Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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case of a Hamiltonian that is a perturbation at the

origin:

ðpÞ¼

j¼1



¼ x

þ y

; j ¼ 1; ...; m

where the symplectic form is

! ¼

j¼1

^ dy

If the eigenvalues 

are assumed to be independent

over the integers (no resonances), then there is

a formal system of symplectic coordinates

j = 1, ..., m, called action-angle variables, in which

the Hamiltonian only depends of the action variables

. Such a coordinate system is generically divergent

because, under generic assumptions on the 3-jet of

the Hamiltonian, the system displays isolated periodic

orbits in any neighborhood of the origin (see Moser,

Vey, Francoise). Normal forms are normally used in

applications (e.g., Nekhoroshev theorem, Hopf bifur-

cation theorem) in their truncated versions. Birkhoff

normal form was conjectured (A Weinstein) to enter

in the asymptotic expansion of the fundamental

solution of the wave equation on a Riemannian

manifold near elliptic geodesics. This conjecture was

recently proved by V Guillemin.

Stability Theory of Hamiltonian Systems,

Nekhoroshev Theorem, Arnol’d Diffusion

The generic divergence of the Birkhoff normal form does

not allow one to conclude about the stability of the

elliptic singular point. In the case where it is convergent,

the motion is trapped inside invariant tori (conservation

of the actions). The KAM theorem (see Gallavotti

(1983)) provides the existence of many invariant tori

but, except in low dimensions, this does not exclude the

existence of trajectories that would escape to infinity.

Arnol’d indeed provided a mechanism and examples of

such situations (this is now called Arnol’d diffusion) (see

Introductory Articles: Classical Mechanics). This diffu-

sion process needs some time, which is estimated below

by a theorem of Nekhoroshev.

Consider the Hamiltonian



ðp; qÞ¼hðpÞþf ðp; qÞ

where h(p) is strictly convex, analytic, an isochro-

nous on the closure

U of an open bounded region U

of R

and the perturbation f (p, q) is analytic on U 

. Nekhoroshev’s theorem tells that there are

positive constants a, b, d, g,  such that for any initial

data p

, q

, the actions p do not change by more

than a

before a time bounded below by e

b=

(see

Gallavotti (1983)).

Bifurcations of Periodic Orbits

Consider a one-parameter family of vector fields X



of class C

, k  3,

x ¼ Fðx;Þ

Assume that X



(0) = 0 and that the linear part of the

vector field at 0 has two complex- conjugated

eigenvalues () and

() such that Re(()) > 0

for >0, Re((0)) = 0 and (Re(()))=dj

 = 0

6¼ 0.

Then, for >0 but small enough, the vector field



has a periodic orbit 



which tends to 0 as 

tends to 0.

This bifurcation of codimension 1 is named Hopf

bifurcation and it occurs in many model s.

When several oscillators (conservative or dissipa-

tive) are weakly coupled, they may display fre-

quency locking (existence of an attractive periodic

orbit) phase locking, and synchronization. The fact

that we always see the same face of the Moon from

the Earth can be exp lained by a synchronization of

the rotation of the Moon onto itself with its rotation

around the Earth. Synchronization also plays a

fundamental role in living organisms (e.g., heart,

population dynamics: see D Attenborough’s movie

‘‘The Trials of Life’’). It is sometimes possible to be

convinced of synchronization via computer experi-

ments, but the main theoretical approach is due to

Malkin. See Bifurcations of Periodic Orbits, where a

full mathematical proof is included.

Homoclinic Bifurcation, Newhouse’s Phenomenon

Homoclinic bifurcation occurs in the family X



the bifurcation value of the param eter  = 0ifX

displays a singular orbit which tends to 0 both for

t !þ1 and for t !1. In dimension 2, if  is

slightly deformed around 0, one periodic orbit may

appear (or disappear). For planar systems, the

Bogdanov–Takens bifurcation is the codimension-2

bifurcation, which mixes the homoclinic and the

Hopf bifurcations. In dimension 3, more complicated

phase diagrams may occur (such as in the Shilnikov

bifurcation) with the appearance of infinitely many

periodic orbits or homoclinic loops (in a stable way:

Newhouse phenomenon). This eventually gives rise to

strange attractors (the Roessler attractor).

The Poincare´ Center-Focus Problem, Local

Hilbert’s 16th Problem, Abel Equations, Algebraic

Moments

Hopf bifurcation theory for two-dimensional sys-

tems deals with the first case of a general situation

592 Singularity and Bifurcation Theory

often referred to as degeneracies of Hopf bifurca-

tions or alternatively Hopf–Takens bifurcations.

Consider more generally a planar vector field,

tangent at the origin to a linear focus:

x ¼ y þ x þ f ðx; yÞ

y ¼x þ y þ gðx; yÞ

The Poincare´ center-focus problem asks for

necessary and sufficient conditions on the perturba-

tion terms so that all orbits are periodic in a

neighborhood of the origin. This problem is still

pending in the case, for instance, where f and g are

homogeneous of degrees 4 and 5. It was solved a

long time ago for degrees 2 and 3. Part (b) of

Hilbert’s 16th Problem asks for finding a bound in

terms of the degrees of polynomial perturbations for

the number of limit cycles (isolated periodic orbits)

in the neighbor hood of the origin. In the case of

homogeneous perturbations, a Cherkas transforma-

tion allows the reduction of both problems to the

so-called one-dimensional periodic Abel equations:

dy=dx ¼ pðxÞy

þ qðxÞy

where p and q are trigonometric polynomials in x.

