. The solution of eq. (3.32) is represented in the form

u(q) =~(q)v(q), where ~(q) is a scalar function '

~(q) =expWe

(p)M(p,q)da~(p).

~

(3.44)

obtain the equation

v + Pov= l,

(3.45)

solvable by iteration. If

A(q) =

diag(at

• ..

an) +Ao(q),

(3.46)

the same method is applied. In this case appears the operator

PlUm

-1 f f r ~ - I (p)Ao(P)~l(p)u(p)M(p,q)doz(p) ,

where ~/(q) = diag (771. . . . .

(3.47)

~n),

71J(q) = exp - -~ 1 ffra j(p)M(p,q)daz(p).

(3.48)

The norm [[P1Hc(r) has to be less than one, t[Pl[lc(r) < 1.

4. I n t e g r a b l e

systems

4.1. Korteweg-de Vries equation. Reflection finite-gap potentials 4.1.1. Boundary value problem statement In this section we consider some physical applications of the Riemann problem. We don't pretend to completeness, and confine ourselves to illustrations of the method. Many of the most important applications

36

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

(Yang-Mills equations, multidimensional scattering) have been omitted; all details have been omitted too. In this section we rely on the finite-gap integration method for the KdV equation ([1, 15]; see also S.P. Novikov [a], B.A. Dubrovin [a], A.R. Its and V.V. Matveev [a], S.P. Novikov and B.A. Dubrovin [a, b], I.M. Krichever and S.P. Novikov [a], P. Lax [a], V.A. Marchenko [a, b], V.A. Marchenko and I.V. Ostrowskii [a], N.N. Meiman [a], H. McKean and P. Van Moerbeke [a]). Consider the one-dimensional Schrbdinger equation,

L(t)~ =

--d2ff + u(x, t)~b = Xtp, dx 2

(4.1)

with a real potential u(x, t). In order that the spectrum of the operator L(t) be independent of time t, the Lax equation

L=[L, AI=LA-AL

(4.2)

has to hold. Here A is a skew-symmetric operator. Indeed, from eq. (4.2) it follows that if A is independent of t

L(t) --- exp ( - A t ) L ( O ) exp (At),

(4.3)

where expAt is unitary. The unitary equivalence (4.3) of L(t) and L(0) entails the coincidence of all spectral characteristics of these operators. In the simplest non-trivial case, A is a differential operator of the third order, A = dx-----7 - 6u

- 3u x

(4.4)

and the Lax equation has the form U t --

(4.5)

6uu x + Uxxx = 0

(the Korteweg-de Vries equation). If the order of A is more than three, we have higher KdV equations. For simplicity we confine ourselves to the case (4.5). Let O(x, X) and q0(x, X) be the solutions of eq. (4.1) determined by the conditions

0(0, X) = ¢p'(0, )t) = 1,

0'(0, X) = ¢p(0, h) = 0.

(4.6)

For every non-real h, eq. (4.1) possesses solutions [10, 11, 14]

fj(x, h) = O(x, h) + mj(h)cp(x, h),

j = 1,2,

(4.7)

that are square integrable over the semiaxes ( - o¢,0) (for j = 1) and (0, o¢) (for j = 2). Here mj(h) are the so-called Weil-Titchmarsh functions, represented in the form

mj(X)=

::[ ~o

1

+1+

"]

mj(•)=(-1)Jix/X - + p j ( - o o ) + ~ ( 1 ) ,

j=1,2, X~.

(4.8)

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

37

The functions pj('r) are the spectral functions of the operator L in the spaces L2(-oo,0) (for j = 1) and L2(0, oo) (for j = 2). The spectral functions are nondecreasing, absolutely continuous (except for finite discontinuities at the points of the discrete spectrum), and are determined by the conditions

f?o~(x,X)~(y,X)dpj(X)=~(x-y), x,y~

[(-oo,0),

ifj=l,

(0, oo),

ifj=2.

(4.9)

Furthermore, if [l~ij(~)[I is the spectral matrix for the operator L in the space L 2 ( - oo, oo), m j ( )k ) =

(-1)2½+M12(A) Mn(A ) '

M U ( A ) = f °°1 d~lj(Z) -o¢ X 7 ~ '

(4.10)

j=1'2

From the formulae (4.8), (4.10) it follows that the functions mj(A) have poles on the real axis at the points of the discrete spectra of the operator L in the spaces L2(-oo,0), L2(0, oo). The functions mj(A) of the complex variable )~ have jump discontinuities across the continuous spectrum E of the operator L in the space L 2 ( - oo, oo), with possible logarithmic singularities at the ends of the spectral bands. Because of (4.5), the functions mj()~) are time independent. Here we consider the Cauchy problem for the eq. (4.5). We assume that the initial value of the potential u(x, 0) is continuous and that the operator L(0) (and hence L(t)) has continuous spectrum n-1

