University of Alberta

MODUL.4TION OF THE HARMONIC SOLITON SOLUTIONS

FOR THE DEFOCUSIXG NONLINEAR SCHRODIXGER EQUATION

A thesis submitted to the Faculty of Graduate Studies and Research in partial fi iWhent of the requirements for the degree MASTER O F SCIENCE

in

APPLIED MATHEM ATICS

Department of Mathematical Sciences

Edmonton, Alberta

Spring, 1999

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ABSTRACT

In this thesis we examine some of the properties of the slolvly decaying

solutions of the nonlinear Schrodinger equation. By the superpositioning slowly

decaying solutions we get the so called harmonic soliton solutions. In particular

we examine some of the modulating properties of the harmonic soliton solutions

for the defocusing Schrodinger equation:

iut + u,, - 2 1 ~ 1 ~ ~ = O

ACKNOWLEDGEMENTS

1 would me to express sincere thanks to my supervisor, Dr. M. Kovalyov,

for the excellent advice and considerable time and effort that he has spent on my

work while I have been completing this thesis.

Thank you Ion for teaching me LaTeX and all your constructive critisism.

FinaIly; 1 would Iike to t h d my brothers (Polly, Teeka, John). sisters (B,

Norma: Mila? Finey, Sandy, ShKmilla, Rita) and magé k d a Ch- for all their

support.

For

Mom and Dad

Contents

1 Beginnings 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . 1

3 1.2 Background and Some History . . . . . . . . . . . - . . . . - . . . -

2 Fourier Transforrn for linear equations 6

3 Origins 9

3.1 Derivation of IïdV and XLS equations . . . . . . . . . - - . . . . 9

3.2 Inverse Scattering Theory for KdV and NLS equationi . . . . . . 12

4 Soliton Theory 17

4.1 Derivation of the Y-soliton solution . . . . . . . . . - . . . . . . . 1 7

4.2 Harmonic So1ii;ons . . . . . . . . . . . . . . . . . . . . . . . - . . 31

4.3 Nonlinear Interference . . . . . . . . . . . . . . . . . . . . . . . . '24

Conclusion 33

Bibliography

List 01 E'igures

FIGURE 1 : The nonlinear interaction of solitons.. ..................................... -3

FIGURE 2 : Solitary waves of various speeds with a dispersive tail ............... 5

FIGURE 3 : Snapshots of the single harmonic soliton solution ................... 23

....................... FIGURE 4 : Snapshots of the two harmonic soliton solution 26

FIGURE 5 :Sime evolution of the wavepacket generatated by

8-harrnonic solitons.. ...................................................... ..28:29 :30

FIGURE 6 : Generation of an IV-wave type profile ..................................... 31

................................ FIGURE 7 : Generation of a 6-function profile..

FIGURE 8 : Time evolution of a N-wave generated by

10-hamonic solitons .......................................................

FIGURE 9 : Time evolution of a 6-function generated by

10-harmonic solitons ............................................................... -33

FIGURE r O: Matlab results.. ................................................................... .34

Chapter 1

Introduction

Most wave motions are modeled by linear hyperbolic equations, which we

can solve by means of the Fourier transform as we will discuss in Chapter 2.

After some simplifications these models often lose their hyperbolicity and more

important their lineasity, [El. As a result the Fourier transform method can no

longer be applied.

In this thesis we investigate the possibility of a replacement of the Fourier

transform for some nonlinear KdV-type equations, in particular the nonlinear

defocusing Schrodinger equation. First we remark, that the Fourier transform

method is a mathematical description of what is known in physics as lineâr mod-

ulation of waves. Linear modulation can be used to construct locdized in space

solutions for linear partial differential equations! so cdled wave packets. The

natural question one can ask is whether the existence of nonlinear modulation

yields exact solutions of a nonlinear partial differentiai equation. The andogy

between Fourier analysis and the Inverse Scattering Theory used for solving the

Cauchy problem for integrable equations suggests that if any nonlinear partial

differential equations d o w nonlinear rnodulat ion, t hen t hey would be the inte-

grable equations. It will be shown that the nonlinear defocusing Schrodinger

equat ion perrnits nonlinear modulation.

