## Abstract

The method of finite-gap integration was created to solve the periodic KdV initial problem. Its development during last 30 years, combining the spectral theory of differential and difference operators with periodic coefficients, the algebraic geometry of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems, had a strong impact on the evolution of modern mathematics and theoretical physics. This article explains some of the principal historical points in the creation of this method during the period 1973–1976, and briefly comments on its evolution during the last 30 years.

## 1. Introduction

After being invited by Vadim to write the introductory article to this volume I started to hesitate. A question was: what should it be about? It is certainly impossible to cover the whole field taking into account thousands of publications and several books completely (Belokolos *et al*. 1994; Gesztesy & Holden 2003) or partially (Marchenko 1977; Wilson 1981; Mumford 1983; Novikov *et al*. 1984; Bogoyavlenskij 1991; Dickey 1991; Cherednik 1996; Kaku 1998) devoted to this topic. Finally, I have decided to discuss in more detail the very first developments of the theory and to sketch briefly some of the most important developments of the last 30 years.

### (a) Inverse scattering method before the finite-gap integration

After the fundamental discovery of the inverse scattering method made by Gardner *et al*. (1967), allowing a solution of the Cauchy initial problem with rapidly decaying initial data for the KdV equation, there was a short delay before the appearance of seminal works by Lax (1968), Zakharov & Shabat (1971, 1972) and Ablowitz *et al*. (1973). In these works, it was realized that the same method might be applied to an infinite number of 1+1 field theoretic models, and that among these there were several models of fundamental importance for the physics of nonlinear phenomena. Soon after, the soliton virus started to propagate across the world. Many important models were solved, in the spirit of the above-mentioned works, by Manakov, Zakharov, Shabat, Faddeev, Takhtajan, Ablowitz, Kaup, Newell and Segur during the period 1973–1974, and the first solitonic teams were formed in the former Soviet Union, the USA, Italy and other countries.

The finite-gap integration method was created in 1974–1976 in the former Soviet Union and the USA. In the next few subsections the development of the method is presented in a narrative form and as such concerns mostly the ‘Russian side’ of the story,1 but even in this part it is very incomplete. It reflects mainly the author's personal recollections in the belief that it might be of some interest for new generations of researchers.

### (b) Ufa 1973: ‘how to solve the KdV periodic initial problem?’

In October of 1973, the first soliton conference in the Soviet Union bringing together 30 participants was organized in Ufa. Among the participants were several people who had already made some crucial contributions into the field of integrable models: Zakharov, Shabat, Faddeev, Manakov and Takhtajan. There were also many people (including the present author) fascinated by the new promising area of activity, but at that moment still having made no contributions in the ‘solitonic sector’. In particular, Arnold, Kirillov, Manin and Novikov were among the participants.

At that conference many new impressive results were first presented, including the solution of the Toda lattice model and the vector NLS model by Manakov, the solution of the sine-Gordon model by Zakharov, Faddeev and Takhtajan, with strong emphasis on the Hamiltonian interpretation and the construction of the related action-angle variables.

One of the topics widely discussed in the couloirs of the conference was how to solve the KdV-like nonlinear equations for periodic initial data and, in particular, what kind of animals the periodic analogues of solitons should be.

To understand the difficulty of the problem and its distinction from the case of the rapidly decaying initial data, it is necessary to emphasize that the basic technical tool for solving KdV with rapidly decaying initial data was the solution of the highly non-trivial inverse scattering problem for the Schrödinger operator on the line obtained by Faddeev 10 years before the appearance of the article by Gardner *et al*. (1967).2 The so-called reflectionless potentials taken as initial data for the KdV equation were generating the famous multi-solitons solutions. The possibility of describing the multi-soliton solutions by explicit determinant formulae was the direct consequence of the degeneracy of the kernel of the related Marchenko Fredholm-type integral equations.

The efficient solution of the inverse spectral problem for the Schrödinger operator with periodic potential at that time was unknown. The only known periodic solution of the KdV equation was the simple cnoidal wave solution constructed almost 100 years previously by Korteweg and de Vries.

Tentative answers to the question of how to solve the KdV initial problem circulating around the conference discussions appeared later to be misleading.

### (c) Akhiezer's work, first breakthroughs: Novikov's equations, trace formulae and moving Dirichlet eigenvalues

This subsection describes what happened after the Ufa conference during the period between November 1973 and January 1974.

Coming back from the Ufa conference to St Petersburg, I started to think about the periodic KdV problem. I was at the time probably the only one to sense the importance of the work of Akhiezer (1961). I learned about the work of Akhiezer in 1964, being a fourth year student of the physical faculty of Leningrad (now St Petersburg) University from the excellent book of Glazman (1963). In his work, Akhiezer claimed the existence of multi-parametric families of periodic Schrödinger operators with a finite number of spectral gaps. Even more, he reduced the reconstruction of some subfamily of the finite-gap potentials to the classical Jacobi inversion problem on hyperelliptic Riemann surfaces.

While coming back from the Ufa conference, some 10 years after taking a first look at Akhiezer's article, I got the strong feeling that his work, after being properly understood, should be the key for solving the KdV periodic Cauchy initial problem.

In general, the spectrum of the Schrödinger operator with a periodic potential contains an infinite number of closed intervals separated by gaps. In particular, this is the case of the Krönig–Penny model with ‘rectangular’ periodic potentials. The Krönig–Penny model was the only solvable periodic model then known in the majority of the solid-state physical community since the creation of quantum mechanics. Therefore, I was rather surprised by Akhiezer's result. In Akhiezer's work, the existence of the class of Schrödinger operators having an absolutely continuous spectrum consisting of a finite number of intervals separated by gaps was established via the inverse problem approach. More precisely, Akhiezer succeeded in reducing the reconstruction of the potential from some special class of spectral functions to the solution of the Jacobi inversion problem on the hyperelliptic Riemann surface, whose branch points were coincident with the boundaries of the gaps. The solution of the Jacobi problem was not achieved by Akhiezer; he presented the final result only in the one-gap case, corresponding to the simplest Lamé elliptic potential.

It was very inspiring, due to a known fact that one of the infinite period limit cases of the simplest Lamé potential was the reflectionless potential with one discrete negative eigenvalue.

In general, the KdV flow of the simplest Lamé potential coincided with the simple cnoidal wave solution, found long ago by Korteweg and de Vries—the only *x*-periodic solution of the KdV equation known for almost a century. It was also well known that one of the infinite period limit cases of this solution coincided with the one-soliton solution of the KdV equation. Therefore, the idea that the finite-gap potentials might represent the kind of initial data for which we could expect to calculate the solution of the Cauchy initial problem explicitly became natural.

Therefore, at that moment, the question of how to attack the periodic KdV problem was reduced to the following more technical questions.

How does one characterize and construct explicitly all smooth, real valued periodic finite-gap potentials?

What is their KdV dynamics?

Taking into account the fact that the Schrödinger equation with the simplest Lamé potential was explicitly solvable in elliptic functions, there was also a natural question: is it possible to solve explicitly the Schrödinger equation with the finite-gap potential?

It is necessary to say that at that time, elliptic functions, theta functions, Abelian functions and integrals were almost excluded from the standard curricula of mathematics and especially of physics faculties of the majority of universities all over the world. Therefore, to understand Akhiezer's work better than he did, it was necessary to learn these things. Many books on Abelian functions written by recognized mathematicians (a typical example is the book of C. Chevalley on algebraic functions) were not adapted for quickly introducing the potential reader into the subject.

Therefore, instead of dedicating all my time in this direction, I started to think about an alternative approach of a ‘dynamical nature’. It was clear from the spectral theory (Titchmarsh 1958; Glazman 1963; Magnus & Winkler 1979) of the one-dimensional periodic Schrödinger operators *H*(1.1)that the spectra associated with different boundary value Sturm–Liouville problems behaved differently with respect to the deformations of the potential generated by simple translations of its argument or by the KdV flow. It was obvious that the boundaries of gaps *E*_{j} coinciding with eigenvalues of periodic or anti-periodic Sturm–Liouville problems(1.2)(1.3)were the integrals of motion: they are the same for initial potential *u*(*x*), for its translations *u*(*x*+*τ*) or its deformations by the KdV flow *u*(*x*, *t*),(1.4)By contrast, the eigenvalues *λ*_{j} of the Dirichlet boundary value problem(1.5)should move inside the gaps [*E*_{2j},*E*_{2j+1}] acquiring dependence on *τ* and *t*. This motion is periodic with respect to the translation parameter *τ*. The minimal and maximal values of *λ*_{j}(*τ*), considered as a function of *τ*, coincide with lower and upper boundaries of the *j*-th gapThis picture concerns any real smooth periodic potential. It follows immediately from Lyapunov inequalities, discovered approximately 100 years ago (Glazman 1963), showing that in the case of the continuous periodic potential there is exactly one Dirichlet eigenvalue inside the closure of any spectral gap.3

Now a question is, what was so special from the point of view of this picture concerning the finite-gap case? It was clear that in the finite-gap case only the Dirichlet eigenvalues lying in the closure of the non-degenerate gaps could depend on the translation parameter *τ* or on the KdV time variable *t*: when the gap is contracted to the point, i.e. *E*_{2j}=*E*_{2j+1} there is no room to move inside it. At this moment I asked myself, what will happen if we take the difference of the famous Gelfand–Levitan–Dickey trace identities (Dickey 1955), written for the Dirichlet eigenvalues of *H*, with the translated and non-translated potential? The result for the first three trace identities (valid for any smooth periodic potential) looks as follows (after subtraction we replace *τ* by *x*):(1.6)

(1.7)

(1.8)

In general, for any smooth periodic potential, using the results of Dickey (1955), we obtain the infinite sequence of identitieswhere the polynomials *P*_{m} can be computed using the simple recurrence relations obtained in Dickey (1955).

