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PROCEEDINGS OF TEE IEEE,VOL. 61, NO. 10, OCTOBER 1973 1443

The Soliton: A N e w Concept in A p p lie d ScienceALWYN C. SCOTT, F. Y. F. CHU, AN D DAVID W. McLAUGHLIN

I& Paper

Abstract-The term sditon has recently been coined to describea pulselike nonlinear wave (solitary wave) which emerges from acollision with a similar pulse having unchanged shape and speed.To tiate at least seven distinct wave systems, represeating a widerange of applications in applied science, have been found to erhibitsuch s o l u ~ n s . his review paper cove= the am en t status of solitonresearch, paying parti& attention to the very important inversemethod whereby the initial value problem for a nonlinear wavesystem can be solved exactly through a succession of linear calcn-&tiOXlS.

CONTENTSI. ntroduction11. Wave Equations that Exhibit SolitonsA. The Korteweg-deVries Equation and Its General-

B. The Self-Induced Transparency EquationsC. The Sine-Gordon EquationD. Nonlinear Lattice EquationsE. The Boussinesq EquationF. The Nonlinear Schrodinger EquationG. The Hirota EquationH. The Born-Infeld Equat ionA. KdV Solitary WavesB. Two-Soliton (Doublet) SolutionsIV. Elementary Spectral ConsiderationsA. Periodic Boundary ConditionsB. Infinite SystemsC. The Manley-Rowe EquationsA. Related Linear Problem for KdVB. Evolution of t he Scatter ing DataC. Th e Gelfand-Levitan (Marchenko) EquationD. Evaluation of + ( x , t ) forKdVby he nverse

izations

111. Elementary Soliton Calculations

V. The Inverse Method

Method: Soliton and Doublet FormulasE. @ O ( X ) - ~ X ~ ( X - X O )F. KdV as a CompletelyntegrableHamiltonianG. The Inverse Method for Self-Induced TransparencyA. Definition of Constants of the MotionB. Const ruction of an Infinite Numberof ConservationC. Applications of Conservation LawsD. Interacting Solitons

SystemVI. Constants of the Motion and Conservation Laws

Laws

This inoikd paper is cu e of a series planned m lopus of g ~ o ls-

VII. Stabi lity of Traveling WavesA. Linear Stability TheoryB. Nonlinear Stability TheoryC. Envelope StabilityD. Inverse MethodA. The Fermi-Pasta-Ulam ProblemB. Elementary Particle Theory

VIII. Fundamental Physical Theory

IX. ConclusionsAppendix A N-Soliton FormulasAppendix B TravelingWaveSolutions for theNonlinearAppendix C The Lagrangian Densi tyAppendix D Th e Gelfand-Levitan (Marchenko) Equat ion

Shriidinger Equation

I. NTRODUCTIONT E CO NCE PT of a solitary wave was introduced tothe budding science of hydrodynamics well over a cen-tury ago by Scott-Russell with the following delightfuldescription [I921 :I was observing the motion of a boat which was rapidly

drawn along a MITOW channel by a pair of horses, when theboat suddenly stopped-not so the maw of water in the chan-nel which it had put in motion; it accumulated round he prowof the vessel in a state of violentagitation, then suddenly leav-ing it behind, rolled forward withgreat velocity, assuming heform of a large solitary elevation, a rounded, smoothand well-defined heap of water, which continued its course along thechannel apparently without change of form or diminution ofspeed. I followed it on horseback, and overtook it still rollingon at a rate of some eight or nine miles an hour, preserving itsoriginal figure some hi r ty feet long and a foot to a foot and ahalf n height. Its height gradually diminished, and after achase of one or two milesI lost it in the windings of the chan-nel. Such, in the month of August 1834, was my first chanceinterview with that singular and beautiful phenomenon. . . .In 1895 Kortewegand deVries 128] provided a simpleanalytic foundation for he study of solitary waves by de-veloping an equa tion for shallow water waves (see (II.A.l)below) which includes both nonlinear and dispersiveeffectsbut ignores dissipation.Travelingwave olutions of thisKorteweg-deVries KdV)equationcan be obtainedby as-suming

44% = h(t) (I.1)

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1444 PROCEEDINGS OF TEE EEE , OCTOBER 1973

Fig. 1. (a)Travelingwave olutions to he KdV equation (II .A . l ) .(b ) A solitary wave. (c) A solitary wave with different asymptoticva l u e s a t E = + - an d [ = - - .

(1.2) then implies

and(1.3a)

(1.3b)so the original partial differential equation is reduced to anordinary differential equation which in turn can be solved byconventionalanalyticorgraphical echniques (see Section111-A, Appendix B, and [197, ch. 41). The qualitative natureof such a solution is indicated in Fig. l (a ). It consists of a ninfinite and periodic train of localized humps for which thevelocity u of the train and the spacing d between humps arecontinuouslyvariable. The solita ry wave,which so stronglyexcited the scientific and poet ic imaginat ion of Scott -Russell ,is obtained simply by lettingd+ m.This review paper studies certain special solutions of dis-persive waveequations which are called solitons. Scot t-Russells solitary wave is an example of such a solution. Inorder to attempt the efinition of a soliton we first define pre-cisely the concept of a traveling wave as follows.Definition: Given an underlying wave equation, a travelingwave h( i ) s a solution which depends upon x and t onlythrough = -ut, where u is a fixed constant.From the class of traveling waves we then pick ou t local-ized solutions, called solitary waves, which satisfy the ollowingdefinition.Definition: A solitary wave @ST([) is a localized travelingwave; or, more precisely, a traveling wave whose transitionfrom one constant-asymptotic state as [+- 00 to (possibly)another as .++ m is essentially localized in f .

with respect to ( of t he wave in Fig. l(c) has the shape indi -cated in Fig. 1 b).The soli tary wave was long considered a rather unimpor-tantcuriosity n hemathematica l tructure of nonlinearwave theory. Since it clearly is a special solution to the partialdifferential equation (PDE), many have assumed that some-what special initial conditions would be required to launch itand, therefore, that ts role inrelation o he nitialvalueproblem would be a minor one at best. Furthermore, it wasgenerally supposed that if two soli tary waves were initiallylaunched on a collision course, the nonlinear interaction uponcollision would completely destroy their integrity and iden-tity. With the development of the modern digital computer itbecame possible to test these assumptions by direct calcula-tion.

The results of the first such test were obtained in 1962 byPemng and Skyrme [181]. They were interested in the soli-tary wave solutionsof the sine-Gordon equation se e (II.C.2)below) as a simple model for the elementary part icles of mat-ter.Computerexperiments were initi ated o see how suchmodel elementary particles would scatter upon collision. Thecomputer solution indicated that the solitary waves did notscatter. They emerged fr om the collision having the same shapesandvelocities withwhich heyentered. From his computerclue, Pemng and Skryme were able to find analytic expres-sionsdescribing collision events which, i t is interesting tonote, h ad e n erived a decade earlier bySeeger, Donth, andKochendorfer [200]. Such is the vicissitudinousprocess bywhich knowledge of nonlinearwave heoryhas beendis-seminated.Shortly thereafter Zabusky and Kruskal published resultsof a completely independent comput er study of the applica-tion of the KdV equation to the investigationf plasma waves[245]. Once again the computer indicated that solitary aveswould emerge from the collision having the same shapes andvelocities with which theyentered.ZabuskyandKruskalcoined the term soliton to indicate this remarkable propertyand launched a deep and many-faceted investigation of t heconditions under which solitons shouldbe observed. The ram-ifications of this investigation are the subject of the presentreview. I t is by no meanscomplete (indeed thenumber ofpapersappear o be entering he exponential growth re-gime), but certain broad outlines are taking shape.

Although the term soliton was originally applied only tosolitary waves of the KdV equation, here are now severalnonlinearwaveequations known toexhibitsimilar effects,and the term is often used in a wider context without formaldefinition. We do not mention this as a criticism; indeed wefeel that outstanding theoretical uncertainties make it prema-ture to establish a final definition. I t is, however, necessaryfor us to indicate to the reader whatwe mean by a soliton inthis paper. Thus we present the following working definition.

Working Definition: A soliton @,(%-ut) is a solitary wavesolution of a wave equation which asymptotically preservesits shape andelocity upon collision with other solit arywaves.Th at is, given any solution @ ( x , t ) composed only of solitarywaves for large negative time,

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SCOTT et al.:THE SOLITON 1445

DlSPERSlONLESS DISPERSION

NONLINEARDWERSIOKESS DISPERSDN

(No SOLrfbRY W V E S (SCUTARY WAVES)

Fig. 2 . Conditions under which solitary waves obtain.

such soli tary waves will be called solitons if they emerge fromthe interaction with no more than a phase shift, i.e.,N+(xl t>- 4 B T ( h ) , as t ++ a

& = x - j t + 6j l 6 j const.j - 1Note that we have not explicitly introduced the conceptof stabi lity into this working definition. This is not becausewe think stability considerations are unimportant. Indeed we

feel that some aspect of stability will play a major role in thedevelopment of a fully satisfactory definition of the solitonconcept. In any ase, the asymptotic sta tements in thisefini-tion imply some degree of stability.Under this definition the simplest exampleof a soliton is apulselike traveling wave solution of the dispersionless linearwave equation1

which is amiliar o all electrical engineers. I t may seemstrange that a nonlinear and dispersive wave equation couldeven exhibita solitary wave solution, muchess a soliton. Thissituation is illustra ted diagrammatically in Fig. 2 . The effectof introducing dispersion without nonlinearity into (1.4) is todes troy the possibility of soli tary waves because the variousFourier componentsof any initial conditionswill propagate atdifferent velocities. Introducing nonlinearity without disper-sion again removes the possibility for solitary waves becausethe pulse energy is continually injected (via harmonic genera-tion) into higher frequency modes. In the time domain thisoften appears as the formation of shock waves. But with bothdispersion and nonlinearity, solitary waves can again form.The solitary wave can be qualita tively understood as repre-senting a balance between th e effect of nonlineari ty and thatof dispersion.Solitary wavesoccur also on propagating systems that arecharacterized by nonlinearity and dissipation.Again a balance

stored per unit length of candle as E (joules per mete r) andthe power required to suppor t the flame as P (joules per sec-ond), the flame travels at the fixed velocityu = P/E.

The rate at which the flame eats energy must equal the rateat which the energy is digested. The most important exampleis probably the axon or outgoing transmission line) of a nervecell. Again the velocity of propagation is determinedby apower balance condition and canbe calculated from measuredgeometric and electrical properties f the axon [197]. For thecandle i t is obvious that two flames started on opposite endswill eventually destroy each other uponollision at the center.A similar property has been demonstrated for the nerve axon[165]. Thus these soli tary waves are not solitons.In this paper we concentrate on nonlinear wave systemsin which dispersion (o r energy storage effects) dominate anddissipative effects ar e smal l enough to be neglected. One wayto characterize such a system is to demonstrate the existenceof an energy function, called a Lagrangiandensity,fromwhich the equation defining the system can be derived [89].For the convenience of the reader, a brief review of this classi-cal theory is included in Appendix c. The paper begins witha catalog of interest ing wave equations including the corre-sponding energy functions, analytic expressionsfor solitarywaves,evidence for soliton behavior, and a brief survey ofexperiments hatrevealsoli ton effects. This s followed bySections I11 and IV which outline elementary solitoncalcula-tionsand ntroduceconsideration of theFourierfrequencyspectrum of a soliton. On this basis, Sections V through VI1discuss various theoretical developments related o the solitonconcept and attempt wherever possible to ndicatedemon-strated or suspected theoretical interrelationships. Particularattention will be given to he mportantinversemethodwhereby the initial value problem for several nonl inear waveequations can be solved by l inear techniques. Finally , we in-clude some comments on the implications of soliton conceptsfor some fundamental theories in physical science.

