PHYSICAL REVIEW A VOLUME 45, NUMBER 4 15 FEBRUARY 1992

Properties of soliton-soliton collisions

David W. Aossey and Steven R. Skinner*Department ofElectrical and Computer Engineering, University of Iowa, Iowa City, Iowa 52242

Jamie L. CooneyDepartment ofPhysics and Astronomy, University of Iowa, Iowa City, Iowa 52242

James E. Williams, Matthew T. Gavin, and David R. AndersenDepartment ofElectrical and Computer Engineering, University of Iowa, Iowa City, Iowa 52242

Karl E. LonngrenDepartment ofElectrical and Computer Engineering, University of Iowa, Iowa City, Iowa 52242

and Department of Physics and Astronomy, University ofIowa, Iowa City, Iowa 52242(Received 6 September 1991)

The amplitude dependence of the phase shift originating in an overtaking soliton-soliton collision isinvestigated for solitons that can be described with a Korteweg —de Vries equation and a nonlinearSchrodinger equation. The size dependence of the interaction regime is also amplitude dependent. Lab-oratory and numerical experiments are compared with a simple model for the elastic collision.

PACS number(s): 03.40.Kf, 42.65.—k, 52.35.Mw

I. INTRODUCTION

The ability of two solitons to survive a collision com-pletely unscathed is frequently used as an identifyingcharacteristic in laboratory and numerical investigationsas a technique to separate a soliton from a large panora-ma of signals that may appear on the screen of a labora-tory oscilloscope or a computer terminal. During thecollision, the solitons interact and appear to exchange po-sitions in a space-time trajectory as if they had passedthrough each other. After such a collision, the two soli-tons may appear to be instantly translated in spaceand/or time but otherwise unaffected by their interaction.This translation will be called a phase shift in what fol-lows. In one dimension, this process results from two sol-itons colliding head on from opposite directions or in onedirection from having a large-amplitude soliton catch upwith a slower small-amplitude one (since the velocity isamplitude dependent).

This survival property during a collision was originallynoted in a numerical study of the Korteweg —de Vries(KdV) equation and led Zabusky and Kruskal to coin theword soliton as a descriptive name for this nonlinear wave[1]. Scott Russell did note a similar collision effect in hisobservations of water waves [2]. Soliton aficionados,however, usually refer to the numerical work as being thegenesis of soliton literature. A mathematical structure,the inverse scattering technique, was used to demonstratethat this survival during a collision is to be expected.This is well documented in several texts [3]. Other non-linear equations such as the nonlinear Schrodinger (NLS)equation exhibit this collision survival property also.

The purpose of the present paper is to describe a seriesof laboratory and numerical experiments dedicated to in-

vestigating the detailed structure of the collision of KdV

and NLS solitons. Both the phase shift of the collidedsolitons after their interaction and the size of the interac-tion region are a function of the relative amplitudes ofthe two colliding solitons and this effect will be investi-gated. The results for KdV and NLS solitons will bedifferent. In both cases, the solitons preserve their shapeafter the collision. This work is presented in Sec. II. InSec. III we suggest that the phase shift can be understoodusing a model that is based on a very simple calculationand certain fundamental properties of the soliton. In ad-dition, it is possible to predict the length of the interac-tion region using the same model. The model describesboth KdV and NLS solitons and can be extended to oth-ers. Section IV is the conclusion.

II. SOLITON-SOLITON COLLISIONS

Solitons that can be described with the KdV equationcan be easily excited in certain laboratory plasma physicsdevices. Using such devices, it is possible to verifyseveral of the expected properties of a KdV soliton andreviews of this work have appeared [4]. Although thecollision property was noted in the original experimentsof Ikezi, Taylor, and Baker [5], a detailed investigationwas not presented.

We have recently uncovered a technique to launch andstudy KdV solitons in a positive-ion —negative-ion —electron plasma [6]. We have studied, among othereffects, the overtaking collision property in order to verifythe soliton nature of an observed signal in the laboratory[7]. The details of the experiment need not concern us

here and only the results of the collision experiment will

be shown.The results of this experiment are shown in Fig. 1(a)

where the timing sequence of the applied signals was such

45 2606 1992 The American Physical Society

45 PROPERTIES OF SOLITON-SOLITON COLLISIONS 2607

that a large-amplitude soliton would be launched after asmall-amplitude one was launched. The large-amplitudesoliton (BB') catches up and passes through the small-amplitude soliton (AA'). The soliton (CC') is excited atthe same point as (BB') but travels slower due to itssmaller amplitude. The trajectories of the three signalsare shown in Fig. 1(b). We note that the two solitons un-

dergo a phase shift at the point of collision, with thesmaller-amplitude one shifting more than the larger one.This "conservation of momentum" for solitons was alsonoted in other experiments [5]. In addition, there is awell-defined region where the solitons overlap and we wi11

define this distance as the interaction length. This lengthalso depended upon the relative amplitudes of the twocolliding solitons. A summary of the experimental resultswill be included later in Figs. 4 and 5.