A slightly more precise form of the trace identities might be obtained in a similar way, completing the Gelfand–Levitan–Dickey (equal to GLD) formulae by similar trace identities written for the eigenvalues of periodic and anti-periodic Sturm–Liouville problem. In general, this leads to the infinite series of identities linking the Dirichlet eigenvalues with eigenvalues of periodic and anti-periodic Sturm–Liouville problems and the potential, which is assumed to be periodic and smooth. It is necessary to emphasize that the same formulae remain valid if we replace *u*(*x*) by *u*(*x*, *t*), where *u*(*x*, *t*) is the solution of the KdV equation with initial data *u*(*x*, 0)=*u*(*x*). The only difference is that the Dirichlet eigenvalues acquire the time dependence *λ*_{j}=λ_{j}(*x*, *t*). In the *g*-gap case, assuming that exactly first *g*-gaps are not closed, the first three of those identities are(1.9)(1.10)(1.11)where prime denotes differentiation by *x*.

Therefore, in order to reconstruct the finite-gap potential with given values of boundaries of non-degenerate gaps (or, respectively, the solution of KdV equation) with finite-gap initial data, it is enough to find the sum of moving Dirichlet eigenvalues.

In the absence of gaps in the spectrum of *H*, the first of the above identities says that the potential *u*(*x*) is constant: *u*(*x*)=*u*(0).

In the one-gap case, assuming that only the *m*-th gap is not closed, we conclude that the motion of the *m*-th Dirichlet eigenvalue determines the potential up to a constant . In the same one-gap case, excluding *λ*_{j}(*x)* from the identity (1.7), and differentiating once, we see that any one-gap periodic potential represents the solution of the generalized stationary KdV equation(1.12)Therefore, at this point it becomes obvious that any one-gap periodic potential has the form , where *c*=−*v*/4 is an arbitrary constant and is a Weierstrass elliptic function. In particular case when *E*_{1}+*E*_{2}+*E*_{3}=0, the one-gap potential represents the solution of the stationary KdV equation. It is clear that, in general, the one-gap potential generates the simple wave solution *u*(*x*−*vt*) of the KdV equation, so that the boundaries *E*_{j},*j*=1, 2, 3 of non-degenerate gaps determine the velocity of its propagation, according to equation (1.12).

Quite similarly, in the finite-gap case, assuming that only *g* first gaps4 are not closed we get from (1.6) the formulashowing that the *τ*-dynamics of the Dirichlet eigenvalues, moving inside of non-degenerate gaps, completely determines the potential. It is also clear that the amplitude of oscillations of the potential, as well as the amplitude of the related solution of the KdV equation, does not exceed the sum of the lengths of the gaps. Therefore, for the narrow gaps, the amplitudes of the oscillations of the potential *u*(*x*, 0) and of the related solution *u*(*x*, *t*) of the KdV equation are always small.

In general, the trace identities mean that any finite *g*-gap periodic potential satisfies some nonlinear ordinary differential equation of the order 2*g*, which can be proved in a same way, excluding the *g* ‘moving’ Dirichlet eigenvalues *λ*_{j}(*x*)−*s* (belongings to the closures of non-degenerate spectral gaps), from the first *g* trace identities.

I was at this point at the end of 1973,5 when L. D. Faddeev told me at the beginning of January 1974 that in Moscow S. P. Novikov had discovered the importance of the finite-gap potentials for solving the periodic KdV problem and was working actively in this direction. I informed Faddeev about Akhiezer's work and gave him Akhiezer's article for Novikov (since he was going to Moscow for a few days) at the middle of January. Faddeev brought back the manuscript of the work of Novikov (1974), which contained many remarkable results, in a sense complementary to those I had discovered for myself up to that moment.

First, Novikov came to the idea of importance of the finite-gap potentials while looking for the spectral interpretation of the simple wave solution. This solutionwhere (*x*) is a Weierstrass elliptic function, is obtained by the direct substitution of the anzatz *u*=*u*(*x*−*vt*) in (1.2) and leads after two quadratures to a problem of inversion of the elliptic integral, which is naturally solved by means of the Weierstrass function. Next, he learnt from Arnold, as he explained to me later, that there existed a work by Ince (1940*a*,*b*), where it was shown (see also Erdélyi *et al*. (1955) where the results of Ince are reproduced) that the spectrum of the 1D Schrödinger equation with Lamé periodic potential contains exactly *n* gaps. Inspired by this information, combined with the well-known fact that, when the imaginary period tends to infinity, the simple wave periodic solution goes to the one-soliton solution, Novikov conjectured that the finite-gap potentials should be the natural periodic analogues of the reflectionless potentials, and their KdV dynamics should generate the natural extension of the multi-soliton solutions to the periodic case. He posed the same question as me but attacked the problem from a different direction. Namely, he proved in his first work (Novikov 1974) the following statement. Let *I*_{j} be the integrals of motion of the KdV equation corresponding to the periodic initial problem (see Novikov *et al*. (1984) for definitions and details). Then, any periodic solution of the higher stationary KdV equation(1.13)represents the *g*-gap periodic potential.6

Therefore, Novikov's theorem gave the nonlinear ordinary differential equations that provided the sufficient conditions for the potential to produce the *g* gaps in the spectrum, while my own approach, based on the Gelfand–Dickey trace formulae, provided the necessary conditions. Despite the different form and the different way to derive them, for *g*=1, 2, 3, the ordinary differential equations (coming from completely different points of view) appearing in Novikov's and my approaches were in fact the same. Therefore, it was natural to conjecture that they should coincide for any value of *g*, although at that moment the proof was still missing.

In Novikov (1974), it was also proved that the variety of solutions of equation (1.13) is invariant with respect to the actions of the KdV and higher KdV flows. The action of those flows on the variety of solutions of equation (1.13) is commutative. Novikov also proved that equation (1.13) represents a completely integrable Hamiltonian system with *n* degrees of freedom depending on *n*+1 parameters (*c*_{1}, …, *c*_{g}, *d*), whereby the collection of *g* commuting integrals of this system and all the parameters (*c*_{1}, …, *c*_{g},*d*) are expressed in terms of the 2*g*+1 boundaries of the spectral gaps.

Another important ingredient of Novikov's work was the study of the evolution of the monodromy matrix of the periodic Schrödinger operator with respect to both a change of the reference point and the KdV flow. The important by-product of this study was the discovery of the zero-curvature representation of the KdV equation with 2×2 matrices depending in a polynomial way on the spectral parameter.

Novikov also mentioned that, for generic choices of the constants *c*_{j}, the solutions of the higher stationary KdV equations (nowadays called Novikov's equations) should be almost periodic functions of *x*, tentatively representing the almost periodic *g*-gap potentials. The correct definition of the integrals of motion in the almost periodic case and the proof of their existence, somehow, was missing in Novikov (1974), mainly based on spectral theory corresponding to the strictly periodic case. The spectral theory for this kind of almost periodic problems was also not yet constructed at that moment.

In Novikov (1974), it was also pointed out that the reflectionless potentials in this picture correspond to degenerate separatrix solutions of equation (1.13), corresponding to the special choice of the constants *c*_{j} and *d*.7

The Novikov (1974) work was extremely important for further development. In his work, the Hamiltonian approach to the finite-gap integration theory was developed for the first time. It was very inspiring for many further results obtained by Novikov himself, and his school, and also by Gelfand and Dickey and many other researchers.

However, the principal questions, concerning the explicit description of the finite-gap potentials and related solutions of the KdV equation, still remained open.

It became clear soon that the attempts to solve explicitly the highly nonlinear ordinary differential equations describing the *g*-gap potentials (independently of the chosen way to generate them) by ‘brute force’ were hopeless and the centre of gravity of further efforts reverted to further understanding Akhiezer's work.

### (d) Return to Akhiezer. Solution of the KdV periodic Cauchy problem and theta-functional formulae

At this moment new actors came to the game: Boris Dubrovin (at that time a second year PhD student of Novikov at the Mathematical Faculty of Moscow University) and Sasha Its (at that time my fifth year student at the Faculty of Physics of St Petersburg University). The problem of finding a complete, efficient, description of the whole class of smooth real-valued *g*-gap periodic potentials and related solutions of the KdV equation was completely solved during the following three months independently, and with certain important variations, by Its and myself in St Petersburg and Dubrovin in Moscow (see Dubrovin 1975; Its & Matveev 1975*a*).

The main ingredients of the solution obtained by Its and myself were the following. First, using the spectral theory of the general one-dimensional periodic Schrödinger operators, we proved that for any *g*-gap periodic potential the pair of Bloch solutions *ψ*_{1,2}(*x*, *λ*) of equation (1.2) can be considered as a single function *ψ*(*x*, *P*), where *P*=(*w*, *λ*) varies on a hyperelliptic curve *Γ* of genus *g*,(1.14)

The related Riemann surface can be realized as a twofold covering of the complex plane. To construct this covering, it is enough to take two copies of the complex plane with cuts corresponding to the spectral intervalsand glue them along the cuts. After this, the cuts represent the transition lines from the upper sheet of the Riemann surface *Γ* to its lower sheet. The local parameters near the branch points *P*=(0, *E*_{j}), *P*∈*Γ* are . At ‘infinity’ the local parameter is . At the other points, we can take the projection *π*(*P*)=*λ* as a local parameter. In Its & Matveev (1975*a*,*b*), we proved that *ψ*(*x*, *P*) is a single-valued analytic function on *Γ* except for *g*+1 points. The only singularities of *ψ*(*x*, *P*) are *g* simple poles *P* (forming the non-special divisor ) whose projections on the complex plane are the Dirichlet eigenvalues: , lying in the closures of spectral gaps. In addition, the function *ψ*(*x*, *P*) has an essential singularity of exponential nature at infinity

The two Bloch solutions *ψ*_{1,2}(*x*, *λ*) can now be considered as the projections of *ψ*(*x*, *P*) to the upper and lower sheets of *Γ*.8

The *x*-dependent part of the product of two Bloch solutions, corresponding to the finite-gap periodic potential, is a polynomial *F*(*x*, *λ*) of *λ* of order equal to the number of the non-degenerate gaps. The inverse statement, saying that the existence of two solutions such that their product is a polynomial of *λ* with *x*-periodic coefficients means that the related potential is periodic with *g*-gaps where *g* is the order of the polynomial, is also true.