E=

[,J (X2k, hEk+l)U(X2,,oo)

(4.11)

k=0

consisting of a finite number of bands. The discrete spectra of L in the spaces L2(-oo,0) and L2(0, oo) are supposed to be finite. This is the class of reflectionfinite-gap potentials. The complete description of this class is still unknown. First of all, this class contains periodic and almost periodic finite-gap potentials [1, 15]. In the periodic case, the Weyl-Titchmarsh functions have the form [10] mj(h)=

Q(A) P(X)

( - 1 ) J i "R(x)] P(X)

j=l,2, '

2n

R ( X ) = I--[(X-X,),

P(~,)= fi(~,-$k),

1=0

k=l

Q(~,)=P(A) i k=l

~/-R(~k) P'(~k)( )~- Xk) '

(4.12)

where the h I are the endpoints of the bands of E, and ~k e ()~2k-1,)~2k) are arbitrary points lying in the gaps. The Weyl solutions (4.7) coincide with the Floquet solutions, i.e. the eigenfunctions of the operator of the translation through one period. The class of reflection finite-gap potentials contains other potentials as well. For example, a disturbance of a periodic finite-gap potential by a rapidly-decreasing potential Uo(X,t) satisfying the condition

38

Yu.L. Rodin/ The Riemannproblemon closedRiemannsurfaces

[a]; for the case n = 1 see also E.A. Kuznetsov and A.V. Mikhailov [a]). A disturbance by an integrable potential generates finite-gap potentials whose discrete spectrum can be located in both gaps and bands. Note that potentials on the semi-axis of the type

u ( x ) = ( q1(x)' q 2( x ) ,

x O,

qj ( x ) are periodic finite-gap functions

(4.13)

have a finite-gap continuous spectrum. Consider the two-sheeted Riemann surface M of the function w 2 = R(X), R ( X ) -- l-lj:o(h 2, have the local coordinate ~(h) at infinity ~= 1,

k=~f~_x2,.

-

~j).

We

(4.14)

The sheets of M are determined by the sign of Im k for Ikl > IX2.1 + I~01. If Im k > 0, we have the sheet M1; the sheet M 2 corresponds to Im k < 0. The functions fj(x, ~) have the following asymptotic behavior:

fj(x,X)=exp(-ikx)epj(t)+o(1),

h~oo,

j=l,2

(4.15)

As long as the functions ~(x, X) are integrable with respect to x, we conclude that ~ ~ Mj ( j = 1, 2). We now introduce the functions fj(x,X)=~(x,h),

X~Mj,

j=l,2.

(4.16)

The operation ~ consists in the passage from the point h to the point ~ followedby projection to the other sheet of the surface. On the cuts E we have the relation

fl(x, X+) = a(X)f2(x, X_) + b(X)f2(x, X_),

X+_~ E.

(4.17)

Here ~, + are boundary points of the sheets M 1 (the sign +) and M 2 (the sign - ). By the gluing rule, if the point ~,+ belongs to the upper edge of the cut on M1, the corresponding point ~_ belongs to the lower edge on 3'/2, and conversely. As follows from (4.6), (4.7)

a(X)+b(X)=l,

a(~k)m2(X_ ) + b ( x ) m 2 ( x _ )=ml(~+).

Eq. (4.17) means that the pairs of linearly independent ( f2 (x, k ), f2 (x, ~,_)) are connected on E by the relation

a(x) = detT(X) =

W(fl,]~)

(4.18) solutions

(fl(x,X+),fx(x,X+)),

(4.19) Im ml(X)

W ( f z , L ) = imm2(X ) , where W(f, g) is the Wronskian of f, g. Consider the row-vectors ~p+(~) = (fl, f2) and ~k-(~) = (f2, fl),

Yu.L. Rodin/The Riemannproblem on closedRiemannsurfaces

39

analytic in the domains M 1 and M E respectively. The behaviour of ff ±(~) at infinity will be studied below. We have on E

~+(x)=(l,(X+),12(x_)), ~_(x)=(/2(x_),/l(X+) ), since X ± =

X ~. We obtain the relation

1

1

= (f2 (X_),fl(X+)) G(X)a---~ = ~k_(h)G(X) a--~'

(4.20)

A(X) = det T(Tt), G(X) = (A(X) -b(X)). t b(Tt) 1 Letting

lk°±(X)=lk±(Tt)exp(ikxa')' °3=(~ -7)'