Solitons that are described with the NLS equation canalso be investigated in the laboratory in relation to laser-matter interaction studies. For the present detailedwork, it is more useful to present results of a numericalinvestigation on dark solitons that have received experi-mental confirmation elsewhere [8]. See also the numeri-

f(g) =sin B tanh(g)

[1—B sech (g)]'

and

/=a(x —v,z) .

Here, B is a measure of the "blackness" of the dark-soliton solution and can take on a value between —1 and1, v, is the dimensionless transverse velocity of the dark-soliton center with respect to the z axis, and ~ is theshape factor of the soliton. The dark intensity (Id ) of thedark soliton (i.e., the depth of the irradiance minimum) isrepresented by B uo. Previous work has shown that the

cal studies in Ref. [9]. For the solitons described by theNLS equation, we will limit the study to dark spatial soli-tons. In this case, the temporal viable t is replaced withthe spatial variable of distance z since time-independentoptical beams are being modeled. The field of dark spa-tial solitons is given by [10]

u (x,z ) =uo[1 —B sech (g) ]' exp[+if(g)],

where

A B C

«) / //

l ~ 8

L

L$

I%i

fflIg5CQ

C~~N

2 p,sBl Al CI

(b)v = 1.04

V—

o 20E

l4

+e

t' +'v = 1.01

X

(b)0 0 0 0 0 0

'X

p10

0I

10I

20I

30

0QQQ 0

0 0 0 0 0 0 0 0 0 0

FIG. 1. Experimental test of an overtaking collision of aKdV soliton. (a) Sequence of the detected signals as the dis-tance between the exciter and the detector is increased in equalincrements. The solitons BB' and CC' are launched from thegrid and AA' from the plate. (b) Time-of-flight diagrams of thesignals ( X 10 cm/s).

FIG. 2. Numerical simulation of an overtaking collision ofequiamplitude NLS dark solitons. (a) Sequence of the waves atequal increments in longitudinal position z. (1) Time-of-flightdiagrams of the signals.

2608 DAVID W. AOSSEY et al. 45

shape factor ~ and the transverse velocity v, are related tothe "blackness" of the soliton by [9]

' 1/2, /n, /uo

U, =+ (1—8')no

(3)

and

K=n n K u where no and n 2 are the linear and Kerr nonlinear indicesof refraction. We have assumed ~n2~uo &&no. When twodark solitons collide, their individual phase shifts aregiven by

1/2no

n2 Qo

[( 1 82 )1/2+( 1 82 )1/2]2+ (8 +8 )2

2konoB/ [(1—8) )' +(1 B~—)' ] +(8 —8 )(4)

Dark spatial soliton interactions can be easily investi-gated numerically by using a split-step propagation algo-rithm which was found to closely predict experimentalresults [11]. The results of a simulation of two collidingequiamplitude solitons are shown in Fig. 2(a) and theirtrajectories are plotted in Fig. 2(b). We note that the twosolitons undergo a phase shift at the point of collision. Inaddition, there is also a we11-defined interaction length inz that depended upon the relative amplitudes of the twocolliding solitons.

III. SOLITON COLLISION MODEL

We suggest that a fairly simple model can be used tointerpret the results of the laboratory and the numericalexperiments on KdV and NLS solitons. The model isbased on the fundamental soliton property that two soli-tons that interact or collide will survive the collision andwill be unchanged by the collision. Rather than employthe exact known functional forms for the soliton (sech. gfor KdV solitons and sech( for NLS solitons), we consid-er the solitons to be rectangular pulses with an amplitude

AJ and a width W where the subscript j denotes the jthsoliton.

An evolution of the collision of two solitons is shownin Fig. 3(a). In this case we show two diff'erent amplitudesolitons just prior to the collision and just after the col-lision. The details of what occurs during the collisionneed not concern us here other than to note that thelarger-amplitude soliton has completely passed throughthe smaller one. In the regions external to the actual col-lision, the solitons cannot overlap as they are noninteract-ing. It is this property that states that the solitons mustbe separated by a distance

D =D)+D2

after the interaction. This manifests itself in a phase shiftin the trajectories depicted in Fig. 3(b). This was noted inthe experimental and numerical results. The minimumdistance that this can be equal to is the half-widths of thetwo solitons:

W) 8'2D& +

2

Therefore,

W) W2D& and D2 ~

2 2(7)

Solitons also have the property that their amplitudeand their width are related. For KdV solitons, this iswritten as

MW2 ~Pw&%~

t =T 2 A 21 A

t=2T

t =4T

L)L2

(b)

FIG. 3 Overtaking collision of two solitons. (a) Model of theinteraction just prior to the collision and just after the collision.After the collision, the two solitons are shifted in phase. (b)Time-of-flight diagrams of the signals. The phase shifts are in-

dicated.