In order to prove the first part of this statement (see Its & Matveev (1975*a*) for the complete proof), it is enough to study the solution *ϕ*(*x*, *τ*, *λ*) of the Schrödinger equation with *T*-periodic *g*-gap potential *u*(*x*+*τ*) fixed by the boundary conditions . There are *g* Dirichlet eigenvalues *λ*_{j}(*τ*) depending non-trivially on *τ*. They are the simple roots of the entire function *ϕ*(*T*, *τ*, *λ*). It was shown in Its & Matveev (1975*a*) that this satisfies the third-order differential equationwhere9From the other side, it is well known (first mentioned by Hermite) that, for any two solutions *f*_{1,2}(*x*,λ) of the Schrödinger equation with any potential *u*(*x*), their product *y*(*x*, *λ*)=*f*_{1}*f*_{2} satisfies the same equation as above:(1.15)Thus, in the periodic case, the product of two Bloch solutions, being the only *x*-periodic solution of equation (1.15), is proportional to *ϕ*(*T*, *x*, *λ*), and hence to the polynomial *F*(*λ*),

This fact has a number of significant consequences. First, it leads immediately (Its & Matveev 1975*a*) to a very compact description of the ordinary differential equations satisfied by the finite-gap potentials in terms of the recursion operator *Λ*(*x*): all *g*-gap periodic potentials are the solutions of the nonlinear differential equation(1.16)

Next, it leads to the simplest (a few lines) proof of the fact that the Lamé potential has exactly *g* gaps (Its & Matveev1975*b*).

But the most important is that the same polynomial structure of the product of two Bloch solutions allows (see Its & Matveev (1975*a*,*b*) for details) to establish the analytic properties of *ψ*(*x*, *P*) described above, and to reduce the calculation of all the symmetric functions of *λ*_{j}(*x*) (and, hence, due to the trace formula (1.9), of all the finite-gap potentials) to the solution of the Jacobi inversion problem on a hyperelliptic surface (1.14). The solution of the Jacobi inversion problem allows (Its & Matveev 1975*a*) one to express the *g*-gap potentials and the solutions of the KdV equation corresponding to the finite-gap initial data by means of Riemann *g*-dimensional theta functions (see the formula (1.23) below).

Letbe the normalized Abelian differentials, associated to some canonical basis of cycles *a*_{j},*b*_{j} on *Γ* (see Its & Matveev (1975*b*, 1976) for details), and let *B* be the related matrix of *b*-periods, that is(1.17)

The Abel map of the divisor *D* and the vector of the Riemann constants, ** K**, associated to

*Γ*are defined by the formulae

Let us consider *λ*_{j}(*x*, *t*), defined in §1*c* (and hence belonging to the closures of the spectral gaps), as the projection of the points coinciding with *P*_{j} when *x*=0, *t*=0, i.e. . Then it can be proved that the Abel map of is a linear function of *x* and *t*(1.18)

Of course, the last equality should be understood as modulo periods of the differentials d*U*_{j}.

The problem of reconstructing *D*(*x*, *t*) from *D*(0, 0) is known as the Jacobi inversion problem. Equation (1.18) shows that the KdV flow is described by the straight line on the Jacobian of the curve *Γ*. There are different ways to derive equation (1.18). For *t*=0, it was derived in Its & Matveev (1975*a*,*b*) extending Akhiezer arguments, i.e. making use of the single valuedness of *ψ*(*x*, *P*) represented in terms of Abelian integrals of the second and third kind. At the same time, Dubrovin derived it in another way, also for the case *t*≠0. Indeed, Dubrovin was the first to obtain equation (1.18) for *t*≠0. The main tool in his derivation was the following system of autonomic differential equations for *λ*_{j}(*x*, *t*), nowadays known as Dubrovin's equations:(1.19)where *γ*(*j*)=±1. The evolution of *λ*_{j}(*x*) with respect to the KdV flow is described in a similar way by the following Dubrovin equation:10(1.20)

The data describe the non-special divisor and the system (1.19) and (1.20) describes its evolution, , with respect to the space translations of the potential and to the KdV flow. The Abel map linearizes Dubrovin's equations. Conversely, Dubrovin's equations follow from equation (1.18): it is enough to differentiate the r.h.s. of equation (1.18), and to use identities following from the Lagrange interpolation formula, in order to obtain the system (1.19) and (1.20).

All the symmetric functions of *λ*_{j}(*x*) might be found explicitly from the solution of the Jacobi inversion problem. It follows from the trace formula (9) that the potential *u*(*x*) (or more generally its evolution with respect to the KdV flow *u*(*x,t*)) up to a factor −2 and up to adding the sum of the spectral boundaries is the simplest symmetric function of *λ*_{j}(*x*), their sum. To calculate this sum, and more generally the power sums,the Riemann theta functions can be used. The *g*-dimensional theta function corresponding to the given g×g matrix *B* is defined by the formula

Below, the matrix *B* is defined by equation (1.17) that guarantees it is symmetric and its imaginary part is positively defined. In this case, is an entire function of .11 Let be the variety with oriented boundary obtained by cutting *Γ* along the cycles *a*_{k},*b*_{k}. The calculation of the integralleads to the following identities:(1.21)

Calculation of the residues (Its & Matveev 1975*b*) in the right-hand side of identity (1.21) (which takes in to account the local parameter in the vicinity of the infinity point on *Γ* is ), leads, for *m*=1, 2, to the following two identities:

(1.22)

Comparing the first of these identities with the first of the formulae (1.9), we obtain an explicit description of the *g*-gap periodic potentials and the related solutions of the KdV equation:12(1.23)where the constant *C* and vectors were defined in equation (1.18). Formula (1.23) for the finite-gap solutions of the KdV equation was first obtained in the works (Its & Matveev 1975*a*,*b*).13

Formula (1.23), with *t*=0, reconstructs the finite-gap potential from the boundaries *E*_{j} of the continuous spectrum (i.e. the hyperelliptic curve *Γ* defined in (1.14)) and the divisor of poles of *ψ*. This divisor fixes the eigenvalues of the Dirichlet problem at *t*=0, , belonging to the closures of non-degenerate gaps.

It is important to mention that the spectral approach was explaining how to choose the parameters in formula (1.23) in order to get all real-valued non-singular *x*-periodic finite-gap potentials and the related solutions of the KdV equation; the branch points *E*_{j} should be distinct real numbers and the projections *λ*_{j}(0, 0) of the points *P*_{j} must belong to the closures of the non-degenerate spectral gaps [*E*_{2j},*E*_{2j+1}].

Formula (1.23), representing the finite-gap solutions of the KdV equation as a second derivative of the Riemann theta function of the hyperelliptic Riemann surface of the curve *Γ*, was very typical; the same kinds of formulae describe the finite-gap solutions of the whole KdV hierarchy and of the KP hierarchies (shown later, respectively, by Dubrovin and Krichever). In particular, the action of the higher KdV flows on *u*(*x*, *t*) boils down to adding vector-valued functions depending linearly on the ‘higher’ time variables (with vector coefficients, depending only on *Γ*), in the argument of theta function in formula (1.23).

### (e) Theta-functional solution of the Schrödinger equation with finite-gap potential: the Alexander Its formula

The next important step was stimulated by the same work (Akhiezer 1961) containing the explicit formula for the *ψ* function in the case of a one-gap Lamé potential. It was natural to conjecture that an explicit solution of the Schrödinger equation with arbitrary genus *g* finite-gap potential should exist. This problem was solved by Its at the beginning of 1975, at first for the periodic case.

Let *Ω*(*P*) be the normalized Abelian integral of the second kind, uniquely defined (modulo *b*-periods) by the conditions

The Abelian integral *Ω*(*P*) can be constructed explicitlywhere *Q*(*λ*) was defined in (1.14). The coefficients *r*_{k} are determined from the solution of the linear system arising from the above normalization condition.

Then, the following formula, found by Its, describes all solutions of the Schrödinger equation:with potential (1.23)(1.24)

Despite the fact that the Abelian integrals participating in this formula are multi-valued functions on the curve *Γ*, the whole thing is a single-valued function of *P* on *Γ*.14 Its projections on the complex plane form a fundamental system of solutions, *ψ*_{1,2}(*x*, *P*) (except at the points *P*=(0, *E*_{j}) to the Schrödinger equation, and their Wronskian is given by the formula

For the proof see Its & Matveev (1976). When *P*→∞,and when *t*=0 its poles coincide with *P*_{j}(0,0). If *t*≠0 the poles of *ψ* coincide with *P*_{j}(0, *t*).15

### (f) From the periodic to the almost-periodic case

For a generic choice of parameters *E*_{j} defining *Γ*, the right-hand side of formula (1.23) is obviously not periodic, but almost-periodic and, in general, is a complex-valued function of real the variables *x* and *t*. Therefore, it was natural to suppose that the right-hand side of formula (1.23) remained the solution of the KdV equation, but the periodic spectral theory could not be applied to prove it.

Soon we obtained two different rigorous proofs of the fact that the formula for the solutions of the KdV equation, which we obtained for the periodic case, can be extended to the case when *Γ* is an arbitrary non-singular hyperelliptic curve of genus *g*, and *D* is an arbitrary non-special divisor on *Γ*, deg(*D*)=*g*. We proved that formula (1.24) for the solution of the Schrödinger equation is valid, not only for the periodic case, with smooth real potential, but also for the same generic data *Γ*.

It was, probably, the first case when the large class of almost-periodic spectral problems was explicitly solved.16

The first proof, which I proposed, used a comparison of ‘linear’ and ‘nonlinear’ trace formulae. Amazingly, in this proof the Schrödinger equation was completely eliminated. The heart of the proof is to establish two relationswhich can be called a ‘linear trace formula’ (following immediately from the comparison of the right-hand side of the second identity (1.22) with (1.23)), and the identity which I called a ‘nonlinear trace formula’

The comparison of the right-hand sides of these two identities proves that the function *u*(*x*,*t*), defined by the formularepresents a solution of the KdV equation.