(4.21)

we arrive at the Riemann boundary value problem ~b° (X) = ~b°_( X ) ~

exp ( - i k x o 3 ) G ( X ) exp (ikxo3),

(4.22)

with singularities at infinity and at the points of the discrete spectrum and at the ends of the spectral bands. 4.1.2. Study of the boundary oalue problem Introduce the operator f*(x,h) = d/(x,X) +ikf(x,)t) dx

(4.23)

and the matrices

~+(~,)=

{f1(x,~,) t.f,.(x,x)

f~(x,~).) [A(x,x)]

'/'_(h)= (fz(x'~) '

f](x,~,) )

t

satisfying the equation (below, we suppose that X2, = 0, i.e., we replace u(x, t) - )t2, by u(x, t)) ( ' / ' ± ) x = ( - i k ° 3 + Q(x, X ) ) * ± ,

Q(x, t) =

(0u 01) "

(4.24)

40

Yu.L. Rodin / The R i e m a n n problem on closed R i e m a n n surfaces

The matrices ~/' ±(4) satisfy the boundary condition

1

• +(x) =

(4.26)

a(4) and for t -- 0 have the asymptotic behavior

• +(4)-

1 0

1 ) e x p ( _ i k x o 3) 2ik

as4~o¢.

(4.27)

The KdV eq. (4.5) is the compatibility condition for the system (4.25) and the system ( x/,+ ) t = [ - 4ik3o3 +

g(x,t,k)]ko ~,

t0n

R(x,t,k)=4k 2

-2ik

(

u, Ux

0 + -- U

(4.28)

-Ux 2 U 2 -- U~x

U

(see [15]). In the space of 2 x 2-matrices a unique integral surface of the system (4.25)-(4.28) passes through every point (4 is supposed to be fixed and different from the discrete spectrum points 71..... */,, and the band endpoints 4 o.... ,42n ). Consider the integral surface containing the curve (4.24) for t = 0. The sections t = const and x---const of this surface are integral curves of the eqs. (4.25) and (4.28), respectively. These solutions, for any given time t, are called g'+(4) and the corresponding matrix elements are called ~(x, 4). These functions coincide with (4.7) at t = 0. Therefore, ko+(4) are the solutions of the system (4.28) with initial values q±(X)lt= o. Since the initial values and the right-side of (4.28) are analytic with respect to 4 on

Mj\{ ~ */k} ( J = l f ° r ' / p + a n d j = 2 f ° r q ' - ) , ~]k E Mj

the matrices ko_+(4) are analytic on the corresponding sheets Mj with poles at the points */k and logarithmic singularities at the points 40 ..... 42n. As follows from (4.25) and (4.28), the matrices q'±(4) have the asymptotic behavior

'/'_+(4)-

(10 2~k ) e x p ( - i k x - 8 i k 3 t ) o 3

as4~oo.

(4.29)

The dynamics of the coefficients a(4), b(~,) can be obtained from (4.28). Fix two arbitrary values x 1 and x2, and consider the integral curves ff'_+(X)l~=xj, j - - 1,2. Using (4.17), we calculate the values a(4) and b(X) for every t. The operator

a(h)) b(4)

(4.30)

= Kx [t°(t)]'

(.d(l) = ( U ( X 1 , l), U~(Xl, has triangular form.

t), U'x'~(Xl,t), U(X2, t), Ux(X2, t), U'x'x(X2,t))

YmL Rodin/The Riemannproblemon closedRiemannsurfaces

41

Set if'° (h) = g'_+(X)exp (ikx + 8ik3t)a3.

(4.31)

We obtain the boundary problem ~ ° ( x ) = ~/'°-(X) exp ( - i k x _ 8ik3t)o3 G,X,exp(ik x t ~ a(X)

x

E,

(10

-(2ik)-l](2ik)_ 1 ]

asX-

+ 8ik3t)o3, (4.32)

,

in the class of matrices that are multiples of the divisor "y = Ek~__l*/k and have logarithmic singularities at the points h o..... ~2,The problem (4.32) can be solved by the method of section 3.2. The Cauchy type integrals over the non-closed contour E have, generally speaking, logarithmic singularities at the endpoints of the bands, ~o ..... ~2,; this is permitted by the hypotheses. Solving the problem (4.31), we obtain the formula for the potentials and its derivatives

-Ux(X't) 2fi2(x, t) - Uxx(X, t)

2u(x't))=~t(h2,)~-x(X2,), Ux(X, t)

~=u-X2,

(4.33)

Letting x = Xl, x 2 in (4.33) we come to the non-linear equation (cf. (4.30)) (4.34)

t~(t) = H[o~(t)],

where H is a triangular operator. Solving the eq. (4.34), we can calculate the values a(X), b(X) by (4.30) and obtain the potential u(x, t). Consider the case of periodic potentials at greater length [1, 15]. Taking into account the branches of the function ~R(;k) and eqs. (4.12), (4.18) we conclude that in this case b(X) - 0 and a ( h ) - 1. Therefore, we have the "reflectionless" case • + ( h ) = g'_(X),

X~E.

