– Letterio Gatto, Silvio Greco –

Algebraic Curves and Differential Equations: An Introduction(Queen’s Paper, 1991, B1–B69)

1 Introduction

In the last two decades an incredible number of papers have appeared, dealing with theinteractions between Algebraic Geometry and the differential equations of ClassicalMechanics and Theoretical Physics. Algebraic Geometry has provided tools to studyand solve several physical systems, and, conversely, the study of the famous “KP”PDE, has led to spectacular results in Algebraic Geometry.

The present seminar is an outcome of our attempts to understand something inthis area, and contains a description of some basic constructions and some examples.It is aimed to common people (as we are) willing to have a glimps of this beautiful butdifficult subject. More expert or brave readers will find only some hints for furtherreading.

Although the material is not new, our presentation might be of some interest.Indeed we make use of the classical theory of Linear Systems of plane algebraic curves,an approach not much exploited in the literature: this point of view is elementaryand concrete, and allows explicit calculations.

The main topic is the construction of the Jacobian flow, which is described inSect. 2. Here we consider a Lax Equation:

d

dtA(t, x) = [A(t, x), B(t, x)],

where A,B are square matrices whose entries are complex polynomials in the inde-terminate x, and depend analytically on the complex variable t.

To any solution A = A(t, ξ) one can attach the algebraic plane curve G:

det(A(t, ξ)− ηI) = 0,

which turns out to be independent on t (whence the name isospectral). The linearsystem generated by the cofactors of one line of A − ηI produce a linear series onthe smooth compactification C of G, and hence an analytic curve on the jacobian of

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C: this is the Jacobian flow. (Unfortunately our presentation is not self contained:we have to refer to the literature for two steps, which would deserve a better under-standing and a more elementary exposition. Perhaps we shall come back to this inthe future).

In Sect. 3 we apply the method to a classical mechanical system, the 3-dimensionalrigid body with one fixed point. The Lax representation comes out easily from theclassical Euler equations, the isospectral curve is elliptic, and the Jacobian flow canbe written explicitly and has constant velocity. Moreover it can be easily integrated,and the corresponding solution of the original system can be explicitly written, sothat all the dotted arrows of the flow chart we draw after this introduction, exist inthis case. It is our feeling that this procedure applies to all mechanical systems whosesolutions can be expressed by using elliptic functions.

In Sect. 4 we do the same for a more complicated system, namely the NeumannSystem (see, e.g., [Mu2]). Here the isospectral curve is hyperelliptic, and the jacobiancan be explicitly described by using the so called Jacobi Polynomials . We giveexplicit calculation of the jacobian flow when the genus is one or two, and again wecan integrate the flow, so that, in principle explicit solutions can be written.

In Section 5 we give a quick account of the theory of Burchnall-Chaundy, laterrediscovered by Krichever. Here the general philosophy

Lax Equation −→ Isospectral curve −→ Jacobian

is the same, but things are much more complicated because one has differential op-erators in place of matrices, and the construction of the isospectral curve is a hardmatter. We give an idea on how this approach allows to deal with particular solutionsof the KdV equation, a PDE from Physics. These methods are extremely importantin Algebraic Geometry, because of its connections with the Shottky problem.

The Appendix contains the calculations concerning the physical example of therigid body motion discussed in Sect. 3 : they are added to convince the reader thatactual calculations follow a precise pattern (as in the flow chart), which could be usedin other situations as well. We are grateful to many people for helpful discussionsand encouragement, first of all to Emma Previato, who introduced both of us to thesubject; Fulvio Ricci for some useful suggestions and then to all partecipants to theseminar on Integrable Systems held at the University of Torino in the year 1990-91,especially Mauro Francaviglia, Franco Magri and Cesare Reina. Finally we thankTony Geramita for giving hospitality to our paper in his Curve Seminar: we hope tohave given him a not too unpleasant guest.

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2 Lax Equations, Isospectral Curves and Jacobian

Flows.

In this Section, we shall describe the main procedure to construct the Jacobian flowin order to apply it in a particular but very significant case which is quite sufficientto describe the general philosophy. To begin with, let us set some notation. Weshall denote by Gln(C) the (complex) Lie group of the non singular n × n squarematrices with coefficients in the complex field C, and by gln(C) its Lie-algebra. Asis well known the latter is isomorphic to the Lie algebra of all n× n complex squarematrices, with respect to the usual commutator: [M,N ] = MN − NM . In thefollowing we shall need to use the term “curve” not only in the algebraic geometricsense, but also in the classical differential geometric sense of “parametrized curve”,i.e. smooth map from connected and simply connected subset of the complex lineC to some manifold M . Although the difference between the two notions of “curve”will be clear from the context, we remark explicitly that we shall consider only onekind of algebraic curves, namely the isospectral curves (to be defined below). Let usconsider, then, a smooth curve A : D ⊆ C −→ gln(C) (parametrized by the points ofa open disk D of C) satisfying a first order differential equation of the following kind:

A(t) = [A(t), B(t)] (1)

where B : D ⊆ C−→gln(C) is another fixed curve (parametrized by the same domainD) on gln(C), which a priori is just a known function of A(t) [i.e., the entries of Bare known functions of the entries of A] (as usual the over dot denotes the derivativewith respect to the ”time” t ). Expression (1) is said to be a Lax equation and a pair(A,B) as above satisfying such an equation is said to be a Lax pair.

In the following we shall deal with matrices A and B depending on a complexparameter x, so that we are led to consider one parameter families of Lax equations:

A(t, ξ) = [A(t, ξ), B(t, ξ)]. (2)

For sake of simplicity we shall assume thet A and B depend polynomially on ξ,i.e. A(t, ξ) and B(t, ξ) are polynomials with coefficients in the algebra gln,C) or,equivalently, are matrices with polynomial entries in the variable x).

The algebraic geometric aspects of the Lax Equation depending on a parameterx as above, are due essentially to the following key result:

3

Proposition 2.1 If A is the (unique!) solution of the Cauchy problem (dependingon a parameter):

A(t, ξ) = [A(t, ξ), B(t, ξ)]A(0, ξ) = A0(ξ)

then there exists a 1- parameter family of (smooth) curves Gξ : D ⊆ C−→Gln(C)such that for each fixed ξ , Gξ(0) = I (identity matrix) and

A(t, ξ) = Gξ(t)A0(ξ)G−1ξ (t) (G−1

ξ (t) = [Gξ(t)]−1),

i.e. the characteristic polynomial of A does not depend on t.

Proof.

Consider the following two Cauchy problems:

(∗)

G = −BGG(0) = I

(∗∗)

H = HBH(0) = I

for a given fixed value of the parameter ξ. They both admit a (unique!) smoothsolution by the Cauchy theorem of existence and uniqueness for ordinary differentialequations. We want show that the solutions Gξ and Hξ are inverse of each other. Infact, by computing the derivative of HξGξ with respect to t and using (*) and (**)one gets:

(HξGξ)· = HξGξ +HξGξ = HξBGξ −HξBGξ = 0,

henceHξGξ is constant (sinceD is connected) and we have: Hξ(t)Gξ(t) = Hξ(0)Gξ(0) =I for each t, proving the claim. The conclusion is that any B ∈ gln(C) can be alwaysexpressed in the form GξG

−1ξ , where Gξ is a suitable curve in Gln(C). Of course

this procedure does not depend on the choice of the parameter ξ. Let A(t, ξ) beGξA0(ξ)G

−1ξ . Then it is straightforward to show that A satisfies the Lax equation

A(t, ξ) = [A(t, ξ), B(t, ξ)]. By uniqueness it necessarily coincides with A. Coming back to the Lax equation depending on a parameter, the above property

tells us that for a given solution A = A(t, ξ) of eq. (2) (i.e. for a given initial condition)the coefficients of the polynomial

Φ(ξ, η) = det[A(t, ξ)− ηI] = 0, (3)

do not depend on “time”. Then to the given solution one can attach the algebraiccurve: Γ : Φ(ξ, η) = 0, in the affine plane (ξ, η), which is independent on t. G

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is called the isospectral curve corresponding to the given solution (or to the giveninitial condition). We observe that if (ξ, η) ∈ Γ, then η is an eigenvalue of the matrixA(ξ, t0), and hence of A(ξ, t) for all values of t. In other words the eigenvalues are timeindependent, whence the name isospectral for Γ. From now on, we shall make anothersimplifying assumption, namely that Γ is irreducible and non singular. This conditionis verified for all sufficiently general choices of the initial conditions A0 = A(ξ, t0) forthe Lax equation). Now we shall consider, for each (ξ, η) ∈ Γ, and each t, the linearmap A(ξ, t) : Cn−→Cn and the eigenspace V (ξ, η, t) corresponding to the eigenvalueη. We note explicitly that in general V (ξ, η, t) does depend on t (although (ξ, η), asthe whole Γ, does not).

