who cares about integrability?

17
Physica D 51 (1991) 343-359 North-Holland Who cares about integrability? Harvey Segur Program in Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA Problems that admit solitons and are completely integrable have been analyzed in detail since their discovery by Zabusky and Kruskal 25 years ago. Their study has influenced the development of both mathematics and physics during that time. The questions addressed here are: (i) Has this subject run its natural course? (ii) Will it significantly influence the nonlinear science that develops over the next decade? Supporting examples will be chosen from several disciplines, but especially from the study of water waves. I. Introduction (b) The nonlinear Schr6dinger equation, Nonlinear partial differential equations that admit solitons and are completely integrable are of interest mathematically because they possess a rich mathematical structure. Rarely do nonlinear equations admit so much structure as is present in all "soliton equations". Some of these equa- tions are also of interest because they model various physical phenomena. This paper reviews briefly the current status of integrable equations, and attempts to identify those aspects of the subject that might be important over the next decade. There is still no adequate definition of what a "soliton equation" is, and I will base my remarks on results from three well-known exam- ples. (a) Solitons were first discovered by Zabusky and Kruskal [32] in the Korteweg-de Vries equa- tion: u, + 6uux + uxx~ = 0. (KdV) Many of the special properties of integrable equations were discovered first for this equation, including the method of inverse scattering [13], and complete integrability for an infinite dimen- sional Hamiltonian system [6, 34]. i4,t + qJxx + 210120 = 0, (NLS) was the second important equation to be solved by inverse scattering methods, by Zakharov and Shabat [37]. Their discovery showed that the KdV equation was not just an isolated curiosity. (There are actually two NLS equations, which differ in the sign of their nonlinear terms. Both are com- pletely integrable, but only this version admits solitons that vanish as Ix l ---' ~.) (c) Kadomtsev and Petviashvili [20] introduced their equation, (u, + 6uu x + Uxxx) x + 3Uyy = O, (KP) to study the stability of KdV solitons to trans- verse perturbations, but it also happens to be integrable; it was solved on x2+y2< oo by Ablowitz, Bar Yaacov and Fokas [1]. (There are also two KP equations, which differ in the relative signs of the last two terms. In the terminology of Zakharov [33], this is KP2.) An important ingredient in the development of the theory of solitons and of complete integrabil- ity has been the interplay between mathematics and physics. The three examples listed above were chosen not only because they are integrable, 0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

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Page 1: Who cares about integrability?

Physica D 51 (1991) 343-359 North-Holland

Who cares about integrability?

H a r v e y Segur Program in Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA

Problems that admit solitons and are completely integrable have been analyzed in detail since their discovery by Zabusky and Kruskal 25 years ago. Their study has influenced the development of both mathematics and physics during that time. The questions addressed here are:

(i) Has this subject run its natural course? (ii) Will it significantly influence the nonlinear science that develops over the next decade?

Supporting examples will be chosen from several disciplines, but especially from the study of water waves.

I. Introduction (b) The nonlinear Schr6dinger equation,

Nonlinear partial differential equations that admit solitons and are completely integrable are of interest mathematically because they possess a rich mathematical structure. Rarely do nonlinear equations admit so much structure as is present in all "soliton equations". Some of these equa- tions are also of interest because they model various physical phenomena. This paper reviews briefly the current status of integrable equations, and attempts to identify those aspects of the subject that might be important over the next decade. There is still no adequate definition of what a "soliton equation" is, and I will base my remarks on results from three well-known exam- ples.

(a) Solitons were first discovered by Zabusky and Kruskal [32] in the Kor teweg-de Vries equa- tion:

u, + 6uux + uxx ~ = 0. (KdV)

Many of the special properties of integrable equations were discovered first for this equation, including the method of inverse scattering [13], and complete integrability for an infinite dimen- sional Hamiltonian system [6, 34].

i4,t + qJxx + 210120 = 0, (NLS)

was the second important equation to be solved by inverse scattering methods, by Zakharov and Shabat [37]. Their discovery showed that the KdV equation was not just an isolated curiosity. (There are actually two NLS equations, which differ in the sign of their nonlinear terms. Both are com- pletely integrable, but only this version admits solitons that vanish as Ix l ---' ~.)

(c) Kadomtsev and Petviashvili [20] introduced their equation,

(u , + 6uu x + Uxxx) x + 3Uyy = O, ( K P )

to study the stability of KdV solitons to trans- verse perturbations, but it also happens to be integrable; it was solved o n x 2 + y 2 < oo by

Ablowitz, Bar Yaacov and Fokas [1]. (There are also two KP equations, which differ in the relative signs of the last two terms. In the terminology of Zakharov [33], this is KP2.)

