Vol. 55, No.2. August 1984Printed in Belgium

Reprinted from JOURNAL OF COMPUTATIONAL PHYSICSAll Rights Reserved by Academic Press. New York and London

Analytical and Numerical Aspects of

Certain Nonlinear Evolution Equations.

II. Numerical, Nonlinear Schrodinger Equation

THIAB R. T AHA * AND MARK J. ABLOWITz

Department of Mathematics and Computer Science.Clarkson College of Technology.

Potsdam. Nell' York 13676

Received August 10. 1982; revised November 15. 1983

Various numerical methods are employed in order to approximate the nonlinear

Schrodinger equation, namely: (i) The classical explicit method, (ii) hopscotch method, (iii)implicit-explicit method, (iv) Crank-Nicolson implicit scheme, (v) the Ablowitz-Ladikscheme, (vi) the split step Fourier method (F. Tappert), and (vii) pseudospectral (Fourier)method (Fornberg and Whitham). Comparisons between the Ablowitz-Ladik scheme, whichwas developed using notions or the inverse scattering transrorm, and the other utilized

schemes are obtained.

INTRODUCTION

The nonlinear Schr6dinger (NLS) equation describes a wide class of physicalphenomena (e.g., modulational instability of water waves, propagation of heat pulsesin anharmonic crystals, helical motion of a very thin vortex filament, nonlinearmodulation of collisionless plasma waves, self-trapping of a light beam in a color-dispersive system 11]). The NLS equation was investigated numerically by Karpmanand Krushkal 12], Yajima and Outi [31, and Satsuma and Yajima 141, Tappert [51and Hardin and Tappert 16]. In the latter two works the NLS equation was integratedby the split-step Fourier method. As discussed in part I Ablowitz and Ladik 171found nonlinear partial difference equations (based on the inverse scatteringtransform) which can be used as a numerical scheme for the NLS equation. Thisscheme has certain desirable properties [81 (see part I).

This work aims to compare the Ablowitz and Ladik scheme and other knownnumerical methods for the NLS equation

iq/=qx..+2IqI2q. (1.1)

Roughly speaking numerical methods for obtaining solutions to initial value problemsfall into two categories 191: (1) finite difference methods and (2) function approx-

* Permanent address: Department of Computer Science, University of Georgia, Athens, Ga. 30602.

2030021-9991/84 $3.00

Copyright ~ 1984 by Academic Press. Inc.All right. or reproduction in any rorm reserved.

204 TAHA AND ABLOWITZ

imation methods. For the finite difference methods we seek approximation q:;' to theoriginal function q(x, t) at a set of points xn' tm on a rectangular grid in the x, tplane, where Xn = hn, tm = km, h is the increment in x, and k is the increment in t. By

expanding function values at grid points in a Taylor series, approximations to thedifferential equation involving algebraic relations between grid point values can beobtained. The function approximation method approximates the exact solution q(x, t)by an approximate solution defined on a finite dimensional subs pace

n

q(x,t)~ij(x,t)= 2: C;(t)f/>;(x).;=1

The CP;(x) are appropriately chosen basis functions. Common choices for these arethe trigonometric functions, leading to a finite Fourier transform or pseudospectralmethod and piecewise polynomial functions with a local basis, giving the finiteelement method.

The following numerical methods were applied to the NLS equation.I. Finite difference methods.

(a) Explicit methods.(i) The classical explicit method.

(ii) The hopscotch method [101.(b) Implicit methods.

(i) Implicit for the linear part and explicit for the nonlinear part (implicit-ex-

plicit).(ii) Crank-Nicolson implicit scheme.

(iii) The Ablowitz and Ladik scheme.2. Finite Fourier transform or pseudospectral methods.

(i) Split step Fourier method 16].(ii) Pseudospectral method by Fornberg and Whitham 1111.

In order to compare schemes, our approach for comparison is to (a) fix theaccuracy (Loo) for computations beginning at t = 0 and ending at t = T; (b) leaveother parameters free (e.g., ift or if.\") and compare the computing time required toattain such accuracy for various choices of the parameters 1121.

These methods are applied to the NLS equation (1.1) subject to the following con-ditions:

(A) THE INITIAL CONDITIONS

(i) I-Soliton Solution

The exact solution of (1.1) on the infinite interval is

q(x, t) = 217e-112tX-4<l2- ,,2/.'+ (1/10+ "f2/1 sech(217x -8f.17t -xo),

where xo, 11, t; and '1'0 are constants.For initial conditions, Eq. (1.3) is used at t = 0, and the constants are chosen to be

.\"0 = 0, '1'0 = 0, t; = I, and 11 = 0.5, 1,2, and 3.