In this formula, the *λ*(*x*, *t*) are defined as the projections on the complex plane of the solutions *P*_{j}(*x*, *t*) of the Jacobi problem (1.18). The initial divisor *D*(0, 0) was supposed to be any non-special divisor of degree *g* on *Γ*, and *Γ* was supposed to be any non-special hyperelliptic curve of the form (1.14). Quite obviously, an explicit formula for *u*(*x*, *t*) might be proved as before. In order to prove the nonlinear trace formula it is enough to write the second Dubrovin equation (which follows from equation (1.18)) as and then perform the summation over *j* from 1 to *g*. The details can be found in Its & Matveev (1976).

The second proof of the fact that the same formulae are valid for a generic hyperelliptic curve and non-special divisor on it amounts to checking that the associated formula (1.24) for the *ψ*-function is valid also in a generic situation, that is, for the potential *u*(*x*, *t*) constructed by the same formula, starting from generic data (*Γ*, *D*). This produces another kind of nonlinear trace formula (Matveev 1975), allowing a check that *u*(*x*, *t*) is again the solution of the KdV equation (Its & Matveev 1976). All these results were first reported in my talk at the Petrovsky seminar in Moscow in April 1975, and published in the last issue of 1975 of the Russian journal *Uspekhi Math. Nauk* (Matveev 1975) known in English translation as *Russian Math. Surveys*. Unfortunately, it was the last issue of the journal for which part of the content reproducing the communications of Petrovsky seminar was not included in the English translation. Its content was reproduced later in a review article (Dubrovin *et al*. 1976), which we wrote with Dubrovin and Novikov, and in a more detailed version in Its & Matveev (1976).

In the latter article, see also Matveev (1976), it was also shown how to get explicit formulae for multi-soliton solutions from formula (1.23), by contracting the branch points of the spectral curve in such a way that . In this limit the diagonal elements of the matrix *iB*, where *B* is the matrix of *b*-periods, tend to −∞. The limit of the non-diagonal elements of *B* is given by the formulain which it is immediate to recognize the asymptotic phase shifts characterizing the large time asymptotics of the multi-soliton solutions of the KdV equation. Owing to the presence of the elements of the matrix *B*, both in the definition of the theta function and in the components of the vector ** l**, in such a limit the theta function in formula (1.23) transforms into a finite sum where the summation is taken over the finite number of

*g*-dimensional vectors . The form of the multi-soliton solution obtained by this passage to the limit reproduces the well-known Hirota representation for the multi-soliton solutions. The related initial data for this solution is a reflectionless potential, having discrete negative eigenvalues at the points α

_{j}. This gives a precise meaning to the claim that finite-gap potentials represent the multi-periodic generalization of the reflectionless potentials, and that the finite-gap solutions of the KdV equation are the natural almost periodic extensions of multi-soliton solutions.

The partial degeneration in the formulae of the finite-gap integration might be described quite differently. It is possible to perform the degeneration in the generic formulae, or to use the Darboux dressings (Matveev 1979*a*) or to use an axiomatic description of the degenerate Baker–Akhiezer function (Krichever 1975). These three approaches lead to different descriptions of the same solutions.

Summarizing, the explicit construction of the *ψ*-function from the algebro-geometric data (*Γ*, *D*), and its application to the integration of the KdV equation, free from the restriction on the spectral data, imposed by the condition of smoothness, reality and periodicity of the potential, was completely worked out in the beginning of 1975.

Getting an explicit solution for the Schrödinger equation with finite-gap potential was an important by-product of this activity, having the same value as the solution of the periodic KdV initial problem. For the solid-state physics community, this provided large classes of periodic and almost periodic potentials, for which the related Schrödinger equation was explicitly solvable. This class is much better adapted to approximating real physical situations than the models of Krönig–Penny type.

The Its formula (1.24) later found many different applications to finite-dimensional dynamical systems (geodesics on an *N*-dimensional ellipsoid, coupled nonlinear oscillators, etc.) in the works by Veselov (1980) and Knörrer (1982) and some other researchers.

The Hamiltonian aspects of the finite-gap integration were developed in a series of beautiful papers by Novikov, Dubrobin, Bogoyavlenskij and also by Gelfand and Dickey. See, for instance, the recollection of review articles from *Russian Math. Surveys* (Wilson 1981).

### (g) Passage to *2+1* integral systems. KP and Krichever's work

The next very important development of the method of finite-gap integration was the passage to the integration of 2+1 KP-like systems, realized by Krichever in 1976. He extended our method of using the *ψ*-function for integrating the KdV equation in two senses. First, he observed in Krichever (1975) that the non-stationary asymptotic conditionwhere and is a local parameter at infinity, specifying the solution of the Lax systemhas an important advantage. Namely, for this asymptotic condition the divisor of poles of *ψ* becomes *t*-independent. Then it is easier to prove that *ψ*(*x*, *t*) (reconstructed from the same kind of data as in our approach) solves the Lax system with coefficients expressed via the first terms of its asymptotic expansion at infinity. We used the stationary normalization (see the text above) of *ψ*(*x*, *t*, *P*), which is less comfortable for working with evolution equations. Our way of checking that *ψ*(*x*, *P*), constructed via the Its formula, satisfies the Schrödinger equation with potential corresponding to generic algebro-geometric data was considerably longer (Its & Matveev 1976), although the main idea was the same.

The next remarkable observation of Krichever was that replacing the first equation in the Lax system by the evolution equationone can solve it via the same kind of formula as in the previous case, replacing the hyperelliptic curve in the previous construction by any non-singular algebraic curve *Γ* of genus *g*. The coefficient *u*(*x*, *t*, *y*) of the related Lax system is the solution of the Kadomtcev–Petviashvily equation and, as in the KdV case, this solution can be obtained easily from the second term of the asymptotics of *ψ* at the marked point of the Riemann surface replacing the infinity point of the KdV case. The data, determining *ψ*(*x*, *t*, *y*, *P*), are the non-special pole divisor *D* (deg*D*=*g*), the marked point *P*_{0}, and the asymptotic condition at the marked point(1.25)where *K*^{−1} is a local parameter at the vicinity of the marked point *P*_{0}. Quite similarly to the Schrödinger–KdV case, *ψ*(*x*, *y*, *t*, *P*) can be reconstructed explicitly from the algebro-geometrical data (*Γ*, *D*, *P*_{0}) in terms of theta functions and normalized Abelian integrals of the second kind, with simple poles at the point *P*_{0}, and principal parts equal to *k*, *k*^{2} and *k*^{3}, respectively:(1.26)where ** d** is chosen in such a way that the poles of

*ψ*coincide with the divisor

*D*, and

**,**

*p***,**

*g***are the vectors of**

*r**b*-periods of the Abelian integrals divided by the factor 2

*iπ*.

Krichever noticed that in order to check that *ψ*(*x*, *t*, *y*, *P*) is a common solution of the systemit is enough to check that *ψ*(*x*, *t*, *y*, *P*) solves it asymptotically,17 when , which can be achieved by taking

Compatibility of this system implies that *ψ*(*x*, *y*, *t*) is solution to the KP equation(1.27)

Combining these arguments, with an explicit representation for *ψ*, Krichever derived the following formula for the solutions of equation (1.27):(1.28)When *w*_{2}(*P*) is a meromorphic function (which automatically means that the underlying curve is hyperelliptic and the marked point *P*_{0} coincides with one of its branch points), *u*(*x*, *y*, *t*) loses its dependence on *y* and transforms into the solution of the KdV equation (modulo obvious rescaling with respect to the definition (4) of the KdV equation) as discussed above. Quite similarly, if *w*_{3}(*P*) is a meromorphic function on *Γ*, which means that *Γ* is a trigonal curve, and *P*_{0} is one of its Weierstrass points, the solution (1.28) becomes *t*-independent and reduces to the solution of the Boussinesq equation.

Of course, this ‘non-spectral’ approach produces in general complex-valued and singular solutions. Isolating the smooth real-valued solutions needed separate non-trivial work, which was mainly done by Dubrovin & Natanzon (1989). The same scheme was applied by Krichever to integrate the whole KP (or Zakharov–Shabat) scalar hierarchy of equations obtained as compatibility conditions of the linear evolution equations, with higher order differential operators (of mutually prime order) in the *x* variable on the right-hand side. The only modification needed was to replace *k*^{2} and *k*^{3} in the asymptotic condition (1.25) by arbitrary polynomials of *k*.

Owing to knowledge of the Baker–Akhiezer functions, it is possible to obtain larger families of solutions to the KP equation, also expressed by means of the Riemann theta functions, by applying the dressing formulae (Matveev 1979*a*; Matveev & Salle 1991).18 Namely, for any *m* densities , defined on the algebraic curve *Γ*, and given solution (1.28), *u*(*x*, *y*, *t*) of the KP equation (1.28), the formula(1.29)(1.30)(where *W* means the Wronskian determinant and *ψ*(*x*, y, *t*, *P*) is a Baker–Akhiezer function associated with *u*(*x*, *y*, *t*)) gives a new solution of the same KP equation depending on *m* functional parameters ρ_{j}(*P*). The obtained family of solutions is very large and is still not completely studied. Its connection with the so-called higher rank solutions, introduced by Krichever & Novikov (1980) and Krichever & Novikov (2003) for the discrete systems, is not completely understood, to my knowledge, until now.

Using equation (1.29), we can assume that *u*=*u*(*x*, *t*) is a given solution of the KdV equation, and the related Baker–Akhiezer function depends on *y* exponentially. In this way, we arrive to generate the solutions of the KP equation starting from a given solution to the KdV equation. A similar remark applies also to the generation of the solutions of the KP equation from the solutions of the Boussinesq equation, given the assumption that *u*=*u*(*x*, *y*) and the dependence of *ψ*(*x*, *y*, *t*, *P*) on *t* is purely exponential, i.e. ** r**=0.19 Further results concerning the periodic closures of the chain of Darboux transformations can be found in Takasaki (2003).

It is interesting to mention that for the cylindrical KP equation, also known as the Johnson equationthere exists a family of theta-functional solutions that are still periodic or almost periodic functions of *x* and *t*, but no longer periodic or almost periodic with respect to *y* (Lipowskij *et al*. 1986; Matveev & Salle 1991), which completely changes the qualitative properties of the solutions with respect to the KP case. The interested reader can find many more details and graphic images of some most interesting solutions describing complicated wave interaction with families of crossed parabolic fronts in a recent article (Klein *et al*. in press), where the gauge equivalence of the related Lax pairs is also explained.