An important ingredient in the development of the theory of solitons and of complete integrabil- ity has been the interplay between mathematics and physics. The three examples listed above were chosen not only because they are integrable,

0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

Page 2: Who cares about integrability?

344 H. Segur / Who cares about integrabilio,?

but also because each has been derived in several

physical contexts as a model of physical phenom-

ena. In fact, each equation describes approxi- mately the evolution of ordinary water waves

under the right conditions. To emphasize this

interplay between mathematics and physics, I now

describe briefly the sense in which each equation

models water waves. The interested reader can

find more details in ch. 4 of ref. [2].

lmagine a long, straight channel, with a flat,

horizontal bottom and vertical sides, containing water. At one end of the channel is a mechanical

device to create waves on the surface of the

water; these waves propagate to the other end of

the channel, where they are absorbed. The waves

that one commonly sees at the beach are caused primarily by gravity, with much weaker effects

from surface tension and viscosity; this will be the

situation of interest here.

According to the standard theory of waves of infinitesimal amplitude (e.g. ref. [24]), gravity-

induced water waves are dispersi~'e, and waves with longer wavelengths travel faster than shorter

waves. The result is that water waves emanating from a spatially and temporally confined source

tend to sort themselves out, with longer waves

moving ahead of the shorter waves. (One can

observe this phenomenon by throwing a rock into a quiet pond: in the wave pattern that radiates

outward, the long waves lead, followed by waves

of decreasing wavelength.) This suggests the fea- sibility of studying experimentally the evolution of water waves with only a narrow range of

wavenumbers, because waves with different

wavenumbers will separate from each other spa- tially.

The KdV equation was first derived by Korte- weg and de Vries [22] to describe approximately the evolution of long waves of moderate ampli- tude propagating in one direction. The KdV

equation applies in a coordinate system moving with the speed of long waves of infinitesimal amplitude; x measures horizontal distance, t rep- resents time, and u represents the displacement of the water surface above its mean level. All

three variables have been scaled appropriately.

The equation implicitly assumes that there are no variations across the (narrow) channel, and it

ignores any dissipation due to viscosity.

The nonlinear Schr6dinger equation (NLS) de-

scribes the evolution of short waves, propagating

in the same channel. Suppose one creates (per-

haps with an oscillatory paddle at one end of the

channel) a packet of waves, all with nearly the

same wavelength ("nearly monochromatic"), which is much shorter than the mean fluid depth.

One follows this packet by travelling along the

channel with the group velocity of the waves, and

NLS describes approximately the evolution of the complex envelope of the wavepacket as it propa-

gates. In NLS, x measures spatial distance in this

moving coordinate system, t represents time, and represents the complex amplitude of the packet.

As with KdV, all three variables have been scaled

properly, no variation across the channel is per-

mitted, and the effects of viscosity are ignored. The KP equation describes almost the same

situation as KdV, except that waves are no longer required to be strictly one-dimensional; the KP equation also allows weak transverse variations.

In the channel discussed above, one could imag-

ine simply moving the sidewalls outward to per-

mit transverse variations.

2. Extra structure of integrable equations

Every integrablc equation has a long list of

special properties that hold for integrable equa- tions, and only for them. We now review this list briefly; more details can be found in any book on

soliton theory, including that of Ablowitz and

Segur [2].

2.1. Solitons

The first surprise is that these equations admit exact, N-soliton solutions. (This statement holds for the examples given, but there are integrable equations that admit no bounded, localized solu-

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H. Segur / Who cares about integrability? 345

\

\

jx , \ I \

, Fig. 1. A typical interaction of two KdV solitons at four successive times.

tions, and no solitons.) A single soliton is simply a

solitary wave: a spatially localized wave of perma- nent form. For KdV, we have

U(x,t)=2K2sech2[K(X--4KZt+x,~)], (1)

where K and x 0 are constants. Solitary waves are

not unusual; what is unusual is that the KdV equation also admits exact solutions in closed

form for 2 solitons, and for N solitons, for N > 1.

Fig. 1 shows a 2-soliton solution for the KdV equation, at four different times, showing how 2 solitons interact. The two isolated waves that

emerge as t ~ + ~ are precisely those that were present as t ~ - ~ , except that each has been

phase-shifted from where it would have been had there been no interaction.

For KP, each soliton is a plane wave, an one-

dimensional, KdV-type soliton, but travelling in

an arbitrary direction in the x-y plane:

u(x,y,t) =2K 2

× sech2(K[x + .Y - (302 + 4K2), + x,,]} (2)

Fig. 2 shows a 2-soliton solution for the KP

equation. The figure is a snapshot taken at a

particular time, but as time changes this entire wave pattern simply translates.