In Chapter 2 we tersely review the ideas behind the Fourier transform method

for linear partial differential equations. We will review how to construct an

arbitrary solution of the form e'(kxcf-w(k)'), which can be viewed as the simplest

components. In Chapter 3 we use the representation of such solutions to derive

the KdV and NLS equations. The solutions to the KdV and NLS can then be

obtained by the inverse scattering method. Chapter 4 contains the main results

of this thesis, namely the generation of the N-harmonic soliton solution for the

defocusing Schrodinger and some time evolution profiles for the N-harmonic

soliton case. These results are new and could possibly be applied to Soliton-

Lasers and Soliton-Based Communicatioo S ysterns , 1131.

1.2 Background and Some History

Solitary waves were f h t observed by J. Scott Russel in 1834. After extensive

experiments in the laboratory, he observed that:

1. solitary waves are long, shallow water woves of permanent form,

2. solitary waves propagate at speed c = Jm,

where g is the acceleration due to gravity, a is the amplitude and h is the uni-

form depth of the waveguide/channel. Further investigations into the existence

of solitary waves were undertaken by Airy (1845): Stokes (1847): Boussinesq

(1871-72) and Raleigh (1576). Korteweg and de Vries (1595) derived a nonlinear

equation, named after them, goveming long one dimensional, small amplitude,

surface gravity waves in a shdlow media, [14]. Further progress came when

Fermi, Posta and Ulam (1955) considered the problem of a one dimensional an-

harmonic lattice of equal masses coupled by nonlinear springs modeled by the

KdV? [15]. This subsequently lead to the discovery of so called solitary waves

i.e. a localized traveling wave solution, of the form f(x-ct), of a nonlinear par-

tial differential equation. Zabuslcy and Kruskal (1965) did extensive computer

simulations for the initial value problem for the KdV:

u(x, O) = cos(~ix), 0 5 x 5 2

where u, u,, u,, are ~eriodic functions on [0,2] for every t. They discovered

some solitary wave-type solutions with some rather interesting properties, (1 61.

They c d e d them solitons. They observed:

1. solitary wave is a stable formation;

2. when solitary waves move with different velocities, the faster one will over-

take the slower one and, after a complicated nonlinear interaction, the

solitons emerge in their original form with possible delays (phase shifts)

due to interaction (Fig. 1).

3. every sufficiently smooth and exponentially decaying solution of the KdV

with initial condition u(x, O ) = p(x), decornposes as t t m into a finite

number of solitary waves of various speeds and a dispersive tail (Fig. 2).

This kind of behavior is expected for linear problems, for exarnple for the wave

equation ut* = c2 u, with u(x, 0) = p(x) on O < x < 1, u(02 t ) = u(l, t ) and

ut (x7 0) = +(x) since each eigenfunction evolves se~arately, but that this codd

happen for a nonlinear problem was not expected. Later on, a number of other

equations with solutions possessing properties similar to that of solitons were

discovered, e.g.

Nonlinear Schrodinger: iut + u, + & ~ u / ~ u = O, u = f 1

e Sine-Gordon: utt - uzZ + sin(u) = O

We start by reviewing the ideas behind Fourier transform which will subse-

quently lead us to the KdV and the NLS.