(4.35)

The function

B(X)=[ fl(x'x)' X MI' [f2(x,

X),

(4.36)

X • M2,

is called the Baker-Akhiezer function. We have

B(h) - exp ( - i k x - 8ik3t) as X--* oo.

(4.37)

The poles of the Baker-Akhiezer function are determined by the initial conditions. From (4.12) it is seen that only one of the functions mj(~) ( j - - 1,2) has a pole at the point ~k (k = 1..... n) depending on the

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

42

chosen branch of ~ - R ( ~ k ) . This means that the function B(X) has only one pole ~/k(~k, + ) over the point ~k, and so has exactly n poles. Therefore,

B(X)=[3j(X)epj(X),

XeMj,

j=1,2,

where qpj(X) is analytic on Mj, q0j(oo) = 1 and

f l j ( A ) = e x p ( - i k x - 8 i k 3 t ) n,~Mj I-I £k +~( - -1 ) ~J ~ /

' X ~ Mj,

(4.38)

where ~/ is an arbitrary point belonging to the upper half-plane, Im 71> 0. We obtain the scalar boundary value problem q0i()k ) = ~l-~q02tl~2()k)/~k), X ~ E ,

qpj(oo) = 1.

(4.39)

The quantities (2.24) are easily calculated:

lj = ixw5(oo) + 4itwf" (oo) + i wj(nr),

j = 1 ..... n,

(4.40)

r=l

where wj(X) are the normalized Abelian integrals of the first kind. From (2.54) it follows that

{xj

epj(X)=O(w,(X)-k,-l~)exp

)

~

X~Mj,

(4.41)

where the functions ~j(X) are defined by eq. (2.54) and are independent of x and t. Note that for hyperelliptic surfaces a = 0 [16]. From (4.1) it follows that

u ( x , t ) = X + B - X ( X ) dzB(A) dx 2

,

for IXl < oo,

E*=OM 1 is E passed twice.

(4.42)

In order to obtain the analogous formula for X = oo we assume X = dln B/dx. The function X satisfies the Riccati equation

d---~X X2 dx +

(4.43)

-u+X=0.

We have the asymptotic expansion X(A) = i k +

x.(x) rt=l

(2k)" "

(4.44)

Then ,,. d 2 In B ( ~ ) ~ .

u(x,t) = 2 i x l ( x ) = z ,

(4.45)

Yu.L. R o d i n / T h e Riemann problem on closed Riemann surfaces

43

The Its-Matveev formula dx 2 ln0 i x w ' ( ~ ) + 4itw" (~) +

w,(nr) - ks

(4.46)

follows from eqs. (4.42), (4.45). The same formula is valid for almost-periodic finite-gap potentials [1, 15]. 4.2. The Landau-Lifschitz equation 4.2.1. Rapidly decreasing potentials In this section we rely on the V.E. Zakharov-A.B. Shabat [a, b] method called the Riemann problem

method. We consider the equation

S,=SXSxx+SXJS,

IS(x,t)l=l,

J=diag(J1, J2, J3), Jl
(4.47)

This equation describes spin waves in a ferromagnet (M. Lakshmanan [a], L.A. Takhtajan [a], V.E. Zakharov and L.A. Takhtajan [a], E.K. Sklyanin [a], A.E. Borovik [a], A.V. Mikhailov [b], Yu.L. Rodin [e, f], A.B. Borisov [a] and the book [9]). This equation is a compatibility condition for the systems 3

i~px= L~,

L = ~ w.(X)S.(x, t)%,

(4.48)

a=l 3 i~t=

3

Mq~, M= E wa(X)SaSvx°~e"#v- E w#(X)wv(X)S~%le~aVl, a=l

(4.49)

a=l

where % are the Pauli matrices, °t=(011),

a2=( 0

1

oi) '

0),

a3=(0-1

e*#v is the completely anti-symmetric tensor of the third rank, e 123 in the period parallelogram R = (IRe)q < 2K, lira h I < 2K'}, wl(X) =

0 sn(X, k) '

~J2-Jt k=

J3 -

w2(X) = 0dn(X, k) sn(X, k)

1 '

p=

"

,

W3(X) =

=

1 and w~()~) are the elliptic functions

pcn(X, ) sn(X, ) (4.50)

If J1 = J2 = "/3, then 1

w~(X) = X'

a=1,2,3,

(4.51)

44

Yu.L. Rodin/The Riemannproblemon closedRiemannsurfaces

and the period parallelogram R degenerates into the complex X-plane. We consider the Cauchy problem for the eq. (4.47) and assume that S(x,O)--, (0,0,1) as x--* +oe sufficiently fast, The Jost functions for eq. (4.48) are determined by the asymptotics

f-+(x, X) = exp ( --iw3()k)X03} +z~(1)

as x ~ + oe,

(4.52)

and are connected by the transition matrix T(X),

f+(x, X)--y_(x, X)T(X),

ImX=0,2K'.