Lemma 2.1 Let the notation and the assumptions be as above. Then for all butfinitely many points (ξ, η) ∈ Γ and all t’s, V (ξ, η, t) has dimension 1.

Proof.

Let π : Γ−→A1 be the projection onto the ξ-axes, i.e. π(ξ, η) = ξ. Since we areworking in characteristic zero, π−1(ξ) consists of n distinct points for all but finitelymany values of ξ (just avoid the zeroes of the discriminant of Φ considered as apolynomial in η). Hence for almost all ξ’s and all t’s the matrix A(ξ, t) has distincteigenvalues, and the conclusion follows.

Let us consider now Γ′, the subset of Γ consisting of all points verifying theconclusion of Lemma 2.1 and define a map:

χt : Γ′−→Pn−1, (4)

by setting:χt(ξ, η) = point of Pn−1 corresponding to V (ξ, η, t).

As noticed before the map χt depends on t. Moreover:

Lemma 2.2 χt is a rational map (and hence extends uniquely) to a morphism χt :C−→Pn−1, where C is the normalization of the projective closure of Γ in P2.

Proof.

Let φ1, . . . , φn be the cofactors of one fixed row of the matrix A(ξ, t)−ηI, and considerthe linear system Σ of plane curves generated by φ1, . . . , φn . If (ξ, η) ∈ Γ′ is not abase point of S, it follows from elementary linear algebra, that

χt(ξ, η) = (φ1(ξ, η), . . . , φn(ξ, η).

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This implies that χt : C ... → Pn−1 is a rational map and for general facts, it extendsto a morphism defined on the whole C.

Definition 2.1 We call χt : C−→Pn−1 the eigenvector morphism or the eigenvectormap, corresponding to the given solution of the Lax equation.

Remark 2.1

1. Once again we remark that χt does depend on t and hence keeps track of thegiven solution.

2. One can show that, for a general choice of the initial value for the Lax equation,the complete normalization C of the isospectral curve Γ has genus:

g =1

2(n− 1)(ne− 2),

where e is the largest degree of the polynomials appearing in the matrix A0 (seee.g.[Be] or [G])

3. Under the same assumption one can show that the eigenvector map has de-gree d = n + g − 1 (independent on t) and that χ∗tOPn−1(1) is non special,

OPn−1(1)being the line bundle corresponding to the linear series cut out by Σ

outside of the base points (See e.g. [R-STS]).

4. We are considering here the (abstract) smooth compactification C of Γ. As amatter of fact one should consider C (or sometimes another suitable compact-ification of Γ) as lying on a certain ruled surface, see e.g. [G], [Be], [R-STS].A description of these ruled surfaces from a classical point of view is in [Gr-P].Let us now denote by Lt the line bundle χ∗tOPn−1(1). By the previous discus-

sion it is clear that Lt is an element of Picd(C), i.e. the degree d piece of thegroup Pic(C) of linear equivalence classes of divisors on C. Recall moreoverthat Picd(C) is isomorphic to an algebraic g-dimensional torus, the so calledJacobian Variety of the curve C (see e.g. [Fo]).

Definition 2.2 The map (i.e. the “curve” in our non algebraic sense) t−→Lt will besaid to be the flow on the Jacobian Variety of C (or the “Jacobian flow ”) associatedto the given solution of the Lax equation A(ξ, t) = [A(ξ, t), B(ξ, t)].

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To be more explicit, the standard way to realize the Jacobian flow is the following.First choose a basis of the g-dimensional C-vector space of holomorphic 1-forms,ω = (ω1, ω2, . . . , ωg), on C viewed as a compact Riemann surface. Let

D(t) = P1(t) + P2(t) + . . .+ Pd(t)

be an arbitrary effective divisor corresponding to a global section of Lt (i.e. a divisorof the linear series cut out on C by the linear system Σ outside of the base points,(see proof of Lemma (2.2), and fix a point P0 on C. Then the Jacobian flow is givenby:

t 7→ J(Lt) :=

(∫ P1(t)

P0

ω1 + . . .+∫ Pd(t)

P0

ω1, . . . ,∫ P1(t)

P0

ωg + . . .+∫ Pd(t)

P0

ωg

)(mod Λ)

(5)Λ being the period lattice in Cg defined by the ωi’s. By Abel’s theorem the abovedefinition does not depend on the choice of the representative D(t) in Lt (see [Fo] fordetails).

Remark 2.2 The Jacobian flow has been used to get remarkable results concerningmany classical mechanical systems, expecially to study linearization of the flow andcomplete integrability of the system. Just to have an idea one can have a look at [G],[A-vM] and [R-STS].

Our aim now is to show how efficient the method is, by exhibiting explicit calcu-lations for two classical mechanical systems. In fact we are interested in determiningin some concrete examples the cases in which the flow, defined above, is linear inthe sense that a suitably defined “velocity” dLt/dt of the point Lt on the jacobian isconstant1. In the cases we shall deal with, it will suffice to think to the velocity of Lt

at the “time” t = t0 as the tangent vector of the “curve” t 7→ P (t) on Jac(Γ), at thepoint P (t0), corresponding to t 7→ Lt, via the Jacobi isomorphism J.

From a purely computational point of view, this simply means to compute the firstderivative of the “moving point” P (t) with respect to the parameter t. At least inprinciple, finding the velocity of the Jacobian flow associated to a dynamical system,allows to explicitly solve the differential equations describing it. This is made possible

1We have to remark that thisi is not the most general definition of linearity, though this is enoughfor all the examples we are going to exhibit (and,by the way, for many others, see e.g. [G]). Fora more general definition of this delicate concept we refer the reader to [G] and [A-vM], also forfurther references

7

using a suitable inversion of the Jacobi map expressed in terms of the theta functions(see, e.g., [Mu1] and [Mu2]). Instead of developping the general theory, which is overthe purposes of this exposition, we shall show this procedure on a particular, butconcrete and relevant example in the next section, by solving explicitly the classicalEuler’s equation for the one fixed point free rigid body. Our calculations are madepossible by the explicit expression of the Jacobian flow given by using the linearsystem Σ.

3 A Remarkable Example:the one Fixed Point Free

Rigid Body

.The purpose of this Section is to show in detail how the procedure shown in the

previous section works for a physically relevant example of Lax equation. In order todo this, define a constant matrix J and a “time” dependent matrix Ω as follows:

J =

a 0 00 b 00 0 c

and Ω =

0 ω3 −ω2

−ω3 0 ω1

ω2 −ω1 0

Setting

M = JΩ + ΩJ =

0 l −m−l 0 nm −n 0

,one can easily check that the Lax equation

M = [M,Ω] (6)

is equivalent to the classical Euler equations for the one fixed point rigid body freemotion (See Appendix). By using a remark due to Manakov [M], eq. (6) can berewritten, in an equivalent form, as a Lax equation depending on a parameter. Infact, setting A = M+J2ξ and B = Ω+Jξ, ξ being a parameter, a simple computationshows that (6) is equivalent to the following:

A = [A,B]. (7)

Referring to the definitions set in Sect.3, the isospectral curve Γ is given by:

Q(ξ, η) ∼= det(M + J2ξ − ηI) = 0

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i.e., by an easy calculation:

(a2ξ − η)(b2ξ − η)(c2ξ − η) + (b2m2 + a2n2 + c2l2)ξ +−(m2 + n2 + l2)η = 0 (8)

By the general theory, the coefficients of (8) are time independent and then, inparticular:

L = l2 +m2 + n2 and K = b2m2 + a2n2 + c2l2 (9)

are prime integrals of the “motion”, according with their physical meaning: in factthe former represents the square norm of the angular momentum while the latter isa linear combination of this and the energy:

T =1

2

(l2

b+ c+

m2

a+ c+

n2

a+ b

).