Figure 1

Figure 2

Chapter 2

Fourier Transforrn for linear

equations

The most general Cauchy problem solvable by the Fourier transform is the

one reducihle to the f o m , ['il

rm-l u = f (2) for t = O

where

P ( D , T)U = O for t > O

k r u=O for k = 0 , 1, ..., m-2 a n d t = O

and P(D7 r) = P(D1, . . . : D,, r ) is a polynomid of degree rn in its n + 1 argu-

ment s,

Consider the simplest case of (2.1)-(2.3) that appears in the theory of waves,

[2] :

A formd solution of the standard problem (2.5)-(2.6) can be obtained using

the Fourier transformation

00

û(k7 t ) = u(z , t ) ë i k Z d z (2.S)

Assuming the validity of the interchange of derivative and integral, by taking

the Fourier transform of (%.5), we get:

which is a linear differential equation with solution:

where

The solution of (2.5)-(2.6) is:

If the system (2.10) is conservative, Le. w ( k ) is a real function and dispersive

(wM(k) # O ) , then the solution may decay into wavepackets which move with

group velocities wl (k ) and decays as t + m. So an nrbitraq solution can be

decomposed into ei(kr-w(k)t), which can be viewed as the simplest components of

that solution.

Chapter 3

Origins

3.1 Derivation of KdV and NLS equations

We f i s t begin by deriving the well known KdV equation and then the NLS

equation siiice the analysis of the NLS equation parallels that of the KdV equa-

tion. tVe showed in the previous chapter that an arbitrary wave can be decom-

posed into harmonics ei(kx-w(k)t) . Consider the plane wave corresponding to one

of these harmonics:

Assume the dispersion relation is of the form:

A partial differential equation for v(x, t ) with this dispersion relation is:

which is often called the linearized KdV equation, [6]. In shdow water theory,

the following conservation law must also hold, [2]:

For the lineasized version of the KdV equation, j is given by :

Combining (3.2),(3.3),(3.4) together ive obtain, [2]:

ut + Qu, + pu,,, + n,v"t = O

Using the rescaling,

and

we can rewrite (3.6) in the standard form

w, + w<<< + 6wwC = O (3.10)

In a similar fashion we derive the nonlinear Schrodinger equation (3.1 1):

iut + u,, + Z Y ( U ~ ~ U = O (3.11)

For a modulate wavetrain with most of the energy in wave numbers close

to çome values ko, f ( k ) is concentrated near k = ko, and one can approximate

(2.12) by:

where wo = w(ko): wh = wf(k0 ) : w{ = w"(ko].

Let k - ko = K , then:

where p(t, z) describes the modulation in (3.13) and satisfies

with the corresponding dispersion relation:

The equation for $ with the original dispersion relation

is:

If the approximation (3.14) to the linear dispersion is combined with a cubic

nonlinearity, [2], we have:

Since p(t ,2) = a d " - w t ) is still a solution to (3.15), we get that the non-

linear correction to the dispersion relation modifies W to:

Equation (3.18) can be normalized by choosing a frame of reference moving

with linear group velocity wh to eliminate the 9, term in (3.15) and then: after

rescaling the variables, [2], we get the nonlinear Schrodinger equat ion (3.11).

There are actually two NLS7s, one with v = 1, the other with v = -1.

The NLS can be considered xi the Hartree-Fock equation for a one-dimensional

quantum Boson gas with 6-point interaction. Then u plays the role of a coupling

constant: the case u > O corresponds to attractive interaction between particles

and v < O is the repulsive case, [6]. The two cases are essentially different in

optical applications, describing selffocusing or defocusing of the light rays in

nonlinear waveguides, [1 O].

3.2 Inverse Scattering Theory for KdV and

NLS equations

To illustrate the method of solving KdV, NLS-type equations using the method

of inverse scattering developed by Gardner-Green-Kruskal-Miura, [17, 181, we

start by considering the simplest of the equations, [5]. Consider the KdV equa-

tion

with

where f (x) is sufficiently smoot h and decays rapidly as lx 1 + m. The basic idea

is to relate the KdV equation to the time-independent Schrodinger scattering

problem:

The time-dependence of the eigenfunctions (3.22) is given by

ivhere 7 is an arbitrary constant pârameter and X is the spectral parameter.

The KdV can then be written using the Lax pair [M,L], as:

where the operators 1l.I and L are given by:

Equation (3.24) is satisfied if and only if u(x, t ) satisfies the KdV.