11. WAVEEQUATIONSHAT EXHIBIT OLITONSIn his section we present a list of equations hatareinteresting from the pointf view of soliton theory. Clearly,

this li st is incomplete; we expect it to grow in future yearsassolitons become better understood. The ultimate limit of thisgrowth and the directions it may take are presently unknown;bu t it is just this uncertainty, this sense of exploring a newcontinent napplied science, which is so intriguingandexciting. We include a relatively full bibliography on poten-tial applications with the hope t ha t knowledge of the fruitsan d resources which have already been discovered will leadto rapid colonization and active exploitation.In most cases for which they are available we include asolitary wave solution and a two-soliton (doublet) solutionto indicate that the solitary waves are solitonss defined inthe Introduction. Various N-soliton solutions are collected in

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1446 PROCEEDINGSOF TBE EEE, OCTOBER 1973A . The Kmteweg-deVrks Equation and Its Generalizations

Theequation [20], [21] , [112], 159], 162], 246], 248]+ t + a++, + I$=, 0, a const II.A.l)

was first derived by Korteweg and deVries to describe thelossless propagation of shallow water waves 128], [70], [1491,[150].* I t is a useful approximation in many studieswhen onewishes to include (and balance) a simple nonlinearity and asimple dispersive effect. Such studies include 1) ion-acousticwavesn lasma 1721, [233], [219-2211;) magnetohy-drodynamicwavesnlasma [8O], [117], [US], [164],[213]; 3) the anharmonic lattice [246], [249], [251]; 4) longi-tudinaldispersivewaves nelastic ods I671 : 5) pressurewaves n iquid-gas bubble mixtures [243]; 6) rotating flowdown a tube [144] and 7) thermally excited phonon packetsin ow-temperaturenonlinearcrystals [217]. In general, arather large class of nearly hyperbolic mathematical systemshas been shown t o reduce to the KdV equation [208], [264].Since(II.A.l)can be derivedfrom heLagrangianden-sity [24O]

L = +est+- a + e+= + ++* ( 1 1 . ~ . 2 )a6where e,=+ and $=On, it ca n be considered lossless in theconventional sense (see Appendix C).As was already noted, Korteweg and deVries showed t ha t(II.A.1) asolitarywaveolutions128],ndn 965Zabusky and Kruskal published numerical results indicatingthe formation of solitons [245]-[247].More recently, hesecomputations have been extended by Tappert [218]. One ex-plicit analytic expression describing a doublet solution (i.e,,two interact ing solitons) is [162], [248]

I$=k )2 3+4cosh 2~-88t)+cosh 4x-64t)[3 cash (~-228t)+co~h (3~-36t)]* * (II.A.3)For arge t , (II.A.3)approaches he uperposition of twosolitons in the form of

1 2 K i 9+=- sech2 [Ki(Z-4Ki2t)+6i], i = 1,2; 6 i C O l l S t (II.A.4)awith ~ 1 =and ~2 = 2.

The firstevidence of solitons nshallowwaterwas re-corded by Scot t-Russel l [192]. Soliton effects have recentlybeen observed in tank experiments by Zabusky and Galvin[250], and Hammack [go]. The natureof these effects is indi-cated nFig. 3 (redrawnfrom hedata of Hammack). Aninitial disturbance of the water waves breaks up into a num-ber of individual solitons with the larger amplitude solitonstraveling at greater velocities. This process can be computednumerically by directly integrating the KdV equation. Theresult of such an integration is also shown in Fig. 3. Experi-mentally,ZabuskyandGalvin [250] how tha t his KdVmodel of shallow water waves is very accurate even for verylarge nonlinearities. We shall see in Section V that the evolu-tion of the solitons can be exactly computed through a linear

:i-CALCULATED FROU Kd YI .

YEASURED FOR WATER

I-Eix0IY INITIAL DISPLACEYENTa

t

Fig. 3. Comparison of KdV predictions and water waveobservations. Redrawn from Hammack [90] .

/ I \

LFig. 4. Nondestructive collision of ion-acoustic plasma pulses.Redrawn from Ikezi et d. 106].

calculation known as the inverse method. Determination ofthe soliton velocities is quite simple: it involves only calcula-tion of the bound state energies of a Schrodinger equationfor which the potential is the initial displacement. Similarsoliton effects have been observed by Ikezi et 02. [46], [106]-[lo8 , [93] in ion-acoustic plasma wave experiments whichcan also be described by the KdV equation [233]. The non-destruct ive collision for on-acousticwavesolitonshasalsobeen observed [46], [106]; for example, Fig. 4 (from the ex-periments of Ikezi et al.) shows he collision of two uchsolitons that are traveling in opposite directions. However, inthiscase, the ion-acousticwavesolitonsshouldperhaps be

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SCOTT et al. : THE SOLITON 1447A generalized form of the KdV equation is

41 + a4Pdz + &*+I = 0,a const; p , r nonnegative integers (II.A.5)

where

If r=2a (where a is a nonnegative integer), the Lagrangiandensity of (II.A.5) is

+ +%- ~ , a $~ , z + +%-I+z + * * +a+l$a + $+a,z2 (II.A.6)wheree, = 4+i et%, i = I, 2.

If r = 2b - , where b is a positive nteger, he Lagrangiandensity of (I I.A.5) is

+ $2&2,z$I,z + * $b&b-I,z + 3 # b 2 * (II.A.7)I t is obvious tha t when r =0, p = O [i.e., when (II.A.5) islinear nddispersionless],inear olitarywavesdo xist.Zabusky [246] has shown that solitary wave solutions existfr = 1. In this case, if p is odd, one gets a solitary wave with

sgn [amplitude of wave] = sgn [a].However, if p is even, one gets either a compressive solitarywave withsgn [amplitude of wave] = sgn [a]

or a rarefactive wave wheresgn [amplitude of wave] = - gn [a].

By phase plane analysis one can also show that no solitarywave solutions existfor r>1.Th e most simple generalization included in (II.A.5) is them o d i f i d KdV equation

4: + + 4zzz = 0. (II.A.8)Thisequationhas beenused to describeacousticwaves ncerta in anharmonic latti ces [246] and AlfCn waves in a col-lisionless plasma114]. In heir ompu ter film, Zabuskyet ul. [247] have shown that the rarefactive and compressivesolitary waves of (II.A.8) are soliton solutions. Miura [160]has shown that f 4 is a solution of (II.A.8) with a= 6, then(&++*) is a solution of the KdV equation with a= 6. Theinverse method for his modified KdV equation has been

a resonant two-level optical medium as if i t were transparent[155]-[157]. This effect has been extensively studied [131]-[138], [SO], [MI, [85] and can be physically explained as fol-lows. The time interval of an ult rashor t pulse (10-9-10-12 s)is less than the phase memory time of the atomic levels in theoptical medium; herefore, he nduced polarization can re-tain a definite phase relationship with the incident pulse. Theleading edge of the pulse then inverts the atomic populationwhile the railingedge eturns it o hegroundstate viastimulated emission. Thus heenergy ransferred from theleading edge of the pulse to the quantum system s recapturedby the trailing edge. Th e result, under proper conditions ofcoherence and intensity, is steady pulse profile which propa-gates withou t attenua tion at a velocity that can be two orthree orders of magnitude less than the phase velocity of lightin the medium.Inorder o model this effect, oneneeds (in principle)Maxwells equations to describe the light wave, and an assem-bly of quantum (two-level) atoms o describe the medium.The ight wave polarizes the atoms which, acting ogether,become a source of the electromagnet ic (EM) field. Supposethe atoms are distributed with uniform density no and heelectric field is

E ( x , t ) = &( x , t ) cos (kox - wot)where the envelope & ( x , t ) is assumed to be slowly varyingwith espect to hecarrier, cos (kox-wot). Then Maxwellsequations reduce toE t + 8, = (6) (1I.B.la)

whereforconvenience we have aken he velocity of light,and other physical parameters, to be unity. For the equationscomplete with physical parameters, we direct the reader tothe excellent survey paper by Lamb [135].To understand (6)etter [135], [139], consider a singletwo-level atom with energy levels separa ted by w =wo-&,anddenote hepolarization at ( x , t ) due o heatom byp ( h , , t ) . Now p(&, x , t ) may be approximately decomposedinto components n phase and in quadrature with he EMcarrierwave.Thus p(&, x , t ) =6(&, x , t ) sin (&x-wot)+Q(h,, t ) cos ( K o x - o & ) , and the Schrodinger equation fortheatom educes o he Bloch equations or heenvelopefunctions 6 nd Q :

6 1 &X- AwQ (1I.B.lb)Q t = A w 6 (1I.B.lc)

3 t t = -&E6 (1I.B.ld)where X(&, x , t ) denotes the population inversion for singleatom. We can now define

(6) no! a g(Aw)@(Ao, x , t )dAwwhere no is he atomic density and g(&) describes the un-certainty in the energy levels.

-sa

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1448 PROCEEDINGS OF THE IEEE, OCTOBER 1973equations that have recently been developed by Lamb. Thismethod is discussed in Section V-G. A doublet solution is

2/r1 sech X1+2/72 sech X Z1- B(tanh X1 anh X z - sech X1 sech X ,)& = A (II .B.2)

whereTq2 - 1A =71%+ 7 2 2

1- 1+ T i tU i (ri2(Aco) + 1).Here (l/[ ~i* (Au )*+ l]) ndicatesanappropriatelyweightedaverage over g(Au). The doublet breaks up into two isolatedpulses of the form

gi = 2/7i sech- t-z), i = 1 , 2 (II .B.3)T i

traveling at velocities ul and ut.Eilbeck, Gibbon, Caudrey, and Bullough [56] have ndi-cated that self-induced transparency equations can be writ-ten for the total (instantaneous) electric field E ( # , t ) , ratherthan for just a slowly varying envelope & ( x , t ) . Nozaki andTaniuti [I711 have discussed a nonlinear interaction of threeplane waves, which is simila r to that described by (II.B.l) ,and have shown how the formalism can be applied to plasmawave propagation.C. The Sine-Gwdon Equation

One can obtain he familiar sine-Gordon equation fromthe self-induced transparency equations (II.B. l) in the limitAw4). Precisely, the transformationA o + O

g ( A 4 + (Au>8 +4t6+- in 4t + t

( t - 4 + (II.C.1)yields

4= - tt = sin 4 (II.C.2)which is the sine-Gordon equation [IS], [186], [195], [197].Sincehis quation an be derived romheLagrangiandensity