~

205NONLINEAR EVOLUTION EQUATIONS, II

(ii) Collisions of Two Solitons (131

The exact solution of (1.1) on the infinite interval is

q(x, t) = G(x, t)jF(x, I),

where

1 + a(l, 1*) eXp("l + "f) + a(l, 2*) exp("1 + ,,?!)

+ a(2, 1 *) exp("2 + "f) + a(2,2*) exp("2 + ,,?!)

+ a(l, 2,1 *,2*) exp("1 +"2 +"f + ,,?!),

F(x, t) =

G(x, t) = exp(17I) + exp(17z} + a(l, 2, 1*) exp(171 + 172 + 17f)

+ a(l, 2, 2*) exp(171 + 172 + 17:}:).

(1.8)

(1.9)

(1.10)

(1.11)

a(i,j*) = (PI + Pj*)'

a(i,j) = (PI- Pj)2,

a(i*,j*) = (PI* -Pj*)2,

a(i,j, k*) = a(i,j) a(i, k*) aU, k*),

a(i,j, k*, 1*) = a(i,j) a(i, k*) a(i, 1*) aU, k*) aU, 1*) a(k*, 1*),

where * implies a complex conjugate, and

n. = iP~.J J"j = PiX -not -"~O)I "1'

where P j and '1jO) are complex constants relating respectively to the amplitude and to

the phase of the ith soliton.For initial conditions, Eq. (1.4) is used at 1=0, and three different sets of

parameters are studied:

(i) PI = 1 -0.25i, P2 = 0.5 + 0.15i, '1~0) = -2 and '1~0) = 0,

(ii) PI = 2 -0.5i, P2 = 1 + O. 75;, 'I~O) = -2 and 'I~O) = 1.0,

( "' ) P _ 4 - 2 0 P _ 3 0 (0)- 904 (0)- 2111 1- " 2 -+ I, '11 --.,'12 -.1.

Iq I

~ I I I I I I I~""!~\' -1.- I I I I I I I J

-20 -10 0 10 w 20

FIG. Initial condition (1.3), ,,=0.5 (i.e., amplitude = 1).

206 TAHA AND ABLOWITZ

Iql

(B) BOUNDARY CONDITIONS

Periodic boundary conditions are used. The period is chosen to be -20,20] in thecase of small amplitudes and [-10, 101 in the case of relatively higher amplitudes.(See Figs. (1) and (2).)

The numerical solution is compared with the exact solution. In addition, two of theconserved quantities are computed, namely, f fql2 dx, and f(lql4 -laqjaxI2) dx.

2. THE REPRESENTATION OF THE NLS EQUATION USING NUMERICAL METHODS

I. FINITE DIFFERENCE METHODS

(A) EXPLICIT METHODS

(i) The Classical Explicit Method

Using the classical emplicit method with central difference in time (for stability),the finite difference representation of Eq. (1.1) is

m+1 m-qn -qn

2 m + q m m'q~+I-qn n-I+2(q2);qn.= (.:fx) 2 (2.1 )2Al

where I n I < p, and m > O.It is easily shown that this method is linearly (dropping the nonlinear term) stable

for At/(A.x-)2 ~ 1/4. The truncation error of this scheme is of order (O«At)2) +

O«AX)2».

(ii) A Hopscotch Scheme

The NLS equation (1.1) can be approximated by(a) an explicit ~cheme:

.q:;'+I_q:;' -q:;'-..t-2q:;'+q:;'+I+ 1(lql:'-1)2q~-I+(lql~+1)2q~+I]'I At -(Ax)2 (2.2}

~

NONLINEAR EVOLUTION EQUATIONS, II

(b) an implicit scheme:

m+l m m+1 2 m+l+ m+1iqn -qn=qn+1 = qn 2 qn-I+[(iql::,~..)2q;:,~..+(lql::,:..)2q::,:..I.(2.3)

Lit (Lix)

~

The hopscotch scheme applies Eq. (2.2) at odd values of (n + m) and (2.3) at evenvalues of (n + m). This combination makes (2.3) explicit. If we write Eq. (2.3) form = m -1, and substitute the resulting equation into (2.2), the explicit scheme (2.2)tafter the first time step) may be replaced by

q;:'+'=2q;:'-q;:,-I. (2.4)

This scheme has truncation error of order (O«At)2) + O«AX)2», and it is uncon

ditionally stable according to linear analysis.

