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Theta functions and non-linear equations

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Uspekhi Mat. Nauk 36:2 (198 1), 11 -8 0 Russian Math. Surveys 36:2 (1981 ), 11 -92

Theta functions and non linear equationsB.A. Dubrovin

CONTENTSIntroduction 11Chapter I. Theta functions. General information 14 1 . Definition of theta functions and their simplest prop erties 14 2. Theta functions of a single variable 17 3. On Abelian tori 18 4. Add ition theorems for the ta functions 22Chapter II. Th eta functions of Rieman n surfaces. The Jacob i inversion problem 24 1. Periods of Abelian differentials on Riemann surfaces. Jacobi varieties 24 2. Abel's theorem 30 3 . Some remarks on divisors on a Riem ann surface 31

4. The Jacobi inversion problem . Examples 34Chapter III. The Baker-Akhiezer function. App lications to non-linear equ ation s 44 1 . The Baker-Akhiezer one-point function. The Kadomtsev-Petviashviliequation and equations associated with it 44 2. The Baker-Akhiezer two-p oint function. The Schrodinger equa tion in amagnetic field 54Chapter IV. Effectivization of the formulae for the solution of KdV and KPequ ations . Recovery of a Riem ann surface from its Jaco bi variety. Theproblem of Riemann and the conjecture of Novikov 58 1 . The KdV equa tion. Genusg = 1 or 2 58 2 . The KP equation. Genus 2 and 3 62 3 . The KP equation. G e n u s g > 2 . Canonical equations of Riemann surfaces. 65 4 . The problem of Riemann on relations between the periods of holomo rphicdifferentials on a Riemann surface and the conjecture of Novikov 70Chap ter V. Examples of Ham iltonian systems that are integrable in terms oftwo-dimensional theta functions 72 1 . Two-zone potentials 72 2. The problem of Sophie Kovalevskaya 75 3. The problems of Neum ann and Jacobi. The general Gam ier system 76 4. Movement of a solid in an ideal fluid. Integration of the Clebsch case.A mu lti-dimensional solid 79Ap pendix . I.M. Krichever. The periodic non-Abelian Toda chain and its two-

dimensional generalization 82References 89


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12 B.A. Dub rovinIntroduction

In recent years the development of methods for the exact solution of non-linear equ ation s of m athem atical physics (the inverse scattering m eth od ; see[13] and the literature cited there) has involved several function-theoreticalco ns truc tion s relate d to Riem ann surfaces. This refers first of all to thetheory of multidemensional theta functions in terms of which one canexpress the so-called ra nk 1 so lutio ns of a wide class of non-linearequations (the theory of solutions of higher rank is connected withholomorphic vector bundles over Riemann surfaces and is presented in thesurvey [ 1 4 ]; a com plete bibliograp hy is given the re). In the prese nt surveywe present the basic principles of the application of theta functions to theintegration of equations, together with important examples of this application.

We recall briefly the histo ry of this que stion . In 1974 in a series ofpapers Novikov, the present author, Matveev and Its ( [ 2 0 ] , [ 5 3 ] , [ 5 4 ] , [ 6 0 ] ,[ 1 7 ] ) , and Lax ([ 5 5 ]) introduced and studied the class of f ini te-zoneperiodic and quasi-periodic potentials of the Schrodinger operator (Sturm-Liouville, Hill). On the basis of this class a prog ram for the con stru ctio n ofa wide class of solutions of the Korteweg-de Vries (KdV) equation waspropo sed and realized. Som e results of these studies were also obta ined byMcKean and van Moerbeke in 1975 [ 5 6 ] . As was later proved rigorously(Marchenko and Ostrovskii [57]), the set of periodic finite-zone potentials isdense in the space of periodic functions w ith given per iod. In these articlesa connection was established between the spectral theory of operators withperiodic coefficients and algebraic geometry, the theory of finite-dimensionalcompletely integrable Hamiltonian systems and the theory of non-linearequ ations of KdV typ e. A generalization of this theo ry to spatially two -dimensional ( 2+ 1 ; x, y, t systems, among which there is the importanttwo-dimensional analogue of the KdV equationthe Kadomtsev-Petviashviliequation (KP)was realized by Krichever [5 9 ], [1 5 ] , [1 6 ]. Krichever 'sapproac h also gives a methodolog ically very conven ient and lucid presen tationof an algebro-geometric procedure of constructing the above-mentionedfinite-zone solutions of the KdV equ ation and its num erous analogues. Inthe case of (2+ l)-systems this method reveals new important connectionswith algebraic geometry, which are used essentially in the statement of theproblem of Chapter IV of this paper.

As Novikov and Natanzon have pointed out to the author, thedetermination of conditions for the selection of real solutions in variousproblems of this kind turns out to be a non-trivial and as yet unsolvedproblem, except for individual cases literally analogous to the KdV equation,which do not, however, include the sine-Gordon equation or the non-linearSchrodinger equation with repulsing interaction, nor all the (2+ l>systems.An essential advance was made by Cherednik in [49] and developed in [14].The results of [49] hardly pretend to completeness and are not effectiveeven in the simplest cases.


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Thetafunctions andnon linear equations 13

Th e first two chapters of this survey are devoted to the most importantresults of the theory of theta functions, their connection with Riemannsurfaces and Abelian varieties. The material of these chapters basically goesback to the classical works of Abel, Jacobi, and Riemann (see the books[1] [ 8] ; for a modern account a detailed bibliography can be found in [4] ) .In th e third chapter we explain Krichever's meth od (see [15 ], [16]) forconstructing exact solutions of non linear equations (in 2 we also useresults of [21]) . This method allows us to express the solutions thusconstructed in terms of theta functions. The basic instrument forconstructing such solutions is the so called Baker Akhiezer functions, that is,meromorphic functions on a Riemann surface with an essential singularity of agivenform (see [ 18] , [ 19]; apparently, thefirst functions of this type wereconsidered by Clebsch andG o rd a n ) . It is essential to note that in theapplications to non linear equations it is not arbitrary theta functions thatarise, but only theta functions of Riemann surfaces. Novikov has conjecturedt h a t the identities on a theta function (or on the Riemann matrix defining it),obtained after making an elementary substitution of it in the KadomtsevPetviashvili (KP) equation, precisely distinguish the theta functions of Riemannsurfaces from all other theta functions. A partial realization of this programt h ederivation of a system of identities connecting the parameters of a the tafunction of Riemann surfaces (without proving the completeness of thissystem)was obtained

( 1) by the author in [23] ) ; the corresponding resultsare presented in Chapter 4. Incidentally we also solve Novikov's problem onth e effectivization of formulae for the solution of non linear equations in the

case of small genera, where restrictions on the theta functions do not yetarise. Moreover, the use of the KP equation allows us to give an explicitc o n s t r u c t i o n for the recovery of an algebraic curve from its Jacobian (that is,t o give a new proof to the classical Torelli theorem, which asserts theuniqueness of this correspondence; see [4] ) , and this construction does notrequire a knowledge of the solutions of the tran scendental equation ( ) = 0.Very recently the author has succeeded in proving Novikov's conjecture in aweaker version: the relations on a theta function that follow from the KPequation distinguish the variety of th eta functions of Riemann surfaces inth e space of all theta functions up topossibly superfluous irreduciblec o m p o n e n t s . The idea of th e proof is also given in Chapter 4. Thus, the useof the KP equation allowsus effectively to solve the classical problem ofR i e m a n n on the relations between the periods of holomorphic differentialson Riemann surfaces. Finally, in the last Chapter we list the dynamical

statementsof some results of [23] were published(andused)in [37]. Wealsomention that after [23] waspublished,theauthorwas showna preprint by H irota[24];the methodsof this preprint intersect with some technical argumentsof [23]. H irotasolvestheproblemof constructing exact solutions of non linear equat ionsof KdVtypeby meansof thetheory of theta functions. Theaccount in [24] doesnot give explicitformulae.


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14 B.A.Dubrovin

systems th at are integrable in terms of th eta functions of genus 2. Exceptfor equations of the theory of two zone potentials, which are interpreted in[20] and [17 ] , and also multidimensional Euler equations, all these systemsare classical, although the structure of their solutions is not well known.The invarian t varieties of these systems are Abelian varieties of genus 2. Thisallows us to obtain explicit formulae of various types for the universalbundle of Abelian varieties of genus 2 (the first such formulae were obtainedin [35]) . As an application we give the work of Krichever on the integrationof the non Abelian Toda chain by algebraic geometric methods.

The author thanks S.P. Novikov and I.M. Krichever for their interest inthis work and for a number of useful discussions. In th e process of writingChapter 4 the auth or consulted A.N . Tyurin, to whom he expresses his deepappreciation.

CHAPTER THETA FUNCTIONS,GENERAL INFORMATION

1 . D e f i n i t i o n of t h e ta f u nc t i onsandthe i r s im p les t p rope r t iesDefinition 1.1.1. A symmetric (g g) matrix = (Bjk) with negativedefinite real part Re = (R eB

jk) is called aRiemann matrix.

Definition 1.1.2. A Riemann theta function is defined by its F ourier seriesof the form

(1.1.1) Q ( z \ B ) =

Here = (z1, ..., zg) G C* is a complex vector. The diamond brackets den otee the Euclidean scalar product:


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Thetafunctionsand non linearequations 15

T h e vectorsfk can also be written in the form(1.1.3') h = Beh,

where is the linear operator corresponding to Bkj.Assertion 1.1.1. When the argument isshifted by the vectors 2mek, theRiemann theta function fk is transformed according to the law

(1.1.4)(1.1.5)

Proof. The periodicity of (1.1.4) is obvious: th e general term of (1.1.1)does not change under the shift >z + 2mek. To prove (1.1.5) we changet h e summation index in (1.1.1), setting = M~ek, GZg. We have

= e x p {^ ( B M , >


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1 6 .Dubrovin

such that0 +2 , ) , } [ , ] ( ).

I tiseasytoshow (butweomit this here) tha t the entire functionsofgvariableszlr ...,zg subject tothe transformation law(1.1.10) formalinearspaceofdimensionng. Asbasis 0 function ofordernwith characteristics[ , ]wecan take,forexample, the functions

(1.1.11)where the coordinatesofy range independently over all values from0to 1.Definition 1.1.3. The characteristics [ , ]forwhichallthe coordinatesah are0 or arecalled halfperiods. Ahalf period [ , ]issaidto beevenif4 , ./V>* 2 the general termoft h e series (1.1.8)ismultipliedby

exp{( + ,4jua>} = exp{4 ( , >}.T h e signofthis factor is completely determinedbythe parityofthe number4< , ). This proves th e assertion.

I tisno t hardtocompute that there are precisely 2*~1(2*+ 1)even halfperiods and 2AT X(2'l)odd ones.


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2 . Theta f u n c t i o n s of a singlevariableIn the classical theory of elliptic functions (the caseg =1) there occur

only theta functions corresponding to half periods. The Riemann matrixhere is thesinglenumberb = Bn, Reb 0), = 2 .