2) Bloch wall motion of magneticcrystals [16], [53] , [59];3)propagation of a splaywavealonga lipid membrane[63]; 4) a unitary theory for elementary particles [ 5 8 ] , [185],[204], [205]; and 5 ) propagation of magnetic flux o n aJosephson line [130], 143], [194],[196],[198].It is interesting to note that by a simple change of inde-pendentvariables, hesine-Gordonequationcan be trans-formed into4=t+ 4tt = sin 4 (II.C.4)

which has beenuseful as a guide to solutions of the self-induced transparency equations (II.B.l) [6]-[8], [54], [104],[131]-[135], or intoq5zt = sin 4 (II.C.5)

which wasstudiedmanyyearsago nconnectionwith hetheory of pseudospherical surfaces (surfacesof constant nega-tiveurvature) [57]. EvidentlyII.C.l),II.C.4),nd(II.C.5) share many common properties since they are equiva-lent except for an independent variable transformation.Solitary wave solutions of the sine-Gordonquation(II.C.2) which correspond to a rotation in 4 by 2 r (as x goesfrom - 1 to += ) have the form4 = 4 tan- [exp ( * - )] (II.C.6)JG -2

in which the + sign corresponds to a positive sense of rotationand the pulse can be considered a soliton, while the - signindicates a negative sense of ro tation and the pulse can beconsidered an antisoliton. Since total rotation mustbe con-served, the difference between the number of solitons and thenumber of anti soli tons must be conserved n any collision.They are created and destroyedn pairs. In addition, the sine-Gordonequation is invariant o a Lorentz ransformation.Thus the solit ary wave solutions f (II.C.2) do indeed exhibitmanyproperties of theelementaryparticles of physics. In1953, Seeger, Donth, and Kochendorfer, using this equationto describe the motion of a crystal dislocation, derived ana-lytic expressionsfor solit onsoli ton and forsoliton-anti-soliton collisions [200], which were independently obtainedby Perring and Skyrme in 1962 [181]. The solitonso litoncollision is given by

4 u sinh ( % / d l - u)tan- (11.C. 7)4 cash (uf/dl- u)and the soliton-antisoliton collision is given by

4 sinh ( u f / d m )t an - = * (II.C.8)4 u cash dl - u)I t is interesting and impor tant to notice that soliton and anantisoliton can, according o (II.C.8), pass through each otherwithoutmutualdestructioneven houghsuchdestructionwould not violate conservation of total rotation.

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SCOTT et a! . :THE SOLITON 1449Physical evidence of such solitons has been obtained fromstudies of magnetic flux propagationon uperconductingtransmission lines shunted by Josephson-type unnel unc-tions 178], [196], [73], 74], 76].Eachsoliton epresents

a single quantum of magnetic flux. Thequantum 'fluxshuttle" described byFulton,Dynes, ndAnderson 75],[77] and recently realized by Fulton and Dunkleberger [78]uses this effect to obtain a shift register. A mechanical analogof (II .C.2) is useful to demonstra te hese pulse interact ions[195], [197].D . Nonlinear Lattice Equations

The electrical engineer is familiar with the lat tice of masspoints connected by springs as-a mechanical analog fo r thelow-pass electric filter. The physicist is interested in such asystem to model the propagation of sound waves through acrystal latti ce. The Toda lattic e" equat ions [222]-[227]

d2ynm- = a[eWbrn - b r*+ l ] , n = 1, 2; a, b const ( I I . D . l )dt2I n = yn - n-1

have been used by Toda [222]-[227] to describe motion on aone-dimensional la ttice of mass points interacting through anonlinear potential

a ab b#(I,,) =- +- + urn-- ( I I .D .2 )

This potential has wide applications since, by varying a an db , one can go from the harmonic limit(ab inite, a+ m , b-0)to hehardsphere imit (a b finite, b - a , a 4 ) . The La-grangian of (II.D.l) ism a aL = - n z -- -brn+- +m+l. ( I I .D .3 )n 2 b b

The function14 = -- / a y n = e - h - 1a

which is the force between the mass points, is a solitary wavesolution in the form ofl m4 ab=-- 2 sech2 [ 3 ( ~ n Bt) + 63, K , 6 const ( I I .D .4 )

whereP2 = 4ab/m sinh2~ / 2 .

Toda [226] has found a doublet solution for4. He shows thatthisdoubletbreaksupnto tw o solitons symptotically.Ooyama and Sait8 [I751 have numerically demonstrated thatthe solitary wave solution for is a soliton. They also discussnumerical studies of stability.Toda [223] hasshown hat II.D.1)has periodic wave

Hirota recently presented N-soliton solutions for the non-linear electric filter equations [99]d 2dt2and [98]-log (1 + V n ( t ) ) = Vn+l(t) - V n ( t ) + J'n-I(t) ( I I*D.S)

Equation (11.D.S) is identical to (II.D.l) and the N-solitonsolutions have the same form as those for the KdV equation(II.A.1). Equations (II.D.6) describea 'self dual" filterforwhich the N-soliton solu tions have he same form as thosefor the modified KdV equation (II.A.8). See Appendix A fordetails.Suzuki, Hirota, and Yoshikawa [209] have described howa two-soliton interact ion can be appl ied to provide coding fora secure communic?tion system. I n this scheme, the two in-coming soliton pulse trains are phase (position) modulated,and then allowed to interact ona nonlinear transmission line.This nteraction wave s ransmitted and can be separatedagain at th e eceiver into two modula tedwaves if i t is appliedto an ia!entical nonlinear line.For urther discussion of nonlinear filters we refer thereader o heaccompanyingpaperbyHirotaandSuzuki[95 , 1961.E . The B owsinesq Equation

From a suitable distributed limit of (II .D .l) , Toda [226]has obtained the Boussinesq equation

4 r r - t t + 6(4')zr + 4rzrr = 0 (II .E.1)which was first derived to describe shallow-water waves prop-agating in both directions [X?], [228]. I t is also used to de-scribe a one-dimensional nonlinear attice [246]. Recently,Hirota [lo01 showed by numerical comput ations that (II.E. l)possesses soliton solutions.He also displaysananalyticalformula for an expression of a doublet solution [1o0]:

~ K I ~ech2X l - k i ~ 2 ~ech2X2-I-A sech2X I ech2 X 2[cosh B+sinh B t anh X I tanh X2I2b ,0 f

K; const; i = 1 , 2 (II .E.2a)wherexi = $ ( K iX - it )

B i = _+ K i ( 1 + K i2 ) '"6% - 2)' - KI - 2 ) ~ ~ 1 ~ ) ~( 8 1 + 82)' - KI + ~ 2 ) ~ ~ 1 + K,)'e4 B = -

A = sinh B [ ~ ( K I ~2 ~ )in h B + 3 ~ 1 ~ 2osh B ] .

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1450 PROCEEDINGSOF THE IEEE, OCTOBER 1973

n1Y

W0zs

where81- = Yl82- = 7 2

B61- = 71 + -2B62 - = 7 2 --2

and the velocity of the soliton is f 1 +~ci)l*.F. The Nonlinear Schrddinger Equation [I91

can be derived from the Lagrangian density

where 4* is the complex conjugate of 4. This equation hasbeen used to describe 1) stationary two-dimensional self-focusing of a plane wave [22], 124], 210]; 2) one-dimen-sional self-modulation of a monochromatic wave [213], 191,11161, [91], [92 ]; 3) the elf-trappingphenomena of non-linear optics [116]; 4) propagation of a heat pulse in a solid[217]; 5) Langmuir waves in plasmas [72], [105], [202]; an d6) is related to the Ginzburg-Landau equation of supercon-ductivity [84]. Envelope solitary wave solutions to this equa-tion are derived in Appendix B and take the form

where u. and uc are the envelope and the carrier veloci-ties, respectively. These must satisfy the inequality u,>2&and hus cannot be equal.Numerical nvestigations of col-lision events between these envelope solitary waves indicatethat they are envelope solitons [ 2 4 ] . 4 An example of such acollision, computed by Tappert, is shown inFig. 5, where theamplitude 141 is displayed.Hasegawa and Tappert [91], [92] have recently discussedthe results of theoretical calculations supported by numericalsimulations which indicate hat a typical glass fiber wave-guide will support envelope solitons of the nonlinear Schrod-ingerequation at power levels of about 1 W. Numericalsimulations indicate that suchpulses are stable under the in-fluence of smallperturbations, argeperturbations,whitenoise, or absorption over distances of the order of 1 km. Thisdevelopment could lead oanenormous increase indatahandling capacity of a fiber optic bundle.G. The Hirota Equation

Very recently Hirota [ loll introduced the equation~ t + i 3 a I ~ 1 2 ~ z + & 1 3 + i u ~ Z Z I + ~ I ~ ( 2 ~ = 0 0 ,

ap = u8 ( I I .G. l )and presented N-soliton solutions which are discussed in Ap-pendix A. Equation (II.G.1) is very interesting because withu=u=O it reduces o henonlinearSchrodingerequation( I I .F . l ) , with p = 8 = 0 it reduces to the modified KdV equa-tion, and with a = 6 = 0 i t reduces to the linear equation

* t +&tz+ b4132 = 0. (11. G.2)Hirotas N-soliton solutions exclude the case p = u =O .The imposing generality of ( I I .G. l ) leads one to suspectsomerelationwith he general form of the nverse methodrecently announced by Ablowitz et al. [2].H. he Born-Infeld Equation

Finally, heequation

was derived in three spatial dimensions by Borns a nonlinearmodification of the Maxwell equations to permit the electronto appear in a natural way as a singularity [3O], [SI] , [M I,[62], [173], [182]. Equation (II .H.1) can be derived romthe Lagrangian density

L = [l + 4.2 - t ] l / 2 (II .H.2)where # J ~ epresents the magnetic ield and r # ~ ~he elect ric ield.Traveling waves of unit velocity

4 = &(z 5 t ) (II .H.3)will render the Lagrangian density constant and thus satisfy( I I .H.1 ) for any suitablycontinuous) unction h . In hiscase, th e velocity of the solitary wave is fixed, but the sk p Cisarbitrary.Recentstudies ndicate hat,after a collision

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SCOTT et al. :THE SOLITON 1451111. ELEMENTARYOLITONALCULATIONS B. Two-SolitonDoublet)olutions

In this section we briefly sketch some of the computationa ltechniques th at have proved useful in ferreting ou t solitons.The first task is to determine whether suitable solitary wavesolutions exist [197]. Th e second task is to see if these solitarywaves are solitons, i.e., whether they preserve their shapes andspeeds after a collision. One way to begin checking for thisproperty is toook for two-soliton r doublet solutions. Eventu-ally, of course, i t will be necessary to check any such solutionsfor stability. Often computer results can indicate stability.A . K d V Solitary Waves

Starting with the KdV equation (II.A.1) and assuming atraveling wave solution with velocityu as indicated by (1.1)-(1.3), we have the ordinary differential equation4rb4 - 4 + 4IIE = 0 ( I I I .A.l )

where ~ = + ( X - U # ) =$([). This equation can be integrated toobtaina4I I = K1+ 244 - 42 (III.A.2)

where K 1 s a cons tant of the first integration. The secondintegration may be effected after multiplying both sides of(III.A.2) by 41.Then

$ 4 1 ~ Kz + K14 + 2 t$2 - 4* (III.A.3)and the general traveling wave solution can be written in theform of an elliptic integral 1381

U a

(III.A.4)where $0 is the value of 4 a t ( x - u t ) =0, and

aP ( 4 ) 2Kz + 2K14 + 244- +*. (III.A.5)3Since a solitary wave is localized, i ts first and second deriva-tives qvst vanish as [+ m. Thiscondition, ogetherwith(III.A.2) and III.A.3), equires K1=0 and K t = O . Then(III.A.4) is readily integrated to yield the solitary wave