The natural question concerning whether one can find an explicit solution to the Johnson equation that tends to some periodic or almost periodic solution to the KdV equation, to my knowledge, remains open.

### (h) Dirac and Baker in Cambridge

Explicit formulae for Baker–Akhiezer functions, and explicit statements similar to formulae (1.23)–(1.28), were not known before, although concerning the solutions of the linear problems in the strategic point of view, saying that it is possible to reconstruct certain kinds of single-valued functions from the algebro-geometric data, was mentioned by Baker (1928) as a comment to remarkable works by Burchnall & Chaundy (1923, 1928).20 Baker noticed that these functions, which can be considered as higher genus analogues of exponentials, can be used for solving linear differential equations with coefficients that can be calculated in terms of coefficients of the asymptotic expansion of these functions in the vicinity of some marked point on the algebraic curve. Somehow, as in the case of Jules Drach, his work was not widely known and in contrast to Akhiezer's work was rediscovered too late to play some role in the creation of the finite-gap integration method, as happened also with the works of Burchnall & Chaundy.21

It is not widely known that around 1926–1927 Dirac was regularly meeting Baker to participate in his scientific tea-parties at Cambridge. Dirac claims, in his inspiring article (Dirac 1977), that projective geometry, a main topic at the Baker tea-parties, was very important for his vision concerning the role of beauty in the mathematical description of reality. He also explains that the projective geometry viewpoint was often hidden behind his own quantum mechanical discoveries owing to the necessity of adapting the presentation to the mathematical background of the physical community of that time. The first scientific communication made by Dirac in Cambridge concerned the proofs of some theorems in Euclidean geometry by projective methods. But soon Dirac left Baker's seminar. He also never attended his lectures ‘since Baker was doing geometry’ and Dirac was fully involved in the development of quantum mechanics. The period during which Baker wrote his article (Baker 1928) was certainly close to the time when Dirac, at first sceptical about Schödinger's formulation of quantum mechanics, became interested in it. He certainly ignored Baker's activity on differential equations and Abelian functions; otherwise the story of finite-gap integration might have been very different.

### (i) Matrix differential operators, zero-curvature representations and finite-gap integration

Parallel developments were coming mainly from several directions. First, it was connected with the extension of the finite-gap integration method to the cases connected with Lax pairs containing matrix differential operators or, more generally, zero-curvature representations with rational or elliptic dependence on the spectral parameter. This extension of theory was relevant for constructing the finite-gap solutions for physically important models like NLS, sine-Gordon equation, Landau–Lifschitz equation (see Belokolos *et al*. (1994) for detailed exposition and further references), vacuum axially symmetric Einstein equations of general relativity (Korotkin 1988; Korotkin & Matveev 1988, 1990, 1999, 2000; Frauendiener & Klein 2001, 2004) corresponding to the special dependence of the moduli of the underlying spectral curve on space and time variables, and for many other models describing quite interesting physical situations.

For developing the general algebro-geometrical point of view on the models connected with matrix-differential Lax pairs, the analysis of the NLS model (first solved in the ‘finite-gap’ way by Its (1976) in the ‘repulsive’ case, and by Its & Kotlyarov (1976) in the attractive case) and the sine-Gordon model (Kozel & Kotlyarov 1976) was certainly very important.22 Analysis of the NLS model and the sine-Gordon model gave the precise idea about the difference between analytical properties of the vector-valued Baker–Akhiezer functions, and their scalar counterparts, by reason of introducing the very first examples of the multi-points Baker–Akhiezer functions. This together with the important work by Dubrovin (1976) on general matrix finite-gap differential operators built a base for the development of the matrix version of Krichever's scheme. In fact, the first explanation of how it is possible to integrate the NLS equation, the modified KdV equation and the sine-Gordon model, in the spirit of Krichever's scheme, was presented in my Polish lectures (Matveev 1976), based in part on some unpublished works by Its.

Soon after, Krichever (1976*b*) generalized his approach to generic hierarchies of the nonlinear Zakharov–Shabat type equations connected with matrix differential operators. This ‘generic integration’ scheme, though very impressive, did not include many of the physically interesting models such as the sine-Gordon equation, the modified NLS equation (Its & Matveev 1984), the Kaup–Boussinesq equation (Matveev & Yavor 1979), the Landau–Lifshitz equation (Bobenko 1985), the Ernst–Einstein equation of general relativity (Korotkin & Matveev 1990; Korotkin 1996) and many other models where very subtle work taking into account the necessary reductions on the parameters has been done.

### (j) Discrete Toda-like models, polynomials orthogonal on a system of intervals and finite-gap integration

The background to solving the discrete periodic problems associated with the Toda-like or discrete KdV-like integrable systems was, in principle, provided by the article by Akhiezer (1960), which, for some mysterious reasons, is cited in the literature much less often than Akhiezer (1961). In Akhiezer (1960), the construction of the system of polynomials orthogonal on the finite number of intervals was reduced to the solution of the Jacobi inversion problem on the hyperelliptic curve, and the analytical properties of the related solution of the discrete Schrödinger equation were also established, although explicit theta-functional formulae for those polynomials were unknown at that time.23 Among the old works written before the creation of the finite-gap integration method, one should mention also the result by Naiman (1962), see also Glazman (1963), describing the spectrum of generalized complex-valued periodic Jacobi matrices and the discrete analogue of the theorem by Burchnell–Chaundy, concerning the description of commutative rings of periodic difference operators. One of the obvious conclusions suggested by these results was the fact that discrete periodic difference operators have always a finite-gap spectrum formed in general by a finite number of arcs in the complex plane. In the self-adjoint situation, these arcs become finite closed intervals of the real line. Therefore, for discrete systems the finite-gap structure of the spectrum corresponds to the generic periodic case.

During the period 1974–1976, many important aspects of the finite-gap integration method for the discrete Toda-like models were worked out (Kac & van Moerbeke 1975; Date & Tanaka 1976; Dubrovin *et al*. 1976; Flaschka & McLaughlin 1976). It was clear from the spectral theory of the Jacobi matrices that for the discrete models the finite-gap solutions contain all solutions periodic with respect to the discrete (lattice) space variable. Again, generic solutions are almost periodic but the isolation of the purely periodic solutions from the generic case is easy, and efficient, contrary to what we have in the continuous models. For instance, in the Toda lattice case, the condition of *n*-periodicity suggested by the spectral theory of difference Shrödinger operator is that the related hyperelliptic curve should be defined by the equationwhere *P* is a polynomial in *z*. Explicit finite-gap solutions of the Toda lattice can be obtained along the same lines as in the KdV case; after the works by Kac & van Moerbeke (1975), Dubrovin *et al*. (1976) and Date & Tanaka (1976), it was finalized by Krichever (1978). It is also necessary to mention here the important work by Flaschka & McLaughlin (1976) where the action-angle variables for the periodic Toda lattice were constructed. Many additional important results, concerning the finite-gap integration for discrete systems can be found in Mumford & van Moerbeke (1979), Krichever (1981) and Krichever & Novikov (2003).

## 2. Some further developments of finite-gap integration after 1976

### (a) Reductions to the lower genera, elliptic solitons, effectivization and Schottky problem

From the very beginning of the appearance of the theta-functional formulae, such as (1.23) for the solutions of the KdV equation, there had been a discussion about their efficiency. I remember well that after the appearance of formulae (Its & Matveev 1975*a*) for the solution of the KdV equation, Novikov's comment was that it was of extreme level of complexity and not efficient for calculations. Nowadays, we understand that formulae of the type (Its & Matveev 1975*a*,*b*; Krichever 1976*a*) are perfectly adapted for performing numerical calculations and for producing the plots of solutions (Bobenko & Bordag 1987, 1989). But, at that time the authority of Novikov pushed people to look for different answers to this challenge.

First, the idea pursued by many people was to look at special initial data, such as the 2-gap or *g*-gap Lamé potential. In particular, Airault *et al*. (1977) had shown that the common action of the KdV and higher KdV flows on the Lamé potentials produced the whole variety of *g*-gap solutions elliptic in the *x* variable. By contrast, the *t* dependence of the same solutions was rather complicated and the evolution of the poles *x*_{j}(*t*) was isomorphic to the trajectories of the classical multi-particle higher Calogero–Moser-like systems. So, despite the appearance of the very interesting connections with finite-dimensional integrable systems there was no gain in efficiency of the solutions. Similar work in the KP case was done by Krichever (1980). In his work, the link between the elliptic in *x* solutions of the KP equation and the trajectories of the elliptic Calogero–Moser-like systems realized by *y*-dependence of poles *x*_{j}(*y*) was discovered.

The solutions of the KdV equation elliptic in the *t* variable and similar questions for other nonlinear integrable equations, including NLS, the Toda lattice and the sine-Gordon equation, were obtained by Smirnov (1994). A new approach to isolating the elliptic finite-gap solutions, based on extensive use of Picard's theorem, was recently proposed by Gesztezy and Weikard (Gesztesy & Holden 2003).

The next idea, revealing many beautiful effects, was to choose as spectral curves compact Riemann surfaces possessing the non-trivial group of birational isomorphisms or having the appropriate covering structure. In many cases, it was leading to a decomposition of the Riemann theta function of genus *g* to a sum of a finite number of terms representing products of a finite number of one-dimensional theta functions, keeping the linear dependence of their arguments with respect to space and time variables. In such situations, it is often possible (Babich *et al*. 1983, 1985) to calculate the matrix of the *B*-periods explicitly, taking into account only symmetry arguments. For some recent, interesting, applications to quasi-periodic vortex structures in two-dimensional flows of incompressible inviscid fluids, see the recent article by Babich & Bordag (2005).24 Another approach to the reductions of the special finite-gap solutions to one-dimensional theta functions exploring the Weierstrass programme of the reductions of the Abelian integrals to elliptic integrals was proposed by Belokolos and Enol'skii (see Belokolos *et al*. (1986) for the related results and references).