A perturbative approach was developed for several

years and yields a theory of algebraic moments

related to Livsic’s generalized problem of moments.

Fast–Slow Systems

Fast–slow systems



x ¼ f ðx; yÞ;

y ¼ gðx; yÞ

are characterized by the existence of two timescales.

Variables x are called fast variables and y are called

slow variables. Different approximation techniques

can be used (averaging method, multiscale approach

(see Multiscale Approaches)). The behavior of

solutions is approximated as follows (when the

scale  is small). The orbit jumps to an attractor of

the fast dynamics. This attractor may eventually lose

its stability and/or bifurcate as time evolves. Then

the orbit jumps to another attractor of the fast

dynamics. Once again, this attractor may evolve/

bifurcate/disappear, depending on the slow variables

y. This explains why bifurcation theory enters in the

process in a crucial way, and it has to be adapted to

this special context where some new phenomena may

occur (e.g., singular Hopf bifurcation theory,

Canards, etc.). Fundamental tools to be used in this

context are Takens theorem, Fenichel central mani-

fold theorem, blowing-up (Dumortier–Roussarie).

Excitability is also an important feature which occurs

in some fast–slow systems. Consider initial data in a

neighborhood of an excitable attractive point. For some

initial data, the orbit goes very quickly to the attractor.

For some others instead (usually below some threshold),

the orbit undergoes a long incursion in the phase

diagram before turning back to the attractive point.

Singular Hopf bifurcation, hysteresis, and excit-

ability can, for instance, occur in the electrodissolu-

tion and passivation of iron in sulfuric acid

(see Alligood et al. (1997)).

Sometimes, the orbit leaves the neighborhood of a

first attractor to jump to a second one and then this

second one disappears and the orbit jumps back to

the initial attractor as the slow varia bles have

undergone a cycle. This is called a hysteresis cycle.

In case one of the attractors is a point while the

other is an attractive periodic orbit, it may lead to

bursting oscillations. These oscillations are charac-

terized by the periodic succession of silent phases

(attractor of the fast dynamics) and active (pulsatile)

phases (periodic attractor of the fast dynamics).

They are ubiquitous in physiology, where they were

first discovered and can be also observed in physics

(laser beams) and in population dynamics.

Example

The Hindmarsh–Rose model displays bursting

oscillations:



x ¼ y  x

þ 3x

þ I  z



y ¼ 1  5x

 y

z ¼ sðx  x

Þz

The fast dynamics is two dimensional. For some values

of the parameters, it displays an attractive node, a

saddle and a repulsive focus. Under the slow variation

of z, the fast dynamics displays a saddle–node

bifurcation, a Hopf bifurcation from which emerges

a stable limit cycle which disappears into a homoclinic

bifurcation. The fast–slow system undergoes a hyster-

esis loop which yields to bursting oscillations.

Conclusions

Over the past three decades, mathematical tech-

niques gathered under the names of singularity

theory and bifurcation theory of dynamical systems

have offered a power ful means to explore nonlinear

phenomena in diverse settings. These include

mechanical vibrations, lasers, superconducting cir-

cuits, and chemical oscillators. Many such instances

are further developed in this encyclopedia.

Singularity and Bifurcation Theory 593

See also: Bifurcation Theory; Bifurcations of Periodic

Orbits; Chaos and Attractors; Entropy and Quantitative

Transversality; Ergodic Theory; Feynman Path Integrals;

Generic Properties of Dynamical Systems; Gravitational

Lensing; Homoclinic Phenomena; Hyperbolic Dynamical

Systems; Multiscale Approaches; Optical Caustics;

Poisson Reduction; Stationary Phase Approximation;

Symmetry and Symmetry Breaking in Dynamical

Systems; Symmetry and Symplectic Reduction;

Synchronization of Chaos; Weakly Coupled Oscillators.

Further Reading

Alligood KT, Sauer TD, and Yorke JA (1997) Chaos, An

Introduction to Dynamical Systems, Textbooks in Mathema-

tical Sciences. New York: Springer.

Alpay D and Vinikov V (eds.) (2001) Operator Theory, System

Theory and Related Topics, The Mosche Livsic Anniversary

Volume, Operator Theory, Advances and Applications

vol. 123. Birkhauser.

Briskin M, Francoise JP, and Yomdin (2001) Generalized

Moments, Cener-Focus Conditions and Compositions of

Polynomials. Operator Theory, Advances and Applications

123 ( in honor of M Livsic, 80th birthday).

Diener M (1994) The canard unchained, or how fast–slow dynamical

systems bifurcate? The Mathematical Intelligencer 6: 38–49.

Francoise JP and Guillemin V (1991) On the period spectrum of a

symplectic map. Journal of Functional Analysis 100: 317–358.