The first half period is odd, the others are even. Consequently, thefunction #i(z) is odd and the others are even. The function 0,(z) vanishes atall vertices of the period lattice = 2mm + bn. From this it is not hard tofind the zeros of the remaining 0 functions. In particular, 3( )vanishes atall points = m + b/ 2+2mm + bn. We claim that there are no other zeros.F or convenience we take 03. We need to prove that

(1.2.3) r cwhere Cis the contour of the parallelogram spanned by the vectors 2vn andb.The general transformation formulae (1.1.9) for 03(z) take the form

( 1. 2. 4) , ( + 2 = 3 ( ) , 3( + ) e x p ( | ) 3 ( ) .We split the integral (1.2.3) into two:

log 93(2) = ^ j [dIo g83( 2) d log 63( 2+ 6)] +c b

+ j [d l o g 9s (z + 2ni) *d l o g 3( )]'From the relations (1.2.4) it followsthat the second integral is zero, and thefirst integral has the form

2

as required.


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18 . Dubrovin 3 . On Abel ian to r i

Assertion 1.3.1. T h e vectors 2 n i e x , . . . , 2nieg, / l t . . ., fg in the spaceQS = R 2 # define(i fry (1.1.2) and (1.1.3) are linearly independent over th efield of real numbers.Proof. We assum e the co n t ra r y : th a t som e l in ea r com bina t ion o f thesevectors vanishes:

(1.3.1) 2 m 2 Xe f t + S l W *= 0 , , G R.F r o m (1 .1 .3 ) we ob ta in immedia te ly tha t the rea l pa r t o f th i s equa l i ty hast h e form

H e n ce, a ll the & are zero because t h e m atr ix Re is no n singula r. F rom(1 .3 .1 ) i t then follows tha t a l l \ k are zero , which proves th e asser t ion .Let be the la t t i ce genera ted b y th e vec to rs ( 2 m ek, fx) . Th e vectors of have the form(1.3.2) 2niN + B M ,

where a n d are int eger vecto rs . Th is is precisely t h e period la tt ice of th e0 fun ct ion in th e sense of 1. I t is con ven ient to associa te wi th an oth ergeom et r ic ob jec tthe qu o t i en t o f C* = R 2* by th is la t t ice . F ro m Assert ion1.3.1 it follows im m edia te ly th a t th i s qu o t i en t C*/ F is a 2g d imen sion a l to rusT 2g. Moreover , T 2g has th e n a tura l s t ruc ture o f a com plex com pact L iegroup , and the express ion

(1.3.3) ds2 = , (R eB ) l\ d zk d z,gives a Kahler met r i c on T 2g, which is even a H odge m etr ic (see [4 ]) . I f akan d , a re rea l coo rdin ates in R2g = Cg, where = 2 + , t h e n t h eim agin a ry p a r t o f th i s m et r i c has th e fo rm

(1.3.3') = Such tori are called Abelian. We den o t e th e Abelian to r us T 2g co ns t ruc t edf ro m t he R i em a nn m a t r i x = (B jk) by T 2g( B ) .T he m ero m o rp h i c f unc t i o ns o n T 2g( B ) are calledAbelian functions. I no t h e r words , Abel ian funct ions are :2g fold per iod ic func t ions o f g co m p lexvar iab les. Th us , th e qu o t i en t o f two 0 fun c t ion s o f th e same o rd er wi th thesam e cha ra cte ristics is single valued on T 2g( B ) an d is ther efore an Abelianfun c t ion . I t is kn own th a t an y Abelian fun c t ion can be ob ta ined in th i sway. A n um ber of exam ples in which cer ta in Abelian func t ion s areexpressed expl icit ly in term s of 0 fun ct ion , will be given i n C h . I I , 4 .


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Thetafunctions andnon linearequations 19L et ( 2nie'j, f'h) b e a n o t h e r basis of ,w h e r e f'h = B ' e 'h a n dB ' = ( B / k) isa n o t h e r R ie m an n m a t r i x. T h e t r an si ti o n fr o m t h e basis(2niej, fh) t o{2nie'j,f'h) ( an d b ac k ) i s given b y an i n t e g e r u n im o d u l a r m a t r i x

2nie[= 2 dlj2mej + 2

where a = ( a i; ) , 6 = ( btJ) , c = ( c i; ) , d = (d{y) are integer ( g g ) m a t r ic es,an d(1.3.5) d e t ( J J ) = l.

H er e( 1.3 .6) ' = 2 { + 2nib){cB + 2nid) 1.

T h e requ i remen t on 5 ' t o be a Riemann mat r ix imposes a s t rong res t r ic t ionon t h e m a t r i x i " dj; i t m ust be symplectic:

(137) ( * bd) Sp(gf Z),that is

b W 0 1 \ / a 1 c < \ _ / ^ \U i / l l ) U

( d'/ ~ l l 0/( t h e s y m b o l f d e n o t e s t r an s p o s i t i o n ) .T w o R i e m a n n m a t ric es a n d B ' c o n n e c t e d b y t r an sf o r m a t io n s o f t h efo rm (1 .3 .6 ) , (1 .3 .7 ) a re sa id to be equivalent. T h e y d e t e r m in e t h e sam eAbel ian tor i T 2g( B ) = T 2g( B ' ) ( even wi th th e same H odge s t ruc tu res

T h e following is the t r ans fo rm at ion law of ^ fun c t ion s un dert r an s f o r m a t io n s o f t h e f o r m ( 1 . 3 . 6 ) , ( 1 . 3 . 7 ) :(1.3.8) [ ' , ' ] ( ' | 5 ' ) =

= K detM exp {1 2 W ) * " ^ ^ } [, ] ( ) ,where(1.3.9) =

[ 1, '] = [ , ] ( _ ^ " J j + i d i a g l c d * . &'];fc is a c o n st an t i n d e p e n d e n t o f an d 5 . H e re t h e sy m b o l diag m e a n s t h a tone has to t ake the d iagona l e lemen ts o f the mat r ix \_cdx, abf] ( a p r o o f o fth is t r ans fo rmat ion l aw i s in [5 ] ) .( 1 ) T h e setH g of all{g ) Riemann matrices is called the Siegel(left) half plane. Thegroup Ag = {Sp(g,Z ) / t 1}acting o n Hg in accordance with (1.3.6) is th e Siegelmodular group. T h efactor groupHJA g parametrizes th eAbelian varieties T 2g withaKahler metric of thetype (1.3.3), (1.3.3 ') .


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20 B.A.Dubrovin

Remark. Let = (B/ k) be a (g g) Riemann matrix of block form = L ), where B' andB" are (k k)and (1 1) Riemann matrices.T h e n the corresponding torus T2g(B) splits into the direct product of twoAbelian tori

(1.3.10) T" (B) = T2h ( ') Til (B").T h e corresponding function also splits into a prod uct: if = (zu . . ., zg),z' = (zl t . . ., zh), an d 2" = (zh+1, . . ., zg), then

(1.3.11) ( | ) = ( ' | 5' ) ( " | ").0 functions with characteristics are of a similar structure. We say that aR i e m a n n matrix isdecomposable if it can be brought to block form bytransformations (1.3.6) , (1.3.7). Correspondingly we use the term"decomposable Abelian t orus" and "decomposable 0 function" (formulae(1.3.10), (1.3.11)). For the opposite case of indecomposable tori T2g(B) th efollowing property ho lds: on T2g(B) there are no (non constant) Abelianfunctions depending on a smaller number of variables. Equivalent is th eassertion: if / (z) is an Abelian function on an indecomposable torus and its

8derivative with respect in some direction vanishes , 2 UIJT~= 0,then/ = const. 1=1

Returning to Abelian tori, we note the following important theorem: anyAbelian torus is an algebraic variety (Lefschetz). We do not prove thist h e o r e m in general (the proof is based on the theory of 0 functions; see[11]), but consider only the caseg = 1. In this case any torus T2 isdetermined by its pair of periods 2 , 2 ' (we assume that Im '/ > 0) .We define the Weierstrass elliptic function ( ),setting

(1.3.12) P(z) = p + Zl ( 2 2 ') ~~(2 + 2 ') _' ' *+ 2 0

I t is not hard to verify that the series (1.3.12) converges uniformly on everycompact set in C\ {2mo f 2 '}, so that it is a meromorphic function of having double poles on each vertex of the lattice. Obviously, this function isdoubly periodic:

(1.3.13)


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Theta functions andnon linear equations 21

Then the Laurent expansions of + 2 '} with the Riemannsurfaceof the algebraic function

(1.3.18) t = Ax3 g2x g3.Proof We construct th e inverse mapping from the complex algebraic curve(1.3.18) into the torus T2. Let = ( , y) be a point of the Riemannsurface (1.3.18) . We set

(1.3.19) Z= Z{P) y J yix3g2xg3 CO

The path of integration in (1.3.19) lies on the Riemann surface (1.3.18) .This path is unique, up to addition of an integral linear combination of thebase cyclesa andb on (1.3.18) (Figure 1; hereAx\g2xt #3 = 0; thedotted line shows the part of the cycle b, lying

on the "lower" sheet):

(1.3.20) aI t is not difficult to verify that

Fig.1. Riemann H 2 1> d* _ & *? 2 'surface of (1.3.18) ( . . ) y y , y . bThus, the mapping (1.3.19) carries the curve (1.3.18) into the torus T2.Obviously, the mappings z* > = ( f ( z ) , '(z)) and P>> f are inverse toeach other. This proves the assertion. Remark. Let us indicate an explicit connection between the Weierstrassfunction (z) and the ^ functions of one variable considered in 2 . There isthe formula

(1.3.22) ( * ) = ;


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22 B.A.Dubrovin

where c is a constant. This is obvious inviewof the periodicity of the righthand side and the information obtained in 2 on the location of the zerosof 0j(z). From (1.3.22) and (1.3.17) it follows that 0,(z) satisfiesa thirdorder differential equation. Later we consider the differential relations formultidimensional 0 functions (see Chapters 3 and 4).

4 . Addi t ion theoremsfor0 funct ionsTheta functions are connected by a complicated system of algebraic

relations, the socalled addition theorems. All of them are relations betweenformal Fourier series. We quote the two most important addition theorems/ 1JFrom a Riemann matrix we construct two sets of 0 functions:

(1.4.1) [ , ]( )E = [ , ]( | ), [ , ]( ) = [ , ]( | 25).Addition Theorem 1.4.1. Let , , and bearbitrary realg dimensionalvectors. Then

(1.4.2) [ , ( Z l+ z2) [ , ] ( , 2) == 2 1

I t suffices to prove th is for the case = = = = 0 ( the genera l casereduc es to this on e via (1. 1.7) ). Th en (1.4. 2) take s th e form

(1.4.2') ( + ) ( w)= 2 [ , ]( 2 ) [ . 0](2w).2 ( 2) #First we consider the caseg = 1. Then (1.4.2') can be written as

(1.4.3) ( + w) ( w)= (2z) (2w)+ [i . 0J (2z) [ , ] (2 ).where

( ) = 2 e xP ( ^

The lefthand side of (1.4.3) therefore has the form(1.4.4) 2 exp[^ b(k^ + l2) + k(z +w)+ l(

k, We introduce new summation indicesm, n, setting

(1.4.0) m =^, n = 2.Thenumbersm and are both either integers or half integers. In thesevariables (1.4.4) takes the form

(1.4.6) Q(z + ^(1)Rela tions for theta functions of an entirely different type can be found in [43].


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= (2z) (2a;) + [~ , ] (2z) [ y , ] (2a;).

