Symplectic methods for the Ablowitz-Ladik model

ii NOgrH.I-I(XLANI) Symplectic

Methods

for the

Ablowitz-Ladik

Model

Yi-Fa Tang, V~ctor M. P~rez-Garcla, and Luis V£zquez

Departamento de F~sica TeSrica L Facultad de Ciencias F~sicas Universidad Complutense 28040-Madrid, Spain

ABSTRACT In this paper, we use two symplectic schemes to simulate the Ablowitz-Ladik model associated to the cubic nonlinear Schr5dinger equation and we compare them with nonsymplectic methods. © Elsevier Science Inc., 1997

1.

INTRODUCTION

As it is well known, the nonlinear cubic SchrSdinger equation (NLS) can be written as

{ iWt ÷ Wx~ ÷ alWl2W=O

(1)

w ( x, o) = Wo( where x ~ R, a is a constant and W(x, t) is a complex function, this equation plays an important role in the nonlinear physics and it is a completely integrable system [1-5]. Over the past several years, various types of numerical schemes have been proposed to simulate the NLS [5-14], including a number of conservative schemes which have constant discrete energy a n d / o r charge [5, 6, 8, 9, 10, 12, 13, 14]. These methods discretized the NLS in space and time simultaneously. Recently, in [15] and [16] a space discretization of the NLS was considered generating a set of coupled ordinary differential equations that can be cast into a Hamiltonian form. Once this is done, and because of the

APPLIED MATHEMATICSAND COMPUTATION82:17-38 (1997) © Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010

0096-3003/97/$17.00 PII S0096-3003(96)00019-7

18

Y.-F. TANG ET AL.

hamiltonian structure, one can use the symplectic methods to simulate it. The integration of (1) with periodic initial data and boundary conditions was considered in [15] and a comparison between symplectic and non-symplectic methods was made. In [16], we considered the system for different initial data: one-soliton and three-soliton solutions and tested the good properties of symplectic methods for this Hamiltonian system by using numerical simulations. However, the spatial discretization of the NLS used there provided a hamiltonian system but a nonintegrable one, which is a difference with the integrable nature of the continuous PDE. The Ablowitz-Ladik model of the Nonlinear SchrSdinger Equation

iWt(0 +

W (~+ ~) - 2 W (° + W (t- 1) h2

a

2

+ ~lW(;)l ( W (~+a) + W (;-~)) = O,

(2) where h is the spatial step size and W ( O ( t ) = W(lh, t) for 1 = . . . , - 1 , 0 , 1 , . . . , is one of the most important discrete models of the NSE because, as it happens for the continuous system, it is completely integrable and eminently suitable for numerical computations [17]. On the other hand, and in spite of this advantage, the Hamiltonian structure of the A-L model is nonstandard, which leads to some technical difficulties. In the present paper, we study the behavior of the A-L discretization of the NLS when integrated in time by symplectic methods. Firstly, we prove the convergence of the discrete A-L model (2) to the original continuum NLS (1) (Section 2), and display some of its invariants by utilizing the technique due to Vekslerchik [18] (Section 3). Secondly, in Section 4 the model is standardized, so that symplectic schemes can be straightforward applied to integrate the standard Hamiltonian system (Section 5). Finally, we try to get accurate solutions and test the conservation properties of the model invariants by the scheme for different temporal stepsizes (Section 6). 2.

CONVERGENCE OF ABLOWITZ-LADIK MODEL

In Theorem 1, we will prove that the solution of (2) converges to that of (1) when the stepsize h -~ 0.

THEOREM 1. The solution of the A-L model (2) converges to that of the continuous original NLS (1) equation when h --) O.

19

Symplectic Methods for the Ablowitz-Ladik Model

PROOF. Let us suppose t h a t w(x, t) is t h e solution of (1) a n d let w(l)(t) = w(lh, t) for l = . . . , - 1 , 0 , 1 , . . . , t h e n w(t+l) _ 2w(O + W( l - l )

iw~ 2) +

+ ~ l w ( ° 1 2 ( w (l+l) + w (z-i)) = M t (3)

h2

where w (l+l)

2w(I) + w ( t - l ) --

M, =

- 4'~ + ~a l w

h2

(l) 2

I (

W(I+ I)

+

w(I-

")

- alwU)12 wU) M l = h2Bl a n d B l = O(1). S u b t r a c t i n g (2) from (3), we o b t a i n

.d

,-~ el

el+l -- 2~1 + e l - I

+ ----

a

+-~{,~'",~(~,+1 + ~,_~ + [ ~'".~, + ~'"~,]I. ,''+~' + w,,-~,j} =M~ (4) where e I = w (l) - W Cl) for l = . . . , - 1 , 0 , 1 , . . . , . Multiplying (4) b y the c o m p l e x c o n j u g a t e of e l s u m m i n g up for all l, a n d t a k i n g the i m a g i n a r y part, we get

1

d

2 dt

y, lell2 + - I r a ~lw")l~(ez+~ + e,_~)~, t

2

a{

2

Im E [ ~ ( %

+ w'"~l](e,+, + e,_~)~lk~l

}

2

I

l

;

20

Y.-F. TANG ET AL.

with some tedious calculations, we get

d [:18~12< C0 dt

18~1

+ C, ZI8,12+

t

C~h~

t

and then

E Is,( T)I ~ ~ e~( c3 T) O4h3.

(6)

l

Therefore,

118( T)II 2 < exp(C3T)

(7)

C4 h4.

where C1, C2, C3, C4 are constants depending on the solution of (1), and H denotes

lie( t)ll 2 = h ~ l s z (

t)l 2,

l

which proves the theorem. The meaning of (7) is that given a simulation time T, we can make h small enough to guarantee that the solution of (2) is close enough to that of the continuum NLS (1). 3.