Of course, solutions obtained in the cases where there exists a non-trivial group of conformal automorphisms of the spectral curve are much simpler for numerical calculations with respect to the generic situation. The first visualization of the special genus 3 finite-gap solutions of the sine-Gordon equation using the aforementioned symmetry reduction approach was realized in the article by Babich & Bordag (1985), using the results of Babich *et al*. (1985). I think at that time it was the very first work where plots of finite-gap solutions of genus greater than 2 were obtained.

The interested reader can consult, for instance, the book by Belokolos *et al*. (1994) for further examples and references concerning applications of the symmetry reduction approach. Among other things, this activity explained how to obtain the Lamé equation as a reduction of the general finite-gap formulae. Some more complicated reductions of the generic finite-gap potentials (1.23) representing the four-parametric generalization of the Lamé potentials were found by Treibich (1989) and Verdier. They called the related solutions of the KdV equations elliptic solitons (in my opinion, a not very much justified name, which is, however, frequently used). Later, it was realized that the same equation was first found and solved by Darboux (1982*b*, 1894) using an absolutely different approach. For further generalizations and more references see Smirnov (2002). More recently, the same potentials were rediscovered within the framework of a SUSY approach in works by Khare & Sukhatme (e.g. Khare & Sukhatme (2005) and the references therein). Additional material concerning this topic can be found in Belokolos *et al*. (1986, 1994), Gesztesy & Holden (2003) and Matveev & Smirnov (1990, 1993, 2006). In the last article, among other results (including the proof of the Baxter *et al*. (1988) normalization conjecture, see also Baxter (2002)), it was shown that the Riemann theta functions corresponding to (non-hyperelliptic) Ferma curves, parameterizing the Boltzmann weights, corresponding to the solvable chiral Potts model of statistical physics (the first quantum model with spectral parameter ‘living’ on the curves of genus greater than 1), can be split into a finite sum of finite products of elliptic theta-functions.

Remarkably enough, the discussion of the efficiency of the general theta-functional formulae also had a very important consequence for algebraic geometry. This emerged in the course of pushing the original ‘effectivization programme’ for finite-gap integration, proposed by Dubrovin and Novikov, which is presented, for instance, in the review article by Dubrovin (1981). The idea there was to consider the finite-gap-like formula for the solution, taking, instead of the matrix of *B*-periods, any symmetric matrix with positively defined imaginary part, not reducible to a quasi-diagonal block structure by the Siegel transformations, and then directly substituting the so-obtained anzatz into the relevant nonlinear equation, and then solving the appearing dispersion equations for the vector coefficients in front of *x, y* and *t* in the arguments of the theta functions.

This programme fails already for genus 3 for 1+1 systems due to the existence of the non-hyperelliptic curves in genus 3 and the absence of efficient criteria to distinguish the hyperelliptic 3×3 matrices of *b*-periods from those associated with non-hyperelliptic curves. For 2+1 KP-like systems where (as we know from Krichever) all the finite genus curves are admissible, the same programme works in the genus 3 case also, but it fails again starting from genus 4, owing to the appearance of new restrictions on the matrix of *b*-periods known as the *Schottky relations*. There are no efficient tools to determine if the given *B*-matrix (satisfying of course the necessary conditions of being symmetric with positively defined imaginary part and irreducible to the block form by the Siegel transformations) is the matrix of *b*-periods of some compact Riemann surface. Indeed, this question constitutes the classical Schottky problem. This inspired Novikov to formulate the following conjecture: *‘The given matrix B (satisfying the aforementioned necessary conditions), is a matrix of B-periods of an algebraic curve if and only if the right-hand side of* *(1.28)* *is the solution of the KP equation.’* This conjecture was proved by Shiota (1986), who thus obtained what nowadays is acknowledged as a solution of the Schottky problem. For a different proof of Novikov's conjecture see Arbarello & de Concini (1987).

Shiota's proof of Novikov's conjecture revealed a new and extremely exciting side of algebro-geometric integration. It showed that soliton theory not only makes use of algebraic geometry but can actually contribute to it. It showed that integrable PDEs, such as the KP equation, provide an adequate tool for studying several highly non-trivial questions in the theory of algebraic curves.

This direction has been pursued further recently in a series of papers by Buchstaber *et al*. (1997), see for example their review article, and has led in particular to a revival of the beautiful classical theory of multi-dimensional Kleinian *ζ* and -functions.

With all its theoretical importance, the solution of the Schottky problem offered by Shiota's proof of Novikov's conjecture does not provide an efficient algorithm for selecting matrices of *b*-periods of algebraic curves out of generic symmetric irreducible matrices with positive imaginary parts. Indeed, one can make sure that a given theta-functional anzatz satisfies the KP equation only up to some numerical accuracy. It is efficient only to check whether a given matrix *is not* a matrix of the *b*-periods of some Riemann surface. In other words, the efficient solution of the Schottky problem is still missing and it is not clear whether it exists at all. In some special cases, however, it is possible to find the matrix of the *b*-periods explicitly (see Babich *et al*. (1983, 1985), Belokolos *et al*. (1994) and the references given there).

By contrast, the computation of the finite-gap solutions became much more efficient than had been expected initially. In this respect, the positive evolution was approximately the following.

First, Bobenko tried to use the Poincaré ideas to compute theta functions and Abelian integrals via general Poincaré theta series. The result was an article (Bobenko 1983), where a new representation of the same finite-gap solutions of the KP and KdV equations in terms of the Poincaré theta-series was obtained, but it was not helpful for performing numerical calculations.

Next, I remembered that Burnside (1890, 1892) used special classes of automorphic functions to solve some concrete problems of hydrodynamics (he considered the two-dimensional flows of ideal liquid in the presence of *N* cylindrical obstacles). The related compact Riemann surfaces were parameterized by classical Schottky groups. The elements of the matrix of *b*-periods and the normalized Abelian integrals were represented there in the form of a convergent Poincaré theta series of dimension 2. At that moment, I had only a copy of the relevant chapter of the book (Baker 1897) that reproduced Burnside's results,25 which I gave to Bobenko, suggesting he look at these works to see if they could give something better than his first ‘automorphic attempt’. It was really the case. Soon after, Bobenko proved, using these works and some other results, that it is possible to parameterize all real non-singular solutions of the KP-II equation by the classical Schottky groups.26 The related numerical algorithms were developed in Bobenko & Bordag (1987, 1989) and Belokolos *et al*. (1994). In particular, the plots of the genus 4 solutions of the KP-II equation using the classical Schottky group parametrization were first obtained in Bobenko & Bordag (1989). From that moment, it became clear that there is no problem in calculating the solutions of the KP equation of any genus and to producing related plots of the solutions.

Later it was understood that the original spectral parametrization for the KdV-like integrable equations works perfectly well for doing the numerics. Finally, the initial scepticism concerning the finite-gap formulae as a tool for doing numerics completely disappeared.

It also became clear later that the automorphic approach is not actually the best and in many cases direct spectral parametrization is better adapted for doing numerics and producing plots. Nowadays, even in much more complicated situations where the moduli of the spectral curve depend on space and time variables (e.g. this is the case of vacuum axially symmetric stationary solutions of the Einstein equations discussed in §2*b*), the finite-gap formulae are still good for numerics, as was shown recently by Frauendiener & Klein (2001, 2004, 2006) and Klein & Richter (2005). The related solution plots can be obtained on a standard PC using the standard Matlab program. More generally, for the whole class of algebro-geometrical solutions of the Zakharov–Mikhailov–Burtsev integrable systems (Burtsev *et al*. 1987) with variable spectral parameter, the same comments are indeed true. Here we have to mention also important progress in computing the objects connected with arbitrary algebraic curves algorithmized by Deconinck and van Hoeij in Maple—see, for example, their paper (Deconinck & van Hoeij 2001). Further developments involving calculations with Riemann theta functions implemented in Maple VIII and higher versions are described in a recent article by Deconinck *et al*. (2004). More detailed references on the computation of finite-gap solutions can be found in Deconinck & van Hoeij (2001), Deconinck *et al*. (2004), Frauendiener & Klein (2004, 2006) and Klein & Richter (2005). In particular, in Frauendiener & Klein (2004, 2006) and Klein & Richter (2005), the interested reader can find the algorithms for computing that are most rapid in the case of the hyperelliptic curves. The later algorithms using the Matlab software allow the production of plots of solutions to the Ernst equation, for which the modular parameters of the underlying spectral curves depend on space and time variables.

Therefore, it is now clear that the finite-gap solutions have the same practical value as standard special functions.

### (b) Deformations of Riemann surfaces in finite-gap integration

Here, we summarize some developments involving deformations of Riemann surfaces and their application to solutions of physical and mathematical problems. Some of these developments were directly influenced by the method of finite-gap integration. The relationship of other developments to finite-gap integration was realized only *a posteriori*, confirming once more an impressive universality of the ideas behind the method.

The first, and most physically important 1+1 differential equation, whose theta-functional solutions involve deformations of algebraic curve (Korotkin 1988; Korotkin & Matveev 1988, 1990) is the Ernst equation from general relativity, which can be written as follows in terms of a complex-valued function :

Theta-functional solutions of the Ernst equation can be written down in terms of the hyperelliptic ‘spectral curve’where *λ*_{j} are constants (immovable branch points) satisfying appropriate reality conditions. Two other branch points, *z*+i*ρ* and *z*−i*ρ* of the spectral curve depend on space–time variables. The theta-functional solutions of Ernst equation look as follows:(2.1)where ** p**,

**are two constant**

*q**g*-dimensional vectors also satisfying certain reality conditions;

*U*is the Abel map; and ∞

^{1,2}are the points at infinity of two sheets of the spectral curve. Dependence on the space variables enters equation (2.1) via the matrix of

*b*-periods and the Abel map.