(B) IMPLICIT METHODS

(i) Implicit-Explicit Method

The Crank-Nicolson scheme is used to approximate the linear part and an explicitaverage is used for the nonlinear part of the Eq. (1.1). The scheme is

1 1 m 2m + m + m+12m+lm+1 127;bJ1 qn+l- qn qn-1 qn+1 -qn +qn-1

+ (!ql::'-1)2 q::'-1 + (!ql::'+1)2 q::'

The scheme is unconditionally stable according to linear analysis. The truncationerror is of order (O«At) + O«AX)2». To implement this scheme, a quasi-tridiagonalsystem of equations is required to be solved at each time step. An optimization of the

Gaussian elimination method is introduced to solve this system. (See Appendix A.)We carry out the elimination procedure only once and use back substitution

thereafter.

(ii) Crank-Nicolson Implicit Scheme

The difference scheme for representing Eq. (1.1) is

.q::,+I_q::'I

1_ 1 m -2 m + m + m+1 2 m+1 m+IJ-2(Ax)2 qn+1 qn qn-1 qn+1 -qn + qn-1

+ (iql::,+1)2 q::,+1 + (iql::')2 q::'

At

(2.6)

This scheme is also unconditionally stable according to linear analysis. The trun.

cation error of this scheme is of order (O«At)2) + O«AX)2)).

208 TAHA AND ABLOWITZ

We briefly remark on how we solve (2.6). Rewrite Eq. (2.6) as

A.

-Tq:.:.t+(i+A.)q:+1A.__ qm+12 n+1

A-T {q::'+ I + q::'-I} + (i -A-) q::' + Llt{(i q::' + 11)2 q::'+ 1+ (iq::'I)2 q::'},

n"= -N,..., N, (2.7)

and

;. = At2W'

where the solution is sought in the region (-NLtx ~ x ~ NLtx) X (t> mLtt,m = 1,2,...). From the periodic boundary conditions, we have q~N = qZ' and

qZ'+ I = q~N+ I for all m. Therefore, by applying Eq. (2.7) at each mesh point (i.e.,n = -N + 1,..., N), we can write the totality of equations as

AQm+I=F, (2.8)

where

).

-2

.J.

-2

m+i+A. q-N+

A

-2

A.

-2 i+A. q-N

Qm+1 =A=

).

-2

).-2 i+).

).

-2

A

-2i+).. qN

and

)..F}=2 {q~ 1 + qj-l} + (i-)..) qj +At{(lqj+ 11)2 qj+ 1+ (lqjl)2 qj},

j=-N+ I,...,N. (2.9)

The right-hand side of Eq. (2.8) is a function of known values of q at the previoustime level (t = mAt) and unknown values of q at the new time level (t = (m + 1) LIt).We use an iteration technique to solve the system (2.8), and we assume (only in theright-hand side) the values of q':+ 1= q': to start with. Therefore the right-hand side

209NONLINEAR EVOLUTION EQUATIONS, ))

becomes known, and we use the same optimization of the Gaussian eliminationmethod for (2.5) to solve the system (2.8). The resulting values of q at the new timelevel are substituted in the right-hand side of Eq. (2.8) to start the new iteration, andwe solve the new version of (2.8) by the previous method. We iterate until the con-

dition

maxlq::,+I,k_q::,+I,k+ll <tolerance

n = -N,..., N

(where k is the number of iterations) is satisfied. The iteration procedure is repeatedat each new time level. The q:;'+ I,k+ I will be the approximated solution at the point

(nAx, (m + 1) At).

(C) THE ABLOWITZ AND LADIK SCHEME

The scheme is

q:;'+I_q:;'

At

(2.10)_2qm+l+ qm+1

/I /I-I

"n AZ'

k=-oo

+qm+l (q m qm' +q m+l qm+I' ) +2q mqm' qm+1/I /I-I /I /I-I /I /I /I /1+1

+ 2qm+ l qm+ l" qmn n n-l

"-I

nk= -00

where

Sm m m' + m m' m+I" ( m+I )*k=qkqk-1 qk+lqk,qn =qn '

AZ' = (1 + (AX)2 qZ'+ Iq';:+ 1')/(1 + (AX)2 qZ'qZ"),

andL1 S m - Sm+ 1- S m

m k- k k

This is a global scheme, unconditionally and nonlinearly stable, and the truncationerror is of order (O«LJt)2) + O«LJX)2)). A local scheme (with the same truncationerror) from (2.10) can be obtained if the sum terms are zero and the product terms

are equal to one.