4(z - ut) = - ech2[$ x - ut)]. (III.A.6)3ua[More general solutions of (III.A.4) include a periodic travel-ing wave with the form indicated in Fig. 1. This may be ob-tained by taking nonzero values for the constants K 1 and K tand suitably adjust ing the locat ion of the zeros of P ( 4 ) ; hedetails of such an adju stm ent are discussed in Appendix B.]From (III.A.4)-(1II.A.6), however, i t is possible to see somebasic properties of KdV solitary waves.1) The amplit ude of a KdV soliton increases with its ve-

1) Numerical computations can provide a direct answer tothe question: Is this solitary wave a soliton? Initial condi-tions can be selected which correspond to two solitary waveson a collision course. If they emerge from the collision withtheir initial shapes and speeds, they cane considered solitons.Earlycalculat ions of this effect were donebyPerringandSkyrme [181] and by Zabusky and Kruskal [245]. A recentexample of such a calculation for the nonlinear Schrodingerequation (II.F.l) by Tappert is shown in Fig. 5 .2) Dependent variabk ransformations provide nterestingand helpful simplifications of many nonlinear wave equationswhich exhibit olitons.Such ransformations eem tobearsomerelation to he Cole-Hopf transformation[47],[103],4= +2(log f).,hich reduces the Burgers equation, &+#.+&=O, to the linear diffusion equation f t + f , , 1.0. For ex-ample, the transformation

4 = - 200g f> zz (III.B.1)reduces the KdV equation, (II.A.1) with a= 6, to

f f . t - z f t +f-f - fzzzjz + 3fn2 = 0. (III.B.2)Onemight well wonderwhether III.B.2) s ndeed a re-duced form of (II.B.1). Note, however, that if f is a solutionof (III.B.2), then any constant timesf is also a solution, andeven more, so is a ( t ) f ( x , t ) . Thus, although (III.B.2) s notlinear, i t shares one of the properties of a linear equation.Hirota [94] has obtained an N-soliton solution of the KdVequation from a stu dy of (III.B.2); this solution is listed inAppendix A. I t is interesting to no te that all of the N-solitonequations in Appendix A are related to dependent variabletransformations similar to (III.B.1).

3) Backlundransformationechniques [131]-11331,[126], 199], 200], 13] haveproveduseful ndevelopingN-solitonsolutionsfor hesine-Gordonequation.Since theturn of the century it has been known tha t if 40 s a solutionof the sine-Gordon equation (II.C.S), then 41,which satisfiesthe first-order pair [57], [69]3 ( h z- JO,+) = a sin (0-O) (III.B.3a)

(III.B.3b)also satisfies (II.C.5). Equations (III.B.3) define a Backlundtransformation which generates a new solution t$l from anold solution 40 .Clearly the new solution can be taken asthe old in a subsequent ransformation and a hierarchy ofsuch solutions can be sequentially developed. For he sine-Gordon equation, each transformation can introduce an addi-tionalsolitonwith a velocitydeterminedby heparticularvalue chosen for a . Thus N-soliton formulas can be explicitlydeveloped for the sine-Gordon equation (see Appendix A and[I31 for details). Recently, McLaughlin and Scott [158] have

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1452 PROCEEDINGS OF THE EEE, OCTOBER 1973IV. ELEMEN TA R YPEC TR A LS I D ER A TI ON S d

Habit uated as we are to the app lication of Fourier trans-form echniques n problems of linear wave dynamics, it isnatura l to consider the frequency spectrum of a soliton. Anunderstanding of spectralproperties is important orbothstabil ity theory and collision theory. One must tread warilyin this direction,however. The re are important ifferences be-tween the spect ra of nonlinear systems and hose of linearsystems; for example, Fourier components do not appear tobe particularly useful for the constructionof solutions to non-linear wave equations. They may, however, be helpful for thest udy of such solu tions. The purpose f this section is to indi-cate some of the difficulties and possibilities of a careful theo-retical development.A . Periodic Boundury C o n d i t w n sConsider first the spectrum of a linear wave system withperiodic boundary conditions shown in Fig. 6(a). Any solu-tion of theformexp [ i (Rx+wt) ] musthave a propagationconstant R which is an integer multiple of 1/R . Thus we canwrite

K = K J = Z/R, Z = 0, f , f2. (1V.A.l)For a linear system, the propagation constantwill be relatedto the frequency bya dispersion equation w = w ( R ) which maybe multivalued. If the number of branches is finite (or evencountably infinite), then the frequency spectrums countablyinfinite;Thesituation squitedifferent for a nonlinearsystemwhich can support a periodic traveling wave with adjustablevelocity and pwwd as indicated in Fig. l(a). Adjus ting d=2.rRgives a periodic traveling wave as an exact solution. If thevelocity of this wave is adjusted to the value u, as indicatedin Fig. 6(b), the corresponding spectrum il l be the countablyinfinite set

Note, however, th at (IV.A.2) indicates he frequency spec-trum of only a single traveling wave. Spectra of other travel-ingwavescan be obtainedbyadjusting hecontinuouslyvariable velocity u. Or, conversely,given any frequency w ,chosen from the continuum of real numbers and a fixed valueof R, we can find values for m an d u which satisfy (IV.A.2).This observation can be expressed in terms of a specificexample. Consider the KdV equation in the form

4 t + a M S + 4 , = O (IV.A.3)with periodic boundary conditions as indicated in Fig. 6(b).For a =0, (IV.A.3) is inear; it ha s a dispersion equationw = P and a Fourier spectrum with the cardinality of the in-tegers. For a>O, any real number can belong to the Fourierspectrum of solut ions of (IV.A.3). (See [23] or a precisedefinition of the concept of cardinal ity.) A similar statementcan be made for other nonlinear wave equations that haveperiodic solutions with adjustable velocity and period as indi-

Fig. 6. Wave systemswith periodicboundary conditions.(a) Linear. (b) Nonlinear.

&Fig. 7. Two-soliton nteraction.

Since 4 is a traveling wave, only terms or which 2 = m appearand we can write4(2 - &) = AIeiZ(*ML)/B (IV.A.4)

Iwhere

A t = - 2 r R S I R 4(z - Ct)e-it(*YC)/Bd(x - ut). (IV.A.5)Consider next the possibility of a two-soliton interactionas shown in Fig. 7, where 4~ is a localized soliton (i.e., 2rR>>soliton width) raveling with velocity UA , and is anotherlocalized soliton with velocity UB. According to our definitionof theerm soliton, nd 4~ must emerge fromheireventual collision with he ame hapesand velocities at

which they entered . The effect of the interac tion is to intro-duce (possibly different) time delays TA and TB. The interac-tion of two solitons of th e nonlinear Schrodinger equation,obtained by Tappert and shown inig. 5 , is a typical exampleof the growing body of numerical evidence ha t such solutionsexist.The basic temporal period of 4~ is [(~~R/uA)+TA]. Thisexpression can be considered the definition of T A . The corre-sponding frequency spectrum is

while the spectrum of #B is

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SCOTT et al. : THE SOLITON 1453periodic function [SI, [48 quencypectra.husheystem is excitednly a t the com-

+ ( x , t ) = Almnei(k+wn*t) (IV.A.8)where kt = /R andG~ w A ~ + w B ~ .We can insure that one ofthe frequency components of t $ ~oincide with those of +B byrequiring that the ratio W A ~ / W B ~ ill be an irrational numberN. The temporal period of t$(x, t ) is then infinite,and

1Armn = lim -$ J-,,, (x , 0T + m 2 r R T - T ~

2,m.n

TR TI 2

. e - i (kz*+unr t )dxdt . (IV.A.9)The ra tio of pulse velocities will be

#A U A ( N T A - B )U B 2 r R_ -N + * (IV.A.lO)

Thus, with an infinitesimal adjustment of R, heatioOA,/OB,, can be maintained rrational while the ratio U A / U Btakes any value from the continuum of real numbers.B . Infinite Systems

Let us now a tte mpt to describe the spectrum of the soli-tar y wave of Fig. 6(b) in the limit as R+ 00.I t is customaryto denote 1/R by dk, and 1/R by k after which, as R+m,(IV.A.4) and (IV.A.5) formally become the Fourier integralpair

+()= -$ A ( K ) e W k (IV.B.1)A (K) = $ + ( t ) e -* U i (IV.B.2)

where we haveubstitutedheravelingwave ariable=x-ut. Equat ion (IV.B.1) can be considered eithera spatialor a temporal transform of the solitary wave depending uponwhether we hold t or x fixed in f .Consider now the case of two interacting solitons describedby the almost periodic function (IV.A.8). For any finite valueof R, (IV.A.8) is periodic in x but not in . As R+ 01, he fre-quency spectrum includes elements fromIV.A.6) representingt$~,lements from (IV.A.7) representing +B, and the sum ordifference frequencies representing the nonlinear soliton inter-action. In the limit we might suppose that one rationally re-lated subse t of the frequency continuum represents t $ ~ ndanother represents t$~.rom (IV.A.lO) these two spect ra willbe rationallyunrelated if the atio U A / U B is an rrationalnumber.In this sense, then,we suggest that the temporal spectrumof a single soliton is not necessarily the frequency continuumas is formally mplied by (IV.B.1) and (IV.B.2), but mightbe a rational ly related subset of tha t continuum. If the non-linearity can be expressed as a power series, this speculationseems reasonable for then the rational numbers form a closedset of frequency components.

1 27r -

-m

bined frequenciesn

G j = m lw1 +m2w2 + * * + mNwN = miwi ( Iv.c .1)i- 1

where the wits on the right-hand side are the fundamental fre-quencies for he ndividual solitons, he ms range over heintegers, and is an index on all such combinationsof the ms.Assuming that the soliton wave equation can be derived froma Lagrangian density insures the conservation law for energy(C.5). Thus a time averaged power flow can be defined asl T(P) lim - J 0 Pdt. (IV.C.2)

T - c TConservation of energy requires that the total time averagedpower flow into any segment of the x axis (say between x1and xq) must be zero. Thus

( P ( x 1 ) )- P ( x 2 ) )= 0 (IV.C.3)which can be written in the form

i w jwhere ( P i ) s the time averaged power flow at frequency Gj.Since the frequencies w i can be independently varied, the indi-vidual sums in (IV.C.4) must be zero if they are independentof the wis. Tha t this is the case for wave systems which canbe derived from a Lagrangian density function was demon-strated by Sturrock [207] and also in a very complete reviewby Penfield [180]. Thu s we can write the ManleyRowe egua-tions in the form

m l ( P j ( x ) ) = c1i ;j. . . . . . . . . . . .

where Cl - * * CN are constants of t he motion.As Weiss 235] and Brown 36]havepointed out , t isoften nteresting to considerManley-Rowe equations fromthe point of view of quantum theory. If we consider that thepower flowing a t frequency G j is carried bya stream of photonsof energy h G j (where h is Plancks constant divided by 2r ),then the first equation of (IV.C.5) becomesm l r j ( z )= const (IV.C.6)

where r j ( x ) is the number of photons at frequency G j whichpass the point x per unit time. Physically (IV.C.6) can be in-i

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1454 PROCEEDINGS OF THE EEE, OCTOBER 1973Thus heconstants Cl * . CN which appear n (IV.C.5)seem to be connected with an accounting for the nergies ofphotons as they mix andmultiplyaccording oquantumrules. Such constantsdo not require that the initialN solitarywaves remain intact after collisions.