A ( WJ ) =const =E, .

For NLS dark solitons ( W= I /1~), we have

8 W =const=EJ' J 2

Using the minimum values in (7) and (8), we find that theratio of the repulsive shifts for the KdV solitons is givenby

PROPERTIES OF SOLITON-SOLITON COLLISIONS 2609

D1

D2

8'1 /2

8'2/2

' 1/22

A1(10)

The interaction length must then satisfy the relation

L =L2 L—, =(u2 —u, )b, T~ W)+ W2 .

In a similar fashion, we find that the combination of (7)and (9) leads to the ratio of shifts of NLS dark solitonsgiven by

Equation (12) can be written in terms of the amplitudeof the two solitons. For KdV solitons, we combine (7)and (12) to yield

D1 82D, a, L +F1 +

' 1/2

1+A2

In Fig. 4(a) we summarize the results obtained in thelaboratory for the ratio of the phase shifts as a functionof the ratio of the amplitudes for KdV solitons. We in-clude the corresponding ratios obtained in the experi-ment of Ikezi, Taylor, and Baker [5] and those obtainedfrom the numerical work of Zabusky and Kruskal [1]andLamb [3]. The solid line is from Eq. (10). In Fig. 4(b) re-sults obtained from the NLS dark-soliton simulation arepresented. The solid line is from Eq. (11).

In addition to predicting the phase shift that resultsfrom the collision of two solitons, the model also allowsus to estimate the size or the duration of the collision re-gion. Each soliton depicted in Fig. 3 travels with its ownamplitude-dependent velocity u . For the two solitons tointerchange their positions during a time hT, they musttravel a distance L1 and L2,

(14)

For NLS solitons, the combination of (9) and (13) yields

L ~E2 +1 1

1 2(15)

In Fig. 5(a) we summarize the results obtained in thelaboratory for the measured interaction length as a func-tion of the amplitudes of the colliding solitons. We havealso included the numerical lengths (which we havescaled} from the paper of Zabusky and Kruskal [1]. Thedashed line is from Eq. (14) with A, =1 and E, = l.

In Fig. 5(b) the results for NLS dark solitons arepresented. The dashed line is from Eq. (15) with B2=1and E2 =6. The calculated interaction time (solid line) isthe sum of the two soliton widths minus their repulsive

L1=u16T and L2=u25T . (12)

4-

D,

D2

3

2-

~ (~)~

'L

o (2}

~IP~

I

2

2- ioo; (b)

D~

p

82Bc

0.01 O. &

B,

FIG. 4. Summary of the ratio of the measured phase shifts asa function of the ratio of the amplitudes. (a) KdV solitons. Thedata are from (I) this experiment, (2) Ref. [1], (3) Ref. [3], and(4) Ref. [5]. The theoretical line is from Eq. (10). (b) NLS soli-tons. The theoretical line is from Eq. (11).

FIG. 5. Summary of the measured interaction lengths as afunction of the amplitudes. (a) KdV solitons. The data arefrom (1) this experiment, and (2) Ref. [1]. The dashed line is Eq.(14) with K, = A, = l. (b) NLS solitons. The dashed line is Eq.(15) with Bz = 1 and Ez =6.

2610 DAVID W. AOSSEY et al. 45

phases shifts divided by the transverse velocity of soliton1. Because the longitudinal velocity is a constant, thisscales as the interaction length.

IV. CONCLUSION

Using a fairly simple model that is based upon thebasic properties of solitons to describe the collision of twosolitons, we are able to predict the resulting phase shift

and the size of the interaction region. The model is inagreement with laboratory and numerical experiments.

ACKNOWLEDGMENTS

This work was supported in part by the National Sci-ence Foundation Grant No. ECS 90-06921, the Office ofNaval Research, and the Defense Advanced ResearchProject Agency.

'Present address: Department of Electrical Engineering,Wichita State University, Wichita, KS 67208.

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[5] H. Ikezi, R. J. Taylor, and D. R. Baker, Phys. Rev. Lett.25, 11 (1970).

[6] K. E. Lonngren, D. R. Andersen, and J. L. Cooney, Phys.Lett. 156, 441 (1991);J. L. Cooney, M. T. Gavin, and K.E. Lonngren, Phys. Fluids B 3, 2758 (1991).

[7] J. L. Cooney, M. T. Gavin, J. E. Williams, D. W. Aossey,and K. E. Lonngren, Phys. Fluids B 3, 3277 (1991).

[8] S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L.Smirl, in Proceedings of the Conference on Lasers andElectro-Optics, Baltimore, 1991 (Optical Society of Ameri-ca, Washington, DC, 1991).

[9] R. N. Thurston and A. M. Weiner, J. Opt. Soc. Am. B 8,471 (1991).

[10]V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 64,1627 (1973) [Sov. Phys. —JETP 37, 823 (1973)]; A.Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171(1973).

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