The isopectral curve C is the projective closure of γ in P2, and hence if the initialconditions are sufficiently general it is a smooth plane projective curve of degree3, hence an elliptic curve. If we consider projective coordinates [ξ, η, ζ] in P2, theeigenvector map

χt : C−→P2

is given by the equations

[x0, x1, x2] = [φ0([ξ, η, ζ]), φ1([ξ, η, ζ]), φ2([ξ, η, ζ])],

where [x0, x1, x2] are homogeneous coordinates in the second P2 and φ0, φ1, φ2 thecofactors of the first row of the matrix A− ηI, namely:

φ0 =∣∣∣∣ b2ξ − η nξ−nξ c2ξ − η

∣∣∣∣ , φ1 =∣∣∣∣ nξ −lξc2ξ − η mζ

∣∣∣∣ , φ2 =∣∣∣∣ −l b2ξ − ηmζ −nζ

∣∣∣∣ .This means that χt corresponds to the linear system Σ : λ0φ0 + λ0φ0 + λ0φ0 = 0. Soto compute χt ∗ OP2(1) we can pull back an arbitrary line of the second plane, e.g.the line x0 = 0. The corresponding divisor is then the intersection of C with φ0 minusthe base points of Σ. So we have to solve the system:

(a2ξ − η)(b2ξ − η)(c2ξ − η)+ (b2m2 + a2n2 + c2l2)ξζ2 − (m2 + n2 + l2)ηζ2 = 0

(b2ξ − η)(c2ξ − η) + n2ξ2 = 0(10)

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We find two time independent double solutions P1∼= P3 = (1, b2, 0) and P2

∼= P4 =(1, c2, 0) and two simple solutions depending on time:

U1(t) =

(n(m2 + l2)

ml(c2 − b2),b2m2 + c2l2)n

ml(c2 − b2)), 1

), (11)

U2(t) =

(n(m2 + l2)

ml(b2 − c2),(b2m2 + c2l2)n

ml(b2 − c2)), 1

). (12)

A straightforward calculation shows that the base points of Σ are precisely P3, P4

and U2(t), so that the “motion of the linear series” on the curve is represented bythe “motion of the divisor” D(t) on the curve, defined by the formal sum P1 + P3 +U1(t). Since P1 and P3 are time-independent , the motion of the divisor D is entirelydetermined by the motion of the point U1(t) on the curve. Since the jacobian varietyof an elliptic curve is the curve itself (modulo the Jacobi isomorphism), the pointU1(t) is in fact describing a curve on Jac(C). Then we can express the Jacobian flowon Jac(C) associated to the Lax equation (7) by means of the Jacobi isomorphism (5),namely:

t−→J(U1(t)) =∫ U1(t)

P0

ω modΛ

where ω is a holomorphic 1-form on C and Λ the periods’ lattice in C with respectto ω.

What we want to show is that actually the above flow is linear, in the sense that(cfr. end of last Sect.) its first derivative with respect to the time is constant. Fromnow on all our discussions will be devoted to the purely technical problem of findingthe best way to perform the required computations. In order to do this, one shouldobserve that a holomorphic 1-form on C is:

ω =ξdη − ηdξ√

(Kξ − Lη)(a2ξ − η)(b2ξ − η)(c2ξ − η), (13)

(where we set , as above, K = a2n2 + b2m2 + c2l2 and L = m2 +n2 + l2), and that wehave to show that:

v =d

dt(J(U1(t)) = ω(U1(t))U1(t),

is constant. Actually, this is the fact, because after performing all the easy, thoughtedious, algebraic manipulations one finally gets:

v = 2

(b2m2 + a2n2 + c2l2

(a+ c)(b+ c)(a+ b)

)=

2K

(a+ c)(b+ c)(a+ b), (14)

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which is clearly constant, being a function of time-independent terms (from aphysical point of view: of prime integrals only). We refer the reader to the Appendixfor the details of the calculations.

We want to show, now, that finding v actually means to be able to explicitly solvethe Lax equation, which is the same as explicitly finding the time evolution of l,m, n(i.e. the entries of the matrix M appearing in equation (6). To understand why, letus recall that any elliptic curve is represented by an equation of the form:

y2 = 4x3 + 20a2x+ 28a4,

and can be parametrized by the points of a complex torus, via the Weierstrass ℘-function, gotten by taking the quotient of C with respect to a suitable lattice Λ,whose points satisfy the relations:

a2 = 3∑

λ∈Λ\0

1

λ4and (15)

(16)

a4 = 5∑

λ∈Λ\0

1

λ6(17)

(see, e.g., [11] for details). Let then:

y2 = Ax3 +Bx2 + Cx+D, (18)

be the equation of a non singular plane cubic curve (i.e. Ax3 + Bx2 + Cx + D = 0has three distinct roots). Set:

a2 =1

203

√4

A

(C − B2

3A2

)and (19)

(20)

a4 =1

28

(D − BC

3A+

B3

9A3

). (21)

General facts ensure us the existence of a lattice Λ such that (15) and (17) are satisfiedwith the values of a2 and a4 given by (19) and (21) (see e.g. [Ca] or [H]). At thispoint, a straightforward check shows that the “parametric equations” of (18) are:

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x = − 3

√A4

B3A

+ ℘Λ(z)

y = ℘′Λ(z)

, (22)

where ℘Λ is Weierstrass ℘-function with respect to the lattice Λ as above. Comingback to our isospectral curve (8), in the Appendix we shall show that by a suitablechange of coordinates, it can be expressed as:

z2 =

(H1

Ky − a2

K

)·(H2

Ky − b2

K

)·(H3

Ky − c2

K

)(23)

where we set:

H1 = a2m2 + a2l2 − b2m2 − c2l2 = a2L−K,

H2 = b2n2 + b2l2 − a2n2 − c2l2 = b2L−K,

H3 = c2m2 + c2n2 − b2m2 − a2n2 = c2L−K,

where L and K defined as in (9).By computing the right hand side, equation (23) takes the form (18), with A,B,

C and D given by:

A =H1H2H3

K3,

B = −a2H2H3 + b2H3H1 + c2H1H2

K3,

C =a2b2H3 + b2c2H1 + a2c2H2

K3,

D = −a2b2c2

K3.

Let Λ be the lattice satysfying (15) and (17) using A, B, C and D as above tocompute the a2 and a4 by means of (19) and (21). Then: y(U1(t)) = − 3

√A4

B3A

+ ℘Λ

(2K

(a+c)(b+c)(a+b)t+ q

)z(U1(t)) = ℘′Λ

(2K

(a+c)(b+c)(a+b)t+ q

) (24)

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gives the “parametric equations” of the “moving point” U1(t), q being a constantdepending on the initial data. But (see Appendix), we also have:

y(U1(t)) = a2n2−Kn2H1

= a2

H1− K

n2H1

z(U1(t)) = ml(c2−b2)n3(a2L−K)

(25)

By comparing(24) with(25), we get the two equations:

a2

H1

− K

n2H1

= − 3

√A

4

B

3A+ ℘Λ

(2K

(a+ c)(b+ c)(a+ b)t+ q

), (26)

ml(c2 − b2)

n3(a2L−K)= ℘′Λ

(2K

(a+ c)(b+ c)(a+ b)t+ q

). (27)

Solving (26) with respect to n, one obtains:

n(t) = ±

√√√√√ KH1

a2

H1− 3

√A4

B3A

+ ℘Λ

(2K

(a+c)(b+c)(a+b)t+ q

) (28)

Once found the expression (28), by substituting into the two prime integrals ofthe motion:

L = l2 +m2 + n2, K = a2n2 + b2m2 + c2l2

one gets:

m2 =K − c2L+ (c2 − a2)[n(t)]2

b2 − c2(29)

n2 =K − b2L+ (b2 − a2)[n(t)]2

c2 − b2(30)

A straightforward computation (keeping in mind the relation (27) shows that thetriple (l,m, n), defined as above, satisfies the Euler Equations (Cfr eqns. (A.2)).The ambiguity of the sign can be decided by using the initial data, and so the LaxEquation (7) have been integrated.