Thetafunctions and non linear equations 23

We split this sum into two parts, the first part containing the terms with mand integers, and the second half integers. In the second part we changem to m + \ and ton + \ . Thenm and are integers, and (1.4.6) takes theform2 exp[bm2+ 2mz]exp[bn2+2nw] +me

m , n e

Thus, the theorem is proved in this special case. In the general casegwe have to repeat this argument for each coordinate separately.Addition Theorem1.4.2. Let [m;] = [m[, m'l] (i = 1, 2, 3, 4) bearbitrary real 2g dimensionalvectors. Then

(1.4.7) [m t] (Zl) [TO2](z2) [m3] (z3) [m4] (z4) == ^ 2 exp( 4ni


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24 B.A. Dubrovin

We divide the sum of exponentials of th e form (1.4.11) in to two parts: ( 1 )and (2). (X) includes only th e integers llt ..., /4 and (2) only the halfinteger ones. In ( 2) , as in the proof of the preceding theorem , we make thechange of variables lt >* lt \ ,after which both ^ and (2) include onlyinteger summation indices lu .... /4. We remark that the set lu ..., /4 isnecessarily even, since ku ..., k4 are integers and (/cl7 . . ., /c4) = (Zl5 . . .. Z4)7\Thus, the summation in ( 1 ) and (2) is on ly over even sets lu ..., / 4. Wesum over all the sets li, ..., l4 and to compensate for the inclusion of theodd sets we also add the sums [ and [2)> in which we have made thechange of variables [> + [ + (i = 1, . . ., 4) in th e general term ; also, tothe whole exponent of the exponential we add the term (n[+ . . . ++ n'Jni = 4jtiwij/2.. After these tran sformations the even terms in ( )+ ' ( ) are doubled, and the odd ones cancel each other. The same holdsfor the sum ( 2 ) + '( 2) . Thus, we obtain

,] ,) . . . TO 4] 2 4 ) = 4 ,,,+ ^ + , + ^) .This coincides precisely with (1.4.7) for g 1. In the general case the proofis similar, only we have to repeat these arguments for each coordinate.

Setting z1 = u + v, z2 = u v, z3 = z4 = 0 in (1.4.7) , we haveu>1= w2 = u, w,3= Wi v. The second addition theorem in this specialcase takes the form(1.4.12) [m (](u + v)Q [m2] (u v)Q [m3] (0) [m4] (0) =

= W [ 4 1. If is theRiemann surface of an algebraic function w =w(z) given by an equation

(2.1.1) R(w, z) = wn +a^w 1 + . . . + an(z) = 0,

here ononly compact Riemann surfaces occur,and we do notstate this everytime.


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Thetafunctionsandnon linear equations . 25

where R(w, z) is a polynomial, t h e n the affine part of coincides with thecomplex algebraic curve (2.1.1) in C , provided th at this curve is non singular( s m o o t h ) . The important example for us is th e class of hyperelliptic curves,which are given by equations

(2.1.2) w*= P2g+1( z ) ,

(2.1.3) u;2= P2g+2( z ) ,where P2g+1(z) andP2g+2(z) are polynomials of degrees '2g+ 1 and 2 g+ 2 , respectively, without multiple roots(and in either case the genus of th e corresponding Fig. 2.R i e m a n n surface isg). We mention th at every Riemannsurface of genusg 1 org = 2 has the form (2.1.2) (or (2.1.3); see [3] ) ;for g = 3 this is no longer so.

Topologically, a surface of genusg is a sphere withg handles attached. Inthe 1 dimensional homology group ( ) Z + ... + (2g factors) one canchoose abasis of cycles (closed contours)alt ..., ag, bx, ..., bg with thefollowing intersection indices:

(2.1.4) at a, = bt bj = 0, at bj = (i,7 = 1, . , #)This basis has the p roperty that under cutting along these cycles, becomesa 4g gon, which we den ote by (see Fig. 2 for g = 2). Each cycle goes toa pair of sides at, oj1,bt, b~xof , which are identified on .Differential 1 forms = a dx + b dy = a dz+ dz (where =x + iy isa complex local coordinate on ) are simply called differentials.Definition 2.1.1. A differential is said to beAbelian of the first kind orholomorphic if in a neighbourhood of any point it can be written in the form

(2.1.5) = f ( z )dz ,where / (z) is an analytic function and a local coordinate.

F o r example, on the hyperelliptic surface (2.1.2) the differentialscoj, ..., of th e form

(2.1.6) Vare all holom orphic.

A holomorphic differential is closed: du> = 0. Its complex conjugatedifferential = f{z)dz is also closed. For any closed differential on itsperiods over the cyclesalt ..., ag, bly ..., bg are defined as follows:

(2.1.7)


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26 B.A.Dubrovin

We fix a pointPo not lying on the cycles at, bj. Then we can define thefunction(2.1.8) / / > = j ,

which is singlevalued on the cut surface . The following lemma is useful.Lemma 2.1.1. Let and be two closed differentials

QL on ; let Ah A'h Bit B\ be the set of their a and

'. b periods; let f(P) =lig. 3. then( 2 . 1 . 9 ) JJr e r i=iHere 3 is the boundary of ',oriented in the positive direction.Proof. The equality f f ' =&/ ' is obvious from Stokes' formula.F u r t h e r , r ep

(2.1.10) $/ ' = 2g

2F r o m Fig. 3 it is clear that f(Pt) f(P'i)= \ = Bt (the cycle PtPl on ishomologous tobt) and that /(Qt) f(Q'i)= Aj for a similar reason. Therefore,t h e sum (2.1.10) can be rewritten in the form

g e s

t = l ft, ' = 2 (AtB'i A'il

t = l/ '= 2 ( 5

er i = 1 " l = l "tThis proves the Lemma.Applying this lemma to a pair , , where is a holom orphic differentialan d its complex conjugate, we obtain the following corollary.

Corollary 1. Let be a non zero holomorphic differential on . Then itsa and bperiodsAk, Bk satisfy the inequality

g _(2.1.11) Im 2 AkBh


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Thetafunctionsandnon linearequations 27

F o r a hyperelliptic Riemann surface wehave produced above (see2.1.6))aset ofg holomorphic differentials , ..., tog. F or anyRiemann surface we canalso constructg linearly independent holomorphic differentialscoj, ...,tOg. F or aproof oftheir existence,see [1] . F rom thisandCorollary2we obtain afurther corollary.Corollary 3. Thespaceof holomorphic differentials on aRiemann surfaceof genusg isg dimensional.

Let T?I, ...,r\gbe a set of linearly independent holomorphic differentialsona Riemann surface . Then th ematrix

(2.1.12)

of their periods is non singular. F orotherwise some non trivial linearc o m b i n a t i o n = c ^ j f . . . + cgr\ g would have zero periods, whichcontradicts Corollary 2. Therefore,on e canchoose another basisofholomorphic differentials coj, ..., cog,

g( 2 . 1 . 1 3 ) 7 = 2 C } h * \ k (y= l , . . . , ) ,A = l

normalizedby theconditions(2.1.14) ..., bg.F r o m thecanonical basis 1, ... , Gigweconstruct th ematrix of b periods

(2.1.15) BJh=Theorem 2.1.1. Bjk is aRiemann matrix.Proof. I tssymmetry follows from Lemma2.1.1applied to thepair of basisdifferentials = }, ' = o^ (obviously, ' = 0 if and ' arebothh o l o m o r p h i c ) . Toprove that th ereal part ReBjk is negative definiteweconsider anon trivial linear combination

(2.1.16) =where all th ecoefficients c1 (...,cg arereal. Theperiods ofth is holomorphicdifferential are


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28 B.A.DubrovinApplying (2.1.11) to it, we obtain

g g _2 ReBjkckCj = Im 2 AhBh


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Thetafunctionsandnon linear equations 29

corresponding principal parts of do not change. Such a normalizationtogether with a specification of the poles and principal parts determines themeromorphic differential uniquely (otherwise the difference of two suchdifferentials would be a holomorphic differential with zero periods).T h e sum of the residues on of any meromorphic differential is zero.Therefore, it can be represented as a linear combination (up to addition of aholomorphic differential) of the following basic merom orphic differen tials:(1)

a) Abelian differentials of the second kind. These are differentials ^with asinglepole atQof multiplicity +1 and with principal part of the form

(2.1.18) ( )= ^ . + . . .

b) Abelian differentials of the third kind . These have a pair ofsimple poles at and Q with residues + 1 and 1 , respectively.We recall th at these differentials are uniquely determined by the

normalization(2.1.19)

(2.1.20) ( 3 = 0 (i = l, . . . , g).ai

We now list the relations between the periods of these differentials thatwill be needed later on:Lemma 2.1.2. Thefollowing relations hold:

. 1 . . 1 ) (V) O n r r ^ ; ( = 1 , . . . , ^ , / = 1 . , . . . ) , d z""1 '

(2.1.22) ( 3 = (

b , (i = l, ...,g),

t Q

where ^ and ojpg a r e t n e normalized differentials of the second and th irdkind introduced above, and 1 ; ..., ? are basis holomorphic differentials,given by , =j}(z)dz in a neighbourhood of Q.

T h e proof of the relations (2.1.21) and (2.1.22) is obtained by integrating

th e expressions ^" and ( , where At(P) = \ at, over the boundaryof the domain obtained from the 4g gon by removing the set of edgesgoing from the initial pointPo to th e poles of th e differentials. We omit th ec o m p u t a t i o n s , which are similar to those in the proof of Lemma 2.1.1.

donotprove heretheexisten ce ofsuch differentials (see [1]) . For ah yperellipticsurface it is notd ifficult toproduce themby explicit formulae. In thegeneral case,theexistence of such differentials can bededuced,for example, from the Riemann Rochtheorem (seebelow3).


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30 . Dubrovin

2 . Abel s theoremL et be aRiemann surfaceofgenusg,and / ( )itsJacobi variety.We

define theAbel mapping A{P) = AX{P), . . ., Ag P)), : / ( ),

setting

(2.2.1) Ak P)=^h (ft=1, . . . , * ) .H e r ePois afixed point; the pathofintegration fromPotoischosento bet h e samefor allk. Ifwe choose another pathofintegration fromPotoin( 2 . 2 . 1 ) , thenwehavetoadd cofttothe integral on the right, where7 is aclosed con tour (cycle). The cycle7canberepresentedas an integral linearc o m b i n a t i o nofthe basis cycles:

(2.2.2) =; iThere fore , the added term on the right hand sideof(2.2.1) has the form

(2.2.3) o>f=2nink +2 Bjhmhy i

b u t thisisthe kth componentofsome vectorofthe lattice {2niN + BM},t h r o u g h whichwefactor. This proves that the Abel mappingis well defined.F o rg= 1(an elliptic Riemann surface)wehave already considered theAbel mappinginCh.I, 3and have shown thatit isan isomorphismofthisR i e m a n n surface on to the two dimensional complex torus thatis itsJacobivariety.We apply the Abel mappingtosolve the following problem . Let/beam e r o m o r p h i c function onaRiemann surface . The numberof itszeros on mustbeequaltothe numberofits poles (including multiplicities).Theques t ion arises: what setsofpointsPl,..., Pn,Q1,...,Qn canbezeros andpolesof ameromorphic function on ? The answerisgivenbythefollowing theorem

Abel's theorem. Forpoints Plt .... Pn,Qu.... Qnon aRiemann surfacetobethezeros andpolesofsome meromorphic function it isnecessary andsufficient thaton theJacobi variety / ( )

(2.2.4) 2 A(Ph)Y A(Qh)^0ft=l A= l the symbol =here and later denotes congruence modulo theperiod lattice).Proof. Suppose thatafunction/on has zerosatPlt ...,Pnand polesatQi, >QnWeconsider the meromorphic differential =d\ o%f.N ow


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Theta functions and non linear equations 31

has on ly simple poles with residue + 1 atPl, ..., Pn and with residue 1 atQ\, Qn Therefore, can be written in the formg22.2.5) = 2 Q + 2

where the copfcQfc are normalized differentials of th e third kind, , ..., ^are basis holomorph ic differentials, and ct, ..., cg are constants (see 1above). Since / is single valued on , the integral of over any closed cyclemust be an integer multiple of 2m. In particular,

2 .2 .6 )ak bh

where nk an dmk are integers. H e n ce , bearing the normalization (2.1.20) inmind,we obtain

akNext, the formulae (2.1.22) for th e 6 periods of the cope give

2 .2 .8 )

Hence we obtainpj

pj

~ AWj)]k= J *= 2nimk+The right hand side of the equality is the kth component of some vector ofthe period lattice, which means that (2.2.4) is valid. Carrying the argumentout in the reverse order, we obtain a differential with the required polesand with periods that are integer multiplies of 2iri. Then the function

f(P) = exp f is single valued on and has the prescribed singularities.This proves the theorem.