Recall that the KP equation describes waves in shallow water; i.e. waves with wavelengths much

longer than the mean fluid depth. Fig. 3 shows a

photograph, taken by Terry Toedtemeier on the coast of Oregon, showing the interaction of two

nearly plane, nearly solitary waves in shallow

water. Note that each wave experiences a phase shift as a result of its interaction with the other

wave. No quantitative information about these

waves is available, but the qualitative resem-

blance of the wave patterns in figs. 2 and 3 is striking.

Fig. 2. A 2-soliton solution of the KP equation, for one choice of the parameters.

Page 4: Who cares about integrability?

346 H. Segur / Who cares about integrability?

Fig. 3. Oblique interaction of two nearly solitary waves in shallow water. (Photograph courtesy of T. Toedtemeier.)

For NLS, a soliton is a localized, complex wave

packet:

0 ( x , t) = aexp[ i (b 2 - a 2 ) t - ibx + c~,,]

× s e c h [ a ( x - 2bt +/3,,)]. (3)

At a fixed time (or at a fixed location), Re ¢) looks

like the solid curve in fig. 4, while 101 corre-

sponds to the dashed curve. However, the solid line in fig. 4 is not simply the graph of Re 0. Fig. 4 actually shows an experimental wave record for a packet of waves in deep water, measured by Joe Hammack in a narrow channel like that described above. The solid line is the time-history of the water surface at a fixed gauge-site. In this experi- ment, he created a packet of water waves corre- sponding to a NLS soliton with an oscillating

paddle at one end of the channel. According to NLS theory, this wave packet should propagate

down the channel without change of form. Fig. 4a

shows the wave that passed a gage 6 m down- stream of the paddle, while fig. 4b shows the same wave packet 30 m downstream. In each

case, the dashed line is the shape of the packet

with the measured peak amplitude, as predicted by NLS theory, (3). The close agreement between

the (dashed) envelope shape and the (solid) mea- sured wave amplitude shows how well NLS pre-

dicts actual waves in deep water. Note that the peak amplitude of the wave

packet in fig. 4b is smaller than that in fig. 4a.

This decrease is due to the viscosity of the water in the experiment, an effect ignored in the NLS approximation. NLS theory describes the evolu- tion of nearly monochromatic wave packets on a

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H. Segur / Who cares about integrability? 347

012

008

0-04

0

-0.04

-0-08

012

(a)

- \

\

time-scale much longer than the period of a wave, but much shorter than the time of viscous decay. Similar statements apply to KdV and KP.

2.2. Quasi-periodic solutions

Equations that admit solitons also admit exact, N-phase, quasiperiodic solutions that can be writ- ten in terms of Riemann theta functions. These solutions are generalizations of N-soliton solu- tions, which they approach in the appropriate limit. For both KdV and KP, they take the form

u = 2a~ In 0 ( 6 , , 6 2 . . . . . 6N)- ( 4 )

00. . /

00

Fig. 4. Measured displacement of water surface, showing the evolution of envelope soliton at two downstream locations. Water depth, h = 1 m; wave frequency, ~o = 1 Hz; wavenum- ber, kh = 4.0. Solid line: measured history of surface displace- ment; dashed line: envelope shape predicted by NLS. (a) 6 m downstream of wavemaker; (b) 30 m downstream of wave- maker. (Courtesy of Joe Hammack.)

Here O is a Riemann theta function of genus N, and "quasiperiodic" means that O is periodic in each of its N phase variables if the other N - 1

variables are held fixed. (Theta functions are de- fined explicitly in section 4.) For KdV,

dpj = I.t jx + ogjt + qSjo , (5a)

while for KP

(5b)

In the simplest case, N = 1, (4) is equivalent to the "cnoidal wave" found by Korteweg and de Vries [22]. Fig. 5 shows a KP solution of genus 1 for one choice of the parameters, while fig. 6 shows a KP solution of genus 2. The solution in

Fig. 5. A typical KP solution of genus 1. The solution is one-dimensional, and it is a periodic generalization of one soliton.

Page 6: Who cares about integrability?

348 tt. Segur / l~7to cares about integrahilio'?

Fig. 6. A KP solution of genus 2, near the 2-soliton limit.

fig. 6 is near the 2-soliton limit, and the sense in

which this solution is a periodic generalization of

that in fig. 2 is apparent. KP solutions of genus 2

reduce to 2 solitons in one limit, to genus-1 solutions in another limit, and to 2 Fourier modes

in a third limit. However, there is no need to be

in any of these limits, and fig. 7 shows a KP solution of genus 2 away from any of these limits.