13

The solution of the KdV with initial condition (3.21) is as follows:

At t = O, we need to determine the spectmm of the Schrdinger equation

(3.23) which consists of a finite number of discrete eigenvalues, A, = 5; > 0;

n = 1, . . . , N and continuous spectrum, A = -k2 <O. The corresponding eigen-

functions are:

CQ 1. For X = (: : $,(x;t) z ~ ( t ) e-cnx as x i 05, with J-= $:(x,f)dx = 1

2. For X = -k2:

(a) +(x, t ) e-ikx + r ( k , t ) eikx as z + oo

(b) +(x, t ) a ( k , t ) e-'kx a s x - + - C O

where r(k, t ) and a ( k , t ) are the reflection and transmission coefficients respec-

tivel y.

At t = O we define the scattering data to be:

The time evolution of the scattering data

Using the scat t ering data define

Shen we recover the potential u(x, t) via:

where K (x, x; t ) is the solution of the Gel'fmd-Levitan--VI archenko equation

The method just described is a concise version of the method of inverse

scattering developed by Gardner-Green-Kruskal-Miwa.

Zakharov and Shabat (1972) showed that the inverse scattering approach

may also be applied to solve the NLS, [l]. In particular they shomed that the

nontrivial operators

and

satisS. Lm's equation (3.24) if and only if u(x, t ) is a solution of the NLS equa-

tion.

If we let

where

a r(E, t ) = r(5, O) eaiCzt

ci(t) = ~ ~ ( 0 ) e4ik:'

Then we obtain the Gel'fand-Levitan-Marchenko system:

which allows us to recover the potential via

( x ) = ( X x), JZw lu(s)12ds = -2K2(xl x).

In the reflectionless case the reflection coefficient is zero, i.e. r(k, t ) = r ( k , O) = 0,

and we can solve the Gel'fand-Levitan-Marchenko equation using the method of

separation of variables, [Hl , [19].

'ü denotes the cornplex conjugate of u

Chapter 4

Soliton Theory

4.1 Derivation of the N-soliton solution

Besides the method described in Chapter 3, we can obtain the N-soliton

solutions for the NLS following the methods of [l], [4], and [6]. We first observe

that the NLS can be written as:

v, - v, = [V, li]

which is a compatibility condition for:

where

and

The field functions q ( x , t ) and r(x, t ) are independent of A. Note that (4.1)

can also be written in Lax's form:

We can easily obtain the field functions for both NLS's with r = +q = O and

For r = q = O and Y = k1 equations (4.4) and (4.5) assumes the form:

and equations (4.2), (4,3) degenerates into

Solving the system (4.8), (4.9) we get,

With Qo given we can now obtain N-soliton solutions, q ( x , t ) , by means of

the N-fold Backlund-Darboux-Matveev transformations [4], [6], in the form:

where ,8, = c X~ = O + i q j , c= constant. +0>02(5, t i X j ) ' - V

For the 'JLS equation, r = f Q, Aj+w = X j ? and @j+w = - for some constants Pj

j cj7 with j 5 N

The 1-soliton solution of the NLS is:

and the corresponding functions given by :

Case 1: v = -1

where:

ail = ((A - ( - iT)es + (-iv + - ~ ) ~ - e ) e-iAf-2i"2'

ai2 = 2i9 e ' ( " ( X - 2 C ) + 2 t ( 2 V 2 - 2 ~ 2 + X 2 ) - 2 ~ € )

- sii) e-i(~(d\-2<)+2t(2~2-2$ + A 2 ) - 2 7 ~ ) azl - - -

Q2* = ((A - E + iq)ee + (iq + E - A) e-O) ei"f+2'"2t

e = 2 ( ~ - p + 4tc)q

Case II: v = +1

where:

The constants q = 3(X1) and 5 = R(Xi) determine the amplitude and velocity

(v = -40 of the soliton respectively, [Il. Unlike the KdV soliton, 7 and f are

independent and can be choosen arbitrary. In the general 3-case. the N-soliton

solution of the NLS equation depends on 4N arbitrary constants; qj? c j , p j , -fi-

For distinct cj7 as t t +oo the Pi-soliton breaks up into individual solitons

in such a way that the fastest soliton is always in front of the slocvest one in the

rear, and vice-versa as t -t -m. In this case, pj (the center coordinates) and

-fi (phase angle) are no longer fixed. For identical tj j: the solitons form a bound

state; [9].