3 . Some remarkson d iv isorson a R iemann sur face1. Adivisor on a Rieman n surface is a set of points of withmultiplicities. I t is convenient to write divisors as formal linear combinations

of poin ts of :(2.3.1) fl = 2 fijPj; nt are integers.t = l


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32 .DubrovinF o r example,for an ymeromorph ic function/ on wedefine itsdivisor ( / )of zerosPu ..., Pnwith multiplicities , ..., pn andpoles Qx, ..., Qm withmultiplicitiesqx, ..., qm:

(2.3.2) (/ )= PlP1 + . . . + PnP n ? 1 & . . . qmQ m.Divisors form anAbelian group:

(2.3.3) D=%n,Pt, D'= ^ \ D + D'= n,Pt+ niP\( t h e zero is th e"em pty divisor").

2. Thedegree of adivisor D =V.nfi is thenumber(2.3.4) degZ) = Y, nt.

F o r example, th edegree of thedivisor of amerom orphic function iszero( t h enumberofzerosisequalto thenumberofpoles, including multiplicities).T h e degree is a linear function on thegroup ofdivisors: deg(D+D') = deg>+ deg.D'.

3. We saythat twodivisors DandD'arelinearly equivalent if theirdifference DD' is th edivisor of ameromorphic function. Thedivisorsofmeromorphic functions arelinearly equivalent tozero (they arealso calledprincipal divisors). Thedegreesof twolinearly equivalent divisors areequal.Example. F or anyAbelian differential = f{z)dz on weconsideritsdivisor ofzeros andpoles ( ) . If = g(z)dz isanother Abelian differentialan d (17)itsdivisor, t h e n ( ) and(17)arelinearly equivalent, sinceth eq u o t i e n t / is ameromorphic function on (dz"cancels"). The equivalenceclass ofdivisors of allAbelian differentials iscalled th ecanonical classof an d isdenotedbyC. The degreeofthe divisors ofthis class is2g2(see [1] ) .

We extend th eAbel mapping (2.2.1) linearly to thegroupof all divisors

(2.3.5) # = > , A(D) = yjnlA(Pl).i iNow Abel's theorem can bereformulated in th efollowing form: twodivisors DandD'arelinearly equivalent if andonly if the following twoconditions hold:

1) degZ)= degD ' .2)A(D) A(D')on theJacobi variety / ( ) (werecall that=means

congruence modulo th eperiod lattice) .4. AdivisorD=2 niPi iscalledpositive (oreffective) if allthe multiplicities

, arepositive. Bydefinition,D>D'for twodivisors DandD' if theirdifference DD' is apositive divisor. Wenoteauseful property of divisorsof degree >g: anysuch divisor islinearly equivalent to apositive divisor.Thiscan bededuced,forexample, from th eresults of thefollowing section.


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Thetafunctions andnon linearequations 33

With each divisor D there is connected the linear space L(D) of thosemeromorphic functions / on for the divisors (/ ) of which th e followinginequality holds:

(2.3.6) (/) > D.I n particular, if D =Y>niPi is positive, then the space L(D) consists of thosemeromorphic functions that can have poles only at the pointsPt of multiplicityn o t exceeding nt. The dimension of the linear space L(D) is denoted byl (D) . It is clear th at if D andD' are two linearly equivalent divisors, thenth e numbers 1{D)and l(D') are the same (the spaces L{D) andL(D') areisomorphic).

F o r a positive divisor D in general position the number l(D) behaves asfollows: 1) If degD 2, which are called special, l(D) > deg, D g+ 1. It turns out (see[ 4 ] ) , that the special divisors D = Px+ ... +PN, degD > g are preciselyt h e critical points of the Abel mapping

(2.3.8)1A(PUt h a t is, those sets of points {Pu ..., PN) at which the rank of the differentialof the mapping (2.3.8) is less thang. HereSN denotes the totality of

unordered sets of points of (the 'W th symmetric power of ") .In the general case, information about l(D) can be extracted from the

R i e m a n n R o c h theorem:(2.3.9) l(D) = degD g + 1 + l(C D),

where C is th e canonical class defined above. We do not discuss thist h e o r e m here (see [1] [ 4] ) .


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34 .Dubrovin

4 . TheJacobi inversion problem. ExamplesWe sawabove (see Ch. I, 3)that forg = 1 theAbel mapping : *

* J(T)isinvertible an d is anisomorphism. F orlarger generag > 1 theproblem of inverting theAbel mapping can bestated asfollows (theJacobiinversion problem): for agiven point = ( , ...,) / ( ) tofind g pointsPu ...,Pg of such that

H e r e , ..., ^ is acanonical basis of holomorphic differentials on ; Poisa fixed point of . Thesystem (2.4.1) must hold on theJacobian/ ( ) ( th esymbol =,asusual, means congruence modulo theperiod lattice).T h e unordered sets ofg points of form theg thsymmetric powerSPYof . I n th elanguage of theAbel mapping theproblem (2.4.1) can berewritten asfollows: toinvert themapping

(2.4.2) A: SeT + / ( ),where

(2.4.3) A(Plt . .., Pg) = A(PJ + . . . + A(Pe).We shall show thatforalmostanypoint / ( ) the set ofpoints (P, ,...,Pg)== '1 ( )exists and is uniquely determinedby thesystem (2.4.1) (withouttaking theorderof these points into account).T o solve th eJacobi inversion problem we use th eRiemann 0 function ( ) = (\B)of . Here = (Bjk)is th eperiod mat rix ofholomorphicdifferentials on (see 1above). Lete (ely ..., eg)=Cgbe a fixed vector.We consider the function

2.4.4) F P)= B(A(P) - ).I t issinglevalued andanalytic on the cutsurface . Weassume thatF(P)isn o t ident ically zero. This happens,forexample, when 6(e) 0.Lemma 2.4.1. If F(P) 0, then F(P) has g zeros on includingmultiplicities).Proof. Tocalculate the numberofzerosweneedtocomputethelogarithmicresidue

2 .4.5) (we assume that thezeros ofF(P)do n ot lie ondf). F orbrevity wei n t r o d u c e thefollowing notat ion (which isalso useful later on ): byF+ wed e n o t e thevalue th atF takes at apoint ondT lying on thesegmentak orbk, and byF~ thevalue ofFat thecorresponding pointonal1 orb^1(see


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Thetafunctions andnon linear equations 35

Fig. 2). Theexpressions A+ andA~have ananalogous meaning. Inthisn o t a t i o n th eintegral (2.4.5) can berewritten in th eform

8(2.4.6) J ^ j dlogF(P)= 1 i r 2

We note thatif is apointonak, t h e n(2.4.7) A](P)= A*(P) + Bjh,

an d if lies inbk, t h e n(2.4.8) A){P)= A]{P) + 2nibjh

(see th eproof ofLemma 2.1.1). F r o mthe transformationlaw of^ functions(formulae (1.1.4), (1.1.5)),weobtain on thecycleak

(2.4.9) logF~ (P)= \Bhh~Ah (P)+ ek+ logF](P);o n bk

(2 .4 .10) logF*= logF .SincedAk(P) = cjfc , then wehave onak

(2.4.11) d logF~(P)= d logF+{P) ah;o n bk

(2.4.12) d log/ 1" = d \ ogF+.T h u s , th e sum(2.4.6) can berewritten in th eform

.where we have normalizedby cofe = 2 . This proves thelemma.

We nowclaim that theg zeros ofF(P) also solve theJacobi inversionproblem for asuitable choiceof thevectore.Lemma 2.4.2. LetF{P) 0and letPu ....Pgbe itszeroson . Thenon/ ( )

(2.4.13) A(PU ..., Pg)==e K,where = (Ku ..., Kg) is thevector of Riemann constants,and

(2.4.14) Kr JL ' V ( ) , ( / = 1 , ...,g).

I = a l P oProof. Wec o n si d e r the f o l lo w i n g i n t e g r a l :

( 2 .4 .15 ) ,= ^ 7< Aj( )d log F( ), ( / , . . . , ^ ) .


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36 B.A.Dubrovin

O n the one hand, it is equal to the sum of the residues of the integrand, that is(2.4.16) } =AjiPJ + . . . +

where Pg are the zeros ofF(P). On the other hand, as in Lemma2.4.1, we have (see (2.4.7) (2.4.12) )

> 2

F takes(2.4

2ft = i

t h e.17)

u ah

+

same

f 1 )( ^N :h

g1 V \

i l f

values, at

log/

[|d 1

1 I I .1 at h e

.(

logF+ (4j"

ftends of ak,d logF+ =

F~\

+ Jk

it **,

therefore2mnh,

F +

b

where nk is an integer. N ext, let Qj and Qj be the beginning and end ofb,.T h e n(2.4.18) [ d logF+ = logF+ (Q}) logF + (Qt) + 2nim} =

bi= log (A (Qj) + f} e) log (A {Qj) e) +2nimj =

where nij is an integer and^ = (Bju ..., Bjg) is a lattice vector.T h e expression for ,can now be rewritten as:(2.4.19) ^ =e,^BJJ AJ(Q]) + r2i J

T h e last two terms in (2.4.19) can be droppedthey are coordinates of somelattice vector. To get rid of the termAj{_Qj) we transform the integral

AjCuj. We have 3


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T h eta functions and non linear equations 37

where Rj is the beginning of ; and Qj is its end (which is also the beginningoibj). Further, A}(Qj) = AJ(RJ) + 2ni. We obtain

\ Ajaj = 2ni (2Aj (Qj) 2ai),hence

This proves the lemma.Remark. The vector of Riemann constants is connected in a simple waywith the canonical class C of :

2 . 4 . 20 ) 2t f = 4 C).Therefore, when the base pointPo is skilfully chosen, the expression (2.4.14)simplifies (see [4]).