Remark 3.1. A careful reading of the above example, together with the com-putational strategy shown in the Appendix, reveal quite clearly how to extend theprevious analysis to all known dynamical systems admitting elliptic solutions. It is

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quite reasonable that all those systems should admit a Lax pair formulation, leadingto elliptic isospectral curves. Some questions naturally arises. Primarily, one can askwether such a Lax pair, supposed to exist, is or not unique. Strictly related with thisproblem is to investigate the relations between the isospectral curves associated totwo different Lax equations equivalent to the given system of ODE. For instance onecan easily find a Lax pair for the Euler Equations, involving 2 × 2 square matrices,rather than the 3x3 used above, leading to the same isospectral curve. All thesequestions shall be investigated in a forthcoming paper, where we shall exhibit a setof interesting classical examples, as, e.g., a Lax pair for the equation describing thenon-linearized pendulum.

4 A Class of Linearizable Systems for any Genus:

the Neumann System.

The linearization procedure described in the example of the rigid body motion, worksalso for Lax equations leading to isospectral curves with genus higher than 1. As amatter of fact, the method we have shown is very general and it is not difficult tobelieve that the most relevant difficulty for general cases is to find explicitly a basisfor the holomorphic differentials. In the rigid body example, which is very particularsince in genus 1 the jacobian variety coincides with the curve itself, the trick had beento change projective coordinates in order to write the curve as a double covering of P1.In fact, dealing with a double covering of P1 one is able to write easily, in a standardway, a basis for the holomorphic differentials. This suggests that the above methodshould work also for hyperelliptic isospectral curves, which by definition are nothingbut double branched covering of P1. The so called Neumann Dynamical System,extensively studied in [27], will provide examples of hyperelliptic isospectral curvesand, precisely, one for each genus g ≥ 1. Hence our next goal can be summarized asfollows:

1. Defining the Neumann Dynamical System;

2. Finding a Lax representation for it leading to a hyperelliptic isospectral curve;

3. Showing that the Jacobian flow is linear by explicitly computing its “velocity”by means of the “time” derivative of the Jacobi map.

However, in order to let the next discussion be entirely self contained, we shall beginto briefly recall the algebraic construction of the Jacobian of a hyperelliptic curve.

14

4.1 Hyperelliptic Curves and their Jacobians

It is well known a hyperelliptic curve C of genus g ≥ 2 is birationally equivalent tothe affine plane curve characterized by the equation:

Γ = (ξ, η)/η2 = (ξ − a1)...(ξ − a2g+1),

where a1, . . . , a2g+1 are pairwise distinct complex numbers. The closure of Γ in P2 hasjust one point at infinity, mamely [0, 1, 0], which corresponds, in the normalizationC, to a unique point we shall call ∞.

The covering of C over P1 is the unique extension to C of the projection π : Γ−→P1

which sends (ξ, η) 7→ (ξ). It is clearly a two-sheeted branched covering where thebranch points are exactly the fixed points of the natural involution of C onto itselfinterchanging the sheets, i.e. the map:

ι : (ξ, η) 7→ (ξ,−η) ι(∞) = ∞.

We have hence exactly 2g + 2 branch points: namely ”∞” and the points whosecoordinates in Γ are given by (ai, 0). What we want to do in the following is todescribe quite explicitly Pic0(C), i.e. the subgroup of linear equivalence classes ofdivisors of degree zero, following essentially the detailed procedure that the readercan find in [27]. The basic idea consists in finding some “canonical” representative ofthe elements of Pic0(C). To this purpose let us denote by C(g) the g-fold symmetricproduct of the curve C (i.e. the orbits of the g-fold product of the curve C by itselfunder the action of the symmetric group in g variables). Then consider the obviousmap:

α : C(g)−→Picg(C),

sending (P1, ..., Pg to the linear equivalence class of the divisor P1 + ... + Pg. Thismap is clearly surjective by Riemann-Roch. Furthermore, each non special class ofPicg(C) admits a unique pre-image. Let us consider now an isomorphism betweenPicg(C) and Pic0(C). For our purposes it is convenient to choose the one sendingD ∈ Picg(C) onto D − g · ∞. The composition of these two maps leads to the map:

I : C(g)−→Pic0(C),

which is clearly surjective, since α was. In C(g) consider now the subset Z of g-upleof points corresponding to non special divisors. The image of the complement of Zin Pic0(C), via I , is the so-called theta divisor Θ (for the proof that Θ is a divisor

15

in Pic0(C), we refer the reader to [26]). Hence, by Riemann Roch, the restriction, I′of I to Z, sets up a 1− 1 correspondence between Z and Pic0(C)−Θ. In the case ofthe hyperelliptic curves, for which one knows exactly the structure of the canonicaldivisors, Z can be explicitly described as the set Div+,g

0 (C) of all effective divisors onC of degree g (i.e. all D of the form

∑gi=1 Pi such that Pi 6= ∞, Pi 6= ι(Pj) if i 6= j,

and Θ turns out to be the subset of Pic0(C) of elements equivalent to divisors like∑g−1i=1 Pi− (g− 1) ·∞. By the above discussion it turns out that Z is a g-dimensional

algebraic variety. We show how to perform the construction, referring the reader to[27] for the proofs. Let D be a divisor of Z = Div+,g

0 (C). The key idea is to attachto D three polynomials U, V,W as follows:

1. a polynomial U(ξ) =∏g

i=1(ξ − ξ(Pi)), which is monic and of degree g.

2. let mi the multiplicity which Pi occurs in D with, and define V (ξ) as the uniquepolynomial of degree ≤ g − 1 such that

(d

dξ

)jV (ξ)−

√√√√2g+1∏i=1

(ξ − ai)

∣∣∣∣∣∣ξ=ξ(Pi)

= 0 for 0 ≤ j ≤ mi − 1.

If the support of D is made of exactly g distinct points, then V (ξ) is nothingbut the unique polynomial of degree ≤ g − 1 such that V (ξ(Pi)) = η(Pi) for1 ≤ i ≤ g, so that its explicit expression is:

V (ξ) =g∑

i=1

η(Pi)

∏j 6=i(ξ − ξ(Pj)∏

j 6=i(ξ(Pi)− ξ(Pj));

3. by the above construction it turns out that∏2g+1

i=1 (ξ − ai)− [V (ξ)]2 is divisibleby U(ξ) (indeed

∏2g+1i=1 (ξ−ai)− [V (ξ)]2 vanishes at each of the ξ(Pi)’s). We can

hence define a third polynomial W (ξ) by means of the relation:

2g+1∏i=1

(ξ − ai)− [V (ξ)]2 = W (ξ)U(ξ).

Since deg V (ξ) ≤ g−1, deg[V (ξ)]2 ≤ 2g−2 < deg∏2g+1

i=1 (ξ−ai) and hence, sinceU(ξ) is monic of degree g, W (ξ) turns out to be monic of degree 2g+1−g = g+1.

16

We just shown that to any D ∈ Div+,g0 (C) we can attach three polynomials as

above with degU = g, deg V = g − 1 and degW = g + 1. Conversely, given threepolynomials U , V , W such that (ξ − a1)...(ξ − a2g+1) − V 2 = UW , with U and Wmonic, having degree as above, we get the divisor (U)0 of g points on the ξ-line.If ξi ∈ (U)0 then (ξi, V (ξi)) ∈ C and

∑i(ξi, V (ξi)) ∈ Z. What just said is certainly

sufficient to convince that there is a bijection between Z and the triples of polynomialsas above. Let now U, V,W be any three polynomials of degree as above and expandthem by power of ξ:

U(ξ) = ξg + U1ξg−1 + ...+ Ug,

V (ξ) = V1ξg−1 + ...+ Vg,

W (ξ) = ξg+1 +W0ξg + ...+Wg,

by imposing they represent an element of Z, one gets something like

f(ξ)− V 2 − UW =2g∑

α=0

aα(Ui, Vj,Wk)ξα,

so that (Ui, Vj,Wk) can be assummed to be coordinates in C3g+1. Using such coordi-nates one can claim that aα(Ui, Vj,Wk) = 0 are the equations of a smooth algebraicvariety. Because of the bijection between Z and Pic0(C)− Θ, we can conclude thatPic0(C)−Θ also can be equipped with a structure of an algebraic variety. By transla-tion, as shown in [27], one can give to Pic0(C) a structure of a smooth g-dimensionalvariety, covered by affine pieces looking like Pic0(C)−Θ. However, for our purposes,it is sufficient to be able to parametrize Pic0(C) − Θ by means of the polynomi-als U, V,W which are the fundamental tools for discussing the Jacobian flow of theNeumann Dynamical System.