Gallavotti G (1983) The Elements of Mechanics. New York: Springer.

Goodstein DL and Goodstein JR (1997) Feynmann’s lost lecture.

London: Vintage.

Guckenheimer J (2004) Bifurcations of relaxation oscillations. In:

Ilyashenko Y, Rousseau C, and Sabidussi G (eds.) Normal Forms,

Bifurcations and Finiteness Problems in Differential Equations.

Se´minaire de mathe´matiques supe´rieures de Montre´al,Nato

Sciences Series, II. Mathematics, vol. 137, pp. 295–316. Kluwer.

Haken H (1983) Synergetics, 3rd edn. Berlin: Springer.

Keener J and Sneyd J (1998) Mathematical Physiology. Inter-

disciplinary Applied Mathematics, vol. 8. New York: Springer.

Malgrange B (1974) Inte´grales asymptotiques et monodromie.

Annales de l’ENS 7: 405–430.

May R-M (1976) Simple mathematical models with very

complicated dynamics. Nature 261: 459–467.

Nekhoroshev V (1977) An exponential estimate of the time of

stability of nearly integrable Hamiltonian systems. Russian

Mathematical Surveys 32(6): 1–65.

Palis J and de Melo W (1982) Geometric Theory of Dynamical

Systems, An Introduction. New York: Springer.

Perko L (2000) Differential Equations and Dynamical Systems, 3rd

edn, Text in Applied Mathematics, vol. 7. New York: Springer.

Siegel C-L and Moser J (1971) Lectures on Celestial Mechanics,

Die Grundleheren der mathematischen Wissenschaften,

vol. 187. Berlin: Springer.

Smale S (1998) Mathematical problems for the next century. The

Mathematical Intelligencer 20: 7–15.

Smale S. Dynamics retrospective: great problems, attempts that

failed. Physica D 51: 267–273.

Sobolev Spaces see Inequalities in Sobolev Spaces

Solitons and Kac–Moody Lie Algebras

E Date, Osaka University, Osaka, Japan

Introduction

Solitons and Kac–Moody Lie algebras were born at

almost the same time in the 1960s, although they

did not have a connect ion at first. They both have

roots in the history of mathematics. From the 1970s

on, they became intersection points for many

(previously known and new) results .

The notion of solitons has many facets and it is

difficult to give a mathematically precise definition;

closely related to solitons is the notion of ‘‘com-

pletely integrable systems.’’ The latter is usually used

in a much broader sense.

The terminology ‘‘soliton’’ was originally used for

a particular phenomenon in shallow water waves.

Now, in its broadest sense, it is used to represent an

area of research relating to this particular phenom-

enon in direct or indirect ways. From the viewpoint

of solitons, particular solutions of differential

equations are of special interest. Although particular

solutions have been studied for a long time, interest

in them was overshadowed by the method of

functional analysis in the 1950s. In the late nine-

teenth century, in parallel with the theory

of algebraic functions, several studies undertook

the solution of mechanical problems by elliptic or

hyperelliptic integrals. Subsequently, however, there

was a drop in activity in this area of work.

Originally it was hoped that this kind of pheno m-

enon could be used for practical applications. No

mention of practical application of solitons will be

made in this article.

First we list several topics which constitute the

main body of the notion of solitons in the early

stages; we will then explain relations with Kac–

Moody Lie algebras.

594 Solitons and Kac–Moody Lie Algebras

Birth of Solitons

The name ‘‘soliton’’ itself was coined by Martin D

Kruskal around 1965. It was originally employed for

the solitary wave solution Korteweg–de Vries (KdV)

equation



ð6uu

þ u

xxx

Þ¼0; u ¼ uðx; tÞ½1

The coefficients here are not important. We can

change them arbitrarily. The unknown function u,

or rather u, represents the height of the wave.

The solitary wave solution in question is given by

uðx; tÞ¼2c sech

ﬃﬃﬃ

ðx  ct  dÞ



½2

This is a traveling-wave solution with the height of

the wave proportional to the speed. This is one

feature of the nonlinearity of this differential

equation.

A reason for this nomenclature comes from the

particle-like property of solitary wave observed via

numerical computations. That is, if we have two

solitons [2] with different speeds, with the faster one

on the left and the slower one on the right, then after

some time they collide and their shapes are distorted.

After a long enough time, they are separated and

recover their original shapes, the only difference

being in the change of the phase shift d in [2].

Solitary waves in shallow water (like a canal)

were first observed by Scott Russell in Scotland in

the middle of the nineteenth century. Differential

equations which possess solitary waves in shallow

water as solutions were sought after Scott Russell’s

report. Boussinesq derived one (now called the

Boussinesq equation, which contains second partial

derivatives with respect to time) from the Euler

equation of water wave; then in 1895 Korteweg and

his student de Vries derived the KdV equation. They

also showed that the KdV equation possesses

solutions expressible in terms of elliptic functions.

In the 1960s Kruskal and Zabusky carried out

numerical computations for the Fermi–Pasta–Ulam

problem; they also came across the KdV equation

and found the aforementioned phenomenon.

Inverse-Scattering Method

Kruskal and his co-workers further pursued the

origin of the particle-like property of solitons and

proposed the so-called inverse-scattering method.