T AHA AND ABLOWITZ210

It is convenient to implement (2.10) as follows. Write the new time level equation

asqm+1

11+1 (2+t:)q:+I+q:~..=B".

where

2i(Ax)2At ' 1t:1~t=

(At is supposed to be of the same order as Ax)

and

n-1

rIk=-ro

Am + 2qm + . qm + .' q mk n n n- Amk+ 2qm qm' qm+ I

" " "+1 II.=-00

m-Qn

(2.11) is solved by a version of the Crank-Nicolson back and forth sweep method forthe heat equation (14, 151. We seek an equation of the form (at the new time level)

(2.13)qn+ 1= aqn + bn

suitable for computing q explicitly by sweeping to the right. For stability we requirelal~ I. Repeated substitution into (2.11) to eliminate qn+1 and qn in favor ofqn-1

gives

(2.14)bn + 10 -(2 + £)] bn-, + 102 -(2 + £) 0 + I] qn-1 = Bn

Requiring the qn-1 term to drop out determines a (uniquely since lal~ I) as a

solution of

(2.15 )a2 -(2 + 6) a + 1 = 0

and leaves for bn a first-order difference equation.

211NONLINEAR EVOLUTION EQUATIONS, II

The corresponding homogeneous equation of (2.14) has a solution of the form

b =knn ,

where the constant k satisfies

k+ (a-(2+e)I=O.

It can be shown that the solution k of (2.17) corresponds to the second root (otherthan a ~ I) of the quadratic equation (2.15) determining a above, and that I k I > I.

It follows that b can be computed explicitly by sweeping to the left,

b"_t=ab,,-aB,,

To obtain the solution qn' first solve for bn from (2.18) then use (2.13) to calculateqn. In order to calculate the b's, we use an iteration procedure. We assumed thatq';;' + I = q';;' in Eq. (2.12) and bN = 0 to start with, and then we apply the Gauss-Seidel

technique 116) (in which the improved values are used as soon as they are computed)to calculate the rest of the b's. The calculated value of the bN (= b-N) is used to startthe new iteration, and the iteration procedure is repeated until the condition

max Ibn-, -(abn -aBn)1 < tolerance

n = -N,..., N

is satisfied. Then we use the above procedure by sweeping to the right by means of(2.13) to obtain the q's. After the calculations of the q's, we substitute their valuesinstead of q;+ I in Eq. (2.12), and repeat the same procedure to calculate the b's and

then the q's. This procedure is repeated until the condition

max I q:;' + I,k -q:;' + I,k + II < tolerance

n = -N,..., N

(where k is the number of iterations) is satisfied. The q::' + I.k + I will be the approx.

imated solution at the point (nAx, (m + 1) Lft).

2. FINITE FOURIER TRANSFORM OR PSEUDOSPECTRAL METHODS

(i) Split Step Fourier Method [6)

For convenience the spatial period is normalized to [0, 21l1. then equation (1.1)becomes

(2.19)

212 T AHA AND ABLOWITZ

where P is half the length of the interval of interest, and X = (x + P) 1C/ P. (Here wetake P to be 20 or 10 depending on the calculation.) This interval is discretized by Nequidistant points, with spacing AX = 21C/N. The function q(X, I), numerically defined

only on these points, can be transformed to the discrete Fourier space by

q(k, t) = Fq =

(2.20)k=-~2 ,...,

N-1,0,1""'2-

The inversion formula is

q(jL1X, t) = F-1q = 7= \ .q(k, t) e2"Uk/,"

yN7(2.21)

Nk=-T"'"

N

-1,0,1'...'2-

These transforms can be performed, efficiently with the fast Fourier transform (FFT)algorithm [171. Following [6] in order to apply the split step Fourier method forEq. (2.19) we (a) advance the solution using only the nonlinear part:

iii, = 21iil2 if. (2.22)

This can be solved exactly,

ij(X, t) = e-2iIQ(X.O)II/q(X, 0), (2.23)

where ij(X, t) is a solution of Eq. (2.22) and q(X,O) is the solution of Eq. (2.19) att = O. (b) advance the solution according to

7r2

iq'=pzqxx (2.24)

by means of the discrete Fourier transform

q(Xj' t + At) = F-1(elk241,,2/P2F(q(Xj' t»). (2.25)

This method is second order accurate in LIt and all order in Llx, and is uncon.ditionally stable according to linear analysis.