Remark 4.1 The description of the jacobian given above has been generalized to alarger class of curves in [30] and later on in [6], by using techniques related to theconstruction of Jacobian flows.

4.2 The Neumann Dynamical System and its Lax Represen-tation.

The Neumann Dynamical Systems (NDS for brief) describe the motion of n particlesof mass m1, ...,mn in a unidimensional simple armonic oscillation whose positions

17

x1, ..., xn are constrained to satisfy the relation∑n

i=1 x2i = 1. The “motion equations”

are hence given by: d2xd2t

= −aiξi∑ni=1 x

2i = 1

, (31)

where we shall suppose the ai’s such that a1 < ... < an. By denoting with a overdotthe derivative with respect to t, and introducing some auxiliary variables yi , theabove “second order system of n ordinary differential equations + constraint” can bere-expressed as a first order system of 2n differential equations as follows:

xi = yi

yi = (α− ai)ξi

, (32)

where we set α =∑n

k=1 akx2k − y2

k. It is very easy to show that the above system isnothing but the system of the Hamilton Equations of the constrained Hamiltonian:

H = 12(∑n

k=1 akx2k + y2

k)∑ni=1 x

2i = 1

We shall analyze in detail this system for two and three particles. Indeed it isenough for our purposes, since nothing changes in the general case, and these twoexample will provide us two more examples of isospectral curves: one in genus g = 1and the other in g = 2. To show how the previous system can be represented ina Lax form we shall assume that we are able to attach to any NDS of n particlesthree polynomials in a certain indeterminate ξ, U, V,W functions of the xi’s and ofthe yi’s only, which fulfill the following requirement: U monic and deg(U) = n − 1,deg(V ) ≤ n− 2, W monic and deg(W ) = n, such that UW + V 2 is a polynomial in ξof degree 2n− 1 with roots all distincts. This allows us to consider the hyperellipticcurve η2 = UW + V 2 = f(ξ). We shall show in the two specific examples how toattach such three polynomials to the NDS. But if this is true, at this point we canexpress a remarkable identity which takes the form of a Lax equation. Define thefollowing two matrices:

A =(−V UW V

)and P =

(1 1

η+VU

−η+VU

)(33)

A straightforward check shows that the following is an identity:

A = [A,B]

18

having set B = −PP−1.Hence, to get a Lax equation equivalent to the NDS it is enough to substitute in

the previous identity the x and y by means of the right side of equations (32). In thisway the equation A = [A,B] will be no more an identity but it will hold just if andonly if (x,y) is a solution of the NDS. Furthermore, the isospectral curve of this Laxequation is exactly η2 = UW + V 2. We shall let work in detail all this machinery inthe next two examples.

Example 4.1 NDS for two particles.The hamiltonian for the Neumann System with two particles is given by:

H =1

2(a1x

21 + a2x

22 + y2

1 + y22), (34)

which has to be considered together with the constraintsx2

1 + x22 = 1

x1y1 + x2y2 = 0

The motion equations are hence given by:x1 = y1

x2 = y2

y1 = (α− a1)x1

y2 = (α− a1)x2

(35)

where we set α = a1x21 + a2x

22 − y2

1 − y22.

As explained in the above general discussion, let us set the function f1(ξ) and thenatural algebraic coordinates on the Jacobian Variety:

f1(ξ) = (ξ − a1)(ξ − a2), (36)

U(ξ) = f1(ξ)

(ξ21

ξ − a1

+ξ22

ξ − a2

)= −

√−1(a2x

21 − a1x

22), (37)

V (ξ) =√−1f1(ξ)

(x1y1

ξ − a1

+x2y2

ξ − a2

)== −

√−1(a2x1y1 + a1x2y2), (38)

W (ξ) = f1(ξ)

(y2

1

ξ − a1

+y2

2

ξ − a2

+ 1

)== ξ2+(y2

1)+y22−a1−a2)ξ−a1y

22−a2y

21, (39)

19

i.e., as in the previous notation:

U1 = −(a2x21 + a1x

22), (40)

V1 = −√−1(a2x1y1 + a1x2y1), (41)

W0 = y21 + y2

2 − a1 − a2, (42)

W1 = −a1y22 − a2y

21. (43)

From (11) and (12), together with (7) it immediately follows that:

U1 = 2√−1V1, (44)

what we shall use in the following.Hence our map from the cotangent bundle of S1 and C4 = C3·1+1 is given by:

((T ∗(S1) −→ C4

(x1, x2, y1, y2) 7→ (U1, V1,W0,W1).(45)

Let η2 = UW + V 2 be the isospectral curve of the Lax equation associated to thesystem (7). The eigenvector map is then given by:

ft : (ξ, η) 7→ (U, V + η).

Clearly, pulling back the hyperplane (line) U = 0, one gets a divisor that representft∗OP1(1). Such a divisor can be explicitly computed by solving the following system:

U(ξ) = 0η2 = V 2(ξ)

(46)

The above equations determine two divisors living in different sheets of C withrespect to the projection π onto P1, and each of them lying in the variety Z =Div+,1

0 (C) (described in general at the beginning of the section). For our purposes,of course, it will suffice to choose one of them, say (ξ0 = −(a2x

21 + a1x

22), η0 = V (ξ0)).

The velocity of the flow on the jacobian variety of C, which is isomorphic to C, isgiven, as usual, by the “derivative” of the Jacobi map, i.e.:

d

dt

∫ ξ0 dξ√UW + V 2

=ξ0(t)

V1(ξ0(t))=

U1(t)

V1(U1(t))=

2√−1V1(t)

V1(t)= 2

√−1,

and this prove that the flow linearizes on the jacobian variety. At this point, it is justmatter of a simple computational exercise to solve explicitly Neumann’s Differential

20

Equations (35), by repeating exactly the same procedure shown for the solution ofthe Euler Equations (Cfr. Sect. 3). Actually, we have integrated the flow on thejacobian of the isospectral curve.

Example 4.2 NDS for three particles. The hamiltonian for the Neumann Systemwith three particles is given by:

H =1

2(a1x

21 + a1x

22 + a3x

23 + y2

1 + y22 + y2

3), (47)

which has to be considered together with the constraints:x2

1 + x22 + x2

3 = 1x1y1 + x2y1 + x3y3 = 0

The motion equations are hence given by :xi = yi

yi = −aiξi + αξi(i = 1, 2, 3), (48)

where we set as before α = a1x21 + a2x

22 + a3)x

23− y2

1− y22− y2

3 . The cotangent bundleof the configurations manifold, S2, is hence represented by the assignment of the sixdata (x,y) = (x1, x2, x3, y1, y2, y3) such that:

x21 + x2

2 + x23 = 1,

andx1y1 + x2y2 + x3y3 = 0.

Let f1(ξ) be the function:

f1(ξ) = (ξ − a1)(ξ − a2)(ξ − a3),

and define:

Ux,y(ξ) = U(ξ) = f1(ξ)3∑

k=1

x2k

x− ak

,

Vx,y(ξ) = V (ξ) =√−1f1(ξ)

3∑i=1

xkyk

ξ − ak

,

Wx,y(ξ) = W (ξ)f1(ξ)

(3∑

k=1

y2k

ξ − ak + 1

).

21

For the sake of brevity, in the following we shall write U for Ux,y, V for Vx,y, Wfor Wx,y.

Explicitly, U , V , W are given by:

U(ξ) = ξ2 + U1ξ + U2, (49)

V (ξ) = V1ξ + V2, (50)

W (ξ) = ξ3 +W0ξ2 +W1ξ +W2, (51)

where the above seven coefficients are given by the following formulae:

U1 = −[(a2 + a3)x21 + (a1 + a3)x

22 + (a1 + a2)x

23], (52)

U2 = a2a3x21 + a1a3x

22 + a1a2x

23, (53)

V1 = −√−1[(a2 + a3)x1y1 + (a1 + a3)x2y2 + (a1 + a3)x3y3], (54)

V2 =√−1(a2a3x1y1 + a1a3x2y2 + a1a2x3y3), (55)

W0 = y21 + y2

2 + y23 − a1 − a2 − a3, (56)

W1 = a1a2 + a2a3 + a3a1 − (a2 + a3)y21 − (a1 + a3)y

22 +−(a1 + a2)y

23, (57)

W2 = a2a3y21 + a1a3y

22 + a1a2y

23. (58)

We have, in this way, a map p : T ∗(S2)−→C7 defined as

(x,y) 7→ (U1, U2, V1, V2,W0,W1,W2).