4.2 Harmonic Solitons

The NLS with = 1 (self-focusing case) has regular solitons that describe

modulated ~ulses . For the NLS with u = -1 (de-focusing case) we get singular

solitons, due to the singularity they can no longer be modulated pulses the way

the regular solitons are. We will show that we c m still obtain modulation for

v = -1 using what we cal1 harmonic solitons. These harmonic solitons appear

when 3 ( X j ) = qj = O, i.e when both the numerator and denominator of (4.11)

are zero. By taking lim of (4.11) we get (4.15): rl+o

where:

where ,Bj = e2i€~(-f i f rf ' E j t ) , p j = -2i(pj - x - 4 t j t ) , yj9s, cj's, pj's are some

constants ,for j 5 n. The (n + j)-th row of A or B is obtained by subtracting

j-th row of (4.11) from the (n + j)-th row of (4.11), dividing the result by vj and

taking limit as qj + 0, [II]. The one harmonic soliton solution, q ( x , t ) , (Fig 3),

Each harmonic soliton is detemineci by- the constan& A: p arid -1 which

corresponds t o frequency, displacement and phase çhift respectively. By onalogy

23

with solitons, we define the nonlinear superposition of N harmonic solitons given

by the the sets of parameters A = ( A l , - - - Y AN)? P = (pi, - Y pn) and =

( T ~ , - - - , Y ~ ) to be the potential obtained by taking r ( k : t ) = r (k , O) exp(4it2t)

in the form:

1 if k, < k < k2 hl Ik) = {

O otherwise

The corresponding potential is given by (4.15).

4.3 Nonlinear Interference

Formulas (1.18)-(4.19) describe the interaction of X-harmonic solitons. Al-

though we cannot prove it, it certainly seems to be true that each N-harmonic

soliton (4.18) is a merornorphic function with at most N real poles. The 2-

harmonic soliton solution of the NLS equation is:

where

Formula (4.20) makes sense only when X2 # Tt&. Yet for X2 = XI + nx, n E Z, the concept of superposition of two harmonic solitons can be naturally

extended to the case Xî = &A1 by taking the limit of (4.20) as X2 t XI and

72X2 t -flX1. This yields a harmonic soliton with:

0 and p satisfying

d For pi's large we can neglect the - (72 X2) term to get,

dX2

Let us tàke a fi-;? region D= t ( izj O < t < T) ~vi;k - - -

1 osci&tory f o m (0.23) with amplituda -:

pk

ing in D is of theform:

Away from the poles the superpositioning of the hamoaic solitons is pract icdy

reduced to adding their absolute values, just like in the lineu case. In Linear

theory this phenomenon is called linear interference so we cal1 its nonlinear

analogue nonlinear interference. It is exactly this phenomenon that is responsible

in Linear theory for the formation of wavepackets. The nonlinear analogue also

leads to the formation of wavepackets, in this case nonlinear. Since there is no

theory for the nonlinear case, we simply construct some of the wavepackets.

Figure 5 shows the time evolution of the 8 harrnonic soliton solution. The

N-harmonic soliton solution does not necessarily vanish outside D as x -+ Ica.

Cornparison with the linear Schrodinger equation suggest that by choosing An's

sufficiently small such that A, 5 A,, + lAAl then we can choose X and T

arbitrarily large. This allows us to create a wavepacket of desired magnitude

and halflife. Equations (4.11), (4.15) have been used to verify this numerically.