2.3. Solution of the initial-t'alue problem

A third surprising feature of integrable prob-

lems, separate from their large families of special

solutions, is that one can actually soh'e these

equations as initial-value problems, with arbitrary

initial data in a specified class. (From this stand-

point, one could define "integrablc" to mean simply that one can integrate the equation for-

ward in time.) The method to accomplish this is called the "method of inverse scattering", or the

"inverse scattering transform". Many nonlinear equations can be solved for short times, but the

method of inverse scattering provides solutions to

integrable equations for arbitrarily long times. Moreover, the solutions that evolve from two

"nearby" sets of initial data (within the appropri-

ate class) can diverge linearly, but not exponen-

tially; i.e. there is no chaotic behaviour in these

solutions.

Fig. 7. A KPso lu t ion o f g e n u s 2 , away f ro m any particular limit.

Page 7: Who cares about integrability?

H. Segur / Who cares about integrability? 349

0.06

0.04

0"02

0

0"06

0.04

0.02

0

I1 0'06

~" 0"04

0"02

0

0"06

0.04

0"02

0

-20

1 2 3

i

(a)

I

(b)

I

(c)

i i

(d) I 2 3

I I I 1 I [ I

0 20 40 60 80 100 120 140 160 180

t(glh)t- (x-b)/h = % - r

Fig. 8. Evolution of a long, nearly rectangular, water wave of elevation into 3 KdV solitons plus radiation. Water depth, h = 5 cm. (a) wave profile at x/h=O. (b) x/h=20. (c) x/h = 180. (d) x/h = 400. Solid line: measured wave profile; dotted line: soliton profile computed from (1) [19].

The solutions that evolve from these initial

data often have simple characterizations. For ex-

ample, if the initial data for KdV are smooth, if u(x , 0) and all of its derivatives vanish rapidly as

x ~ _+~, then the solution evolves into N soli- tons, which separate f rom each other as t--* ~,

plus " radia t ion" , which decays in ampli tude as t ~ ~. (Evolution into solitons plus radiation also

occurs for more general initial data. The delicate question of exactly what initial data are allowed

has been examined by Cohen [8], Delft and

Trubowitz [9] and others.) This evolution into solitons plus radiation can

be seen in the water wave records shown in fig. 8. In this experiment, a long-wave disturbance was created at one end of a long channel, then the

evolving shape of the disturbance was measured

as it passed four successive gages down the chan-

nel. The front of the disturbance is to the left in

each of these figures, and one clearly sees the evolution of an approximately rectangular initial

disturbance into three solitons, plus decaying ra-

diation.

2.4. Addit ional structure

The list of miraculous propert ies goes on and

on. It includes: (i) the idea that a soliton equat ion is an infinite

dimensional Hamil tonian system that is com-

pletely integrable in the sense of Liouville (cf. ref.

[331); (ii) B~icklund transformations;

(iii) Lax pairs;

(iv) the Painlev~ property, and more. To summarize this section, we reiter-

ate that equat ions that admit solitons and are

integrable possess a t remendous amount of struc- ture that is absent f rom most problems, but is

present in integrable problems. All of this miraculous structure might suggest

that the study of integrable problems obviously

has a bright future, in the next decade and be- yond. In fact, its future is more uncertain, as we

now discuss.

3. Shortcomings of integrable problems

The quickest indicator of possible difficulties with the study of integrable problems is simply

scientific fashion. Fig. 9 shows the number of articles published each year that reference the

famous papers of Zabusky and Kruskal [32], and Gardner , Greene, Kruskal and Miura [13]. Amidst

all the scatter, one sees a clear downward trend over the last decade in references to these pa- pers. Twenty r e f e r ences /yea r is still far above average for scientific papers, but even so, the t rend in fig. 9 is unmistakeable. What is causing it?

Page 8: Who cares about integrability?

3511 H. Segur / Who cares about integrability?

70

60

3 0

2o " 4 '> x /x

10

l I I J I I I

1976 1978 19BO 1982 1984 1986 1988 1990

Fig. 9. Pub l i shed r e f e r e n c e s / y e a r to the pape r s by Zabusk~., and Kruskal ( lower curve) [32], and Gardne r , Greene , Kruskal

and Miura (uppe r curve) [13]. Da ta t aken from the Science Ci ta t ion Index.

I am aware of two substantive objections to

soliton theory, especially in terms of their mod-

elling physical phenomena .

(a) Most known integrable equat ions of physi-

cal interest involve only l ( s p a c e ) + l(t ime) di-

mensions. The one-dimensional solitons that

emerge from these (1 + 1)-dimensional theories

may be unstable to transverse per turbat ions in higher dimensions, and may even "col lapse" (i.e.

blow up in a finite time).

An example of a one-dimensional soliton that is unstable in two dimensions can be seen in the

experimental wave records shown in fig. 10. The wave packet in fig. 10 had the same carrier fre-

quency, the same peak amplitude, and the same

shape as that in fig. 4; both packets were in deep water. The main difference was that the tank

used in fig. 10 was wider than that in fig. 4.