We want to compute the expression UW + V 2. After some easy, although tediouscomputations one gets:

U(ξ)W (ξ) + V 2(ξ) = f1(ξ)f2(ξ),

where:

f2(ξ) = (ξ − a1)(ξ − a2)F3 + (ξ − a1)(ξ − a3)F2 + (ξ − a2)(ξ − a3)F1

having set, for notational convenience:

F1 = x21 +

(x1y2 − x2y1)2

a1 − a2

+(x1y3 − x3y1)

2

a1 − a3

,

F2 = x22 +

(x1y2 − x2y1)2

a2 − a1

+(x2y3 − x3y2)

2

a2 − a3

,

F3 = x23 +

(x2y3 − x3y2)2

a3 − a2

+(x1y3 − x3y1)

2

a3 − a1

22

Although this is not very interesting for the following, we incidentally remark thateach of the Fi’s is a prime integral of the motion. The very easy check consists just incomputing the derivative with respect to the time t, and to re-express the ”dotted”quantities by means of the motion equations (7). We remark also that the Fi’s arenot all independents, because of the relation

∑3i=1 F

2i = 1. By generically choosing

the initial data, the polynomial f(ξ) = f1(ξ)f2(ξ) has 5 = 2.2 + 1 distinct roots,so that η2 = f(ξ) represents an affine open part of a hyperelliptic curve of genus2. Hence the six-tuple (x,y) defines a point on the jacobian of the algebraic curveη2 = f1(ξ)f2(ξ); such a jacobian is embedded in C7 by means of the procedure shownbefore. As explained in the previous discussion, h2 = f1(x)f2(x) is the isospectralcurve of the Lax equation associated to the Neumann Dynamical System followingthe method shown in [13]. The eigenvector map has the same expression seen beforein the g = 1 case:

ft : (ξ, η) 7→ (U, V + η) and f ∗t OP1(1) ∼=η2 − UW + V 2 = 0

U = 0

we get η = ±V

U = 0

Since we want a divisor belonging to the open affine set Jac(C) − Q of Jac(C), weshall choose only one of the two possible determinations of η, for example η = V .

NowU = 0 ⇐⇒ ξ2 + U1ξ + U2 = 0.

Let ξ1 and ξ2 be the solutions not corresponding to branch points. If one of them,or both, were a branch point, one should study the same procedure we are goingto show and would discover that from a conceptual point of view, nothing changes.Saying that ξ1 and ξ2 are not branch points is the same as saying that V (ξ1) 6= 0 andV (ξ2) 6= 0. Let P1 and P2 two general points of C. Then the Jacobi map will be givenby:

((ξ1, η1), (ξ2, η2)) 7→(∫ (ξ1,η1)

P1

ω1 +∫ (ξ2,η2)

P2

ω1;∫ (ξ1,η1)

P1

ω2 +∫ (ξ2,η2)

P2

ω2

)(modΛ)

where (ω1, ω2) is a basis of the vector space of the holomorphic differential 1-forms,and Λ is the period lattice Per(ω1, ω2). In the case of the hyperelliptic curves it is

23

very easy to write down explicitly a basis of holomorphic differentials. In our case abasis is given by: (

ω1 =dξ√

UW + V 2, ω2 =

ξdξ√UW + V 2

).

The Jacobian flow is thence:

t 7→(∫ (ξ1,η1)

P1

ω1 +∫ (ξ2,η2)

P2

ω1;∫ (ξ1,η1)

P1

ω2 +∫ (ξ2,η2)

P2

ω2

)(modPer(w1, w2)),

and by the local nature of the problem, the “velocity” of the above flow is nothingbut:

v = (v1, v2) =

(ξ1

V (ξ1(t))+

ξ2V (ξ2(t))

,ξ1ξ1

V (ξ1(t))+

ξξ2V (ξ2(t))

).

By simple computations and using expression (50) one gets:

v1 =x1(V1ξ2 + V2) + ξ2(V1ξ1 + V2)

(V1ξ2 + V2)(V1ξ1 + V2)=

=V1(ξ1ξ2)

· + V2(ξ1 + ξ2)·

V 21 ξ1ξ2 + V1V2(ξ1 + ξ2) + V 2

2

.

Using the fact that ξ1ξ2 = U2 and ξ1 + ξ2 = −U1 one gets, at last:

v1 =V1U2 − U1V2

V 21 U2 − U1V1V2 + V 2

2

.

Now, by using the same substitutions, we shall compute the expression for v2:

v2 =ξ1ξ1(V1ξ2 + V2) + ξ2ξ2(V1ξ1 + V2

V 21 U2 − U1V1V2 + V 2

2

=

=V1ξ1ξ2(ξ1 + ξ2)

· + 12V2(ξ

21 + ξ2

2)·

V 21 U2 − U1V1V2 + V 2

2

,

getting finally:

v2 =−V1U2U1 + 1

2V2[U

21 − 2U2]

·

V 21 U2 − U1V1V2 + V 2

2

=(V2U1 − U2V1)U1 − V2U2

V 21 U2 − U1V1V2 + V 2

2

24

At this point we can use equations which say that U1 = 2√−1V1 and U2 = 2

√−1V2.

By substituting, we get at last the result ensuring us that the flow is linear:

v1 =2√−1V1V2 − 2

√−1V1V2

V 21 U2 − U1V1V2 + V 2

2

= 0,

v2 =2√−1(U1V2 − V1U2)V1V2 − 2

√−1V 2

2

V 21 U2 − U1V1V2 + V 2

2

= −2√−1.

5 Algebraic Curves and PDE: the Theory of Burchnall–

Chaundy–Krichever–Mumford–Verdier.

This theory follows the general philosophy as outlined in section 2, but the isospectralcurve arises in a fairly more complicated way, by using rings of differential operators.We outline here only some general facts, expecially in connection with two famousPDE coming from Physics, namely, the KP (Kadomtsev-Petviashvili) equation:

∂

∂x(6uux − uxxx − ut) + 3uyy = 0, (59)

and the KdV (Korteweg-de Vries) equation:

ut = 6uux − uxxx. (60)

We observe that the KdV equation is a particular case of the KP one, and that KPis only the first of a full “hierarchy” of PDE’s of the same nature. For a descriptionof these equations from a physical point of view see e.g. [5] and/or [29]. See also [12].

We shall be here very sketchy and for the reader’s convenience we shall divide thesubject in seven subsections.

5.1 The soliton solution of the KdV equation.

We begin with a “classical” exercise on KdV, which shows how the so-called solitonsolutions are related to elliptic curves. A soliton solution is a solution of the form

u(x, t) = ϕ(x− ct),

where c is a constant.

25

Substituting in ut = 6uux − uxxx , we get the following equation for ϕ:

cϕ′ = 6ϕϕ′ − ϕ′′′. (61)

By integrating, one has

cϕ = 3ϕ2 − ϕ′′ +1

2e.

In a neighborhood in which ϕ′(z) does not vanish, we can multiply the previousequation by ϕ′, getting:

cϕϕ′ = 3ϕ2ϕ′ − ϕ′′ϕ′ + eϕ′,

having finally:(ψ)2 = 3ϕ3 − cϕ2 + eϕ+ h, (62)

where ψ = ϕ′. Now, for a sufficiently general choice of the initial conditions, (62)represents the equation of an elliptic curve, whose “parametrization” is given by thefunction ϕ and ψ. By imitating what we have done in the last part of Section 3, itturns out that ϕ and ψ are necessarily given by:

ϕ(z) = 3

√34· c

9+ ℘Λ(z)

ψ(z) = ℘′Λ(z)

,

where ℘Λ is the Weiestrab ℘-function associated to a suitable lattice. Hence thesolution of the KdV equation of the required form is given by:

φ(x− ct) =3

√3

4· c9

+ ℘Λ(x− ct).