Thus, if the function d(A(P) e) is no t identically zero on , then itszeros solve the Jacobi inversion problem (2.4.1) for the vector = eK. Westate without proof the following criterion for { { )~ e) to vanishidentically (see [2], [4]).Theorem 2.4.1. Thefunction ( (P)e) is identically zero on if and onlyif the point e can be written in the form

(2.4.21) e= A(Q1) + . . . + A(Qg) + K,where the divisor D Ql+ ... + Qg isspecial.

We recall (see 3 above) that a divisor D Qx + ... +Qg is special if thereexists a non constant meromorphic function on whose poles can lie only att h e points Qu ..., Qg. If D is in general position, there are no such functions.Now we can prove the following important proposition .Theorem 2.4.2. Let ?= (,, ..., &)be a vector such that F(P) = ( ( )~ ) does not vanish identically on . Then

a)F{P) has g zeros Pu ..., Ps on , which giveasolution of the Jacobiinversion problem

(2.4.22) Aj ( )+...+A} (/\ ,) = V \ co, = , (/ 1, . . ., g).A= Inb) Thedivisor D = P

r+ ... + P

g isnon special.

c) Thepoints , ..., Pg are uniquely determined from the system (2.4.22)up to a permutation.Proof, a) follows immediately from Lemmas 2.4.1 and 2.4.2. F urther, ifD = / , + ... + Pg were special, then it would follow from Theorem 2.4.1


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38 B.A.Dubrovint h a tF(P) =0,which is acontradiction. IfPi, ...,P'gisanother solutionof(2.4.22), then th erelation

A(PX) + .. . + A(Pg)= A{P[) + . . . + A(P'g)must hold on theJacobi variety / ( ). ByAbel's theorem this means thaton there exists a meromorphic function with zeros atP[,... , P'gandpolesatP , , ..., Pg. SinceDisnon special, such a function must be aconstant,sot h a t after renumbering P,= Pi. This proves th etheorem.

Later th efollowing corollary will be useful.Corollary. For anon special divisorDP1+ ... + Pgofdegreeg thefunction F{P) = ( ( )~ ) ) hasexactly gzeros = Px, ..., = Pgon .

We also give forreference some information on thezeros of a0 functionon / ( )(asocalled divisor).Theorem 2.4.3. T he zeros of the function d(e) =0admit aparametricrepresentation in the form

(2.4.23) ewhere Px, . .,Pg^ is a set of g1arbitrary points of .Proof. Let 0(e ) = 0. We setF(P) =d{A{P) e). Twocases arepossible.1)F(P) 0 on . Then,by Theorem 2.4.2,

(2.4.24) e ewhere th e setP, , ...,Pg isuniquely determ ined. Since6(e) =0,thissetcontains th epointPo ( thelower limit in th eintegrals); say,Pg Po. ThenA(P0) =0 an d(2.4.24) implies th ate^A(P1) + . . . + A(Pg.1) + K.

2) LetF(P) =0 on . Then,byTheorem 2.4.1,we canrepresentein theform

(2.4.25) e= A(Qi) + . . . + MQg) + K,whereD Qx+ ... + Qg is aspecial divisor. BecauseDis special, thereexists ameromorphic function/ on having poles atQx, ..., Qg,such that/ C P 0) = 0. LetD' = l\ + . . . fPg^ + 1\ be thedivisor ofth ezerosof/ . Then,byAbel's theorem, ') = ). Substituting ') for ) in(2.4.25) an dagain using th eequalityA(PQ) =0, wecomplete th eproofoft h etheorem.

I t hasalready been noted that th efunction F(P) = ( ( ) e) (wheree + )doesn o tvanish identically when d(e) 0. Thezeros of a function ( thepointsof a divisor) form asubvariety ofdimension 2 g 2(forg > 3with singularities) in the2g dimensional torus / ( ). If th e divisor can bedropped from / ( ), then weobtain aconnected


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2g dimensional domain. So wefind thatfor allthe points of/ ( ) the Jacobiinversion problemis soluble, and uniquely foralmostall ofthe points.

T h u s , theset ofpoints ( , ...,Pg)= '1( )of (without counting theorderofthose points)is asinglevalued functionof (, ...,) of/ ( )(having "singularities" atthe pointsofthe 0 divisor). Tofind ananalyticalexpression forthese functions,wetakeanarbitrary m eromorphic functionf(P) on . Then, the assignment ofthe quantitiesf1; ...,$guniquelydetermines theset ofvalues

(2.4.26) / ( ), . . , f(P8), (Pl t . .,Pg) = ).Therefore, any symmetric functionof these valuesis a single valuedmeromorphic functionof=( ,, ...,) / ( ) , th atis, anAbelian functionon / ( ) (see Ch.I,3). Allthese functions canbeexpressed intermsof aR i e m a n n 0 function. In anespecially simplewayone can express thefollowing elementary symmetric functions ("N ewton polynom ials"):

(2.4.27) .( )= 4 f{P}) ( s= l , 2,. . . ) .F o r themweobtain from Theorem 2.4.2 and the residue formula thefollowing representation:

(2.4.28) .( )= fs(P)dlogQ(A(P) tK)~

J /s( p ) d l o g (A p) K)( the second termonthe right hand sideisthe sumofthe residuesofth eintegrandat allthe polesof/ ( / > ) ). As inthe proofofLemmas 2.4.1and2.4.2, one can transform the first termin(2.4.28)byusing (2.4.11) and(2.4.12). Then (2.4.28) canberewrittenasfollows:

2 ^J Jh ah

(2.4.29) as(lft c

2 R e sf*(P)d\ogQ(A(P) ).H ere th e first termis aconstant independentof . Weconsider thec o m p u t a t i o nofthe second term(ofthe sumofthe residues)in twoexamples.Example 1. is ahyperelliptic Riemann surface ofgenusg,givenbytheequation

(2.4.30) w2= P tg+1( z ) ,where P 2 g + 1 ( z)is the polynomialofdegree2g+1 without multiple roots.We consider the following functionon :f(w, z) = z (the projection on the


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40 B.A.Dubrovin

z plane). This function on has a single two fold pole at the branch point( t h a t is, the Weierstrass po int) = . We obtain the analytic expression fort h e functions ax and 2 constructed by (2.4.27). In other words, ifP\ (wi> zi), Pg ~ (wg> zg) is t n e solution of the inversion problemA(P1)+ ... + A(Pg) = , t h e n

(2.4.31) ( ) = Zl + . . . + zg,(2.4.32) 2( ) = zl + . . . + z\.

F o r the basis pointPo (the lower limit of the integrals in the Abel mapping)we take z = . By (2.4.29), the functions , and 2 have the form(2.4.33) ( ) = c, Res zd log ( ( ) ),Zoo(2.4.34) 2( ) == c2 Res z2d log (4( ) ),

where cx and c2 are constants depending only on the Riemann surface ando n the choice of the basis of cycles on it. Let us compute the residue in(2.4.33). We take = \ \z as a local parameter in a neighbourhood of z = .By the definition of the Abel mapping we immediately obtain

(2.4.35) d\ag${A{P) l K) = Y. [\ogQ ( ( )~ )] (P) =i l

where [...],denotes the partial derivative with respect to the i th variable,an d ; = fi(x)dx (i = 1, . . ., g) is the canonical basis of holomorphicdifferentials on . We expand the vector valued function A(P) in a series inpowers of r (for ). By the choice of the basis pointPo = , thisexpansion has the form

(2.4.36) A(P) = xU + x2V + O(x3),where U = (Ult ..., Ug) and V = (Vu ..., Vg) are the vectors of the form

(2.4.37) Uk =(2.4.38) yfi = 4

(we recall that = 0 corresponds to the point on ).Lemma 2.4.3. a) J7 is //ze vector of b periodsof anormalized differential ofthe second kind with asingledouble pole at z =and principal part dr/ i2.

b) V is the null vector.


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Proof, a)follows immediately from Lemma 2.1.2((2.1.21) for = 1) .F r o m th esame lemmawe seethat V(up to an unessential factor) is th eperiod vector of adifferential of th esecond kind with apole of thethirdorder at. Butthis differential isexactit isequal todz = 2 / 3,therefore, it haszero periods overall thecycles. This proves th elemma.

Thus, th eexpansion (2.4.36) has th eform(2.4.39) A(P)= xU + 0 ( 3 ) .

Therefore, in aneighbourhood of log (A(P) )= log ( + ) Vus [ log ( + ')],.i

(we have used thefact that the0 function iseven),/ ( ) = Ul + ( .H e n ce , for ( ) weobtain theexpression

(2.4.40) ,( )= I UtUj [log &+ K)]ti+ c ,.i, 3

We introduce theoperator d/dx= ^. ^ of th ederivative along U. Then(2.4.40) can berewritten in th eform

(2.4.41) ,( )= : | L log ( + ) +c,.An analogous (butlonger) calculation of theresidue in (2.4.34) leadst o

the following expression for 2: 2 . 4 . 42 )

Heredldt = ^ Wid/ is th edifferentiation operator along th evectorW = (Wu ...,Wg), where(2.4.43) W^ l fUO).I t follows from Lemma 2.1.2that Wis th evector ofZ> periodsof anormalized differential of th esecond kind with asingle poleof thefourthorderat with principal part 3dr/ r4.

The formulae (2.4.41) and(2.4.42) completely solve th eJacobi inversionproblem forRiemann surfaces of genus 2 (allthese surfaces arehyperelliptic).

We shall show inChapter IIIthat th efunction u = 2 1 is asolution of theKorteweg de Vries equationdu 1 Ir. du , d3u

( G U +Using this equationit iseasy toreduce (2.4.42) to theform

(2.4.42') 2( )= _ 2 ( ) i i ^ g?which only includes differentiation operators in theJC direction .


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42 B.A.Dubrovin

Remark. We recall th at the basis holom orph ic differentials , ..., oog on theR i e m a n n surface (2.4.30) have the form 2 .4 .44 ) , = LOO

H ere the matrix c/ fe has the form (cjh) = 2ni(Ajh)~ 1, where(2.4.45) AJh =

Therefore, the vector (XJX, ..., Ug) (see above) has the following coordin ates:(2.4.46) Uk = 2chl.

We do not derive the analogous expression forW.Example 2. N ow let be th e hyperelliptic surface of genusg given by theequation

/O / AH\ , ,, 2

We consider again the fun ct ion / (w, ) = and the function ( ) (the sumof the projections on the z plane) constructed from it. In this case hastwo points at infinity: P+ = (, + ) and P_ = (, ) at each of which hasa simple pole. F or (2.4.49) can be written in the form

(2.4.48) 1( ) = 1 [Res + Res ]zd log ( ( ) ).We can take \\z as a local parameter at both pointsP+ and P_. Heret h e holom orphic differentials , , ..., have the form

(2.4.49) a = . dz.V ^ 2 g + 2(* )

Let ^ = 11( ) in a neighbourh ood of P+ (which corresponds to the value = 0) and let a)fe = fa(x)dr in a neighbourhood of J_ (which corresponds tot h e value r = 0) . F r o m (2.4.49) it follows that(2.4.50) / ( 0 ) = / ( 0 ) .