Figures 6-9 shows some other interesting properties exhibited by the harmonic

soliton solutions of the NLS. The values of p, are chosen to be of the forms

pon3, pon2, pan and po so that we can mimic the 6 function with its antideriva-

tives. Figures 8 and 9 shows the time evolution of sorne N - wave type and

&type solutions respectively.

CL.

Z'L [ i - - * .-e.7

I . - ,.A.. 2 .-

i - I

Figure 5 (continued)

Figure 5 : T h e evolution of thz w~vepac~ker gerierâied b y 8 harmooic solitors witb P = (JI , , .. ., p z ) , p, = -65000 eo(3(X, - &,,)');

A = ( ) A, = 3 +0.066(n - 1) and r = (0 ,..., 0).

Fig .6 (a )

a/ 4-

- 2 4 . .

a " O : s E s 9 +

Fig. S ( a )

Figure 7 : The 13-ht-monic Iofibo~? solutior! af t = O with p = (P 1: . . - : P ~ ~ ) ~ pn = -,2.8n2x1O7, I? = (2, ..-, qj and (c) X = .-Zn (b) X = . l Ïn .

Figure 8 : T h e evdu tion CE the wavepah t generated by i O haraonic solitoro m'th P = ( p l , . . . :p~& p, = -1.jn3 x 10'; A = (XI , ...; &O),

A, = O.id(, - 1) and l? = (0, ..., 0).

"i-

Figure 9 : Tine evolution of the wavqaciiet ,o+nerated by IO h ~ m o n i c oolitons with P = ( p l , .... p,,), p, = -2.2n x 1 0 4 ; 11 = ( x ~ ;...,

A, = 0.13(n - 1) and I' = (5: ...., $).

Conclusion

Although we could not apply the Fourier transform method to solve the NLS,

we were still able to obtain exact solutions using the concept of nonlinear mod-

ulation. Using the analogy between Fourier analysis and the Inverse Scattering

Theory used for solving the Cauchy problem foi integrable equations me were

able to show that the nonlinear defocusing Schrodinger equation permits nonlin-

ear modulation. Also using the N-harmonic soliton solutions we were also able

to construct localized in space solutions for the nonlinear defocusing Schrodinger

equation, the so called wave packets.

Some other possible extensions that can be made in this thesis is the further

simplification of the the determinants found in (4.15), so ive could possibly write

the general N-harmonic soliton solutions in an applicable form for large N. One

can also try producing an analytic proof for the existence of N-real poles for the

N-harmonic soliton solution or question the existence of a generalized formula

similar to (4.22) for .N 2 3 that would ultimately link the KdV and the NLS.

Appendix

Matlab Program (1)

clear xp up M disp('started')

% Input Area

r e m f ake=(\pi/3)*2n(1/8)*1.012345678901*exp(1)/2 .71;

delxi=O.l9*remfake; half dim=8 ; PO=-11000 xio=f:halfdim; for n = l:halfdim, xio (n) =delxi* (n-1) ; end

halfgam=l : half dim; half gam= (pi /2) *ones (1, half d i m ) ;

halfp=i:halfdim; for n = Lhalfdim, half p (n) =-pO*n' (0 .5) ; end

% End of Input Area

d e l t ax= (xf -x0) /nu= ;

nx=i : numx; xp=xO-deltax + nx*deltax;

d e l t a t = ( t f - t 0 ) / n u t ; n t = l : numt ; tp=tO-deltat+nt*deltat ;

l a = i : dim ; for n=l :ha l fd im, lam(2+n-l)=i*epss+xio (n) ; lam(2*n) =-i*epss+xio (n) ; end

gam=l :dim; f o r n=i :ha l fd im, gam(2*n-l)=halfgam(n); gam (2*n) =halfgam(n) ; end

p=1 :dira; for n=l:halfdim,

p (2+n- 1) =ha l f p (n) ; p (2*n) =-half p (n) ; end

d i s p ( ) p l o t t i n g loop s t a r t e d ) )