Zakharov and Rubenchik [36] had predicted that the NLS soliton was unstable to long transverse

perturbations. Their unstable transverse mode was too long to fit into the narrow channel used

in fig. 4, where the soliton was evidently stable.

Their unstable mode did fit into the wider chan- nel used in fig. 10, which shows clearly an insta-

bility of the NLS soliton. Thus, if one thinks of NLS as a model of nearly

monochromat ic waves in deep water, and re- quires that the deep water be confined to a

narrow one-dimensional channel, then the theory

Fig. 11). Evolu t ion of an enve lope sol i ton ot wa te r waves in a w i d e t a n k , showing the t ransverse instabi l i ty tha t was absent i n i ig . 4. W a t e r depth , h = 1.50 m: f requency ~o = 1 Hz; tank width, b = 2.44 m. (Cour tesy of Joe t l a m m a c k . )

Page 9: Who cares about integrability?

H. Segur / Who cares about integrability ? 351

Fig. 11. Satellite photograph, showing the sun reflecting off the surface of the Andaman Sea. Each of the long, nearly parallel, curves across the lower part of the photograph is caused by a very large amplitude internal wave below the water surface. These internal waves are described approximately by the KdV equation [25].

predicts exper imental results very well. However,

there is no evidence that e i ther the one-d imen-

sional NLS theory or the cor responding one-

d imens iona l exper iments predict deep-wate r

waves on the ocean 's two-dimensional surface.

Fully two-dimensional waves in deep water are

more complicated than one might guess simply

from NLS theory; the in teres ted reader can con-

sult the review by Yuen and Lake [31] for more

details.

Not all one-d imens iona l solitons are uns table

to two-dimensional per turbat ions . Fig. 11, taken

from ref. [25], shows a satellite photograph with

the sun reflecting off the surface of the A n d a m a n

Page 10: Who cares about integrability?

352 tt. Segur / Who cares about inte,~rabilio'?

Sea. The long, dark lines across the lower part of

the photograph are the surface-signatures of very large amplitude internal waves. Osborne and

Butch showed that each of these internal waves is

approximately a KdV soliton (used in this context

to describe the evolution of long internal, rather than long surface, waves), and that the five dark

lines shown represent five solitons that evolved

from localized initial data of tidal origin. Fig. 11

shows clearly that these internal wave solitons are stable to transverse perturbations. Each soliton in

this figure is about 1 km wide, the spacing be-

tween solitons is about 5 kin, and and the lateral

extent of each soliton is about 150 kin.

Thus, some solitons (like those in fig. 11) are stable to transverse perturbations, and some (like

that in fig. 10) are not. Whether a particular soliton is unstable depends on the integrable

equation that generated it, and on the higher

dimensional problem in which it is embedded,

but the possibility of transverse instability limits the validity of many (1 + 1)-dimensional inte-

grable equations as models of physical phenom- ena. The KP equation was included in the list of

examples given above precisely because it is one

of the few known integrable equations in more

than 1 + 1 dimensions. In 3 + 1 dimensions, things can get even worse:

solutions may cease to exist after a finite time ("collapse"), signalling a complete breakdown of

the assumptions that led to the integrable prob- lem. For further discussions of collapse, see refs.

[3, 14, 351.

(b) The property of complete integrability is structurally unstable; i.e. an arbitrarily small

change in an integrable equation can destroy its integrability.

This fact is not surprising if one thinks about how many special properties are possessed by integrable equations, but it is worrisome if one thinks of the integrable equation as an approxi- mate model of some physical phenomena. If the original problem can be approximated by an inte-

grable model, it is likely that it can also be

approximated to the same accuracy by a nearby model that is not integrable. This suggests that

any results that depend fundamentally on inte-

grability cannot be very important, and it returns

us to the original question: Who cares about inte-

grability? It also brings us to the main theme of

this Conference: Will the study of integrable

problems significantly influence the development of "nonlinear science" over the next decade'?

4. A possible answer

These questions invite speculation, and the an-

swers proposed here are speculative. They begin with the following proposed definitions:

Mathematics is the study of abstract structure and

relationships.

"Abstract" means that the study is done without

concern about whether the structure studied helps to build a better widget, even though that may

have motivated the study originally. The second

definition is similar.

Physics is the study of the structure of the uni- verse we inhabit.

According to these definitions, a subject can move from mathematics to physics, or vice versa, de-

pending on current evidence about whether the structure in question can be obscrved experimen-

tally. Movement in each direction has been known

to happen.

Sciences is a search for structure which, when

found, is encoded in "laws".

These are not the usual definitions, and I was unable to find anything like them in three or four dictionaries. For this discussion, it is necessary only for the reader to concede that they contain some truth.