(ii) Pseudospectral Method (Fornberg and Whitham) [III

This is a Fourier (pseudospectral) method in which q(x, t) is transformed intoFourier space with respect to x, and derivatives (or other operators) with respect to xare then made algebraic in the transformed variable. Again for convenience thespatial period is normalized to 10, 2n]. With this scheme, qx,'( can be evaluated as

NONLINEAR EVOLUTION .EQUATIONS, II

F-1jrk2F(q)}. Combined with a leap frog time step the NLS equation (2.19) is then

approximated by

-4iLft Iql2 q. (2.26 )

Using the ideas of Fornberg and Whitham we make a modification in approximating

Eq. (2.19),

(2.27)-4iAt Iql2 q.

The difference between Eqs. (2.26) and (2.27) is in the approximation of the linearEq. (2.24). The linear part of Eq. (2.27) will be exactly satisfied for any solution ofEq.(2.24) (see [Ill). Also it turns out that Eq.(2.26) is linearly (dropping thenonlinear term) stabe for At/(Ax)2 < 1/7t2, while Eq. (2.27) is unconditionally stable

according to linear analysis.

3. CONCLUSIONS

Various numerical methods are used in order to approximate the NLS equation(i.i), namely; (i) The classical explicit method (2.i), (ii) hopscotch method (2.2),(2.3), (iii) implicit-explicit method (2.5), (iv) Crank-Nicolson implicit scheme (2.6),(v) The Ablowitz and Ladik scheme (2.iO), (vi) The split step Fourier method (2.23),(2.25) and the (vii) pseudo spectral method of Fornberg and Whitham (2.27). Weobtain a comparison between the Ablowitz-Ladik scheme and the other utilizedschemes. Our approach for comparison is to (a) fix the accuracy (L",,) forcomputations beginning at t = 0 and ending at t = T; (b) leave other parameters free(e.g. At, or Ax), and compare the computing time required to attain such accuracy forvarious choices of the parameters. For the comparison two sets of initial conditionsare studied: (A) i-soluton solution with different values of the amplitude, (B)Collisions of two solitons with different values of the parameters. According to this

approach we have made the following conclusions:

(i) The schemes explicit (i), implicit-explicit (iii), and the hopscotch (ii) takemore computing time than the other schemes «iv, (v), (vi), (vii») and the difference inthe computing time increases as the amplitude increased. The hopscotch method (ii)takes less computing time than the other two methods «i), (iii)) for the i-solitoncases, while the explicit (i) method took less computing time than the hopscotch (ii)and the implicit-explicit (iii) methods for the 2-soliton cases.

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219NONLINEAR EVOLUTION EQUATIONS, II

(2) The previous three methods; explicit (i), hopscotch (ii), and implicit-ex-plicit (iii) do not appear in Tables I, IV, and VII since extremely long computing time

would be required.(3) The Crank-Nicolson implicit method (iv) takes more computing time than

the Ablowitz-Ladik (local and global) method, the split step Fourier method andFornberg-Whitham method in the case of I-soliton, and it becomes comparable withthe Ablowitz-Ladik local scheme for high amplitudes. In the case of 2-solitons,Crank-Nicolson takes less computing time than the Ablowitz-Ladik local scheme.

(4) The Ablowitz-Ladik global scheme takes more computing time than thelocal scheme in the case of small amplitudes, but for high amplitudes it is roughlyfive times faster than the local, and the Crank-Nicolson schemes.

(5) The pseudospectral method is roughly two times faster than theAblowitz-Ladik local scheme for small amplitudes, but it is much faster for highamplitudes. This method is roughly two times faster than the Ablowitz-Ladik globalscheme for high amplitudes. This method only proves to be faster than the split stepFourier method for small amplitude 2-soliton cases.

(6) The split step Fourier method is faster than all the utilized methods forsmall and large amplitudes for the I-soliton case. In the average it is three timesfaster than the Fornberg-Witham method. Also it proves to be faster than theFornberg-Whitham method for high amplitude 2-soliton cases. The tables and figures

exhibit the results.

As a conlcusion we find that the split step Fourier method is the best method forthe NLS equation, followed by the pseudospectral method then the Ablowitz-Ladikglobal scheme. However we believe that if we were able to go to very high amplitudes(our machine capability prevented this) the Ablowitz-Ladik global scheme wouldimprove dramatically and would prove to be better than the other methods. Howeverit should be noted that the NLS equation is quite unusual in the sense that thenonlinearity is especially simple. This has a dramatic effect in the split step Fouriermethod-see Eq. (2.22}-which means that both steps admit to essentially exactmethods. Generally this will not be true (see paper III). We also note that whereas theAblowitz-Ladik scheme is O«At)2, (AX)2) the split step Fourier and Fornberg-Whitham methods are of order O«At)2, (Ax)P) for all p. (See also the calculations forthe KdV equation in paper III.) It is also worth mentioning that we have tried thesweeping technique in implementing the implicit-explicit and the Crank-Nicolsonmethods, and found that it did not affect the overall conclusions.