AN=eye(dim) ; AD=eye (dim) ; b =l:dim; for it=l :nunt , t = t p (it) ; xO=xO+frameshift ; xf =xf +f rameshif t ; xp=xO-deltax + nx*deltax; end

up=l : nurmr; upl=l:numx; for ix = l:nunx,

f o r n= l:halfdim,

b (2+n) =e~p(l*i*gam(n)+2*e~ss*~ (2*n) ) *(exp (2*i*lam(2*n) *xp (ix) + 2*i*t* (lam(l*n)) -2)) ; lamf ac=l ; f o r k=i:halfdirn,

if kd=n lamf a c = l a m f ac*(lam(2*n-1) -lam(2*k-1)) (lam(2*n) -lam(2*k) ) * (-1) ; else qqqq=l ; end

end b (2*n-1) =b(2*n-1) *lamfac ; b (2*n) =b (2*n) *con j ( l amf ac) ; end

for n = i:dim,

f o r m = i :ha l fd im, AD (n ,2*m-1) =lam(n) (m-1) ; AD(n,Z*m) =(lam(n) ^(m-1) )*b (n) ;

end end AN=AD ; f o r n=l:dim, AN(^, dim) =lam(n) - (dim) ; end

up ( i x ) = real (de t (AN) /de t (AD) ) ; up 1 ( i x ) = real ( i*det (AN) /de t (AD) ) ; end

f i g u r e ( i t ) plot (xp , up) ax i s (CxO,Xf ,y~ ,y f I ) v a l u e o f t = t print -dps -append pusa.ps

figure (i t+numt) p l o t (xp a up l ) =is(CxO,xf ,yO,yfl) v a l u e o f t = t p r i n t -dps -append pusa .ps

Matlab Program (II)

% Input Area

xiO=3 ; half dim=10 ;

gam=i: half dim;

gam= (O) *ones (1, ha l f d i m ) ;

x i = l : half dim; for n = 1:halfdim, xi (n) =xiO+delxi* (n-1) ; end

p=i:halfdim; for n = l:halfdim, p (n) =p~*exp (2* ( x i (n) -xiave) -2) ;

end

% End o f Input Area

d e l t ax= (xf-x0) /numx; d e l t a t = ( t f - t 0 ) /nurat;

nx=l:numx; n t = l : numt ;

b '1: h a l f dim; mu =i:halfdim;

frameshift=framespeed*deltat; disp( ' p l o t t i n g l o o p s t a r t e d ' )

f o r it=l:numt, t = t p ( i t ) ; xp=xO-deltax+nx*delta;

for ix = i:numx,

for n= i:halfdim, b (n) =exp (i*gam(n) +2*i*xi (n) *xp ( i x ) +4*i*t* (x i (n) ) -2) ;

mu (n) =-2*i* (P (n) -xp (ix) -4*t *xi (n) ) ;

end

f o r m = Lhalfdim, for n = L h a l f d i m ,

AD (m ,2*n- 1) =xi (m) " (n-i ) ; AD(m,2+n) =(xi(m) ̂ ( I I - l ) )*b(m) ;

AD (half dim+m, 2*n- 1) = (n- 1) *xi (m) (n-2) ;

AD (half dim+m, 2*n) = (n-l+mu (m) *xi (m) ) * . . .

(xi (m) A (n-2) ) *b (m) ; end

AD(halfdim+m,2 ) = mu(m)*b(m) ; end

AN=AD ; f o r m=i:halfdirn,

AN (m, dim) =x i (m) " (half dim) ;

AN(ha1f dim+m, dim) =ha l f dim*xi (m) (half dim-1) ; end

up (ix) = imag(-2i*det (AN) /det (AD) ) ;

end

f i g u r e ( i t ) p l o t (xp , up) a x i s ( C x 0 , ~ , , 0 , y f 1 ) valueof t=t p r i n t -dps -append pusa .ps

xO=xO+f rameshift ; xf=xf+frameshi f t ;

end

disp ( ' f inished ' )

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