Page 11: Who cares about integrability?

H. Segur / Who cares about integrability? 353

If one accepts that science is a search for structure, then one is led to ask where the struc- ture might be found. This is where integrable equations enter the picture. We have seen al- ready that integrable equations possess extraordi- nary structure. They make good hunting grounds for additional structure. This idea shows up in two, essentially unrelated, ways.

(i) The property of integrability is structurally unstable, but some of the structure of an inte- grable problem persists under perturbations. In order to discern the structure of a nonintegrable problem, therefore, one strategy is to understand completely the structure of a nearby integrable problem, and then to determine which aspects of this structure persist in the nonintegrable prob- lem. In this way, integrable models provide hypotheses about the structure of nearby non- integrable models.

(ii) Problems with hidden or unexpected struc- ture often turn out to be related to integrable equations, and often in mysterious ways. Why this should be true is also mysterious (to me), but several examples are known. One is the surpris- ing connection between link invariants in knot theory, and the Yang-Baxter relations (cf. ref. [29]). Another example is the equally surprising connection between the KP equation and the theory of Riemann surfaces, to be discussed be- low.

These two ideas, (i) and (ii), are the main points of this paper. The rest of the paper con- sists of examples to illustrate them.

Example: KdV. One example of the first point can be seen in the work of Bona et al. [4] on the various perturbations of the KdV equation. They find it convenient to write the KdV equation in the form

u, + u,. + uu~ + u,.,.,. = 0. (6)

They compare this equation with two others:

u, + u x + uu x - uxx , = 0 (7)

and

U t + Ux + bl3/3x + ~xxx = O. (8)

Neither (7) nor (8) is integrable. Which proper- ties of (6) also apply to (7) a n d / o r (8)?

(i) The simplest question to ask is whether each equation admits a solitary wave. Here one substitutes

u ( x , t ) = f ( x - c t )

into the equation, and determines whether the resulting ordinary differential equation admits a localized solution. Existence of solitary waves does not imply integrability, and all three equations admit solitary waves.

(ii) A more delicate question is whether two solitary waves, initially widely separated, interact elastically like those in fig. 1. For (6), of course, the answer is affirmative. For either (7) or (8), the answer is negative, as shown numerically by Bona, Pritchard and Scott [4] and Fornberg and Whitham [12], respectively. This property is not preserved under perturbations, and it is often used as a numerical test of whether a particular equation might be integrable.

(iii) A third question is whether smooth, local- ized initial data evolve into N distinct solitary waves, plus decaying radiation. In work to appear soon, Bona and co-workers (Dougalis, Karaka- shan, Lucier) claim to have shown numerically that both (7) and (8) resolve initial data into N solitary waves plus radiation. In other words, both integrable and nonintegrable models can resolve initial data into N solitary waves plus radiation; this property is preserved under per- turbations. We might never have thought to ask about such a property without the integrable models to guide us, but the property itself does not require integrability.

In fact, this result is already implicit in the experimental data shown in fig. 8. The initial data

Page 12: Who cares about integrability?

354 H. Segur / Who cares about integrability?

there were resolved into three distinct solitons

plus radiation, but this is not the solution of some

integrable equation; it is water wave data.

Fig. 11 also shows that initial data can be

resolved into N solitary waves plus radiation

without integrability. Osborne and Burch [25] ar- gued that the internal waves shown in fig. 11 are

described with fair accuracy by the KdV equa-

tion, and that the five lines shown in the figure

are five solitons that have evolved from localized

initial data. However, the water depth in the

photo is really too large for the usual derivation

of the KdV equation to apply, the depth is cer-

tainly not uniform, and the evolution is clearly not one-dimensional. In other words, the KdV

equation is not really the governing equation

here, and these are not really KdV solitons. Even so, initial data are resolved into solitary waves

plus radiation, showing again that this property

was first discovered in an integrable system, but it

does not require integrability. The integrable model provided hypotheses about the structure of nonintegrable problems, which are confirmed ex-

perimentally in both figs. 8 and 11.

Example: the KP equation and theta functions. The last example of interesting structure in an

integrable problem is the KP equation and its

quasiperiodic solutions, (4), that are defined in terms of Riemann theta functions. The fact that this huge collection of exact solutions even exists is an example of extra structure in this integrable

equation. These solutions make the KP equation

more interesting, because they have such a wide variety of applications: to the theory of Riemann

surfaces (in mathematics), and therefore also to

string theory (in physics); quite independently, they also describe quasiperiodic waves in shallow

water. To make this concrete, it is necessary to define

a Riemann theta function. (A more complete discussion can be found in ref. [! 1].) Construction of a theta function of genus N begins with a Riemann matrix, B, an N × N symmetric matrix with negative definite real part. Next, let 4~ de-

note an N-component vector of phases,

= ( + , , ' / ' 2 . . . . . 6 , , , ) ,

for KP solutions, each phase variable should have the form given in (5b). Then let m denote an