This drastically changes the properties of theta-functional solutions of this equation as compared with solutions of KdV-like systems. In particular, this class of solutions contains a big supply of asymptotically flat solutions. In genus 2, these solutions were successfully applied to solve the boundary-value problems corresponding to an infinitely thin rotating dust disc (Neugebauer & Meinel 1995; Klein & Richter 1998, 2005). Although, originally, solutions (2.1) were derived in Korotkin (1988) and Korotkin & Matveev (1988) using the framework of the Riemann–Hilbert problem, it was realized recently (Klein *et al*. 2002) that these solutions can be also derived directly from Fay's bilinear identities if one uses in addition Rauch's variational formulae that describe the dependence of holomorphic objects on Riemann surfaces on their moduli (Rauch 1959).

Similar solutions (Korotkin & Matveev 1990) can be constructed for other integrable systems belonging to the class of systems with ‘variable spectral parameter’ (Burtsev *et al*. 1987; Matveev 1994; Kokotov & Korotkin 2008).

A simple example of an integrable system in dimension 1, where modular dependence of theta-functions plays a principal role (which means that the underlying ‘spectral curve’ is deforming), is given by the classical Halphen system for three functions *w*_{j} of variable μfor *i*, *j*, *k*=1, 2, 3. The solution of the system is given by logarithmic derivatives of theta-constants: for *k*=1, 2, 3. The Halphen system together with its solutions was used by Atiyah & Hitchin (1988) in their description of the two-monopole solution space. A generalization of the Halphen system,(2.2)which is equivalent to *SU*(2) invariant self-dual Einstein's equations, was shown by Hitchin (1995) to be a special case of the Painlevé 6 equation and solved in a rather sophisticated way. Later, simple formulae in terms of ‘theta-constants’ for corresponding solutions were found by Babich & Korotkin (1998).

The possibility of solving the Ernst equation, the system (2.2), in terms of theta-functions turns out to be the corollary of the explicit solvability of an arbitrary inverse monodromy problem with quasi-permutation N×N monodromy matrices in terms of the Szegö kernel on Riemann surfaces (Korotkin 2004); the simpler 2×2 case was solved earlier in Korotkin & Kitaev (1998) and Deift *et al*. (1999*a*). The new derivation of the solutions of the Ernst equation, based on the solution of this monodromy problem, was obtained by Korotkin & Matveev (1999, 2000).

Even earlier than the explicit deformations of algebraic curves described above, the rather sophisticated implicit deformations of spectral curves entered the scene. These deformations arise from the study of so-called Whitham modulations of theta-functional solutions of KdV-type system (Krichever 1988; Dubrovin 1992). The implicit deformations are described by a system of Whitham equations on periods of some meromorphic differentials. Later, this formalism was applied to the solution of integrable systems of hydrodynamic type by the so-called generalized hodograph method, see Dubrovin (1990) and Kokotov & Korotkin (2008), and found an unexpected application the Seiberg–Witten theory of the low-energy effective action of supersymmetric gauge theories (see Gorsky *et al*. (1995) for more details).

Another important area of applications of deformations of algebraic curves is Dubrovin's theory of Frobenius manifolds (Dubrovin 1996, 1999), which provides a geometrical formulation of Witten–Dijkgraaf–Verlinde–Verlinde equations from two-dimensional topological field theory. The same framework of flat potential diagonal metrics, which forms the basis of the theory of systems of hydrodynamic type, provides the geometrical background of the theory of Frobenius manifolds. The most well-studied class of Frobenius manifolds is related to Hurwitz spaces, i.e. the spaces of deformations of Riemann surfaces represented as branched coverings of the Riemann sphere (Dubrovin 1996; Shramchenko 2005).

Finally, we would like to mention the massive interference of the algebraic curves and methods of algebraic geometry with physics related to the development of perturbative string theory in the mid-1980s. After 1986, the theta functions, Abelian integrals and Teichmuller spaces became standard tools in the community of high-energy theoretical physicists due to the works by Alvarez-Gaumé *et al*. (1986), Beilinson & Manin (1986), Belavin & Knizhnik (1986), Knizhnik (1987) and many others.

It is worthwhile mentioning that the same objects, spectral curves, Abelian integrals, theta-functions, and more specifically the finite-gap integration method, play an increasing role in the theory of matrix models (e.g. Beisert *et al*. 2005, 2006; Eynard *et al*. 2005; Chekhov 2006).

I think that implicitly the experience of applying the algebro-geometric methods to integrable systems played an important role for the stringy activity from the very beginning. The precise connections are still far from completely explored. As an example, we mention the relationship between the isomonodromic tau-function, the G-function of Frobenius manifolds and determinants of Laplacians over Riemann surfaces established in recent work (Kokotov & Korotkin 2004). The reader can find many references in Kaku (1998) and some first links between quantum strings and finite-gap integration topics in Saito (1987). (See also van Moerbeke (1994) for a comprehensive review.)

Finally, we would like to mention the works by Smirnov (1993, 1994, 2000*a*,*b*) where the role of the finite-gap sector in quantizing integrable models and in analysing the quasi-classical limit of conformal field theory has been pointed out explicitly.

### (c) Some other important developments

#### (i) Integrable tops and other finite-dimensional integrable systems

A few examples of finite-dimensional dynamical systems integrable in terms of two- or multi-dimensional Riemann theta functions were known already at the end of the nineteenth century and the beginning of the twentieth century. The most famous examples are connected with the works by Kowalyewsky, C. Neumann, Klebsch, Schottky, Steklov, Garnier and some other researchers. The development of the finite-gap integration method provided many new remarkable integrable cases as well as the systematization, and certain improvements, of the classical results. The references concerning this development can be found in various books (Bogoyavlenskij 1991; Belokolos *et al*. 1994; Reyman & Semenov-Tyan-Shansky 2003) including, in particular, important results by Bobenko, Kuznetzov, van Moerbeke and many other authors.

#### (ii) Integrable surfaces

The differential geometry of integrable surfaces represents another field, where the methods of finite-gap integration appear to be very powerful. It was stimulated by the discovery of Wente of the constant mean curvature tori which contradict a famous Hopf conjecture, claiming that a sphere is the only possible compact constant mean curvature surface in *R*^{3}. Later works by Bobenko, Pinkall and some other authors (see Babich & Bobenko (1993) and Bobenko (1993) for references) provided further progress in this direction which allowed, in particular, the finding of a complete explicit description of the finite-gap Willmore tori in certain cases. Some, again incomplete, reviews of the related development also concerning the finite-gap theory of the integrable discrete surfaces can be found in Bobenko & Pinkall (1999) together with further references.

#### (iii) Phase transitions and Peierls–Frölich models

It turns out that the finite-gap potentials corresponding to the Schrödinger operator and its discrete version are connected with important problems in the theory of quasi one-dimensional conductivity, namely with Peierls–Frölich phase transitions. In particular, one-gap periodic potentials realize a minimum of the Peierls free energy functional, which was first proved by Belokolos in 1980 (see Belokolos *et al*. (1994) for further references, and also Krichever (1982) for the review of results connected with the discrete Peierls models studied by Krichever and Dzyaloshinskij).

#### (iv) Finite-gap potentials and asymptotic problems of mathematical physics

There are numerous and quite non-trivial applications of formulae and methods of finite-gap integration to various asymptotic problems of mathematical physics, including some applications to the Schrödinger operator with decreasing potential. Here we mention only the works by Dobrokhotov & Maslov (1980), Deift *et al*. (1997, 1999*b*), Kamvissis (1997), Kriecherbauer & Remling (2001) and Kamvissis *et al*. (2003). In Kamvissis (1997), it was shown how to get the Dyson formula, providing the Fredholm determinant solution for the inverse scattering problem, from the Its–Matveev formula.

#### (v) Infinite-dimensional theta functions and related solutions to the KdV and KP equation

There exists an infinite-dimensional generalization of the formula (1.23). On this topic, the reader can consult articles (Müller *et al*. 1998; Schmidt 1996) where the important convergence problems are thoroughly analysed. Müller *et al*. (1998) contains also the references on important earlier works by McKean and Trubowitz, Levitan and Knörrer concerning the hyperelliptic Riemann surfaces of infinite genus. A similar extension of Krichever's solution (1.28) of the KP equation to the infinite genus case forms the content of a very recent book (Feldman *et al*. 2003). Unfortunately, in the infinite-dimensional genus case, the theta-functional formulae are indeed less efficient, and their practical applications, contrary to the finite genus case, are still missing for the moment.

#### (vi) Perturbations of finite-gap solutions and KAM theory

Here we mention only two recent books (Kappeler & Pöschel 2003; Kuksin 2000) that are completely devoted to the topic of this subsection where additional literature including references on the important contributions of Kuksin, Bikbaev, Bobenko and many other researchers can be found.

#### (vii) Application of the finite-gap integration in the theory of nonlinear ocean waves

The interested reader can find the relevant information in the review article of Osborn (2002), where the interesting application of finite-gap solutions of KdV, NLS and KP equations can be found.

## 3. Conclusion

I hope that even this brief and incomplete description of the evolution of the finite-gap integration method and its links with different branches of mathematics and physics, presented above, show that after 30 years of intensive development this area is still a very important and progressing branch of modern mathematical physics. Around this method were formed scientific schools in Moscow, New York and St Petersburg (though now dispersed across the different continents). Later, the activity of these schools was extended by numerous individual researchers all over the world. I hope that this volume will also contribute to further extensions of the method of finite-gap integration and its applications. Concluding I wish to thank Alexander Its and Dmitry Korotkin for many valuable comments on the draft of this article, and my wife Nina Matveeva for useful critical remarks.

## Acknowledgments

This work was supported by grant ANR-05-BLAN-0029-01.

## Footnotes

I wish to dedicate this article to the bright memory of Vadim Kuznetzov.

One contribution of 15 to a Theme Issue ‘30 years of finite-gap integration’.

↵Parallel developments in the USA started in the works of Kac & van Moerbeke (1975), Lax (1975), McKean & van Moerbeke (1975), Flaschka & McLaughlin (1976).

↵Faddeev's solution was based on previous works by Gelfand, Levitan, and especially by Marchenko, on the inverse scattering problem on the half-line (see Marchenko (1977) for more detailed references).