V. THE NVERSEMETHODOne of the mpor tantcontributionsmadebyGardner,Greene,Kruskal,andMiura osoliton heorywas hede-velopment of an inverse method whereby the initial valueproblem for the KdV equation can be solved through a suc-cession of linearcomputations [79]. Thismethodwas soonexpressed in an elegant and general form by Lax [141].Wewill begin by summarizing the general formulation of Lax un-

encrusted with analytical details. We trust this brief surveywill make the detailed constructions of this section easier tofollow.We are nterested in a general nonlinear wave equationrepresented abstractly by41 = K(4) (V. )

where K denotes a nonlinear operator on some suitable spaceof functions. Suppose we can find linear operators L and Bwhich depend upon 4, a solution of PDE (V.l), nd satisfythe operator equation

iLf BL - LB. W . 2 )When B is self-adjoint, (V.2) utomatically implies that theeigenvalues E of L, which appear in

L$ = E$ W.3)are independent of time. This is true even though L dependsupon time through 4. (We will see that there is a correspon-dence etweenhese onstant igenvalues ndhe fixedvelocities of the solitonswhich evolve from the ata. ) Further-more, the eigenfunctions $ may be shown to evolve in timeaccording to

i$ t = B$. W.4)I t is sometimes possible to associate a scattering problemwith the linear operator L. When this is the case, given the

data + ( x , 0) , we can find 4 ( x , t ) through the following pro-cedure.1) DirectProblem: Calculatescatteringparameters suchas the reflection and transmission coefficients of L ) for $ a t1 x I = m and t = 0 rom a knowledge of +(x, 0).2) Time Evolution of the Scattering Data: Use (V.4), o-gether with the asymptotic form of B a t x = m, to calculatethe time evolution of the scatt ering data .3) Inverse Problem: From a knowledge of thescattering

data of L as a function of time, construct + ( x , t ) .The inverse method can be represented diagrammaticallyas in Fig. 8. The idea is to avoid path d , Le., solving (V.l)directly, by traversing instead paths a, b, and c which involve

1 z z lSCATTERINGAT X . Q. MRECT

( V.3)

Fig. 8. Diagram of the inverse method.

In spite of these difficulties, the inverse method solves avariety of evolution equations. Before considering the inversemethod in some detail, we summ arize the equat ions that itcovers to date. At the time of this writing, its breadth is bestcataloged in thepreprint of Ablowitz et d . [2]. Stronglymotivated by the work of Zakharov and Shabat 12521 on anonlinearSchrodingerequation, heseauthors consider thelinear problemLv = T v

where

and he coefficients ( q ( x , t ) , r ( x , t ) ) are arbitrary. Assumingthe operator B takes the general form

where the coefficients (a , b , c) are also arbitrary, (V.4) akesthe form

Take the t derivative of (V.5) nd the x derivative of (V.8),while demanding that the eigenvalues 1: be independent of t(and x ) . Consistency of the cross derivatives yields three con-ditions on the arb itrary coefficients:aaa x_ - qc - b (V.9a)

a b a9a x at+ 2icb = i- 2 q (V.9b)

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SCOTT et d . THE SOLITON 1455the conditions (V.9) yield

qt - 6rqqz + qzzz = 0 (V.lOb)~t - 6rqrZ + Y Z ~ Z= 0. (V.0c)

When I = -1, (V.10) educes to heKdVequation;whenI = &q,(V.10) reduces to the modified KdV equation. On theother hand, the choices

yield the sine (sinh)-Gordon equation(V.l lb)

Th e choicea = 2{2 + rq (V.12a)

yieldsiqt + q,, - 2q3r = 0irt - ,, + 2q2r = 0. (V.12b)

When I = q* , (V. 12)reduces to henonlinearSchrodingerequation.Thus, through their approach, Ablowitz et a2. have set upa general inverse framework encompassing the KdV equation,the modified KdV equation, a coupled pair of KdV equa-tions, hesine-Gordonand generalizedsine-Gordon equa-tions, and a nonlinear Schrodinger equation. I t is interestingto note that there is a striking similarity between their ap-proach to he nverse method and he Backlund formalismof classical transformation heory [69]. This imilarity spart icularly apparent in Loewners st udy of the infinitesimalBacklundransformationuringhearly 1950s [147],[148]. I t appears conceivable hat his similarity will yieldageometr ical nterpretation of the inversemethod. In hisconnection, woparticularlygood eferences o n Backlundtheory are [MI, [153].In view of these esults, we see tha t these volutionequations possess remarkably similar propert ies. Along theselines, we remark that Hirota [ lo l l has recently found an N -soliton formula for the Hiro ta equat ion which encompassesthe nonlinear Schrodinger and he modified KdV equation:$ t + 3a I 4 124z+ PL + i c ~6 4 124= 0;

a, , u, 2 0; ap = 6u (V.13)by a dependent variable ransformation (see Section11-G).In this section we analyze the original inverse method forthe KdV equation in detail. I t is the simplest and the bestunderstood of the inverse methods. Corresponding results forthe nonlinear Schrodinger have been published by Zakharovand Shabat [252], or hemodifiedKdVbyWadati 230],[231]andTanaka [211], or thesine-GordonbyAblowitzet al. [ l ] , and for the equations of self-induced transparency

A . Related Linear Problem for K d VConsider the initial value problem for the KdV equation(II.A.l) with a chosen to be -6:

4: = 6 4 4 2 - zz zd x , 0) = 4 0 ( 4 . (V.A.l)We seek solutions+(x , t ) with an infinite number of x deriva-tives, all of which tend to zero as x+* m . It can be shown[141],[203],[254] that KdV has a unique solution of thistype provided 4 0 ( x ) is sufficiently smooth and vanishes rap-idly enoughas x+ * 00.Guided by Gardner t al . [79], we nowpull from the ai r a family of t he second-order differentialoperators L(t)

a=ax2L ( t ) = -- 0 . (V.A.2)Here t indexes the family and &(x , t ) is a solution ofKdV.6[Technically the operators act in the Hilbert spaceof Lebes-guesquare ntegrable unctionsover ( - < x < a).]At present it is an unfortunate characterist ic of the in-verse method that the appropriate linear operator must bepulled out of the air. Only in a recent example (see SectionV-G) s the choice of the inear operator motivated by hephysical problem.The eigenvalue problem for the operator L s-922 + $ ( x , t ) $ = E$ (V.A.3)$ = $ ( x , E ( 0 , 0 . (V.A.4)

where

Note hat for fixed t we have he well-knownSchrodingerequation for a particle n he potential 4 ( x , t ) . Since thesolutions + ( x , t ) ofthe KdV equation are smooth, bounded,and tend to zero as x+ m , this Schrodinger equation willhave at most a finite number of negative energy bound states( E = -K,,*, n = 1, 2, * , N ) and a continuousspectrum forpositive E (E=k2, R real). For fixed t , we define scatteringsolutions of (V.A.3) as shown in Fig.9by heboundaryconditions$ ( x , K , ) = e--ik0+ R(K, )e*, as x ++ m

= T ( k , )e--iLz, as x + - 0 (V.A.5a)and the bound state solutions y the boundary conditionsha(%,K n ( f ) , 4= e-xn(f)z as x + + m

= C , ( K , , ( ~ ) ,)e+r*( t )z , as x--+ - a. (V.A.Sb)Th e direct probkm may be stat ed as follows. Given 4, findthe catteringparameters [ N ; K,,, C,, (for A = 1, * , N ) ;R(K) , T(K) (for O < R * < a) ] . For heSchrodingeroperator,the following inverse problem has been solved. Given the scat-tering data [ N ; ,,, C,, for n= 1, - * * , N ) ; R(R) ( O < k 2 < m)],find 4 for all x .

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1456 PROCEEDINGS OF THE EEE, OCTOBER 1913B . Evolution of the Scattering Data

In this section we consider n detail he calculations forpath b of Fig. 8. This nvolvesshowing hat (V.2) mplies(V.4) and requires the eigenvalues of L to be independent oftime. We will continue to use the KdV equation as an explicitexample of the method.Consider the Schrodinger operator L (V.A.2) where 4 isthe solution of the KdV equation (V.A.1). Notice first tha t ifB is chosen to be the linear self-adjoint operator

then heoperatorequation (V.2) is atisfied. To proceedfurther, it is convenient to define an evolution operator U ( t )through the equation

iU t = B U (V.B.2)with he nitialcondition U ( 0 )=I, he dentityoperator.Since B is self-adjoint ( B = B t ) , U is unikrry ( U t U = U U t=I). This unitarity will yield the fundamental resul t hatthe bound state energies ( - ~ , , ' ( t ) ) are independent of time.To obtain this result,we represent the operatorL n termsof the evolution operator U ( t ) :

L( t ) = U(t)L(O)U t ( t ) . (V .B .3 )T o verify this representation oneneedsonly to show tha tU ( t ) L ( O )U t ( t ) satisfies (V.2). This follows directly rom(V.B.2) and its adjoint.We now use the representation of L ( t ) in (V.B.3) to calcu-late the time evolution of the scattering data. First consideranybound state at time f = O , $, , (x , K ~ ( O ) ,0), withboundstate energy (-~,,'(0)). By definition, the function $,, solvesthe eigenvalue problem

L(o)$n(z, Kn(O), 0)= - K n v n ( z , Kn(o), 0).Operatingwith U ( t ) givesU(t)UO)$m(z, Kn(O), 0)

= - n*(O) U(t)$'n(Z, K n ( O ) , 0 ) (V .B .4 )which can be written as[ U O ) L ( o ) ~ t ( t ) l [ ~ ( t ) ~ k ( z ,n(O), 011

= L(O[U(Wn(z, Kn(O), 0)l= - n2[U(f)$n(X, Ks(O) , O)]. (V .B .5 )

Thus if $, , (x , K , , (O ) , 0) is a bound state with energy ( -K , , * (O) )at time t =0, then U(t)iG;(x,K, , (O) , 0) is a bound state with thesame energy (-~,,*(0)) for t # O . Running he ame proofbackwards shows th at if +,,(x, K,,, t ) is a bound state withenergy (-K,,') t time t , then Ut(t>$,,((r, ,, , t ) is a bound statewith the same energy at t = 0. The number N of bound statcsand the bound st& c i g e n o d u w ( - K,,')o not c h a n g e with t as4 ( x , t ) molacs according to the K d V e q u d i e n ; they are deter-mined by the initial data &(%). These eigenvalues may be

- eJk

Fig. 9. Scattered wave8 of the related linear operator.

be representedby$n(% Kn, t ) = u(f)$n(z, Kn, 0).

Calculating the time derivative f $,,((x,K ,,, t ) , we obtain

Notice that as x+ - m ,B = -4iaa/ar' since 4 and & vanish.This implebehavior of B , togetherwith heasymptoticbehavior of $"(z, ~ ,) [in (V.A.Sb)], shows that as z+- m ,then (V.B.6) reduces toaat- n(Kn, f ) = - Kn'Cs(Kn, f ) .

This immediately yieldsCm(Kn, t ) = Cn(Kn, o)e-Gn't (V.B.7)

where C,,(K,,, 0) is specifiedby the initial data (n ) .Finally, we mustdetermine he imeevolution of thereflectioncoefficient R(R, t ) for the scattering (plane wave)solution defined by boundary condition (V.A.5a) and shownin Fig. 9. Consider the function$b, k, 0 = U(O$(z, , 0)

which satisfiesL(t)$(z, , = W ( z , , 0 (V .B .8 )

andaati- (z, K, t ) = B$(x , k, ) . (V .B .9 )

Since $ ( x , K, t ) satisfies (V.B.8) , it has he asymptotic be-havior$(z, k, ) = u+(k, t)e* + u-(k, t)c* (V.B.lO)

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SCOTT et al .: H E SOLITON 1457

Fig. 10. Left-handscattered waves of the related linear operator.