The inverse problem of scattering theory of the

one-dimensional Schro¨ dinger operator

L ¼



þuðxÞ

was studied by Gelfand–Dikii, Marchenko, and

Krein in the 1950s, motivated by scattering theory

in quantum mechanics.

It gives a one-to-one correspondence between rapidly

decreasing potentials u(x) and scattering data which

consist of discrete eigenvalues 

and normalization

, j = 1, ..., n, of the eigenfunctions corresponding to

them and the reflection coefficient r(). The reflection

coefficient represents the ratio of reflection of the unit

plane wave e

ix

by the potential field. The scattering

data {r(), 

, c

, j = 1, ..., n} are a mathematical ideali-

zation of observable data in quantum scattering. The

procedure of reconstructing a potential from given

scattering data is called the inverse problem. The heart

of this procedure is solving an integral equation (the

Gelfand–Dikii–Marchenko equation). In the reflection-

less case (r() = 0), this integral equation reduces to a

system of linear algebraic equations.

Kruskal and co-workers found that the scattering

data of these operators with solutions of [1] as

potentials depend very simply on t:



ðtÞ¼

ð0Þ; c

ðtÞ¼c

ð0Þe

2i

rð; tÞ¼rð; 0Þe

2i

½3

It was realized at the same time that soliton solutions

correspond to a reflectionless potential (r() = 0) with

only one discrete eigenvalue, while reflectionless

potentials correspond to a nonlinear ‘‘superposition’’

of soliton solutions (called multisoliton solutions) and

describe the interaction of solitons.

As was pointed out by Zakharov and others, the

inverse-scattering method has an intimate relation

with the Riemann–Hilbert problem.

Lax Representation

Looking at this invariance of the spectrum, Lax

reformulated the KdV equation [1] as an evolution

equation for the one-dimensional Schro¨dingeroperator:

¼½A; L; L ¼



þu

A ¼



½4

Here we have changed the sign of the operator for

later convenience. This form of representation

together with the inverse-scattering method gave a

framework for finding nonlinear differential (differ-

ence) equations that have solutions with properties

similar to solitons (soliton equations).

Among such are the sine-Gordon equation

 u

¼ sin u

Solitons and Kac–Moody Lie Algebras 595

the nonlinear Schro¨ dinger equation

þ u

þjuj

u ¼ 0

the modified KdV equation



þ u

xxx



¼ 0

the Toda lattice equation

¼ P

¼exp Q

 Q

nþ1

ðÞþexp Q

n1

 Q

ðÞ

½5

and so on. The first three are obtained by replacing

L bya2 2 matrix differential operator of first

order. For eqn [5], the linear operator corresponding

to L in the case of the KdV equation is a difference

operator of order 2 and has a connection with the

theory of orthogonal polynomials in one variable as

well as with the theory of moment problems.

Later it was remarked that the differential

operator A in eqn [4] is nothing but the differential

operator part of the fractional power of

L: A = (L

3=2

)

. By replacing A in [4] by (L

(2nþ1)=2

)

we obtain higher (nth) KdV equations.

Basic Representations of Affine

Lie Algebras

In the 1960s Kac and Moody introduced indepen-

dently a class of infinite-dimensional Lie algebras

which are in many respects close to finite-dimensional

semisimple Lie algebras. Each of them is constructed

for a given generalized Cartan matrix (GCM),

C ¼ a



; a

¼ 2; a

 0 for i 6¼ j

and if a

¼ 0 then a

¼ 0 ½6

There is a special class of Kac–Moody Lie algebras

that are now called affine Lie algebras. They

correspond to positive-semidefinite GCM and are

realized as central extensions of loop algebras

(current algebras)

C½; 

1

g

of finite-dimensional semisimple Lie algebras g.

They have many applications in physics, in parti-

cular as current algebras. The Sugawara construc-

tion in current algebra plays an essential role in

conformal field theory. Note that finite-dimensional

semisimple Lie algebras correspond to positive-

definite GCMs.

In the late 1970s, there was interest in construct-

ing representations of these algebras after the

general theory of representations was constructed.

Among them was the work of Lepowsky–Wilson,

who constructed basic representations of the affine

Lie algebra A

(1)

) using differential operators of

infinite order in infinitely many variables. These

operators were called vertex operators by Garland,

in view of the resemblance to objects in string

theory. Character formulas for these new Lie

algebras were intensively studied and many combi-

natorial identities were (re)derived.

Geometric Interpretation

How do Kac–Moody Lie algebras enter into this

picture?

In the early stages of the history of solitons

Kac–Moody Lie algebras appeared rather artifi-

cially. Some authors tried to understand solitons

from geometric viewpoints. A typical example is the

sine-Gordon equation. This equation appears as the

Gauß–Codazzi equation in the theory of embeddings

of two-dimensional surfaces of constant negative

curvature into three-dimensional Euclidean space,

while the Gauß–Weingarten equation is the linear

equation that appears in the Lax representation of

the sine-Gordon equation. Another approach of a

geometric nature, involving the prolongation struc-

ture, was the direction initiated by Wahlquist–

Estabrook. In this approach, the Lie algebra

appeared in a natural way, although the nature of

such Lie algebras was not so clear. This direction of

research is close in spirit to the method of Cartan for

treating partial differential equations.