↵In Akhiezer's original construction (Akhiezer 1961), the related

*λ*_{j}were assumed to coincide with boundaries of spectral gaps. The fact that the generic choice of these boundaries leads to the almost periodic potentials (first time mentioned by Novikov (1974) in a context of discussion of the finite-gap solutions of the KdV equation) was also not observed by Akhiezer.↵Of course, the same results remain valid in the case when any

*g*-gaps are not closed, with obvious modifications of enumeration of*λ*_{j}.↵Slightly later, I found the article by Hochstadt (1965) where he proved in a similar way a more special result, saying that any smooth one-gap potential is expressed by means of the Weierstrass function. Hochstadt probably ignored the existence of the infinite series of GLD identities stopping him at the one-gap level.

↵The same statement was proved in a different way by Lax (1975).

↵It is necessary to mention here also the works by Marchenko (1974) and by Marchenko & Ostrovskij (1975), where some prescriptions to construct the boundaries of the gaps, corresponding to the strictly periodic potentials, were given in terms of Schwartz–Christoffel integrals. Marchenko & Ostrovskij (1975) also proved an approximation theorem, establishing the density of finite-gap periodic potentials in the space of all periodic potentials; see also Marchenko (1977).

↵Akhiezer in his seminal work (Akhiezer 1961) proved a similar uniqueness statement in a more special context, corresponding to a more restricted variety of pair finite-gap potentials. Later, Its found an explicit formula for

*ψ*(*x*,*P*) in terms of theta functions using the fact that for given non-special divisor of the degree g the Riemann theta function, for which*D*represents its divisor of zeroes always exists (see the formula (1.24) below). From this formula it is possible to find the corresponding potential*u*(*x*), thus obtaining the solution to the inverse problem stated above. This strategy has a far reaching extension for solving nonlinear integrable PDEs, especially emphasized later by the appearance of Krichever's scheme for solving the KP equation. However, historically it was not the first way of getting an explicit description of the finite-gap potentials.↵It is worth mentioning that the same

*Λ*operator plays many other fundamental roles in the theory of the KdV equation (especially in its Hamiltonian aspects), providing a very compact description of the conservation laws. It is also very important in the description of versal deformations of Hill's equation obtained by Lazutkin & Pankratova (1975). Their work was later used by Witten (1988) in his further investigation of the coadjoint orbits of the Virasoro group.↵Historically, the fact that the existence of two solutions of the Schrödinger equation, whose product is a polynomial of

*λ*, leads immediately (and with no other assumptions about the properties of potential*u*(*x*)) to the trace formula (1.9), to the system (1.19) and to the Jacobi inversion problem to reconstruct the related potential, was first discovered by Jules Drach (1919); see also Belokolos*et al*. (1994) and the article by Brezhnev (2008). Drach, somehow, completely ignored the spectral aspects of the problem. He also did not obtain the final explicit theta-functional representation for the potential. It was not clear from his work that all smooth real-valued finite-gap periodic potentials are included in his construction. Probably, for this reason, his remarkable work became known to the integrable community only around 1980 and, unfortunately, played no role commensurate with its value in the modern evolution of the theory. More specific structure, corresponding to the existence of solutions of the Schrödinger equation, having the form*ψ*(*x*,*k*)=*P*(*x*,*k*)e^{kx},*λ*=*k*^{2}, where*P*(*x*,*k*) is a polynomial in*k*-variable, corresponds to a degenerate case of finite-gap potentials, known as Bargmann potentials. Bargmann studied only the case of smooth reflectionless potentials. In fact, the class of Bargmann potentials is pretty large. It contains, in particular, all the reflectionless potentials, singular periodic and multi-periodic potentials, slowly decreasing and oscillating singular positon potentials (Matveev 2002) and rational singular potentials. For the Bargmann potentials, the product*ψ*(*x*,*k*).*ψ*(*x*, −*k*) is indeed a polynomial of*λ*. Therefore, all the Bargmann potentials form a degenerate subfamily in the space of all finite-gap, almost periodic, potentials.↵Of course the matrix

*B*depends on the choice of canonical basis of cycles*a*_{k},*b*_{k}.↵In our derivation of the identities (1.21) and (1.22), we followed closely the methodology of an excellent review article (Zverovich 1971), where the solution of the classical Jacobi inversion problem was clearly explained. Novikov told me later that when Dubrovin in Moscow independently from us arrived to the Jacobi inversion problem to reconstruct the finite-gap periodic potentials, they asked desperately the prominent algebraic geometrists how to solve efficiently the Jacobi inversion problem, and the typical answer was that there exists the birational isomorphism between the

*g*th symmetric power of the algebraic curve and its Jacobian, etc. The answer was certainly correct but far from enough to explain how to translate it to the language of formulae. At that time the majority of the algebraic geometry community was living their own life very far from physical applications, and many classical values had been completely forgotten.↵Formula (1.23), being our invention, was first published in Dubrovin & Novikov (1974), with a reference to our work (Its & Matveev 1975

*a*), which was actually written before Dubrovin & Novikov (1974), but it was published only in 1975. May be, for this reason, in some articles (e.g. Airault*et al*. 1977, p. 137), it was attributed to Dubrovin & Novikov (1974).↵Indeed, the factors gained by the exponential and theta-functional parts of the right-hand side of (1.24) when passing along the basic cycles cancel each other. Therefore, to determine the right-hand side of (1.24) it is enough to assume that the contours of integration in none the Abelian integrals involved cross the basic cycles.

↵Formula (1.24) for the

*ψ*appeared to be even more generic than the formula (1.23) for the potential. With small variations, it appears in representations of the matrix elements of the*ψ*-function associated to more complicated models (Belokolos*et al*. 1994). It is also worth indicating the following important difference of formulae of the type (1.24) when compared with the traditional theta-functional representations of rational functions on algebraic curves (Mumford 1983). The latter assume a prior knowledge of both poles and zeroes of the function, while in (1.24) only the poles are explicitly involved. The knowledge of zeros is equivalent to knowledge of an explicit solution of the Jacobi inversion problem. This explains why the traditional formulae are not of great help in finite-gap integration.↵Novikov told me at that time that the same formulae in the almost periodic case might be proved rigorously using the Liouville algorithm. However, this kind of proof was published eventually only in 1979 by Gelfand & Dickey (1979).

↵In fact, the same strategy was proposed, but not finalized, for linear ordinary differential equations of any order, by Baker in his article (Baker 1928), which became known later. Following the proposal of Novikov, nowadays expressions similar to (1.24) and (1.26) and their vector or matrix valued generalizations are called Baker–Akhiezer functions.

↵I proposed to call them Darboux dressing formulae, although the original Darboux results (Darboux 1882

*a*) were valid only for dressings of the Sturm–Liouville equation. The extension of his approach to the hierarchies of linear and nonlinear PDEs and their lattice and non-Abelian versions was proposed in Matveev (1979*a*,*b*) and Matveev & Salle (1991), where numerous applications to the theory of solitons can be found.↵In the one-dimensional Schrödinger case one of the ways to isolate the finite-gap solutions (discovered relatively recently by Shabat & Veselov (1993) and Yamilov) is to look for periodic closures of the chains of Darboux transformations. A similar theory for the case when the Sturm–Liouville equation is replaced by a non-stationary Schrödinger equation or by a heat equation that does not exist, to my knowledge, although the formalism of the Darboux transformation method was generalized long ago to much larger classes of linear and nonlinear PDEs and their non-Abelian and difference versions (Matveev 1979

*a*,*b*; Matveev & Salle 1991). For possible generalizations of Shabat & Veselov (1993) to the PDE case, see Novikov & Dynnikov (1997), where the simplest case of the period 2 closure of the Darboux-dressing chain for the heat equation was discussed.↵As a matter of fact, in his monograph Baker (1897, 1907) introduced certain multi-valued objects on a Riemann surface, which he called ‘factorial functions’, and which are characterized by multiplicative jump conditions across a and b cycles. Moreover, Baker showed that the factorial functions can be expressed in terms of Riemann theta functions using essentially the same arguments as the ones used to derive explicit formulae of the type (1.24) and (1.26). Indeed, the theta functional part of the right-hand side of Its formula (1.24) is a factorial function.

↵The tendency, appearing in certain works, to call the Baker–Akhiezer functions just Baker functions should be understood as a late recompense for his pioneering ideas, although I do not think that elimination of Akhiezer's name is a good idea. Akhiezer's work played a crucial role in the creation of the modern finite-gap integration method. In fact, Akhiezer first found the connection between inverse spectral problems and the algebraic geometry of Riemann surfaces, relevant to finite-gap Schrödinger operators. He was also the first (Akhiezer 1960) to produce similar work in the discrete case in connection with the construction of polynomials orthogonal on a system of intervals, relevant to the integration of Toda-like lattice systems.

↵These works were written in the middle of 1975 but related results were published only in 1976.

↵For more complete references on the Akhiezer and Tomchuk works, and the related advanced results in this direction, relevant in particular to the theory of matrix models, see Chen & Its (2008).

↵In some more exceptional cases, like the famous genus 3 Klein curve, the related Jacobian is not only isogenous but also isomorphic to a product of three elliptic curves. Jean Pierre Serre in his unpublished letter to M. V. Babich of 22 July 1985, commenting on our article (Babich

*et al*. 1983) pointed out that until now it is not known if a similar phenomenon takes place for infinitely many values of genera, and gave some other examples where the same effect occurs (the Briggs curve of genus 4, and some other curves of genus 26 and 43). Generic Riemann surfaces have no conformal automorphisms for genus*g*≥3. Any finite group can be realized as a group of conformal automorphisms of some Riemann surface. For a given genus*g*, it was shown by Hurwitz that the order of this group does not exceed 84 (*g*− 1). There is an infinite number of genera for which this maximal order is realized. It is reasonable to call the related curves as Hurwitz curves. The first instances where Hurwitz curves do appear are*g*=3 (Klein curve) and*g*=7 (Fricke curve). Actually, it is known that for any given number*n*it is possible to find the genus*g*for which there exist*n*conformally inequivalent Hurwitz curves. I believe such beautiful objects as Hurwitz curves should play a special role in the theory of integrable systems and this still has to be understood.↵In my opinion, Burnside's original text was better written compared with its exposition by Baker.

↵For the KdV case, the realization of the same approach was much easier.

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