We denote/ fc+ (0) by Uk(k = 1, ...,g).The calculation of the residues in (2.4.48), similar to the one consideredin Example 1,yields

(2.4.51) a l ( 0 = C i+ 2 ff f[ lo g ^ 1 \ %% \.By introducing the differentiation operator d/dz = 2 Uidld%i in thedirection of the vector U = (Uu ..., Ug), we finally obtain

(/ .4.W) ( ) _ log


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Theta functions and non linear equations 4 3

where cx is a constant. The com pon ents of Uare the periods of a normalizeddifferential of the second kind with a double pole atP+ this follows fromLemma 2.1.2.The method explained above for solving the Jacobi inversion problem isdue to Riemann. We should also mention here the approach of Weierstrass,

who studied the system of differential equations resulting from (2.4.1) bydifferentiation . We only analyse the caseg = 2, where is given by theequation

2 .4 .53) w* = P 5( z ) .As abasis (non normalized) of ho lomorphic differentials we take thedifferentials

dz dz2 .4 .54 ) f t to = ypt( z ) ' ypt(z)We consider two systems of differential equations:

dz1 Pb (zj) dz2 yPr~


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44 B.A.Dubrovin

Proof. We have

This proves the lemma.By (2.4.58) and (2.4.59) the variables and t are (complex) coordinates

o n the Jacobi torus 4 = / ( ) . Therefore, the integration of the systems(2.4.55) and (2.4.56) i>, = Px{x, t), P2 =P2(x, t) solves the Jacobi inversionproblem.

We mention tha t these systems can easily be integrated by quadratures(for explicit formulae see the survey [17] ) . The expression of theirsolutions in terms of the ^ function of can be obtained from the formulaeof Example 1 (see above).

CHAPTERMITHEBAKERAKHIEZER FUNCTION. APPLICATIONS TO NONLINEAR EQUATIONS

1 . The Baker Akh iezer o n e p o i n t f u n c t i o n . The KadomtsevPetv iashvi l ie q ua t i o n and equat ions assoc ia ted wi th it .

Let be a Riemann surface of genus g. We fix on a point and alocal parameter = z(P) in a neighbourhood of it (such thatz(Q) = 0). It isconvenient to introduce the inverse k 1/z, k(Q) = . Suppose, further,t h a t an arbitrary polynomial q(k) is given.Definition 3.1.1. Let D =P t + ... +Pg be a positive divisor of degree g on .A BakerAkhiezer function on corresponding to Q, to the local parameter = \ jk at Q, to the polynomial q(k), and to the divisor D is a function ( ) such t h a t :

a) ( ) is merom orphic everywhere on , except at = Q, and has on \ ( ? poles only at the pointsPit ..., Pg of D (more precisely, th e divisor ofthe poles is \T\Q> D; see Ch. II , 3) ;b) the product ( ) exp [~q(k)] is analytic in a neighbourhood of = Q.

Instead of b) we also say that ( ) has at Q an essential singularity ofthe form ( ) ~ c exp q{k) (c is a con stan t). F or agiven divisor D suchBaker Akhiezer functions form a linear space (we fix the point Q, the localparameter l/k, and the polynomial q ( k ) ) , which we denote by ( >) (byanalogy with the space L(D) considered in Ch. II , 3) .


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Theorem 3.1.1. LetD=Pt ...+Pgbe anon special divisorofdegreeg.Then the spaceA(Z>)forapolynomial qingeneralposition isone dimensional.I n other words,for anon special divisorDandageneral polynomial q(k),th e conditionsofDefinition 3.1.1 uniquely determine the Baker Akhiezer

functions, uptomultiplicationby aconstant.Proof, a)Uniqueness. The Baker Akhiezer function hasg zeros on \ g,an dfor ageneral polynomial q(k) the divisorofzerosDisalso non special.Suppose now that ( ) and ( ) are two Baker Akhiezer functionscorrespondingtothe same divisor D. Then their quotient ( )/ ( )is ameromorphic function on (the essential singularity cancels) with polesatth e pointsofD. Since th is divisorisnon special suchafunctionisnecessarilya constant.b) Existence. Suppose that is adifferential ofthe second kind onwith principal partatQofthe form dq(k), normalizedbythe conditions

(3.1.1) ^> = 0 (7=1, . . . . # ) .'ai

LetU= (U\, ..., Ug)be its vectorofft periods:(3.1.2) Uh=Q.

Wefixan arbitrary pointPo Qon ,take the corresponding Abel mappingA ( P ) , and construct the function

Q(A(P) A(D) K)

whereDisthe given non special divisor. The pathofintegrationinth e

integral f andinthe Abel mapping A(P)= ( \ 4,. . .,\ ischosento P o Po Po

be the same.We claim that (3.1.3)isthe required Baker Akhiezer function. F irst,we

verify thatit isunique on . Ifwe choose another pathofintegration from

Po toP, then we havetoaddto f atermofthe form & , where7 is aJP aclosed contour (cycle). Similarly, the vectorf

5. . ., u>

g)isaddedto

A{P). We split over the basis cycles:g e

(3.1.4) 7 = nhah+ Vm.b},ftl j l


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46 B.A.Dubrovin

where nk and w; are integers. Then on changing the path of integration wehave

3.1.5) JQ| + / / / , = [ + ,U), Po (3.1.6) A(P) > *A(P) + 2 + BM.

H ere = (mu .... mg) and iV = (n l f .... y) are integral vectors. Under sucha transformation, by (1.1.6) , the quotien t of the 0 functions is multiplied byexp f . )~{M, A(P) A(D) + UK)]_ =_

exp [ j (BM, M) {M, A (P) A (D) K)an d the exponentional term acquires the inverse factor exp (M, U). Thisproves the uniqueness.

Next, because the divisor D Px+ ... +Pg is non special, the poles of(3.1.3) (arising from the zeros of the denominator) lie exactly at the pointsPu ...,Pgseethe Corollary to Theorem 2.4.2. Moreover, the function(3.1.3) has an essential singularity of the kind needed, by th e choice of: = dq{k)+ . . ., [ = q(k)+ . . . in a neighbourhood of Q (the dotsd e n o t e regular terms). This proves the theorem.We analyse in more detail the Baker Akhiezer function constructed fromt h e polynomial

(3.1.7) q{k)= kx + k2y + kH,where x, y, and t are the parameters. We denote this function correspondingt o some non special divisor D of degreeg by ( , y, t; P). Then ( , y, t; P)can be normalized so that in a neighbourhood of Q its expansion takes theform

(3.1.8) *( , , ; )H e r e the coefficients &, |2> a r e functions of x, y, and t (which we shallcalculate below).

Let us leave Riemann surfaces for a while and regard the expansion (3.1.8)as formal (without being interested in its convergence). We have a simplebut important lemma.Lemma 3.1.1. For afunction of the form (3.1.8) the formal equalities

(3.1.10) [lr + *+*uL


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Thetafunctionsandnon linear equations 47hold, where thefunctions u and w can be found from the condition for thevanishing of the coefficients of knehx+h' v*kH for = 3 , 2 , 1 , 0 . Thesefunctions have the form

(3.1.11)

The proof is by direct calculation.We d e n o t e byLandA th eordinary differentialoperators inx:

(3.1.13) L = + u,(3.1.14) A = . + . +,.Theorem 3.1.2. Lei = ( ,j ,t; P) be aBaker Akhiezer functionconstructed from thepolynomial q(k)=kx f /c2;/ + /c3ia


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48 B.A. Dubrovin

Proof of the Corollary. The condition of compatibility of the equationsd\p/dy= L\p and 3///9 = has the form

(3.1.19)(By [..., ...] we den ote the commutator of the operators). By computingthis commutator we obtain (3.1.17).

T h u s , for each Riemann surface of genusg, each point Q on it, andeach local parameter k~l we can construct in a neighbourhood of Q a familyof solutions of the KP equation, parametrized by the non special divisors ofdegreeg on (the points in general position of the Jacobi variety / ( ) ) .

T h e change of the local parameter(3.1.20) k

(where , a, b are arbitrary complex numbers and 0) leads to anotherfamily of solutions of the same KP equation . It is easy to verify that theseo t h e r solutions are obtained by means of the following transformations,which preserve the form of the KP equation:

(3.1.21)y > wyt XH,

1 26

Of course, the fact that the KP equation admits the group of transformationsof the form (3.1.21) is trivial toverify (without resort to the theory of0 functions).

We now express the thus constructed solutions in terms of the 0 functionof . To do this we make use of the formula (3.1.3) for the Baker Akhiezerfunction, which in our case is as follows:

(3.1.22) ( ,y, i;P ) = exp ( * JQW + y [ +t \ >)

Po h PoX Q(A(P) A(D) K)

H ere 1*, ^2), and (3^ are normalized differentials of the second kind,with poles only at Q and with principal parts also at Q of the form

(3123) 0 = d f t + . . . , Q> = d ( f c a ) + . . , =d(#>)+ . . .( t h e dots denote the regular terms; k'1 is the local parameter) ; U, V, andWare their vectors of fe periods:

(3.1.24) / i= ^ Q < 1 ) . 7 ; = (>


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Theorem 3.1.3. The thus constructed solutions of the KPequation have theform

(3.1.25) u (x, y, i)= 2 jjL.log 0(xU+yV + tW+ z0) + c,

where = 0(z) is the function on , the vectors U, V, and W aredetermined by (3.1.24), z0 = A(D)~ is anarbitrary vector,and c is aconstant.Proof. By (3.1.11), it suffices to find the coefficient ^ of the expansion(3.1.8). We note that the expansion of the logarithm of the Baker Akhiezerfunction (not necessarily normalized) has the form

(3.1.20) log = kx+mj + kH +10 +^+cx +ay+ bt+ _( 0 is some function of x, y, t; a, b, and c are constants). Consequently,^ + cx + ay + bt is the coefficient of l/k in the expansion of the function

, , . Q(A(P) A(D) K { xU \ yV+lW ) (1 ) lo g . . .. .. T T J T \ F ;' v ' B (/I (/') A (I)) A)in a neighbourhood of = Q. N ext, we recall that it follows from (2.1.21)t h a t the vector valued function A(P) as * Q has the expansion:

(3.1.27) A(P) = A(Q) yU +(We choose Q as the initial point of the Abel mapping. ThenA(Q) = 0, andt h e required coefficient ^ has the form

(3.1.28) , 4 ex4ay \ bt= ^ log 0(xU + yV + tW A (D) ~ K)4 . . .,where the dots denote terms independent of x, y, t. F r o m the formulau = lid^/ dx) we obtain the proof of the theorem.

(3.1.25) gives additional information about the properties of the solutionsof the KP equation: u{x, y, t) is a conditionally periodic (generally speaking,meromorphic) function of the variables x, y, and t. For the secondlogarithmic derivative (92/ 9x2) log0(z) (z is a point of the Jacobian / ( )) is ameromorphic function on the torus / ( ) (an Abelian function). To obtaint h e solution u(x, y, t) we have to restrict this function to the linear( xyf) winding, spanned by the vectors U, V, and W. We do not discusshere the problem of isolating the real and the bounded solutions amongthose obtained.