C. Th eGelfand-Levitan (March enko) EquationHaving foundthe scattering data at time t , we must nowconstruct the corresponding potential 4(z, t ) . This inverseproblem for the Schrodinger equation can be solved throughuse of the Gelfand-Levitan integral equation [83]. This im-

portant equation is derived in Appendix D ; here we merelysta te the result s tha t are necessary for calculations. With theexception of (V.C.5), we omit explicit reference to time de-pendence for typographical convenience.g l ( x , r) + K ( z + r) +s K ( y + r)h(z7 ldy O7

y > x (V.C.1)is an integral equation that is to be solved for g,(z, y). Th epotential can then be determined by

and the linear independence of the exponentials then impliesa@+ aa-at at--- 4ik%+ and - - iRaa-

which immediately integrate to yielda+@, t ) = e 4 * t u + ( ~ , 0)

anda_@, ) = e-4*ta-(k, 0). (V.B.1)

Comparing (V.B.lO) with the boundary condition (V.ASa),we obtain R(R, t)=a+(R, )/a-(K, t ) ; thus, rom V.B.ll),we findR ( K , ) = e * * t R ( k , 0). (V.B.12)

T o recapitulate the results of this section, we have foundthe time evolution of the scattering dat a for the Schradingerequation when the potential +(z, ) evolves in time accordingto the KdV equation.1) Th e number of bound states N does not change withtime. N is determined by the initial potential &(z).2) From the initial scattering data as determined by thepotential &(z), the scatte ring data evolve according to theseformulas: K,,(t)=G(O); C,(a, t )=exp ( -4&*t)C, , (~, 0), forn-1, - * , N ; and R(R, t)=exp (8iPt)R(R, 0).We have traversed path b of Fig. 8 for the KdV equation.In order to negotiate patht and obtain 4(z, t ) , it is necessaryto consider how one obtains the potential of a Schrodingeroperator from the scattering data.We do this in Section -C.Later it will be useful to have analogous formulas for themirror mage of the scattering problem (V.A.5) as shownin Fig. 10. Here we summarize these formulas. The subscrip t1 refers to impingement from the left.#l(z, k, ) = e* = + R l ( k , )e-*, as z + - 0

= TI@,) e * , as x + + 00 (V.B.13a)

The kernel K , which appears in (V.C.l), is defined asNK ( x + y) = &(x + y) + %e-(=*) (V.C.3)n-1

where a ( x + y ) is the Fourier transform of R ( R ) :1 &( x + y) E -~-,R(R)e*(z*)dk. (V.C.4)

Thesum n (V.C.3) is specified by henumber of boundstates N and the eigenvalues K ~ * or the Schrodinger eigen-value problem (V.A.5). Th e m, s defined by&(t) = e&*tm,,(O), n = I, 2, * ,N . ( v .c .~ )

Here m , ( O ) is a normalizationconstantdeterminedby heinitial data & for the KdV equation [ s e e (D.19b) in Appen-dix Dl.D . Evaluation of @(x, t ) fw K d V b y the I n v e r s e Method:Soliton and Doubkt F w m h

Following the prescriptionoutlined nconnectionwithFig. 8, we now insert scat tering data for the KdV equation.From (V.C. 1), (V.C.3), and (V.C.2),h(z, y, t ) + K ( x + y, ) + K ( y + y, t)gl(z,y, t)df = 0,

y > x (V.D.la)INK ( Z + y , t ) = m,,(O)e8~-*te-~~(z*)n-1+-!- R ( k , O)Pte*(+*)dk (V.D.lb)2* --m

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1458 PROCEEDINGSOF THE IEEE, OCTOBER 1973that solveor hl and hz as

NK ( x + y , 1 ) m,(0)e8gmte-c*(r+u) (V.D.2)where we should recall that the individual terms in the sumare determined by the initial data & ( x ) . In particular, eachbound state at energy ( K,,*) of theSchrodingerequationwith potential & ( x ) gives rise to a.corresponding term in thekernel (V.D.2). In order to study this dominant effect in thelimit of large t , we begin with special initial data support ingone bound state (at E = K ) ) with no reflection, R ( k , 0) O.In his case, the Gelfand-Levitan equation V.D.la) e-duces todl(x, y , t ) + m&te-g(r*)

n-1

+ m e 8 c a t ~ m-z(Y+ul)&(x, y , f)dy = 0, y > x. (V.D.3)To solve this equation, note the explicit dependence upon y.Differentiating with respect to y yields

so&(x, y, t> = e - W z , 0 . (V.D.4)

Substituting (V.D.4) into (V.D.3),one caneasilysolvefor& to obtainme8zte-c (z+u)

& ( x , y , 0 = - 9 y > x1+- 8cate-2crm2K

and taking the derivative indicated in (V.D.lc), we have+ (x , t ) = - K 2 sechZ K(Z - Kt) - 1 (V.D.5a)

6 = 4 og [m/2K]. (V.D.Sb)Equation (V.D.5a) indicates a soliton with velocity equal to4 ~ )nd position of themaximumamplitude at x = ~ / K ort = 0. Th e speed is four times he eigenvalue K I .Consider now the doublet solutionobtained rom nitialconditions tha t support two bound states at energies ( - K I * )and ( - K z * ) , butagainnoreflection, so R ( k , 0) O . In hiscase, the Gelfand-Levitan equation (V.D.la) reduces tog l ( Z , y , t ) + mle8glte-g1(z+u) + m2e8zsate-gl(z+v)+ mle8rlat Jzm e-zl(tt+u) tdx, Y, W y + m 2 e 8 K * 3 t ~ i-g*(v+U)~l(x,, t)dy = 0, y > x . (V.D.6)Again taking advantage of the explicit y dependence, we look

where

Then from (V.D.lc) we have the doublet solution

The twobound states of & ( x ) a t energies (-~1*) and( K Z * ) yield, as t++ 00, tw o soliton waves with the form of(V.D.5). To see this, we evaluate (V.D.8) asymptoticallyand obtain, as ++ ,# ( X , f ) N - { ~ 1 ~ech2 K ~ ( x ~ 1 ~ t ) 11+ 2~ sech2 [KZ (Z- K&) - 2 } ] (V.D.9a)where

Th e derivation of these solutions from the inverse methodwas first carried out by Gardner, Greene, Kruskal, and Miura[79] and aterbyMiura,Gardner,andKruskal 161].Ageneral derivation for the N-soliton case is iven by Gardner,Greene, Kruskal, and Miura [SI], Segur [201], and Wadat iand Toda [229] who also showed tha t the center of mass,defined by$-;*(x, t )dx

p x , )d xx0 = (V.D.lO)

moves with constant velocity during an N-solution collision.The formulas for this general case are listed in Appendix A.

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SCOTT et a!.:THE SOLITONWhile i t is historically correct tha t the N-soliton formulasfor KdV were first obtained through the inverse method, wewish to emphasize tha t he power of themethoddoesnotrest with his accomplishment. In fact, as was ndicated nSection111-B,Hirotahas hown 94] tha t he N-soliton

formulas for KdV can be obtained through a simple changeof variables without the inverse machinery. In our opinion,the real power of the inverse method is tha t i t exactly solvesthe initial value problem by inear methods. Study of (V.D.l)will show exactly how arbitrary initial datawill break up intoN solitons plus some additional "background effect" associ-atedwith R(k, ) # O in(V.D.lb).Reference o hehydro-dyna mic data of Fig. 3, the nonlinear electric filters discussedin the companion paper by Hirota and Suzuki [96], and themany other applications noted in Section I1 should convincethe reader that these investigations are of considerable inter-est. The quotation from Scott-Russell included at th e begin-ning of this paper is perhaps the first description of such aneffect. We now consider an application of these more generalcalculations to the special initial condition of a n impluse or"delta"function.E . +o (x )= - 2 X 6 ( ~ - ~ o )

Th e special initial da ta+(x, t ) Ig-0 = - 2 X 6 ( ~- SO), X > 0 (V .E . l )

provide a particularly explicit example of the inverse theory.[While these data are notP(-a a), hey are a limitof theP unctions; the inverse theory may be extended to includesuch data.]The physical scattering problem (V.A.3) is quickly solvedat time t = 0 to yield

Such a deltafunctionsupportsoneboundstate at energyE = -X*; the bound state wave function is given by

[I t is nteresting to note hat he ransmiss ion coefficientT(k)=(l--th/K)-l,hich may be readromV.E.2),sanalytic in theupper half k planewith the exception of asimple pole at R = A ]Similarly, the solution fl(x, k ; 0) [as defined by (D.l a)in Appendix D] is found to befib, k ; 0)

1459Thus the scattering data at time t = O are given explicitly by

N = 1; K = X; m ( 0 ) = XeZAw;iX

k - XR ( k , 0) =- d2*wAt time t these scattering data evolve, accordingand (V.B.12), intoN = I ; K = X; m ( t ) = e a " [ ~ e * ] ; (V.E.6)to (V.C.5)Withhese cattering ata at time t , the Gel'fand-Levitan equation takes the form

& ( x , y ; t> + K(x + y ; t>+ JzaK(y + y'; t)&(x, y'; Ody' = 0, Y > zwhere~ ( xy ; t ) = ~ e a ' t e a ~ l e - ~ ( " + ~ )

dd x+ ( x , t ) = - - ( x , x ; t ) .

At t =0, a short calculation confirms th at (V.E.8) is consis-tent; &(x, x ; 0) =2M(x-x0) [where O ( t ) = + 1 for t > O andvanishes for t < O ] is the solution.Notice that for large t>>O, it seems clear that the kernel Kin (V.E.8) may be approximated by K O :K O ( %+ y ; t ) X e * ' t e ~ e - A ( z * ) . (V.E.9)

If we replace K by KO, single-soliton solution results:t d x , y ; t> = &O(Z, y ; 0 , as t -+ + a

Calculating the solution 4 ( x , t ) from (V.E.10), we obtain+ ( x , t ) N - 2X2 sechz [X(% - 4Xzt) - 1,

as t ++ 00 (V.E.11)where

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1460 PROCEEDINGS OFHE IEEE, OCTOBER 1973from the Gelfand-Levitan eauation.n Darticular. we in- TABLE Ivestigate the structure of the solution for >>O and the mannerin which the soliton evolves from impulse initial data. Region L (Left)Region M (Middle)RegionR Right)First, we musttudyheernel Eo>tl* threeubregions

K([;) = K&; t ) + K c ( [ ; ) (V.E.12a) I ( E o * X; is I& , X; t ) ishe Airy I& , X; t ) ha s a termhat de-oscillatoryntegral cays exponentially plus asolitonlike erm in sub-regions ii) and iii)here

IC,([; ) 5 Aest-uo EquationV.E.13L)quationV.E.13M)quationV.E.13R)

this pole divides the region Fo>>tl/ into three subregions:an d 50 5 0 f 0

24t 24t 24ts k - X In thesehreeubregionsheriticaloint K+ lies i) belowthe pole, ii) on the pole, iii) above the pole. Using the methodsubregions:

i)- x 2 ; s)- A t ; iii)- X2.dk1(t0, ; t ) E - it(a+MO/t)t o = [ - 2 0 . of steepest descent, we evaluatehentegralnhehreeFor large t>>O, the integrand of I is highly oscillatory; hencethe method of stationary phase shows that the integral willbedominatedby hevicinit ies of its wocriticalpoints K*: i) l ( 5 0 , X i ; f ) v i( I k+ I - )