N-component vector, where each component

ranges independently over the positive and nega-

tive integers, and zero. Then a theta function of genus N is defined by an N-fold Fourier series:

O = Y'~ ... Y'~ exp{m • B . m / 2 + im .q~}. (9)

Because O is constructed as a Fourier series, it is periodic in each phase variable separately, so it

is quasiperiodic. Because B has negative definite real part, the Fourier coefficients, exp{m. B .

m/2}, decrease so rapidly that O is an entire function of each phase variable. In the simplest

case, N = 1, B is simply a complex number, b,

with Re b < 0, and (9) reduces to

0 = Y'~ exp(bm2/2) cos(re&).

This is the case considered by Whittaker and Watson [30].

Now we turn to Riemann surfaces. A Riemann

surface of genus N is a compact surface that is topologically equivalent to a sphere with N han-

dles, and that supports a complex structure (i.e. the surface is locally indistinguishable from the

complex plane); see fig. 12. Because of its han- dles, there exist on the surface 2N independent closed curves, called cycles, and none of these

cycles can be shrunk to a point. A standard procedure in algebraic geometry starts with a Riemann surface of genus N, integrates a partic- ular set of N functions around these cycles, and constructs an N × N Riemann matrix (e.g. see ref. [11]).

Thus, associated with each Riemann surface is a Riemann theta function, constructed in this way. In fact, Krichever's [23] construction of

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H. Segur / Who cares about integrability ? 355

f

\

~--.____

,,.c___ae ",, r', Y )

Fig. 12. A Riemann surface of genus 2. The closed cycle, shown on the Riemann surface, cannot be shrunk to a point. Integration of certain holomorphic differentials around the independent cycles leads to a Riemann matrix for this surface.

quasiperiodic KP solutions starts with a Riemann surface, goes to the associated theta function, and from there to the KP solution.

The converse statement is false: not every theta function can be associated with a compact Rie- mann surface. This led to a famous problem in algebraic geometry, originally posed by Schottky [26]:

Given a Riemann theta function o f genus N, can it be associated with a compact Riemann surface?

A striking example of the structure of an inte- grable problem showing up in an unexpected way is implicit in a conjecture of S.P. Novikov: that the theta functions associated with compact Rie- mann surfaces are precisely those that generate KP solutions when substituted into (4). This re- markable conjecture turns out to be correct; its final proof was given by Shiota [28]. Its conse- quence is that the KP equation "recognizes" ev- ery Riemann surface, of every genus. Discovery of this connection has not yet led to a complete solution of Schottky's problem, in which the al- lowable Riemann matrices are defined explicitly, but it has provided an entirely new set of mathe- matical tools for the problem. One can hope that Schottky's problem will be solved explicitly within the next decade.

Next we turn to string theory. There is not enough room here, nor am I qualified, to develop adequately the basic ideas of string theory; a

standard reference is Green, Schwarz and Witten [15]. Briefly, in the path-integral formulation of quantum mechanics, one constructs a Green's function by integrating the action along a path, and then summing over all possible paths. In string theory, one replaces a (zero-dimensional) particle with a (one-dimensional) string, and in- stead of integrating over "all possible paths", one integrates over all possible Riemann surfaces. In practice, this prescription cannot be carried out at this time, because no adequate catalogue of "all possible Riemann surfaces" is available. In the future, one could imagine using KP theory to construct such a catalogue, in which case KP theory would play a fundamental role in string theory.

The structure of integrable problems has ap- peared in quantum field theory in two other ways, which we now mention.

(i) Gross and Migdal [16], Br6zin and Kazakov [5], and Douglas and Shenker [10] have made progress recently by considering a special limit of string theory, in which knowledge of "all possible Riemann surfaces" is unnecessary, and the entire theory reduces to (a discrete form of) an ordinary differential equation. Amazingly, this ODE hap- pens to be the first Painlev6 transcendent, so it is integrable!

(ii) In a review paper, Chau [7] shows that the supersymmetric Yang-Mills equations exhibit at least some of the structure of an integrable prob- lem.

Finally, we turn to the last application of the quasiperiodic solutions of the KP equation: as models of quasiperiodic waves in shallow water. Segur and Finkel [27] noted that the nondegener- ate KP solutions of genus 2 are fully nonlinear, genuinely two-dimensional, and periodic in two spatial directions. Because these solutions are the simplest KP solutions with these properties, they proposed that the 8-parameter family of KP solu- tions of genus 2 provides the simplest model of periodic waves in shallow water that is not degen- erate in some sense (i.e. not one-dimensional, not infinitesimal, etc.). Every KP solution of genus 2

Page 14: Who cares about integrability?