If for U we have the commensurability relation

where 2nie^ . . ., 2neff, / x, . . ., }g are basis vectors of the period lattice,t h e nk and nij are integers, and 0, t h e n u(x, y, t) is periodic in with


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50 B.A.Dubrovin

period T .If inadditionwecan find a second (complex) period T ',wherel m ( T ' / T ) 0such that

(3.1.30) (U)= 2ni (n'fr+... +n'geg)+ m'ijl+...+ m'efg,where then^ andm)areintegers, thenu(x,y, t),as afunction ofx,is adoubly periodic meromorphic function (with the periods and T ')and canbe expressed interms ofelliptic functions.Example, (see[53]). Forth eRiemann surface of the second kind,givenby th e equation

(3.1.31) u , 2= z5_ 2Z3_ 3Z2 2 ^ ^ 3 ia n d forth e Weierstrass pointQ = on , thedependence onydisappears,a n d the KP equation reduces to theKdV equation (see below) whosesolution canbeexpressed interms of elliptic functions by th eformulae

(3.1.32) u(x,t)= 2f (x x ^ t ) ) + 2 f x x 2 t ) ) +2f (x x 3 t ) ) .H eref (x) is th eWeierstrass function (seeCh.I , 3) ,

4 i =

I n particular,if = 0,const = 0,then u(x,0) = 6 ( ) . Anelegant methodof finding th eRiemann surfaces forwhich the function u= 2d2/ dx2 log canbe expressed byelliptic functions has recently been foundbyKrichever [22 ].

Suppose that th eRiemann surface and th epointQo n it aresuch thatt h e r e exists ameromorphic function ( )on with a unique double poleatQ. I t iseasy t o seethat then must behyperelliptic andQmust be aWeierstrass point (abranch poin t). Wechoosek^iP)= [ ( )]"1/ 2 as thelocal parameterof Ar1= k~l(P) in aneighbourhood of = . Thenth eBaker Akhiezer function i//(x,y,t\P)with asessential singularityexp(fcr jft2;/ + k3t) at Q hasthe form

(3.1.33) (,y, i; P ) = (/ ( ) ( ,t; P)),where isth e Baker Akhiezer function with th e same divisor ofpoles as \ p,a n d with anessential singularity ~exp(kx +k3t) atQ. This followsimmediately from th euniqueness Theorem 3.1.1. Then th edifferentialequations (3.1.15), (3.1.16) for canberewritten asfollows:

(3.1.34) L


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Thetafunctions andnon linear equations 51Now (3.1.34) means that is th e"eigenfunction" of th eSchrodinger

(Sturm Liouville) o p er at o r ( 1 ) l with eigenvalue ( ), depending on theparameter t,by(3.1.35). Thecoefficients of th eoperators LandAareindependent ofy, and the KPequation turns into theKorteweg de Vries(KdV) equation

(3.1.3G) u , = l nmx+ uxxx).The commutation condition (3.1.19) gives the"L~A pair" for th e KdVequation

(3.1.37) ^L=[A,L],and finally, (3.1.25) gives thestandard formula (ofMatveev Its) for finitezone solutions of theequation

(3.1.38) u x,t) = 2 r logwhere Uan dWwere defined above.

Similarly, if for acurve and apoint Qon itthere exists a meromorphicfunction ( )with asingle poleof thethird order atQ,then thedependenceon t disappears and the KPequation turn s in to aversion of theequationofa non linear string ( theBoussinesq equation)

(3.1.39) 3yy+ ^ (Guux+uxxx) = 0.which hassolutions of theform

(3.1.40) u (x, y)= 2 ?L log 0 xU fyV + zo) + c.We nowindicatea set of conditions that aresufficient for th econstructed

solutions of thefollowing equation to bereal \ 1 / n . x .' ) rU, n, = \ Ut 7 (buux^ rUVXT) \ ,

which is obtained from the KPequation by thesubstitut ion y> iy. Let rbe ananti involution (that is, ananti holomorphic automorphism r: >> ,2 = 1) on ofgenusg = 2p+n~ 1,having/ : fixed cyclesAo, ... An_x.When we cut along thecyclesAo, ..., An X, weobtain twoconnectedcomponents + an d ^ = ( +), each ofwhich is anopen Riemann surfaceof genus with boun dary consisting of thecyclesAo, ..., An^ x. Then in thehomology group ( ; ) we canthen choose thecanonical basis ofcycles

o i,bi, . . ., , Op, , O p + i , . ,p+ 7l _j, o p + n _ j , a r o 1 , . . . , a p , fcp( 1 )U n d e r suitable conditionson theRiemann surface and thed ivisorDon , thepotentialuofthe Schrodinger operator is areal almo st periodic function. Inthis caseLisanoperator with regularanalytic properties, thatis, hasaneigenfun ction thatismeromorphicon (see [17] ) .


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52 B.A. Dubrovin

s u c h t h a t a p + f e=A k ( k =1 , . . . ,n 1 ) , a n dat, bt T +, al b'iT; ( , )= , (fe;)= 6; (i = l , . . . , p ) ,

f ( a p + f t ) = a P + f t , t ( 6 p + f t ) = bp+h (/ c= l , . . . , 1 ) .To obtain real solutions, the essential singularity Q of the Baker Akhiezerfunction must be chosen fixed with respect to (that is,T(Q)=Q),and thelocal parameter in a neighbourhood of Q (where z(Q) = 0) so that{) = . Such a point Q and local parameter determine the vectorsU, V,and W(see above). Then the smooth real solutions of (3.1.41) haveth e form

(3.1.42) u(x, y, t) = 23* log Q(xU + iyV + tW + zo)+ c,where z

0 has the form

(3.1.42') zo= (z'o, zl, z'o), z'o C , zi' G R 1 ,and c is a real constant (this can be deduced from [8], for example).Remark. The Baker-Ahkiezer function ( , y, t; P) having poles on \ (> ,also hasg zeros there (for almost all x, y, t). These zeros depend on theparametersx, y, and t. We denote them by Q\(x, y, t), ...,Qg(x, y, t). Thedependence of these zeros onx, y, and t can be determined from thefollowing proposition (Akhiezer).Lemma 3.1.2. For the zeros Qx = Qx(x, y, t), ...,Qg = Qg(x, y, t) andpolesPu ..., Pg of the Baker Akhiezer function ( , y, t) the followingrelationholds on the Jacobi varietyJ(T):

(3.1.43) A(Q1) + . . . + A(Qg) = A(PX) + . . . + A(Pg) f Ux ++ Vy+ Wt,where the vectorsU, V, and W aredetermined above.The proof of the lemma repeats almost word for word that of Abel's

theorem. The periods of the meromorphic differential dlog must beintegral multiples of 2m. This differential can be represented as a linearcombination of elementary ones (see Ch. II, 1) :

(3.1.44) d. log^= $ + # >+ / )+ ?>+ j] ^ (.Integrating this expression over all the cycles a/andbk, we obtain (3.1.43).F r o m this lemma one can derive another proof of Theorem 3.1.1.

Differentiating (3.1.43) with respect to x, y, t we obtain differentialequations describing the dynamics of the zeros (Qu ..., Qg). For example,for the hyperelliptic surface of genus 2 {ur = Ps(z)} and for the Weierstrasspoint Q = on the dependence on of the zeros (Qlt Q2) is defined by(2.4.55) (up to a factor of 2). The dependence ony disappears and thedependence on / is given by a linear combination of (2.4.55) and (2.4.56).


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We complete this section by the following general observation. Thepolynomial q{k)= kx f k2y + k3t from which th e Baker Akhiezerfunctionwas constructed was th e simplest possible. I t is easy to generalize th ecalculationsand to prove the following proposition.Theorem 3.1.4. Let be a Riemann surface of genus g, Q a point on , andk'1 a local parameter in a neighbourhood of Q. We fix a non special divisorD of degree g.

a)Each polynomial q(k) of degree determines an ordinary differentialoperator L q (in x) of the n th order, according to the following rule.Suppose that \ pq= \pq(x, y; P) is the BakerAkhiezer function with divisor Dof the poles and with the essential singularity &xp(kx+q ( k ) y ) at Q. Then Lqis uniquely determined by

(3.1.45) ^ = L94VThecoefficients of Lq can be expressed recurrently in terms of the coefficientsof the series ? exp(kx q ( k ) y )in reciprocal powers of k .b)Every pair of polynomials q{k) and r(k) generates a solution of a nonlinearequation in coefficients of Lq and Lr of the form

(3.1.46) [~ + L " 1 +Thissolution can be expressed in terms of the function of .

c)In particular, every meromorphic function X(P) with asingle pole at Qgivesan operator of the form L =Lx and a set of eigenfunctions ,

(3 .1 .47) , = ,if we take q(k) to be principal part of the Laurent series of this function atQ and L = L q. In this case any other polynomial r(k) determines a solutionof a non linear equation in the coefficients of L of the form

(3.1.48) ^ =[A,L], A = Lr,which can also be expressed by functions.

d)A pair of meromorphic functions ( ) ,( ) with asinglepole at Qgivesrise to a pair of commuting ordinary differential operators

(3.1.49) IA, L] = 0.Theequations with the coefficients of these operators are called the Novikovequations and can also be integrated by means of functions.

O n e can also reformulate c) and d) of this t h e o r e m as follows: every nonspecial divisorD of degreeg gives rise to a homomorphismof the ring of themeromorphic functions with a single pole at Q i n t o a commutativealgebra ofordinary differential operators

(3.1.50) ( ) H ^ Lx.


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54 B . A .DubrovinAll the operators of the form have a com m on e igen func t ion ,

(3.1.51) | = ,which is th e Baker Akhiezer fun ction with divisor D and essential singulari tyexpkx at Q .

F o r a p r o o f o f t h i s t h e o r e m a s well a s o t h e r a p p l i c a t i o n s o f i t a n d ag e n e r a l i z a t io n t o m a t r i x a n d d i ff er e n c e o p e r a t o r s , se e [ 1 5 ] a n d [ 1 6 ] .

2 . The BakerAkhiezer two po int fu nc t io n. The Schrodinger equ at ion in amagnetic f ield

T he simplest gen eral ization of th e Baker Akhiezer function we havestudied in th e preced ing section con sists in ad din g superfluous essen tialsingulari ties. Th e general definition of th e Baker Akhiezer 1 point fun ction s("o f ra n k 1") is as follows.Definition 3 . 2 .1 . Le t be a R ieman n su r face o f genus g, Q x, . . . , Q l p o i n t son ; k '\ L , . . . . 71 l oca l paramete r s in a ne ighbourhood o f these po in ts(where &,(?,) = )> > ? i W a se t of polynomials , D a divison on \ ( ( ? U U Q i ) , an d = ( ) a Baker Akhiezer / poin t funct ionspec ified by these da ta : it is m erom orph ic on \ ((? i U . . . U Q i) a n d sucht h a t a) the divisor of >~ D; b )as/ Q u t he pr o d uc t i|i(P )oxp(qXkiiP)) )is analytic ( i = 1, . .., / ) .

T he Baker Akhiezer funct ion s given by the condi t ions of th is def in i t ionform a l inear space , which we d en ot e by A7(Z)) . By analogy wi th Theorem3.1.1 on e can prove th e fol lowing th eo rem .T heorem 3.2 A. L et be a Riemann surface of genus g and D anon specialdivisor on \ ( Q x \ j . . . \ J Qi ) . T hen the dimension of the space A/(Z>)isdegD g+ 1. I nparticular, if D P l+ . .. + P g is anon specialdivisor ofdegree g, then the corresponding Baker Akhiezer l pointfunction exists andis uniquely determined up to a factor.