For fixed finite t o , these two critical points coalesce at K+ =O .Theact that tworiticaloints of thentegrand of I pinch t >> 0, t o >> tilr, 50 = 24X2tat 0 indicates a similarity between I and the Airy function.T o be more precise, we changevariables of integrationand iii) I ( 5 o ,x; t ) &(24t I k+ l ) - l / zobtain e - t t ( ~ o / t ) / ~ + 2*ie8~~t-~0i< l +I - >k = k t1 /* . (V.E.12b)

For t>>O but [O fixed and finite, I(&,, X ; t ) may be approxi-mated by the Airy integral

as t >> 0; to ixed. (V.E.13M)On the other hand, if 50 0; [O >0, 50 >> t l / * , o > 24X2t. (V.E.13R)Th e additional terms in cases ii) and iii) are due to the pole;in case ii) we have defined the integral by the Cauchy princi-pal value. Thus, for large t , the kernel behaves quite differ-ently n he hree regions. The ituation s abulated nTable I .With this information about the behavior of the kernel,we tur n to consider the long-time behavior of the Gelfand-Levitan olution itself. Firs t, fix x arge nough so that(x+y-2xo) lies in region R (see Table I) and to the right of thesoliton line 8X*t-(x+y-2x0)=0. In thi s region, the entirekernel K(x+y : t ) is exponentially small. Since the variableofintegration y > r , the kernel K(y+y: t ) in the integrand isalso exponentially small in the region. Writing the Gelfand-Levitan equation abstractly as g+K = -Kg, we can see that,in his region, -K,!(x, y; ) + xe8xat-A(Z+U-2ep) +- 2&8Xt-A(~+U-Wx2* 1

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SCOTT et d. THE SOLITON 1461an d

ddx+ ( x , t ) = - - ( x , x ; t )

+ O(#) (V.E.14b)for t>>l,nd r>4Xzt.To the lef t of the soliton line, the kernel is exponentiallylarge due to the term exp[8Xat-X(x+y-2~) ]. However, byexplicitly subtracting out the soliton, we can eliminate thisexponentially large term. More precisely, we define

d l = g10 + ewhere g,, is defined by (V.E.10). Then the Gelfand-Levitanequation may be written symbolically as

K, = - K,(1 + 610) - 1 + K J e (V.E.15)where we have used the denti ty K,+g~o+K&o = 0. Inregion R to the k f t of the soliton line it is easy to see from(V.E.15) that 6 is exponentially small.Th e analysis in regions M and L is similar but far moretedious. Segur [201] has shown that in thecase of one solitonthere s no asymptotic (in t ) effect on hesoliton from itsinteraction with the oscillatory tail (the restf the solution),and vice versa. In th e case of no soliton (K,=O, for example,X>+1 that in egion M , + decaysas t-2r. In region M they find that the asymp totic state isone of theBerezin-Karpman imilarity olutions [21]. I tconnects heasymptoticstates of regions L and R. In hecase of no soliton, Ablowitz and Newell give explicit formulasfor all asymptotic (in t ) states. This completesour descriptionof t>>O.

Finally, we describe the mechanism by which the solitonsevolve into a solitonbyexamining the kernel K . Here wefix x finite and considersmall t . We assume tha t he be-havior of K c is still dominated by thecriticalpoint k+ inthe upper half k plane and thepole at k = 21. At time = 0, thecriticalpoint K+ is at f i m ; and, as t increases, it travelsrapidly down the imaginary axis at a speed proportional tota/2.s long as Ik+l >X , an analysis as in egion R iii) showstha t the pole in K c cancels K.. The kernel contains no solitonterm; thus o soliton s present in+ ( x , t ) . As soonas I k+l

--(D + PzS(x)+ )P,z2((z)]dz. (V.F.l)Treating the functional H [ P ,Q] s a Hamiltonian, we calcu-late Hamiltons equationsof motion:

dQ 6 8 dQdt 6 P dt- = - a -= - P + 6PzP,. (V.F.2b)

This pair of equat ions possesses he solutionQ = + ( x , 0 (V.F.3a)P = s-:+(d,)dx (V.F.3b)

where + (x , t ) is a solution of KdV. Thus KdV may be viewedas a Hamiltonian system with canonical coordinates (Q-9,P - /! ,@x) and Hamiltonian (V.F.1). This result is attrib-uted oGardner [82], [141].ZakharovandFaddeev define another set of variables(p,q) in terms of the linear inverse problem associated withKdV, and they how that this set f variables is also canonical,i.e., that they are related to PIQ) by a canonical transforma-tion. To be more recise, they consider the Schrodingerequation (V.A.3) subject o he left boundary conditions(V.B.13). The potential + ( x , t ) is assumed to evolve in timet according to KdV. The new canonicalcoordinates (p, q )are defined in terms of the scattering data. Define ( a @ , ) ,b(K, t ) ] through certain scattering solutions1 and f e of theSchrodingerquation [whichhemselvesreefinedyboundary conditions (D.la) and (D.lb) of Appendix Dl :f l ( x , k ; ) b(k , t ) f z ( x , K ; ) + a ( k , t)f&, --k;0 . (V.F.44Also define cln by#zn(x, t )N e, a s x + -

(czn)-1 = s-:#lnZdx, n = 1, 2, * ,N . (V.F.4b)Comparing (V.F.4a) and D.la) with V.B.13a)yields therelationship

W - k , > = b ( k , t)/a(-k, 0 . (V.F.5)Recall that the scatt ering data are iven by the triple (Rz,.,cln). In erms of thesequantities,ZakharovandFaddeevdefine the new coordinates ((p, q ) :

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1462 PROCEEDINGSOF TEE IEEE, OCTOBER 1973may be considered as a canonical transformation connectingthe canonical variables (P, Q) an d (p, q ) . The ransformedHamiltonian takes the formh (q , p ) :

Since this Hamiltonian is only a function of the "canonicalmomentum," hevariables (p, q) ar e of the "action-angle"type. Hamilton's equationsof motion for HamiltonianV.F.7)are given bydP 6hat 69_ - - - - 0

(V .F .8)These immediately integrate to yield

P W = P 11-0p.(t) = K , ~ , n = 1,, - ,N

q(t) = arg ( b ( k , ) ) = 8k'tqn(t) = - xnat, n = 1, , * * * ,N. (V.F.9)

Viewed in this manner, in order to solve for # ( x , t ) , one needonly invert the canonical transformation mapping (P, ) toFinally, Zakharov and Faddeev connect this theory withan infinite number of constants of the motion (see SectionVI). To do his, hey first observe hat In (a@, 1 ) ) can berepresented in terms of the reflection coefficients &(k, t ) andthe bound state energies K,:

(P,4).

On the other hand, it is easy to show [253] that In a(R, t )can also be expressed asIn a ( k , t ) = - d m ( x , k , 1 ) (V .F .12)m

-m

where u satisfies the Ricatti equation:U. + S - 4 + 2iku = 0. (V .F .13)

Treating k as a largeexpansionparameter, we obtai n heexpansion of u(z, k ; ) asUn(X, 1 )u(2, ; ) = -

,,-I (2ik)"

Placing (V.F.14) i n (V.F.12) and quatingwith (V.F.11)yields an infinite number of constants'of the motion in termsof integral s of polynomials of the solu tion 4(z, t ) .Thus heKdV quation s a "completely integrable"Hamiltonianystemnhe sense thatheHamiltoniansystem can be reduced to an obviously integrable form by acanonical ransformationanalytic n he position and mo-mentumvariables.Ford,Stoddard,andTurner [68] haverecently presented computer evidence that the Toda lattice(II.D.l) is also completely integrable in this ense.G . The Inverse Method for Self-Induced Transparency

Recently,Lamb, nconnectionwith his study of ultra-short optical pulses, has shownhow self-induced transparencycan be described by an inverse method [138]. His work seemsparticularly important because the appropriate linear opera-tor [ L in (V.3)] arises directly from the physics rather thanbeing "pulled out of a hat" as in all other applications of t heinversemethod.Startingwiththeuantumquat ions" ( I I .B.lH),Lamb first observes that solutions satisfy the integral

' X z + P 2 + Q Z = 1. (V. G. 1)Following Eisenhart [57], Lamb hen ntroduces wo newvariables 4 an d - /# =+* asX + i 6 1 + Q-=-= 4 (V ..2a)1 - Q X-&'

' X - - @ 1 + Q 11 - Q X + i 6 #- = - = - - - - 4*. (V .G.2b)

Inverting (V.G.2) yields

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SCOTT et al. : HE SOLITON 1463Inserting (V.G.3) into he self-induced transparencyequa-tions (II.B.l) , we obtain equations for 4 and $ which willreplace (II.B.l). These equations are uncoupled; both $ an d$ satisfy the same Ricatti equation:

i2$ t = &$ - - A@)[$' - 11. (V.G.4)

The transformations

w = u(t) ex p {- J m t E ( t t ) d t t }linearize the Ricatti equat ion, reducing it t o

w t t + +[(Au)' + E) ' + 2iEt]w = 0. (V.G.5)Thus the problem reduces to the coupled set

a&- = (P)ax (V . G.6a)azwa r 2- [(AW)'- V ]W= 0 (V. G.6b)

where r = t - x an d2) = - %(E' + 2&,). (V. G. 7 )

Lamb now uses an inverse method to define 93 in erms ofscatteringdataappropriate o V.G.6b). 2) provides the rdependence of E . Th e x dependence is obtained from (V.G.6a).The mathemat ics of Lamb's procedure is not as systematicas with the KdV inverse method for example, Lamb does notset up and solve an initial value problem).However, his workseems important since the linear problem arises directly fromthe quantum structureof th e physical problem.Finally, note that as h4, II.B.l) reduces to the sine-Gordon equation as indicated in (II.C.l) and (II.C.2). Thisequation asecently been analyzedhrough nnversemethod by Ablowitz et al. [ l ] ndependently of Lamb's work.Thisgroupsystemat ically solves the nitialvalue problemby their inverse method. We have shown (unpublished) tha ttheir work is in agreement with Lamb's and that there existsa correspondencebetween the eigenvalues of theirinearproblem and he uncertainty in the quantum energy levels(Au).Ablowitz et al. [ l ] , [ 2 ] have also shown that the samelinear problem gives rise to a large class of nonlinear evolutionequations, a result that suggestsome underlying mathemati-cal structure. However, their linear problem (and hence theirunderlying structure) is 'pulled out of a hat." On the otherhand, in Lamb'swork, he inearproblem arises directlyfrom the quantum physics. Acomparison of these wo a pproaches should prove fruitful and exciting.VI. CONSTANTSF THE MOTION ND CONSERVATION LAWS

the relation between conservation laws and constants of themotion, and construct an infinite set of conservation laws forthe sine-Gordon equation using as a tool the Backlund trans-formation. Finally, several applications of conservation lawsarepresented.A . Definition of Constants of th e Motionequation in 63:Let 63 denote a linearspaceand consider the abst rac t

4: = m 4 . (VI.A.1)Here K is a (nonlinear) operator mapping 63 into 63 and, for0 < t < m $ ( t ) denotes an element in 63. Consider two exam-ples of (VI.A.1).Example 1: 63 is the six-dimensional phase pace for aNewtonian particle of unit mass and (VI.A.1) becomes