(4.53)

It has the form T(X)=

a()k) b(X)

-b(X)) a(X) ']

a(•+2K)=a(h),

b(X+2K)=-b()~),

a(X+2iK')=a(~t),

b(X+2iK')=-b(X).

(4.54)

Since det fj(x, 2~) is independent of x, detT(X)=la(X)12+lb(X)12=l,

ImX=0,2g'.

(4.55)

The matrices

rp +_(x, X) = f ±(x, X) exp (iw3(h)xo 3}

(4.56)

satisfy the equation i(rp _+)x = Lqo ± - w3(X)rp _+03.

(4.57)

We have the integral equation cp _+(x, X) - if~ -+°°exp { - i w 3 ( X ) ( x - y ) o 3 } [ L ( y , X) - w3(h)o3] q~_+(y, X) × exp {iw3(X)(x - y ) o 3 } dy = 1.

(4.58)

This representation entails the analytic continuability of the first column qol+ of the matrix qo÷ and of the second column q02 of the matrix qo into the upper half R÷ of the period parallelogram R. The second column tO2+of q)÷ and the first column cpx_ of q0_ are analytically continuable into the lower half R_ of R (see, for example, Yu.L. Rodin If]). Assume ~k+( X ) = (q)x+, q)2_), ~_(X) = (qo~_,qp2).

(4.59)

These matrices are analytic in the domains R _+, respectively. Since det ~k÷--- a(X) this function is analytic in R ÷ too. Its zeros determine the discrete spectrum of the problem.

Yu.L. Rodin / The Riernann problem on closed Riemann surfaces

45

On the contour £ = F 1 u/'2, £1 = R (q {Im 2~= 0}, F2 = R tq {Im h = 2 K ' ) we obtain the relation ~k+(?~) = ((P1,£02-) =

=

a(

1

(opt_+ bop2+ exp{2iw3(X)x},~£01 - e x p ( _ 2 i w 3 ( X ) x } + (p2_)

a(x, t, x),

G(x't'X)=ei~(x)~°~(l,b(h)

(4.60)

b ( ~ ) ) e-iw~(x)x°3"

Here we limit ourselves to the case a(h)4= 0 in R÷ (i.e. we assume that there is no discrete spectrum). A solution of the boundary problem (4.60) is determined up to a constant multiplier depending on x and t. In order to separate a particular solution we use the symmetry of the problem. Taking into account eq. (4.57) and the solution's behaviour at infinity, it is easy to verify that the doubly periodic solution ~ ±(~) possesses the following symmetries:

~±(h+2K)=o3~±(h)o3,

~b±(~+2iK')=o3~±(h)o3,

~b±(~,)=(+~(?,)) ÷

(4.61)

Here + is the symbol for Hermitean conjugation. These properties fix the solution. Indeed, if ~ ±(?~) and C~k ±(~) are two solutions satisfying (4.61), then 03Co 3 --- C,

03Ca 3 = C,

C + = C -1,

whence it appears that C = + 1 o r + 0"3 If ~k±(~) is a solution of the problem (4.60), the solution possessing the properties (4.61) can be obtained by averaging over the symmetry group. Let ( g~ } be some set of transformations of R generating a finite transformation group ~, and let T(g) be 2 x 2 matrix representation of ~ (the reduction group, A.V. Mikhailov [a]). Then we assume 1

~b(?~) = n ~" T-l(g)~(g?~ )T(g).

(4.62)

g

In the case considered, gt?~ = ?~ + 2K, g2h = ~ + 2iK'. Since Ib(),)l < 1 and the operator A is an orthogonal projector, a particular solution of the problem (4.60) can be obtained by the method of sect. 3. The dynamics of the coefficients (4.60) follows from eq. (4.49) as x --* oo and has the form

d()~)=O,

b(Yk)= -4iwl(Y~)w2(~,)b(~ ).

(4.63)

The potential can be obtained from eq. (4.57). The regularity of the solution at ?~ = 0 entails the formula

3

S(x, t) ~- Y'~ a.S. = a=l

(

$3

Sl + iS2

$1 - iS2 -S3

) = ~(0)03t~+(0)"

All details can be found in A.V. Mikhailov [b], Yu.L. Rodin [e, f].