5.2 The Lax Equation associated to KdV.

The approach to KdV via Lax equations was stated by Lax ([L]) and goes as follows.Let ∂ = ∂

∂x, considered as an operator on the ring Cx of germs of holomorphic

functions near the origin. Then the theorem of Lax states that a function u(x, t)s isa solution of KdV if and only if the differential operator, known as the Hill Operator,L = −∂2 + u(x, t) obeys to the Lax Equation:

L = [L,M ],

26

where we set M = 4∂3 − 3(u∂ + ∂u) and the commutator is defined, as usual, as[L,M ] = L M −M L, being the (non commutative!) composition of operators.The above equation implies primarily that each of the eigenvalues of the Sturm-Liouville problem L(f) = λ ·f , are prime integrals of the Korteweg de Vries equation,in the sense that λ is time-independent if evaluated on a solution u. From this facta question arises quite naturally: how can we attach, if is it possible, an algebraiccurve to the Lax equation corresponding to the KdV system? Indeed, it turns outthat as a consequence of the Lax equation, one can show that u(x, t) is a solution ofthe KdV equation iff there exists a differential operator:

P = a0(x) + a1(x)∂ + . . .+ ar(x)∂r,

where r is odd and ar(0) 6= 0, such that P and L commute, with respect to theordinary composition of operators.

5.3 The Ring of Differential Operators D = Cx[∂].

The above considerations on KdV lead in a natural way to consider the ring

D = Cx[∂] = n∑

i=0

ai(x)∂i;n ∈ N

consisting of all formal polynomial in ∂ (viewed as an indeterminate) with coefficientsin Cx, where the sum is obvious, and the multiplication is the composition ofoperators (hence D is a non commutative ring). The main property of D, whichallows to bring algebraic curves into play is the following:

Proposition 5.1 . Any commutative subring of D is a finitely generated integralC-algebra ([28]).

So, for example, if u(x, t) is a solution of the KdV equation, the commutators ofL = −∂2 +u, i.e. R = P ∈ D : [P,L] = 0 is the coordinate ring of an affine integralcurve Γ, which turns out to be independent on t, and plays the role of the isospectralcurve. But in order to see the Jacobian flow, one needs the full force of the theory.

5.4 The Burchnall-Chaundy-Krichever-Mumford-Verdier The-ory.

The key concept is given by the following:

27

Definition 5.1 A Krichever Data (C, p, z, L) is the assignment of

1. an algebraic irreducible complete curve C;s

2. a non singular point p ∈ C;

3. a local coordinate z around p;

4. a non-special torsion-free invertible sheaf L of degree g − 1.

The main result is expressed by the following theorem, due to Krichever, whoseproof is rather hard.

Theorem 5.1 There is a one-to-one correspondence between a Krichever datum andthe commutative subrings of D, containing at least a monic operator and at leasttwo different operators with coprime orders modulo the conjugation relation [i.e. tworings A and A′ have to be considered the same iff there exists f ∈ Cx[∂] such thatf(0) 6= 0 and A′ = f−1Af ].

For the proof and further comments see, e.g., [28]-[33] and [24]. We notice also thatwe are dealing with a particular case of the theory : indeed more complicated variantsof the Krichever data correspond to all commutative C-algebras of D (see [33]). Theonly thing we remark here is the following very nice property of the correspondencestated in the following:

Theorem 5.2 Let D be the commutative ring of differential operators correspondingto a Krichever datum (C, p, z, L). Then there exists a C-algebra isomorphism:

ψ : Γ(C \ p, OC) 7→ D

such that each f ∈ Γ(C \ p, OC) is mapped via ψ into a differential operator, ψ(f),whose order is exactly the order of pole of f at p.

5.5 The Jacobian flow for “finite-gap solutions” of KdV.

Coming back to the discussion of the Hill operator, given any smooth function u(x, t),consider its corresponding operator L(t) = −∂2 + u(x, t). Set:

A(t) = set of all operators commuting with L(t),

28

we say that L is a finite gap operator if A(t) contains an element of odd degree ≥ 3.If this is the case, then u(x, t) is a solution of KdV, and is called a finite-gap solutionand we get the corresponding Krichever datum (Ct, Pt, zt, Lt) where C = Ct is ahyperelliptic curve. Then only Lt depends on time, which is the same as saying thatthe solutions evolves on the jacobian variety of C.

5.6 Finite-Gap Solutions for KdV.

Let us now see how to recover the finite-gap solutions of KdV for the Jacobian. LetC be a hyperelliptic curve in the Weierstrass form:

C : y2 = x(x− a1)...(x− a2g) g ≥ 2,

the ai’s being all distincts. Let p be a Weierstrass point of C and z a coordinate aroundp. Let us choose g sufficiently general points P1, . . . , Pg ∈ C, so that L = P1+...+Pg−pis non special. These data give a ring as prescribed by the theorem of Krichever. LetL ∈ J(C) and Lt a linear flow on J(C) starting by L0 = L. We get the correspondence:

(C, p, z, Lt) 7→ [At] (A0 = A),

where [, ] denotes the equivalence class modulo conjugation, [Cf. Thm 5.2]Let f : C−→P1 be, now, the rational function having in p a pole of degree 2. This

corresponds to a monic operator in each At of the class. By suitably choosing therepresentative, we may assume that Lt = −∂2 +u(x, t) and one can show that u(x, t)turns out to be a solution of the KdV equation.

5.7 The KP Equation and the Shottky Problem.

The B-C-K-M-V theory is a powerful tool also for the study of the KP equation.But perhaps the most striking facts in this study is the discovery of deep relationsbetween KP and the famous Shottky problem. This problem is, roughly, to detectjacobians of curves among all abelian varieties. One of the most interesting precisestatement of this problem was the Novikov conjecture which, in a very sloppy way,can be formulated as follows: an abelian variety is the jacobian of a curve if and onlyif its theta functions satisfies the KP equation. Spectacular proofs of this conjecturehand on the K-B-C-M-V theory, are due independently to Shiota [32] and Arbarello- De Concini [3]. For a good reading on this subject we recommend the notes byBeauville [7], and Arbarello-De Concini [4].

29

6 Appendix. Euler Equations and its Jacobian

Flow. Computations.

As is well known, the motion of a rigid body with a fixed point in the euclideanR3-space without external forces, obeys to the Euler equations:

(I1ω1 + (I2 − I3)ω2ω3 = 0(I2ω2 + (I2 − I3)ω3ω1 = 0(I3ω3 + (I1 − I2)ω1ω2 = 0

(63)

Here the coefficients I1, I2, I3, which we suppose to be all strictly positive, have thephysical meaning of the principal inertia momenta, while ω1, ω2, ω3 can be interpretedas the spatial components of the angular speed in a principal inertia frame. A solutionof the system (63) satisfying given initial data on the ωis’s, exists (at least locally)and is unique. Such a solution will give us a curve in R3, t 7→ (ω1(ts), ω2(t), ω3(t))which we call the mechanical flow of the rigid body motion. Our goal is to show howto realize the above flow as a flow on a Jacobian variety of a suitable isospectral curve.For notational convenience we change slightly the expression of the above system, bydefining some constants a, b, c and some functions of the “time” t, l,m, n as follows:

a+ b = I3a+ c = I2b+ c = I1

andl = (a+ b)ω3

m = (a+ c)ω2

n = (b+ c)ω1

The system (63) takes thence the following expression:l = a−b

(a+c)(b+c)mn

m = c−a(a+b)(b+c)

ln

n = b−c(a+c)(a+b)

lm

, (64)

which can be hence expressed as in (6) or, equivalently as in (7) (by introducing theparameter x). The main goal of this appendix, is to compute explicitly the velocity ofthe Jacobian flow s of the rigid body motion described by Lax equation (7). In orderto do this, it suffices to substitute in the above formula the computed expression forthe moving point U1(t). The computations are easy although very long. In fact oneof the most relevant computational problem arises from the fact that the isospectralcurve actually is a covering 3 : 1 of the projective line P1. For technical reasons itis easier to do computations in a different homogeneous frame (X,Y, Z), which is

30

related to the “old” one (ξ, η, ζ) in the following way:X = (b2m2+ a2n2 + c2l2)ξ − (m2 + n2 + l2)ηY = ηZ = ζ

, (65)

the inverse relations being: ξ = X+(m2+n2+l2)Y

b2m2+a2n2+c2l2

η = Yz = Z

, (66)

so that the equation of the spectral curve, in the inhomogeneous projective coordi-nates (y, z) =

(YX, Z

X

)of the open affine set X 6= 0, takes the following form:

z2 =

(H1

Ky − a2

K

)(H2

Ky − b2

K

)(H3

Ky − c2

K

)(67)

where we set, for sake of brevity:

H1 = a2m2 + a2l2 − b2m2 − c2l2 (68)

H2 = b2n2 + b2l2 − a2n2 − c2l2 (69)

H3 = c2m2 + c2n2 − b2m2 − a2n2 (70)

K = a2n2 + b2m2 + c2l2. (71)

(72)

Using this representation of C as a double covering of P1, a holomorphic form ω canbe expressed as follows (see e.g.[14]):

ω =dy√(

H1

Ky − a2

K

) (H2

Ky − b2

K

) (H3

Ky − c2

K

) . (73)

We notice that the expression (73) is exactly the pull-back of the expression (13)in the old projective inhomogeneous frame (ξ, η), so that from the use of the sameω to denote them no confusion will arise : they are two different expressions of thesame intrinsic object!