2 UZ; X = . (VI.A.2)yu =

z V IHere V ( x ) represents he prescribed potentialenergy unc-tion.Example 2: 63 is defined by the space of all infinitely dif-ferentiable unctions f(x) on - m < x < + 0 which vanishas I X I --t m together with all their derivatives. Here we let(VI.A.1) denote the KdV equation (I I.A.l), with a = +1, or

$ t = - M z - 9.Iz. (VI.A.3)Returning to the abstract equation (VI.A.l) , we considera functional I [ . ]mapping 63 into he complex numbers. If

(VI.A.4)for all solutions $( t) in 63 of (VI.A.l), the functional I is saidto be a constant of the motion (also frequently called an integral)of (V1.A. 1).Consider Example 1. Here heenergyfunctional, whichmaps phase space into the real numbers by the rule

E ( x , u ) = V(X ) + u * u (VI.A.5)is a constant of the motion. When the potential energy V ( * )is zero, we can find seven constants of the motion: the threecomponents of the inear momentum u and he hree com-ponents of the angular momentum X X U n addition o heenergy E ( x , u ) = ~ ( v Z 2 + v u * + v , * ) . Clearlyone of these sevenconstants is redundant since the energy is expressed in termsof the linear momenta. The six tha t remain are often used a san alternative parameterization of the six-dimensional phasespace. Any additional constant of the motion, therefore, mustbe functionallydependentupon these si x since theycom-pletely parameterize the phase space. Through this examplewe see how the dimensionality of 63 can place an upper boundon the number of constants that are functionally independent.With these comments in mind, we consider Example 2 in

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1464 PROCEEDINGSOF THE IEEE, OCTOBER 1973integral have been discovered. Each of these takes the formof a constantcoefficient polynomial in $ and its spatial deriva-tives, integrated over all space. While he concept of inde-pendence has not been adequately formulated for the PDEcase, hese mus t be independent since they consis t of poly-nomials in+ of progressively higher degree and order. In fact,Miura, Gardner , and Kruskal [161] use this idea to define aconcept of independence.We have already seen several constants of th e motion forthe PDE case. If the equation has a Lagrangian density thatdoes not depend explicitly upon time, theenergy functional"isconstant ( s e e AppendixC).For heKdVequation, theeigenvalues (of the associated linear problem and hence th espeeds of the solitons which emerge as t++ a) may be con-sidered as functionals of the data. These are constant.

In practice, constants of the motion arise most natural-ly through the concept of a conservation law, an example ofwhichs introducednAppendix C. For moreprecisedefinition we restrictourselves to he PDE casewhere @consists of a set of functions of the continuous real variablex E(- a, a . Consider a pair of (nonlinear) operators map-ping @ into @, D[(-)]nd F[(.)], such that D [ + ( x , ) ] andF [ + ( x , t ) ] depend locally on x an d t for any trajectory in@labeled by t . Trajectories + ( t ) need not be solutions of(VI.A.l), but if

(VI.A.6)for all solutions + ( t ) of (VI.A.1) lying in @, hen (VI.A.6) issaid to be a conservation law where D ( a ) ] is the conserueddcnsity and F[(.)] s the conserved flo w or flux. Clearly thefunctional

I [ + ] =J D D M X , OIdx (VI.A.7)--ODis a constant of the motion provided the integral exists andthe ntegrand satisfies appropriateboundaryconditions atx = a. Thu s a standard procedure is to determine a set ofconservation laws and thense (VI.A.7) to obtain constantsfthe motion.B . Construction of an Injinite Number of Conservation La ws

As an example that llustrates hecons truc tion of aninfinite number of conservation laws, we consider the sine-Gordon equation in characteristic coordinates+z t = sin 4. (VI.B.1)

Given a solution + of (VI.B.l), it is indicated in Section11-Bthat another solution $ may be generated from + through theBacklund transformation [1581(VI.B.2a)

[148]. We seek $ in the formLo

$ ( x 7 4 a)- $j(Z, t)ai, as a+0. (VI.B.3)j= O

Inserting (VI.B.3) into (VI.B.2b) yields

As a 4 , he foregoing demands 0 =+ and = 2&. Equatingthe coefficients for higher powers of yields$0 = 4$1 = 241$2 = 2411$3 = 2+ttt+4+ta$4 = 2 4 t t t t + 24t"tt$6 = 24ttttt + 3 + t 2 4 t t t + 5+t+tt2 + &)+t6 . (VI.B.4). . . . . . . . e t c . . . . . . . . . .

A little edious algebra establishes hat his series for $ isconsistent with (VI.B.2a).From any fixed conservation law, this series enables us toderive an infinite number of conxrva tion laws. Since (VI.B.1)can be derived from the Lagrangian density

L = + + & I t - c o s + + l (VI.B.5)the considerations of Appendix C imply conservation of energy

t W ) t + (cos $ - ) z = 0 (VI.B.6)and, since (VI.B.5) is symmetric in x and t , a correspondingexpression with x and t interchanged. Substituting (VI.B.4)int o (VI.B.6) and equating like powers of a gives an infiniteset of conservation laws. Here we list the first ew densities.Do = ++=% (VI.B.7a)DI = 24tttdz + & b t t d t z + + r 2 + t d z (VI.B.7b)D2 24tttttdz + & b t t t t A t z + & b t t d t t t z +Wttdtddz+ %ttt&t24z + lwttdtddz+ 5+tt4+&z+WtdtzVt + W t 2 + t d t t z + &t '+ tdz . (v1 .B.7~). . . . . . . . etc. . . . . . . . . .

This infinite sequence of conservation laws was first ob-tained by Kruskal and Wiley [168] without specific referenceto he Backlund ransformation heory.Lamb I361men-tionshe onnectionbetweenBacklundheory nd on-servation aws n his recentpaper ntroducingan nversemethodforsolution of the self-induced ransparency equa-tions. In fact, Schnack and Lamb first found the conserva-tion aws orsine-Gordon and hen, using hese results asamodel, ooked or and ' found an infinitenumber of con-servation laws for the equation of self-induced transparency

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SCOTT e t al .:THE SOLITONsystem. For future reference we lis t the first few densities for(VI.A.3).

Dl = 4 (VI.B.8a)Dz = 44 (VI.B.8b)Da = i4a- 4, (VI.B.8c)Dl = i$4 3 w z 2+#aa. (VI.B.8d). . . . . . e t c . . . . . . .

1465

Miura [I621 hasnvestigatedhe Conservation lawsassociatedwith he generalized form of theKdVequationgiven in (II.A.5). He shows that the to tal number of distinctpolynomial conservation laws is nfinite or: a ) p = O (thelinear case), b) r = O , c) p = 1 and r = l (KdV), and d) p = 2an d r = 1 (modified KdV). Otherwise the num ber of distinctpolynomial conservation laws is only three.Recently, Benney [ l a g ] has found an infinite number ofconservation laws for a nonlinear wave equation describingwater waves in one time and two spatial dimensions. Thesetake the form of an integral over one of t he spa tia l variables.Miura [163 ] has extended this approach and has been ableto eliminate the spatial integral. Zakharov and Shabat [ 2 5 2 ]have found an infinite set of conservation laws for the non-linearSchrodingerequation.More ecently Ablowitz et al.[ 2 ] have extended this approach toa wider class of physicallysignificant evolution equations.C . Applications of Conservation Laws

Given the existence of an infinite sequence of conserva-tion laws, one might well ask sowhat? Besides some theoret-ical implications for the description of solution manifolds, ofwhat practicaluse are these conservation laws? The answer isthat they provide simple and efficient methods to study bothquantitativeandqualitativeproperti es of solutions.Theseproperties include stability, evolution of solitons, and decom-position in to solitons. Indeed there seems to be a close rela-tionship between the existence of a sequence of conservationlaws and the existence of solitons, although this relationshipispresentlyfar from understoodand s a topicofcurrentinterest.As an example we indicate how the set (VI.B.8) can beused to study emerging solitons of the KdV equation. Ap-plications to he stab ility problem are discussed inSectionVI I. We begin with the initial value problem (VI.A.3) where$(z,0) =$o(r)-+O as 1x1+a. Denote the sequence of con-served densities by D l [ + ] and the associated constants from(VI.A.7) by I, , , [$]. We assume that he nitial data & de-compose int o N solitons as t + + w .Thus

N+ ( x , t ) - ~ j [ t j ] , as t ++ a (VI.C.1)j-1where

+ ( x , t ) - ~ j [ ~ j ] , as t + + m ; =- otherwise.This nterpretation is valid since sech*[ - ] vanishes ex-ponentially with its argument. This fact guarantees that theintegrals I,,, split into a sum over j(= 1 , 2 , . ,N) as t + CQ .Thus

N(VI.C.3)

j-1where

ODI n [ S j ] E J D m [ S j ] d ~ . (VI.C.4)-

Equations (VI.C.3) and (VI.C.4), togetherwithhe con-stancy of the I,,, along solutions, yield a set of equations forthe ~j in terms of the initial data40:

= s-. Dm[4o(z)]dx. (VI.C.5)ODBy studying such equations, Berezin and Karpm an [ 2 1 ] haveobtained necessary conditions on the initial data to permit abreakup nto Nsolitons, and have estimated he speeds ofthe emerging solitons. Thei r work is summarized by Jeffreyand Kakutani [ 1 1 2 ] . Wadati and Toda [229 ] use a similarcalculation to determine the speed of the center of mass de-fined in (V.D.10). Weemphasize th at all suchcalculationsare based upon the assumption thatbreakup into N solitonsoccurs. The inverse method, when it applies, provides a toolfor establishing this assumption and, in addition, enables oneto make similar estimates.D. nteracting Solitons

In thissection we indicate a means hrough which theexact evolut ion of N-soliton solut ion may be studied by theuse of conservation laws. We restrict ourselves to the initi alvalue problem for the KdV equation:

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1466 PROCEEDINGS OF TEEEEE, OCTOBER 1973ZabuskyandKruskal [245]noticed the nterestingfactthat the variational problem aIa[+] O subject to the con-straint I2 [+] = constant has a soliton as it s sol ution. To eethis, use Lagrange multipliers and calculate theirst variationof I , [+]- I* +] :

-$-: [+*(x, os+- +&. - C W ] d z= $ [+*+ 2& - C+]G+dx = 0 . (VI.D.1)

TheLagrange-Eulerequation for thisvariational problemis seen to be+2 + 2422 - c+= 0

which is the differential equation satisfied by a soliton travel-ing at speed u= C/2. Thus a soliton traveling at speed +umakes the functional Is[+] an extremum, subject to the con-straint I*[+]ixed. Interestinglyenough, Boussinesq (1877)was awareof this factas Benjamin [I71 points out.The fact that a soliton is critical for the variational prob-lem (VI.D.1) led KruskalandZabusky oconjecture hatN-soliton solutions were critical for variational problems ofthe higher conservation laws. Th ey considered the varia tionalproblem

wave or nypai r of positive speeds u1 an d u2. Earlier,Zabusky and Kruskal [245] had observed numerically inter-actions between twosuch solitons. Theyfound hat, whenu1>>uz, he faster-bigger wave swallows the slower-smallerone and reemits i t later. On the other hand, they observedthat, when u1 =u~(t,he bigger wave shrinks as it approachesthe smaller, while thesmaller grows. Thiscontinuesuntilthey have interchanged roles. Lax establishes these observa-tions analytica lly and gives a precise estimate for the ratioof the speeds that separate the types f interactions. I n addi-tion, he found a third ype of interaction ntermediate b