(4.64)

46

Yu.L. Rodin/ The Riemann problem on closed Riemann surfaces

4.2.2. Reflection finite-gap potentials The general method of the construction of explicit solutions of eq. (4.47) in the framework of the finite-gap integration scheme was proposed by M.M. Bogdan and A.S. Kovalev [a], A.I. Bobenko [a, b], R.P. Bikbaev and A.I. Bobenko [a], R.P. Bikbaev, A.I. Bobenko and R.A. Its [a]. Here we follow mainly the last paper. Let kO(X) be a 2 x 2-matrix meromorphic in R, doubly periodic, with the asymptotic behaviour at X=0: ~P(X)=

(j~l ~j(x,t)X j ) exp ( ---~--o3+ ipx 2ip2t ~ --~--03}.

(4.65)

The matrix 'P(X) has to satisfy the following additional restrictions: '/'(X) has singularities at points X1..... X2g ~ R which are independent of x and t, ~/-t(X) = ~ j ( X ) ( X - - X j

On the

o)(

0 1/2

--I

1

- -

I)'

j= 1 .....

(4.66)

2g.

cuts Fj(X2j_I, X2j ) ( j = 1..... g) the conditions

q,+(x) =

(x)ol

(4.67)

have to be satisfied. It is clear that the matrix 'P(X) is single-valued on the two-sheeted covering surface/} over the torus R with the branch points hi,..., X2g. Moreover, at certain points/~1 ..... ~tn ~ R ~ / , ( X ) = ~ ' j ( X ) ( X - / ~ j ) ( o 1 0°), j = l ..... n,

(4.68)

where ~j(X) are non-degenerate holomorphic matrices. Further, the following reduction conditions: o3'/'(X + 2 i K ' ) o 3 = q'(X),

(4.69)

o3'/'(X + 2 K ) o 3 = 'P(X),

have to be valid. Finally, on the segments Flj= ([REX[ _<2K, I m X = 0 } and F2j= {IReX[ _<2K, [ImX[ = 2 K ' } belonging to the corresponding sheets Rj ( j = 1, 2) of the surface R, the boundary condition

1 ~rAt+(X) = ql ( X )

b ( X ) e2iw~(x)x

b(X)e -2iw3(x)x 1

1

--, a(X)

[a(X)12+ Ib(X)12=l, 2

x

r=

(4.70)

U rkj, k,j=l

must be satisfied. The matrix ~P(X) is a solution of eqs. (4.48), (4.49). Indeed, the matrices '/,x,P-1 and g,t,/t-1 are single-valued on R, continuous on the contours Ftj ( l , j = l , 2 ) and holomorphic at the points

Yu.L. Rodin/The Riemannproblem on closedRiemann surfaces

47

)~x. . . . , )~2g,/~1. . . . . #,. The further development can be carried out according to the standard scheme (see the extensive paper by R.F. Bikbaev, A.I. Bobenko and R.A. Its [a], where b - 0). In conclusion, note that instead of the boundary condition (4.70) one may suppose that the matrix q()~) is not analytic but satisfies the differential equation o5# = # ( X ) A ( A )

(4.71)

on R, since in this case the matrices qx g'- 1 and g'txo- 1 are analytic. 4.3. D-bar and similar problems 4.3.1. D-bar problem In recent papers by R. Beals and R.R. Coifman [a, b], A.S. Fokas and M.J. Ablowitz [a, 20], M.J. Ablowitz, D. Bar Yaacov and A.S. Fokas [a] and others, fundamental applications of the D-bar problem to the inverse scattering problem (first of all, for a multidimensional case) were investigated. Following the paper by R. Beals and R.R. Coifman [b] with some modification, we state one approach to this problem. We limit ourselves to the simplest aspects, illustrating methods described in our paper. Consider the system (4.72)

~kx = [)~J + Q(x)] ~k,

where J -- d i a g ( J 1, -/2) is a constant matrix and Q ( x ) is a 2 × 2-off-diagonal matrix that is continuous and bounded at infinity. A solution of eq. (4.72) is sought in the form

(x, x) =

(4.73)

x) exp XJx.

We obtain the equation Dxcp = Qtp,

0 D x = -g~ - )~ ad J,

ad J = [ J , . ].

(4.74)

If cpl and ~P2 are solutions of (4.72) and cpl is invertible, then Dx(qo~-lqo2) -- 0.

(4.75)

Let cp(x, A) have a finite number of poles at the points h =/~j ( j = 1. . . . , n) and the branch points ~1 . . . . , ~,,, at which the matrix qo(x, A) is represented in the form

¢p(x,A)=~j(x,h)(A-)~j)r~Cj,

j - - 1 ..... m.