We want to computed

dt

(∫ U1(t)

P0

ωmodΛ

),

31

where the y-coordinate of U1(t) is given by :

y(U1(t)) =b2m2 + c2l2

n2(b2m2 + c2l2 − a2m2 − a2l2)). (74)

Now,

d

dt

(∫ U1(t)

P0

ωmodΛ

)=

= limt→t0

1

t− t0

∫ U1(t)

P0

ωmodΛ−∫ U1(t0)

P0

ωmodΛ

=

= limt→t0

1

t− t0

∫ U1(t)

U1(t0)ωmodΛ (75)

Let us begin by supposing that t0 is any point such that U1(t0) is not in thebranch locus of the curve C given by the equation (67). There exists then an opendisk centered in t0 and radius δ, Sδ(t0), such that for any t1, t2 belonging to Sδ(t0),U1(t1) and U1(t2) lie in the same sheet of the (double) covering of C on P1. Inthis distinguished neighborhood we can thence choose the y-coordinate as a localparameter of C. In this way we have just to compute the following expression givingthe velocity v of the Jacobian flow :

v =d

dt

∫ y(U1(t))

y0

dy√(H1

Ky − a2

K

) (H2

Ky − b2

K

) (H3

Ky − c2

K

) =

(76)

=y(U1(t)√(

H1

Ky − a2

K

) (H2

Ky − b2

K

) (H3

Ky − c2

K

) . (77)

By applying well known results of integral calculus, from (74) we get:

v =dy/dt√(

H1

Ky − a2

K

) (H2

Ky − b2

K

) (H3

Ky − c2

K

)

=

√K3

H1H2H3

dy/dt√(y(t)− a2

K

) (y(t)− b2

K

) (y(t)− c2

K

) (78)

32

Wewant to compute the final expression (78) by substituting the value of y(t) givenby (74).

It is convenient to show the computations in several steps.

Step 1. - computation of y(t)− a2

H1

y(t)− a2

H1

=b2m2 + c2l2

n2(b2m2 + c2l2 − a2m2 − a2l2)+

− a2

a2m2 + a2l2 − b2m2 − c2l2=

=a2n2 + b2m2 + c2l2

n2(b2m2 + c2l2 − a2m2 − a2l2)= − K

n2H1

.

Step 2.- computation of y(t)− b2

H2

y(t)− b2

H2

=b2m2 + c2l2

n2(b2m2 + c2l2 − a2m2 − a2l2)+

− b2

b2n2 + b2l2 − a2n2 − c2l2=

= − 1

n2H1H2

(b2m2 + c2l2)(b2n2 + b2l2 − a2n2 − c2l2)+

− b2n2(n2(b2m2 + c2l2 − a2m2 − a2l2)

after some easy manipulations one gets:

= − 1

n2H1H2

(c2 − b2)l2(a2n2 + b2m2 + c2l2) =K(b2 − c2)l2

n2H1H2

Step 3 - computation of y(t)− c2

H3

y(t)− c2

H3

=b2m2 + c2l2

n2(b2m2 + c2l2 − a2m2 − a2l2)+

− c2

(c2m2 + c2n2 − b2m2 − a2n2)=

33

= − 1

n2H1H3

(b2m2 + c2l2)(c2m2 + c2n2 − b2m2 − a2n2)+

− c2n2(b2m2 + c2l2 − a2m2 − a2l2)

and, after some easy manipulation:

= − 1

n2H1H3

(b2 − c2)m2(a2n2 + b2m2 + c2l2) =Km2(c2 − b2)

n2H1H3

.

We collect the three previous relations in the formulae (79), (80), (81).

y(t)− a2

H1

= − K

n2H1

, (79)

y(t)− b2

H2

=K(b2 − c2)l2

n2H1H2

, (80)

y(t)− c2

H3

=Km2(c2 − b2)

n2H1H3

. (81)

By means of the previous formulae, we get:

K3

H1H2H3(y(t)− a2

H1

) (y(t)− b2

H2

) (y(t)− c2

H3

) =

=K3

H1H2H3

n2H1

K

n2H1H2

K(c2 − b2)l2n2H1H3

Km2(c2 − b2)=

=n6H2

1

(b2 − c2)2l2m2.

hence1√(

H1

Ky − a2

K

) (H2

Ky − b2

K

) (H3

Ky − c2

K

) =

∣∣∣∣∣ n3H1

(c2 − b2)lm

∣∣∣∣∣ . (82)

Our next step will be to compute the expression y(t) = dy/dt. To this purpose,we observe that:

a2n2 + b2m2 + c2l2 = const. ⇒ (b2m2 + c2l2)· = −a2nn (83)

m2 + l2 + n2 = const ⇒ mm+ nn+ ll = 0. (84)

34

We also remind that the Lax equation A = [A,B] is equivalent to the first ordersystem of ordinary differential equations (64). Hence, by computing the derivative,appearing in formula (78), with respect to the “time” t, one has:

y(t) =(n2(b2m2 + c2l2)·(b2m2 + c2l2 − a2m2 − a2l2)

n4(b2m2 + c2l2 − a2m2 − a2l2)2+

− (b2m2 + c2l2)[n2(b2m2 + c2l2 − a2m2 − a2l2)]·

n4(b2m2 + c2l2 − a2m2 − a2l2)2,

and by using in an obvious way (83) and (84) :

y(t) =−2a2n3n(b2m2 + c2l2 − a2m2 − a2l2)

n4(b2m2 + c2l2 − a2m2 − a2l2)2+

− (b2m2 + c2l2)[2nn(b2m2 + c2l2 − a2m2 − a2l2)]

n4(b2m2 + c2l2 − a2m2 − a2l2)2+

− 2a2(b2m2 + c2l2)(mm+ nn+ ll)

n4(b2m2 + c2l2 − a2m2 − a2l2)2,

now, the last addendum is zero, because of (84). After some manipulation, y(t) canbe finally expressed as:

y(t) =−2nn(b2m2 + c2l2 − a2m2 − a2l2)(a2n2 + b2m2 + c2l2)

n4(b2m2 + c2l2 − a2m2 − a2l2)2=

= − 2(a2n2 + b2m2 + c2l2)n

n3(b2m2 + c2l2 − a2m2 − a2l2).) (85)

Now, one just have to substitute the expression (64) for n into (85), to get thefollowing nice relation:

y(t) =2(c− b)lm(a2n2 + b2m2 + c2l2)

n3(a+ c)(a+ b)(b2m2 + c2l2 − a2m2 − a2l2)=

2lm(c− b)

n3

K

(a+ b)(a+ c)H1

.

(86)Combining formula (78) with (79), (80), (81) and (86), we get at last the searched

expression for the velocity v of the Jacobian flow:

v =

∣∣∣∣∣ n3H1

(c2 − b2)lm

∣∣∣∣∣ · 2lm(c− b)

n3

K

(a+ b)(a+ c)H1

= sgn(c− b) · a2n2 + b2m2 + c2l2

(a+ b)(a+ c)(b+ c)

35

which does not depend on the time.It remains to consider the case in which t0 is a value of the parameter such that

U1(t0) is a branch point of the covering. In this case the integral in (77) can bethought as the limit of J(U1(t)) for t → t0 . We can consider both left and rightneighborhoods of U1(t0), and we can use again the local coordinate y to compute thederivatives. The two limits, the right and the left one, converges to the same value,whose velocity at that point, for continuity reasons, is the same as (14) . This lastremark conclude the proof, by explicit calculation, that the Jacobian flow associatedto the Lax equation (7) is linear. In [15], Griffiths describes a criterion able to decideif the Jacobian flow is linear without computing explicitly its velocity.

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