All the numerical calculations are inspected at every step by using the conservedquantities flql2 dx, and f (iq14 -laqjaxI2) dx (Tables I-VII). The two conservedquantities are calculated by means of Simpson's rule [18]. In the finite differenceschemes we have discretized U x by using a central difference approximation. In theFourier methods the derivatives are calculated using Fourier method. TheAblowitz-Ladik global scheme is the only utilized scheme which has an infinitenumber of conserved quantities. It is worth mentioning that we calculated the L2error norm and found that it reflects the same conclusions as the Loo norm.

220

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8. Fornberg and Whitham method

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FIG. 3. Displays the computing time (E) which is required by each utilized method given in Table II-soliton. amplitude = I.

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cI ~

c

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4. Hopscotch.5. Split step Fourier ..ethod.6. The Ablowitz and Ladik local scheme7. The Ablowitz and Ladik global scheme8. Fornberg and Whitham method

1 2 3 4 5 6 1 8 HetMd

FIG. 4. Displays the computing time (E) which is required by each utilized method given inTable II. I-soliton, amplitude = 2.

225

-~~

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FIG. 5.Table III.

.III 2 J 4 5 Method

Displays the computing time (E) which is required by each utilized method given in-soliton, amplitude = 4.

!-C

~!

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--I 2 J 4 5 Method

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FIG. 6. Displays the computing time (E) which is required by each utilized method given inTable IV. I-soliton, amplitude = 6.

226

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FIG, 7, Displays the computing time (E) which is required by each utilized method given inTable V, Two solitons with amplitudes 0,5 and 1.

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1 2 3 4 5 6 7 8 Method

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FIG. 8. Displays the computing time (E) which is required by each utilized method given inTable VI. Two solitons with amplitudes I and 2.

227

228 TAHA AND ABLOWITZ

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*: Eacll unit in time ; 5Minuto'.

FIG. 9. Displays the computing time (E) which is required by each utilized method given inTable VII. Two solitons with amplitudes 3 and 4.

APPENDIX A: AN OPTIMIZATION OF GAUSSIAN ELIMINATION

This method seeks to optimize Gaussian elimination by eliminating unnecessarystorage and multiplication by zeros. To begin we have the following quasi-tridiagonal

system

bl

b2

dl

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XI

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UN-I

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Instead of storing the N X N matrix, we store the augmented matrix in an N X 4matrix whose elements are the tridiagonal elements and the hi's. We then performGaussian elimination on the N X 4 matrix keeping in mind their original locations inthe matrix. When this is done we have an upper triangular matrix and an originalsystem is of the form

NONLINEAR EVOLUTION EQUATIONS, II

dJ hi

hi:UI

d'tXI

X2

a

au*2 =d* *N-IUN-I

d*N b*NXn

where * values are the updated elements. Using back substitution we obtain the

solution. The total number of operations required to obtain the solution using thismethod is (1IN-15). The same idea can be applied to quasi-pentagonal system ofequations and so on.

ACKNOWLEDGMENTS

This research was partially supported by the Air Force Office of Scientific Research, Grant78-3674-C, the Office of Naval Research Grant NQOOI4-76-C-O867 and the National Science Foun-dation Grant MCS79-O3498-O1. We are grateful for useful conversations with Dr. H. Segur.

REFERENCES

I. N. YAJIMA. M. OIKAWA, J. SATSUMA, AND C. NAMBA, Modulated Langmiur waves and nonlinearlandau damping, Rep. Res. Inst. Appl. Mech. XXII, No. 70 (1975).

2. V. I. KARPMAN AND E. M. KRUSHKAL, Modulated waves in nonlinear dispersive media, Sol'iotPh}'s. JETP 28 (1969), 277.

3. N. Y AJIMA AND A. OUTI. A new example of stable solitary waves, Prog. Theor. Ph.vs. 45 (1971),1997.

4. J. SATSUMA AND N. Y AJIMA, Initial value problems of one-dimensional self-modulation of nonlinearwaves in dispersive media, Prog. Theor. Phys. Supp. 55 (1974).

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