(4.76)

Here the matrices ~j(x, 2~) are non-degenerate, the matrices Ti, Cj are independent of x and ~ and the Tj are diagonal and rational. The matrix function q~(x, X) is single-valued on a multi-sheeted surface M over the )~-plane. One can suppose that h is a point of some Riemann surface M 0 (for example, in sect. 4.2, M 0 is a torus). In this case M is a covering of M 0 (see V.E. Zakharov and A.V. Mikhailov [a]).

Ytt L. Rodin / The Riemann problem on closed Riemann surfaces

48

Since the operators Dx and 0/02~ = 0 commute, the matrix 0~(x, )~) is a solution of eq. (4.74) and hence

O~ = epa, Dxa = 0.

(4.77)

From Dxa = 0 it follows that

a(x, X) = exp ( )~Jx } w()~) exp ( -)~Jx ).

(4.78)

The matrix w()~) is an arbitrary matrix defined on the surface M since the matrices exp()~xJ) and ()~ - h j ) ~ commute. As ~ ~ ~ , ~p(x, ~) = exp ()~Jx) + o ( 1 ) ,

(4.79)

and consequently cp(x, X) = 1 + o ( 1 ) ,

as X ~ ~ .

(4.80)

We obtain the integral equation

1 + C(~pa)=-~p(x,)~)+ 1

ff:(x, r ) a ( x , ~')M(~', )~) do, = ~()~).

(4.81)

Here ~ ( ~ ) is an analytic matrix on M that is multiple of the divisor - A formed by the images of the points/~1,-.-,/~, on M and the points of the characteristic divisor of the kernel M(% 2~); also ~(oo) = 1. If the points/~x . . . . . /% are absent, ~(2~) --- 1 since dim8 = 1. The matrix w(~,) plays a role of the scattering data. The reflectionless case corresponds to w()~) =- 0. It can be verified that

Q(x) = [D x, C](~0a).

(4.82)

Introduce now two independent variables x~ = x and x 2 = t and that 0 O~=~-XadJj,

j=l,2,

(4.83)

where J1 = diag(JJ 1, jj2) are diagonal constant matrices. The compatibility condition for the systems

D ~ = Q:ep, j = 1 , 2

(4.84)

is the Zakharov-Shabat equation 1-

2 + [ 0 1 , Q 2 ] = o.

(4.85)

A solution of the system D~a = 0, j = 1, 2 has the form

a(x, t, X) = exp { XJlx + XJ2t } w( X ) exp { - XJlx - •Jzt )

(4.86)

Yu.L. Rodin / The Riemann problem on closed Riemann surfaces

49

and the solution of the inverse problem is QJ = [D J, C](qoa). One can show that it has the form

QJ= [Jj, q],

q ( x , t ) = f f M ~ ( x , t , ' r ) a ( x , t , ~')do,.

(4.87)

Eq. (4.75) is reduced to the form Oq aq ad J~-~- - ad-J2~--~ + [ad Jlq,ad J2q] = 0.

(4.88)

4.3.2. The dressing method In some cases the Zakharov-Shabat "dressing method" [a, b] leads to the Riemann problem on a Riemann surface (for example, A.V. Mikhailov [c], A.I. Bobenko [a]). The general scheme generating Riemann surfaces was proposed by V.E. Zakharov and A.V. Mikhailov [a]. We follow this paper for the case of a hyperelliptic surface. Let 3~ be the hyperelliptic surface of genus n defined by the equation

w2=R2n(~.),

R2n(~k)--~k2n+a2n_l~.2n-X + "" +a o.

(4.89)

Introduce the projective coordinates w = Zo/Z 1, Xk = Z,+I/Z1, k = 1..... n, and represent the algebraic curve 3~ as an intersection of quadrics, Zo2 _-- Zn+l 2 + a z n - l Z n + l Z n + "'" + a ° Z 2 '

ZjZk-ZtZ,=O,

j+k=l+s,

(4.90)

l
The Zakharov-Shabat equation

L t - M x+ [L, M] = 0

(4.91)

for the systems

xox = L ( x , t, h)~I',

~Jt M(x, t, X)~,

(4.92)

=

where A is a point of a Riemann surface M and L, M are 2 × 2-matrices, L=Lo+

n+l

Zk

k=l

Z°Lk'

•-

M=Mo+

n+l

E

k=l

Zk

(4.93)

Z---~Mk,

takes the form

OL k Ot

OM~ Ox = [ Lk' M°] + [L°' Mg],

OLo Ot

OMo = [L0, Mo] + [L,+t, M,+I], Ox [L,,Mt]=aj_2[Ln+I,M,,+I ],

I+s~j

l , s > l,

(4.94) j<2n-1.