Several authors considered generalizations of the

Toda lattice equation. Bogoyavlenskii and others

observed that the original Toda lattice equation [5]

is related to the Cartan matrix of the affine Lie

algebra of type A. Viewed in this way, it was

straightforward to generalize the Toda lattice

equation to Cartan matrices of another type of

affine Lie algebras and also to ordinary Cartan

matrices. These were typical appearances of Kac–

Moody Lie algebras in the theory of solitons; they

were used to produce soliton equations. The climax

of this is the work of Drinfel’d–Sokolov.

It needed some time to understand another role of

affine Lie algebras in the theory of solitons.

Ba¨ cklund Transformation

In the theory of two-dimensional surfaces of

constant negative curvature, a method of obtaining

another surface of constant negative curvature from

the given one with some parameter was known by

the work of Ba¨ cklund. If we apply this to the trivial

solutions u = 0 of the sine-Gordon equation, we

596 Solitons and Kac–Moody Lie Algebras

obtain a one-soliton solution of the sine-Gordon

equation. From this fact, the transformation of

solutions of soliton equations to other solutions is

called a Ba¨cklund transformation. The original

Darboux transformation is a special case of a

Ba¨cklund transformation.

Hamiltonian Formalism

Another discovery of Gardner–Greene–Kruskal–

Miura was the Hamiltonian structure of the KdV

equation. In the process of showing the existence of

infinitely many conservation laws, they used the

so-called Miura transformation, which relates the

KdV and the modified KdV equation. Faddeev–

Zakharov showed that the transformation to

scattering data is a canonical transformation, and

conserved quantities are obtained from the expan-

sion of the reflection coefficients.

Gelfand–Dikii studied Hamiltonian structures of

the KdV equation using the formal variational

calculus they initiated.

M Adler was the first to try to study the KdV

equation by using the orbit method known for

finite-dimensional Lie algebras. It was known by the

works of Kostant and Kirillov or even earlier by Lie

that the co-adjoint orbits of Lie algebras admit

symplectic structures (the Kostant–Kirillov bracket).

Adler considered the algebra of pseudodifferential

operators in one variable. This acquires the structure

of Lie algebra by the commutation relation. This

algebra admits a natural triangular decomposition

by order. He showed that the KdV equation can be

viewed as a Hamiltonian system in the co-adjoint

orbit of the one-dimensional Schro¨ dinger operator

with the Kostant–Kirillov bracket. By introducing

the notion of residue of pseudodifferential operators

he rederived conserved quantities. The work of

Drinfeld–Sokolov can be regarded as a thorough

generalization of this direction. Hamiltonian struc-

tures of the KdV equation and other soliton

equations are now understood in this way.

The method is also applicable to finite-dimensional

Lie algebras. Symes, Kostant, and others treated the

finite Toda lattice in this way.

The motion of tops, including that of Kovalevs-

kaya, was also studied in this way.

Hirota’s Method

There was another approach to soliton equations, quite

different from the above. This was the method initiated

by Hirota. He placed stress on the form of multisoliton

solutions of the KdV equation, the sine-Gordon

equation, and so on. He made a dependent-variable

transformation of the KdV equation [1],

u ¼ 2



log f

This form naturally arises when we reconstruct the

potential of the one-dimensional Schro¨ dinger

operator from the scattering data by solving the

Gelfand–Dikii–Marchenko integral equation. In this

new dependent variable, eqn [1] takes the following

form:

 4D



f ðx; tÞf ðx; tÞ¼0

where the operator D

is defined by

ðf  gÞ¼

f ðx þ x

Þ gðx  x

Þj

¼0

½7

This operator is called Hirota’s bilinear differential

operator. In such transformed form, he tried to solve

the resulting equation in a perturbative way,

f ¼1 þ

j¼1

expð2p

x þ 2p

t þ q

1j<kn

expð2ðp

þ p

Þx

þ 2ðp

þ p

Þt þ q

þ q

Þþ ½8

It is rather miraculous that in the soliton equation

case we can truncate such a perturbative procedure

at a finite point. The number of steps corresponds to

the number of solitons.

Most of the soliton equations are rewritten in

bilinear form with such bilinear differentiation after

a suitable dependent-variable transformation. (Some

equations need several new dependent variables.)

Once we have a differential equation in Hirota’s

bilinear differential form, it always has two-soliton

solutions.

Up to 1980, keywords characterizing solitons

were; inverse-scattering method, Ba¨ cklund trans-

formation, multisolitons, Hirota’s method, quasi-

periodic solutions, etc. No explicit mention was

made of representation theory.

Hierarchy of Soliton Equations

As was stated above, soliton equations viewed as

Hamiltonian systems have infinitely many conserva-

tion laws. This implies that we can introduce infinitely

many independent time variables consistently. From

this viewpoint, it is natural to consider the KdV

equation and its higher-order analogs simultaneously.