T h e use of th e Baker Akhiezer m ul t ipo in t funct ion s allows us to in tegrateavery la rge num ber o f im po r ta n t no n linear equa t ion s. We ana lyse a s impleexam ple of the use of the Baker Akhiezer two point funt ion , wi th o ut givinghere a genera l theory o f app ly ing m ul t ipo in t fun c t ion s to the in tegra t ion o fequa t ion s ( see [15] an d [1 6] ) .Le t be an a rb i tr a ry R iema nn su rface o f genus g, Q+ , Q _ a pai r of poin tson , an d k+ an d k ~ l oca l paramete r s in ne ighbourhoods o f Q + an d


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Theta functions and non linear equations 55

Such a function exists and is uniquely determ ined up to normalization. Itsexplicit expression in terms of the function of is as follows:

(3.2.1) ( , ; P) = exp ( z \ + + j _) ( ( ) A D ) )Po Pa

H ere + and _ are normalized differentials of the second kind with adouble pole at Q+ and (?_, respectively, and principal parts at these pointsof the form

(3.2.2) = d(k) + ...;U+and U are the vectors of the periods of these differentials,

(3.2.3)In neighbourhoods of Q the function ( , F ; P) has an expansion of theform: as Q+ ,

(3.2.4) Tp= c+ eha n d as > Q_,

3 . 2 . 5 ) = The coefficients c and %fare functions of and F. We normalize so t h a tth e coefficient c+ in (3.2.4) is 1. Then we denote the coefficient c_byc(c = cjc+). By analogy with Theorem 3.1.2 we can prove the next theorem .Theorem 3.2.2. The Baker Akhiezer function ( , F ; P), normalized by thecondition c+ 1,is asolution of the equation

(3.2.6) = 0,where(3.2.7) H ^ ^ ~Ldzand the coefficients a, b of the operator have the form (c e_/ c+)

/ Q\ f, dlogc u / ,( . . ) a = , 0 = 4~

dz dzProof. The form of H, by analogy with Lemma 3.1.1, is selected from thefollowing two conditions: for the expansion (3.2.4) (with c+ = 1):

(3.2.9) # = 0 (for the expansion (3.2.5):

(3.2.10) Hx\= e' Cc (z, z) + O ( 1 ) )


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56 B.A. Dubrovin( t h e explicit form ofc"(z,F ) is n otimportantfor us). From (3.2.9)and(3.2.10) it follows that isagain aBaker Akhiezer function,but at Q+theproduct (H\Jj)exp( k+z) vanishes. By th euniqueness, = 0.This proves th etheorem.Theoperator constructed from , thepair ofpoints Qon , and thedivisor Dcan beinterpreted as th eSchrodinger operatorfor a twodimensional electron in themagnetic field B(x, y) =dxAy dyAx, where( A x, Ay)is avector potential,in thepresence of apotentialu{x, y) (of theelectric field):

(3.2.11) H =____ , J_ J_(J. L\ dx~ dz+ fz ' dy ~ l { dz fz)'eis theelectron charge; h = 2m = c= 1 (themagnetic field B{x, y)is

directed along thethird axis). F or theoperator (3.2.7) wehave(3.2.12) | x i y ^ a ^

(3.2.13) w = = f e + ^ = 4 ^ + 2& dz dz dz dzWenowdeduce formulae expressing thecoefficients a, b (andthus,the

magnetic field = dxAv dyAx and theelectric field u{x, y))intermsof0 functions.Theorem 3.2.3. The function

(3.2.14) ( , ; ) = ( [ +ct+ ) fz ( f _ _ )1

J^

Q(A(P) \ zl ( ( ) + ) ( ( +) + ++~ + ) '

where theconstants +aresuch that \ +a+= /c+ + O(y J . >Q+ra_= \ _, / o rany 0 asolution of theequation 0,where is theSchrodingeroperator of atwodimensionalelectron in amagneticfield,

(3.2.15) H= (i JL +eAx)2+ (i L+eAy)2 + u(x, y).Thecoefficients of this operator have the form

(3.2.16) u(x,y) == 2 ^ log [ (A(P+) + zU++zU_+ 0) (A (P.) + zU++~ z U _+ 0)]+ const,dz dz

3 .2 . 17 ) ^ , =M i f log


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Thetafunctionsandnon linear equations 57

Themagnetic field ( ,y) =dxAy dyAx isdirected along the third axisandhas the form(3.2,18) B(x,y)(,y) ^e dzdz

Proof. (3.2.14) isobtained from (3.2.1) bynormalization (wedivide byc+),where 0= A(D)~ is anarbitrary pointof theJacobi variety / ( ). Thenfor th efunctions c{z, I) and f wehave

(3.2.19) togC(z.i)= log

where thedots denote linear combinations of thevariables z and F ,arisingfrom theexpansion of theexponent of the exponential curve in(3.2.14).This and(3.2.12) and(3.2.13) immediately prove thetheorem.Remark. Thegroup ofperiodsof themagnetic field B(x, y) is thesameast h a tofthe vector potential(Ax, Ay)this follows from theexplicit formulaeobtained. Therefore, in a doubly periodic field (atwo dimensional crystallattice with periods Tx andTy)theconstructed solutions have azero fluxthrough thenucleus of thelattice \ \ (x, y) dx dy= 0. Thecase ofnono zero flux hasbeen integrated in [3 7] .

There is asimple sufficient condition for theconstructed operators tobe real (see [49]).Lemma 3.2.1. Suppose that there is ananti involution on V,permutingQ+andQ_

(3.2.21) : > , 2 = 1, ( < ? + ) =


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58 B.A. Dubrovin

C H A P T E R I V

E F F E C T I V I Z A T I O N OF THE FORMULAE FOR THE SOLUTION OF KdV AND KPEQUATIONS. RECOVERY OF A R IE M A N N SURFACE FROM ITS JACOBIVARIETY. THE PROBLEM OF R I E M A N N AND THE CONJECTURE OF NOVIKOV

1 . The KdV e q u a t i o n . Genusg = 1 or 2In the preceding chapter we have obtained "explicit formulae" (3.1.25)an d (3.1.38) expressing the solutions of a number of important non linear

equations in terms of 0 functions. These formulae are of little use forcalculations within the framework of the theory of ^ functions for thefollowing two reasons:

1) The Riemann matrix Bjk is not arbitrary;

2) The connection between the vectors U, V, Wand the Riemann matrixBjk is transcendental (this connection has never been discussed explicitly).We recall (see Ch. II, 1) that th e set of period matrices (Bjk) of Riemannsurfaces of genus g > 1 depends on 3g 3 complex parameters (for g = 1 ono n e parameter) . However, the general (g g) Riemann matrices form afamily of dimensiong(g+ l)/ 2. Riemann raised the question: whatconditions have to be imposed on a Riemann matrix Bjk so that it is aperiod matrix of holomorphic differentials on a Riemann surface ? For thegenerag = 1, 2, 3 the Riemann matrix can be any indecomposable matrix.Riemann's problem is non trivial for g > 4 and an effective solution of it forallg > 4 has not yet been obtained (see [26], [38], [39]).

Novikov suggested obtaining a full set of necessary relations on the matrixBjk and the vectors U, V, Wvia a simple substitution of (3.1.38) and(3.1.25) in the KdV and the KP equations, where the 0 function is determinedby its Fourier series (1.1.1) . Then one obtains a system of algebraicequations from which the vectors U, V, Wcan be determined; thecom patibility conditions of this system give a full set of relations on theR i em a nn matrix Bjk.T h e present chapter is devoted to a realization of this problem. Inparticular, for g = 1, 2, 3 (where the Riemann matrix B/ k is arbitrary) acomplete effectivization of (3.1.25), (3.1.28) is given to solve the KPequation and the equations connected with it.

We start with the case of small genera g = 1 or 2 for the KdV equation.H er e the Riemann matrix Bjk is arbitrary (in general position). We look fora solution of the KdV equation(4.1.1) Ut = (Suu

x+ u

xxx)

in the form(4.1.2) u(x,t) = 2^


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Thetafunctions and non linearequations 59

where the 0 function is con structed from some Riemann (g x g) matrix Blk,th e vectors are unknown, and z0 is an arbitrary g dimensional vector. Thisformula differs from (3.1.38) by the fact that c = 0, which can always beachieved by the change W>* W U. Substituting (4.1.2) in (4.1.1) weobtain

where = 0(z), = Ux+Wt + z0 is an arbitrary vector, = V.Ut j,and = 2 Wi gjr This expression can be written in th e form

1 \~l dxat +d I a** J + 2Suppose that 5 / fc is indecomposable (see Ch. I, 3) . In this case itfollows from (4.1.3) that the expression in curly brackets is a constant,

which we denote by Ad. An elementary calculation leads us to the relation(4.1.4) , 4 , + 3 ^ 4 ( + 49,6, + 8dW=

( t h e subscripts and t denote derivatives), where ( ), which must holdfor any z. To obtain a finite system of equations in U, W, and the constantd, we use the addition Theorem 1.4.1. In our special case it has the form

(4.1.5) ( 1 ) ( 2 ) = 2 [/]{wl)$[n\ (w2), i ( z2)*

(4.1.5') z' + z ' ^ w1 , z1z2= w2.H ere we use the abbrevation

(4.1.5") Un](w) = 9[ra, 0](u; | 25).T h e notation G {7L2Y means that the summation in (4.1.5) is over allof the half periods {nlt ..., ng), nt 0, \ .T h e values of the functions [ ] (w) and their derivatives

Qu. . . [ ] (w) ^ . . . [ ] (w)J OW OW ja t w = 0 are called constants. We agree to om it a zero argum en t of th e0 constants: Qu . . . [n] = ; [ ] (0) .Lemma 4.1.1. (4.1.4) is equivalent to the following system of 2g equationsin the vectors U (Uly ..., Ug), W = (U^, ..., Wg), and the constant g:

(4.1.6) dfjQ[n] dvdw [n]+dQ [n]= 0,where the equations are numbered by G2{"Z 2)g. Here we introduce thenotation

dbQ [n] =(4.1.7)


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60 B.A.Dubrovin

Proof. We introduce operators

(4.1.8)

a n d , by analogy, operators Xwi, Xw^, Tw,, Twz, where all are substituted forw, connected by the relations (see (4.1.5')) XW2, 1 zl = wl | 1 K)2,

X*Tzi

= = yW,

d vdz) 'd r z 2= 2

d1 az? '

( 419) . v v v T T xJ z2 = L lyl * Ul2 Now (4.1.4) can be rewritten in the form

(4.1.10) 4 , +8d) (Zi) ( 2 ) ] 2 = .= 0.

We express the operator in square brackets in term s of theXwt and Tvi in(4.1.9) and apply it to the right hand side of the addition Theorem (4.1.5)for w1 = 2z, w2 = 0. Since 6[n](w) is even, it suffices to leave the evenpowers of Xw .andTw% in the resulting expression. If we take this intoconsideration, the calculation becomes very simple and leads to the relation

Xw2T wi + d) 0In] (u:1)6[n] (w*)]w^ 2z = 0.4I t is easy to see that the 2g functions 0 [ M ] ( 2 Z ) , % ( Z2) g, are linearlyi n d e p e n d e n t (they form abasis in the space of the functions of the secondordersee Ch. I, 1) . Equating the coefficients of these functions equal tozero we obtain the system (4.1.6). This proves the lemm a.T h e system (4.1.6) is invariant under the following scale transformations:

(4.1.11) > (7, W>>

dubrovin_1981_umn.pdf

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