50

Yu.L Rodin/ The Riemann problem on closed Riemann surfaces

ro

Fig. 5.

The system (4.94) is compatible. A particular solution of (4.94) is L i = aio 3,

M i =

flio3,

i = 0 , . . . , n + 1.

(4.95)

The dressing procedure [15] for this solution leads to a 2 x 2 matrix Riemann problem on M. 4.3.3. DiscontinuOus Riemann problem and the Painlev~ equations The classical problem formulated by B. Riemann is the following. Consider the matrix system on the complex h-plane

dh

Aj h-h+'

(4.96)

where the A/ are constant n × n matrices. The fundamental solution if(h) of eq. (4.96) is multi-valued. Indeed, let the point h move on a path [h - hjl < e encircling the point h/. Then the solution changes, and its new value Tjq~ is equal to Tj~b = Gj~k(h),

j = 1,..., m,

(4.97)

where Gj is a constant matrix. The transformations { Tj } generate the monodromy group of eq. (4.96). Draw the cuts ~ ( h o, )~j), F = UjFj, where h 0 is an arbitrary point (see fig. 5). We obtain the Riemann boundary value problem ~+(h)=G(h)~_(h),

G(X)=Gj

ifh~Fj.

(4.98)

It allows one to calculate the coefficients of the eq. (4.96). The constants G/are called the Stokes multipliers of eq. (4.96). For n = 2 the system (4.96) and the corresponding groups have been studied very intensively.

YmL. Rodin / The Riemann problem on closed Riemann surfaces

51

This is the Painlev~ problem. Many physical problems, the problem of isospectral deformations of differential operators, self-similar solutions of non-linear wave equations and so on (see H. Flaschka and A.C. Newell [a]) are reduced to it. Therefore, there arises the necessity to study the Riemann boundary problem with discontinuous coefficients and contours having self-intersections. Such scalar problems on the plane are studied in detail in [4, 12, 19]. The case of Riemann surfaces is similar (Yu.L. Rodin [h]). The matrix case generates many troubles. H. R~Shrl [a] studied this problem on Riemann surfaces for contours of the type (4.96) ("stars" of the plane curves). In order to obtain the corresponding complex bundle it is necessary to construct the "germ" {epj} (see section 3.1). Let, in the domain Uj, z = z(p), the problem have the form

cp+(t) = G(t)ep_(t),

t ~F,

G(O) = 1,

F = UFj,

Fk= (O, tk)

(4.99)

J

and

G(t) = 1 + Glk t + G2kt 2 + . . . ,

t ~ Fk.

(4.100)

We seek the solution in the form qoj(z) = 1 + q01z + ~pzz2 + . . . .

(4.101)

Placing (4.101) and (4.100) in (4.99) we can calculate the functions ¢p± j(z). 4.3.4. Generalized analytic functions The D-bar problem may be generalized in the form

Ou = Au + B~

(4.102)

called the Carleman-Bers-Vekua (CBV) system. Its solutions are called generalized analytic (or pseudoanalytic) functions [17]. By a quasi-conformal mapping any elliptic system with two unknown functions on the plane can be reduced to the form (4.102). On a Riemann surface eq. (4.102) is equivalent to the integral equations

u(q) +

1

ff

u(q) = ~ ( q ) e x p

[A(p)u(p) + B ( p ) u ( p ) ] M ( p , q ) d o z ( p ) = ~ ( q ) ,

"'~Lff"[A(P)+"

B ( p ) u--~p) M(p,q)doz(p) '

(4.103) (4.104)

where ~/i(q) and cp(q) are analytic. The eq. (4.104) makes sense only for scalar functions. It is possible to build a theory including analogs of the Riemann-Roch and Abel theorems, the Riemann boundary value problem, and so on, for generalized analytic functions (see the monograph by Yu.L. Rodin "Generalized analytic functions on a Riemann surface," published in the Springer Lecture Notes in Mathematics). The matrix eq. (4.102) was studied in detail in the plane case in [18]. Non-local Riemann problems and CBV equations are used to solve Zakharov-Shabat equations. This is a special and extensive theme (S.V. Manakov and V.E. Zakharov [a]).

52

Yu.L. Rodin/The Riemann problem on closed Riemann surfaces

Acknowledgements The author would like to express his sincere gratitude to R.N. Abdulaev, S.J. Alber, A.V. Mikhailov, V.E. Zakharov and L.I. Volkovyskii for numerous stimulating discussions on Riemann surfaces and integrable systems, and H. Flaschka for his help in the removal of many shortcomings.

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Yu.L. Rodin//The Riemann problem on closed Riemann surfaces

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