They have many properties in common. For example,

the t-dependence of the scattering data of the higher

Solitons and Kac–Moody Lie Algebras 597

KdV equation is given by replacing 

by 

2nþ1

and 

by 

2nþ1

in eqn [3]. The totality of soliton equations

organized in this way is called a hierarchy of soliton

equations; in the KdV case, it is called the KdV

hierarchy. This notion of hierarchy was introduced by

M Sato. He tried to understand the nature of the

bilinear method of Hirota. First, he counted the

number of Hirota bilinear operators of given degree

for hierarchies of soliton equations. For the number of

bilinear equations, M Sato and Y Sato made extensive

computations and made many conjectures that involve

eumeration of partitions.

Kadomtsev–Petviashvili Hierarchy

Although it was included in a family of soliton

equations slightly later, the Kadomtsev–Petviashvili

(KP) equation is a soliton equation in three

independent variables, which first appeared in

plasma physics:







ð6uu

þ u

xxx



¼0 ½9

For this equation we have to replace the Lax

representation by





þu 

;



þ v 



¼0 ½10

This form of representation was introduced by

Zakharov–Shabat. Sometimes it is referred to as

the zero-curvature representation or the Zakharov–

Shabat representation. The KP equation is universal

in the sense that it contains the KdV equation [1]

and the Boussinesq equation as special cases. If u

does not depend on y, resp. t, this gives the KdV,

resp. the Boussinesq equation.

Work of Sato

Sato stressed the importance of the study of the KP

equation. He first introduced the KP hierarchy.

Instead of the one-dimensional Schro¨ dinger operator

in the KdV case consider a pseudo- (micro)

differential operator of first order,

L ¼ @ þ u

ðxÞ@

1

þ u

ðxÞ@

3

þ

@ ¼

; x ¼ðx

; x

; ...Þ

½11

Setting B

= (L

)

, the KP hierarchy is defined by

the Zakharov–Shabat representation

 B

;

 B



¼0; m ; n ¼ 2; 3; ...

If we assume that L

is a differential operator, we

have the KdV hierarchy and the constraint that L

a differential operator gives the Boussinesq

hierarchy. This process is called reduction.

Sato found that character polynomials (Schur

functions) solve the KP hierarchy and, based on

this observation, he created the theory of the

infinite-dimensional (universal) Grassmann manifold

and showed that the Hirota bilinear equations are

nothing but the Plu¨ cker relations for this Grassmann

manifold.

Sato also gave an (infinite-dimensional) determi-

nantal formula for Hirota’s dependent variable and

called the latter the -function. Using this

-function, the wave function (the eigenfunction

corresponding to the KP hierarchy) is expressed as

wðx; kÞ¼exp

n¼1

ðx  ðk

1

ÞÞ

ðxÞ

ðkÞ¼ k;

;

; ...



Lw ¼kw

½12

where L is given by eqn [11].

Affine Lie Algebras as Infinitesimal

Transformation Groups for Soliton

Equations

Date–Jimbo–Kashiwara–Miwa found another rela-

tion among soliton equations and affine Lie alge-

bras. After noticing some similarity between the

formula in the paper by Lepowsky–Wilson on the

Rogers–Ramanujan identity using the vertex opera-

tors for A

(1)

and the formula in the computation of

numbers of bilinear operators in Sato’s paper, they

applied the vertex operator for A

(1)

XðpÞ¼ exp

j¼1

2j1

 exp 

j¼1

2j1

to 1 (which is the simplest -function for the KdV

hierarchy), where p is a parameter. They found that

the result is the -function corresponding to the one-

soliton solution of the KP hierarchy. They also

found that successive application of X(p)’s to 1

produced all multisoliton -functions. Therefore,

applications of vertex operators are precisely

598 Solitons and Kac–Moody Lie Algebras

Ba¨cklund transformations. This implies that the

affine Lie algebra A

(1)

is the infinitesimal transfor-

mation group for solutions of the KdV hierarchy.

After this discovery, it was realized that the

totality of -functions of the KdV hierarchy is

the group orbit of the highest weight vector (=1)

of the basic representation of A

(1)

The vertex operators for the KP hierarchy were

also found:

Xðp; qÞ¼ exp

j¼1

ðp

þ q

 exp 

j¼1



If we put q = p, the vertex operator for A

(1)

([12])

is recovered.

Viewed in this way the Lie algebra corresponding

to the KP hierarchy is gl

(=A

). And an embed-

ding of A

(1)

into A

was also found. Subsequently,

the method using free fermions (Clifford algebras)

was established. Frenkel–Kac had already used free

fermions to construct basic representations. In this

approach, the -functions are defined as vacuum

expectation values. Based on this connection with

affine Lie algebras, many conjectures of Sato on the

number of bilinear equations are (re)proved by using

specialized characters of affine Lie algebras.

The use of free fermions was exploited by

Ishibashi–Matsuo–Ooguri to relate soliton equations

with conformal field theory on Riemann surfaces.

This aspect was further studied by Tsuchiya–Ueno–

Yamada using D-modules.

Once such a viewpoint was established, it was

easy to construct soliton equations corresponding to

other affine Lie algebras. Hierarchies similar to the

KP hierarchies (the simplest equation contains three

variables) were also found, which correspond to Lie

algebras like go

, sp

(the BKP hierarchy, the CKP

hierarchy, and so on).