356 fl. Segur / Who cares about inte,~,,rabilio,?

that is genuinely two-dimensional also has the

proper ty that it is s tat ionary in some uniformly translating coordinate system.

Hammack, Scheffner and Segur [17] tested ex-

perimental ly a subset of these solutions that they

called "symmetr ic" . Fig. 13 shows a typical sym- metric KP solution of genus 2; it is character ised

by three dynamic parameters (wavelengths in the

x- and y-directions, and maximum amplitude),

and two phases. These authors used a large tank with a complicated, segmented paddle to create

waves in shallow water that approximated these

symmetric KP solutions. The waves were ob-

served to propagate as waves of nearly pe rmanen t

fl)rm, as predicted by KP theory, and to be quite stable to per turbat ions in the tank.

Fig. 14 shows a mosaic of two overhead pho-

tographs of one of these wave patterns; bright

lights, located near the wavemaker, illuminate the

water surface, so the front face of each wave crest appears dark in the photo, while the rear face

appears light. The sharp contrasts in gray levels

(light to dark) near the wavecrests indicate that the waves are quite steep at their crests. KP

theory requires small ampli tude waves, but these

waves are nearly breaking. Even so, the qualita-

tive similarity between the KP solution in fig. 13

and the observed wave pat tern in fig. 14 is clear.

Y

(a)

Y

,X

(b)

Fig. 13. A symmetric KP solution of genus 2. Every two-dimensional, symmetric solution propagates in the x-direction without change of form: (a) Contour map of surface topography: (b) perspective view.

Page 15: Who cares about integrability?

H. Segur / Who cares about integrability? 357

Fig. 14. A mosaic of two overhead photographs, showing water surface patterns. The bright, sharp lines are wavecrests, which create shadows in the wave troughs ahead of them. This pattern was observed to propagate in the x-direction as a nearly permanent wave [17].

Quantitative agreement between KP theory and these observed waves was good, but suffered be-

cause the tank bottom was not flat. This problem

was corrected in a second set of experiments,

reported by Hammack, Scheffner and Segur [18], and the agreement between theory and experi-

ment was greatly improved. Fig. 15 shows the

output from nine wave gages placed in a linear

array across the wave tank, for one of the experi- ments; it also shows the appropriate KP solution

at the same nine locations. In honesty, it should

be noted that the experiment in fig; 15 had the best agreement between theory and experiment, but the agreement was reasonably good in all of

the experiments. The reader can find more de- tails in the original papers.

The point to be made here is that the symmet-

ric KP solutions of genus 2 exhibit a simple spatial structure (a periodic array of interlocking hexagons, like those shown in fig. 13), and that structure is observed in actual water waves (as in fig. 14), even outside the putative range of validity

of KP theory. This structure apparently does not require integrability for its validity, although it

was suggested by an integrable model.

Hammack, Scheffner and Segur [18] also prop-

agated these hexagonal waves of genus 2 up a

broad, uniformly sloping beach. They found that the spatial structure of these waves was preserved

as the waves moved up the beach, even beyond

where the waves broke, and that the spatial struc- ture of the incoming waves created a spatially

periodic array of return flows along the beach,

called "rip currents". They showed that the spa- tial structure of the incoming waves determined

both the periodic spacing of the rip currents, and the narrow width of each current.

To summarize, integrable models like KP pos-

sess extraordinary amounts of structure. When one of these models is perturbed, integrability is typically lost, but not all of its structure is lost. In this way, integrable models provide hypotheses about the structure of nonintegrable models, and about actual experiments. In the sense that sci-

Page 16: Who cares about integrability?

358 H. Segur / Who cares about integrability?

I Gauge

o l

o 2 . . . . . . _ . .

0 ~ i i ~ ' ~ " ~ 3

/-/max 0 4

o

o 6

O - . . . . . . ~ _ _ : : = = - _- _- - _- - ~ ~ _ _ ± ~ m a ~ a a . . a 9

_ l z

z

I I i I

0.0 0.2 0.4 0.6 08 1.0 OJt

2:t

Fig. 15. Nine one-dimensional slices through two-dimensional water waves of genus 2, like those shown in figs. 13 and 14. Dotted lines: experimental wave profiles at nine gauge sites; solid lines: KP solution at the same nine locations [18].

ence is a search for structure, integrable models are guaranteed to play an important role in the science that develops over the next decade.

A c k n o w l e d g e m e n t s

The author is indebted to Joe Hammack, whose careful experiments have helped to shape the author's thinking about the role of integrable models in mathematics and in physics. He also thanks T. Toedtemeier for permission to use the photograph in fig. 3. This work was supported in part by NSF Grant # D M S 8822444.

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