Summarizing these developments, we can say that

affine Lie algebras, or slightly larger ones like gl

appear naturally as infinitesimal transformation

groups for soliton equations and the solution spaces

are the (completed) group orbits of highest weight

vector -functions of level-1 representations. The

Hirota bilinear equations are the equations describ-

ing these orbits (analogs of Plu¨ cker relations).

Soon afterwards, the notion of -functions was

introduced in the study of Painleve´ equations by

Okamoto, revealing Hamiltonian structures in

Painleve´ equations.

The Method of Drinfeld–Sokolov

The KdV or the KP hierarchies are related to scalar

linear differential operators. A parallel treatment

using matrix differential operators is also possible.

In fact, the nonlinear Schro¨ dinger equation, modi-

fied KdV equation, the sine-Gordon equation, etc.,

are treated in this way.

Drinfel’d and Sokolov gave a general framework

along these lines. The first step is to choose the starting

(matrix-valued) linear differential operator of order

one. For that they use the language of Lie algebras.

Let us start with a matrix realization of a Lie

algebra (for an affine Lie algebra, the elements are

Laurent polynomials in one variable). Consider a

linear differential operator of the following form:

L ¼

þ qðxÞþ

where q(x) is an element of the Borel subalgebra and 

is a sum of positive Chevalley generators in the case of

affine Lie algebras. By using gauge transformations

(adjoint group), they consider several normal forms.

One normal form is obtained by choosing a node of

the corresponding Dynkin diagram. The resulting

matrix system is equivalent to the one obtained by

scalar Lax representation (or a slight generalization of

it). In this way, the generalized KdV equations for

affine Lie algebras are obtained. Another normal form

is to make q h-valued. Soliton equations obtained in

this way are called the modified KdV equations. This is

a generalization of the Miura transformation. They

also comment on the construction of partially mod-

ified soliton equations, which correspond to taking

various parabolic subalgebras. The Hamiltonian

formalism is also treated from their viewpoint.

In summary, in their approach affine algebras are

used to construct soliton equations, or one can say

that they consider the space of initial values of

soliton equations.

They also discuss two-dimensional Toda lattices

in their setting and show that modified equations in

their sense are symmetries of the two-dimensional

Toda lattices.

Common Features of the Roles of Affine

Lie Algebras in Solitons

In -function approach as well as in the method

of Drinfeld–Sokolov, the existence of triangular

decomposition of Lie algebras was essential. In the

former case, it was basic when considering highest-

weight representations and, for the latter, it was

used for the setup.

Solitons and Kac–Moody Lie Algebras 599

Special Solutions of Soliton Equations

(Multisoliton and Rational Solutions)

One of the characteristic features of soliton equa-

tions is that they allow rich special solutions.

Multisoliton solutions were the starting point of

the whole story. They directly relate to vertex

operators of affine Lie algebras.

Rational solutions (in terms of -function poly-

nomial solutions) can be viewed as degenerations of

multisoliton solutions. Motions of poles (or zeros) of

the solutions are interesting. Airault–McKean–Moser

studied the motion of poles of rational solutions of

the KdV equation and found that they are identical to

the motion of particles on a line (Calogero–Moser–

Sutherland system). This viewpoint has now been

generalized by Veselov and others.

Another discovery of Sato was that polynomial

-functions of the KP hierarchy are precisely Schur

functions (character polynomials).

In accordance with the process of reduction,

polynomial -functions of the KdV hierarchy are

Schur functions of special type.

Quasiperiodic Solutions of

Soliton Equations

As mentioned above, the KdV equation admits

solutions expressible in terms of elliptic functions.

Dubrovin–Novikov and Its–Matveev, almost at the

same time, studied solutions of the KdV equation

with periodic initial condition.

To the Sturm–Liouville (i.e., one-dimensional

Schro¨ dinger) operator with periodic potential

L ¼



þuðxÞ; uðx þ lÞ¼uðxÞ

there corresponds the discriminant, which is an

entire function of the spectral parameter. Its zeros

represent the periodic and antiperiodic spectrum 

of the operator:

ðxÞ¼

ðxÞ; f

ðx þ lÞ¼f

ðxÞ

It turns out that, except for a finite number of zeros,

other zeros are double. Such a potential is called a

finite-zone potential. These zones correspond to the

spectrum of the operator in the L

-sense. To a finite-

zone potential u(x) there corresponds a hyperelliptic

curve



j¼0

  



with simple zeros 

of the discriminant as zeros of

polynomials defining the curve. If we consider the

Dirichlet boundary value problem for the operator L,

Lf ¼ ; f

f ðs;Þ¼0 ¼ f ðs þ l;Þ

the eigenvalues are discrete and each eigenvalue 

located in a zone:



2j1

 

ðsÞ

So, for the double zeros (

2j1

= 

), the corre-

sponding Dirichlet eigenvalue 

(s) does not depend

on s.

Dubrovin–Novikov also showed that a finite-zone

potential is a stationary solution of the higher-order

KdV equation (the order being equal to the number

of nontrivial zones) and the n-zonal potentials form

a finite-dimensional integrable system. In other

words, the linear operators L, A

defining the nth

order KdV equations commute,

L; A

½¼0

In passing, it was later found that such a pair of

commuting linear differential operators was first

studied by Burchnall–Chaundy in the 1920s.

H F Baker remarked on the corresponding simulta-

neous eigenfunctions by relating them to multi-

plicative functions on algebraic curves.

The Work of Krichever

Krichever reversed the above argument, utilizing the

properties of corresponding eigenfunctions as a

function of the spectral parameter. In this approach,

we start with a compact Riemann surface C

(= nonsingular algebraic curve) of genus g. Here

we apply his method to the KP hierarchy. Take a

point P

on C together with the inverse of a local

parameter k

1

. Also take a general divisor  on C of

degree g. Consider a function (x, P), x = (x

, x

, ...),

with the following properties:

1. is meromorphic on CnP

with the pole divisor

,and

2. near P

, behaves like

ðx; PÞ¼exp

j¼1

1 þOðk

1



Such a exists uniquely and can be constructed

using the theory of abelian integrals and the Jacobi

problems on algebraic curves. Such a function was

called the Baker–Akhiezer function, since Akhiezer

constructed it by using abelian integrals and Jacobi’s

600 Solitons and Kac–Moody Lie Algebras

problem in his study of moment problems (ortho-

gonal polynomials).

It was later realized that Schur had much earlier

considered such functions in the study of ordinary

differential equations.

It is easy to show that such a function satisfies the

following linear differential equations:



n1

j¼0

ðxÞ



; n ¼ 2; 3; ...

In this way, we obtain a solution of the KP

hierarchy.

If there exists a rational function f (P)onC with

poles only at P

with singular part k

, can be

factorized as

ðx; PÞ¼exp f ðPÞ

ðx

; PÞ

where x

indicates the set of variables other than x

Consequently, we have

ðx; PÞ¼f ðPÞ ðx; PÞ

In this way, for a hyperelliptic curve C and a

branch point of it, viewed as the double cover of

, we recover the case of the KdV hierarchy.

Multisolitons correspond to rational algebraic

curves with ordinary double points, while rational

solutions correspond to further degeneration.

The study of quasiperiodic solutions of soliton

equations revealed an intimate relationship with

the theory of algebraic curves. One particular out-

come was the characterization of Jacobian varieties

among abelian varieties. This was originally posed

by Schottky and subsequently reformulated by

S P Novikov using soliton equations (Schottky

problem, Novikov conjecture). This problem was

solved through studies by Shiota, Mulase, and

Arbarello–De Concini.

Another aspect was finding commutative subalge-

bras in the ring of linear differential operators. This

problem is related to the theory of stable vector

bundles on algebraic curves.

Similarity Solutions of Soliton Equations

Ablowitz and Segur have shown that the Painleve´

transcendent of the second kind solves the KdV

equation as a similarity solution. This was the

starting point of the study of similarity solutions of

soliton equations.

Flaschka and Newell tried to construct the theory of

multisimilarity solutions. As a by-product, they

discussed modulation of the KdV equation by using

the averaging method of Whitham. This opens the

way to study the quasiclassical limit of soliton

equations. This aspect was further studied by Dubro-

vin and others in connection with topological field

theory.

Quite recently, Noumi and Yamada gave a general-

ization of the Painleve´ equation in many variables by

using the idea of similarity solutions of soliton

equations. In the work of Noumi–Yamada, the affine

Weyl group and -functions play an essential role in

constructing generalizations of the Painleve´equation.

The shift or the unit of difference corresponds to

imaginary null roots of affine Lie algebras. The idea is

further applied to elliptic Painleve´equations.

Integrable Many-Body Problems

As mentioned in relation with the rational solutions

of soliton equations, the theory of integrable many-

body problems has an intimate relationship with

the theory of solitons. Recently, Veselov and his

co-workers introduced the notion of Baker–Akhiezer

functions of many variables. This concerns a

commutative subring of differential operators in

many variables. The structure of vector bundles on

algebraic varieties of higher dimensions is quite

different from that of algebraic curves. For this

reason, a naı¨ve generalization of soliton equations to

higher dimensions is not possible. Veselov and

others have set up a class of functions which they

call multidimensional Baker–Akhiezer functions.

They are defined by giving a finite set of vectors in

a Euclidean space. The first problem is the existence.

For the existence of the multidimensional Baker–

Akhiezer function the set must satisfy several

constraints. This is quite different from the case of

solitons. Root systems satisfy these constraints and

the corresponding Baker–Akhiezer function becomes

the common eigenfunction of linear differential

operators appearing in the Calogero–Sutherland–

Moser model corresponding to root systems.

Ball–Box Systems

Satsuma–Takahashi found a soliton-like phenom-

enon in cellular automata. It took much time for a

mathematical explanation of this. Now it is under-

stand that these systems are obtained by a limiting

procedure from soliton equations. Sometimes this is

called ultra-discretization. The system thus obtained

can also be obtained from the theory of crystal bases

of affine Lie algebras. They are now called ball–box

systems.

Solitons and Kac–Moody Lie Algebras 601