-

INVARIANTS OF A-L MODEL With the transformations W (0 = V ( ° e z ' p ( - ( 2 i t ) / h ~ ) , 1, 0, 1 , . . . , (2) becomes an equation for { V(°}l+=~_

iVt( 0 = -

(1 + ~IV(012 a )( V (1+1)+ V(l-1)), ~-~

l=

l = ..., -1,0,1,....

...,

(8)

The original NLS (1) has infinite conserved quantities [1], such as energy, norm and momentum. The Ablowitz-Ladik Model (2) has also infinite conserved quantities. In [18] the first three for the special case h = i and a = 2 are given; similarly one can easily check that the following three

Symplectic Methods for the Ablowitz-Ladik Model

21

quantities are invariants of (8) '~1 = E ~(l) w(l+l)

l S2 = E-V(l-1)v(l)v l

a

1)]2 [ V ( / ) ] 2

(l+l) -It- _ E l ' V ( / 4 I

$3 = E V(l-2)U(l-1)U(l) V(I+I) "[- -a- E -V(l-1)"U(l)U(l)[ v ( l + l ) ] 2

l

(9)

2 z a2 a

+ -~ E [~('-

1)] 2 u(l) V(l) v(l+ 1)

l

+ - - E [ ~(,)13 [ v(l+

1)] 3

12 i

where U (0 = 1 / h 2 + a/2l V(°[ ~. When we denote V (° = p(O + iq(t), (9) can be rewritten R1 = ~ [ p(Op(~+l) + q(0q(t+l)] l

11 = ~_. [ p(Oq(~+ 1) _ q(l)p(,+ 1)] 1

R2 = • U(0[ p( l- l) p( l+ l ) q. q(l-i)q(t+l)] l

+ ~ E ([( p..)2- (q..)21 [( ¢,,)2- (~,,)2] +4p(,_l)q(,_l)p(Oq(O} 12 = E U(')[ ;(,-1)q(t+l) _ q(,-1);(l+l)] l

+2 ~/ ([( P(l-I')2- (q('-1))2]p(t,q(,,_

[( p , , , ) 2 (q(,,)2] p(t-l,q(,-1))

R3 = ~ U(I-1)U(0[ p(l-2)p(Z+l) + q(l-2)q(l+l)] I + a~-'~ U(0[ p( l-1) q( t) + q(l-1)p(O]p(l+i)q(t+l) l

+ a 2 U<0[ p<,-1)p<,)_ q
(10)

22

Y.-F. TANG ET AL. + a E u(')¢'-')q('-'[ ¢')4 '÷1) + l

+.~ ~ [( p(,))3- 3p(,)(q(5)2][(p(l+1))3_ 3p(,+l)( q(t+l))2]

+ a~--~[3(p(O)2q(t)-(q('))3][3(p(t+l))2q('+l)-(q('+l))3] 13 = E U(I-1) U(')[ p(l- 2)q(l+ i) _ q(l- 2)p(l+ l) ] l +

a~-~ U(t)[ p(t- l)p(,)

_

q(t- 1)q(/)]

p(l+

1)q(/+ 1)

l a

--~ ~ U(')[ p('-')q(t) + q(t-1)p(t)][( p('+l))2- ( q('+l)) 21 +a

2~u(Ol(p (t-

-

,))2

-(q('-

1))2][ p(Oq('+l) q(0p('+1)]

+

a~ U(Z)p(t- 1)q(/-1)[ p(l)p(t+ 1) _ q(Oq(t+1)] l a2

+ 1-~~ [( p(,,)3 _ 3p(0( q(t,)21 [3( p(l+l))2q

(l+l) --

( q(,+,,)3]

a2

12 ~ [3( p('))2q (t) -(q(t))3] [( p(,+,))a _ 3p(l+l)( q(l+l,)2 l where S a= R a+ ila, a = 1,2,3 and U(~) = 1/h 2 + a/2[(p(l)): + (q(t))2] for l= ..., - 1 , 0 , 1 , . . . . 4. STANDARDIZATION OF A-L MODEL The equation (8) can be rewritten as

{

p~0 = _ U(0[ q(l+1) + q(l-1)] q~t) = U(l)[ p(l+l) + p(l--1)] l= ..., - 1 , 0 , 1 , . . . .

(11)

23

Symplectic Methods for the Ablowitz-Ladik Model

Assuming the boundary condition V(0, t) = V((n + 1)h, t) -- 0, the (11) can be transformed into the following general Hamiltonian system d --Z = K( Z)VH dt

where ZX---[ q(1),...,

, Z2n IT ~" [pT, qT]T,

[ Zl,...

q(n)]T,

and

(12)

p _ [p(1),..., p(~)]T q =

is anti-symmetric and satisfies

ako (z)

+ %(z) - - + - ak -

0

Oz~

D = diag[dl,..., d~} and

o(z)

Oz b

0, a , b , c =

1,...,2n

d l = h2U (0 = 1 + (ah2)/2[(p(°) 2 + (q(0)2] for

l -- 1.... , n,Vis the gradient operator and 1

H = - ~ Z [ p(l)p(t+l) + q(0q(Z+l)].

(13)

l

In the context of the Doubarx Theorem [19, 20], (12) can be standardized. In fact, the transformations ah2 1 +

P(0 = !

(v(°) 2 2 ah~ tan

_ _ ( v(°) 2

l= 1,..., n

2

q(0 = v(

(14)

24

Y.-F. TANG ET AL.

change (12) into d

- - Y = JVG dt where Y = [ Y l , . . - , [vO),..., v(")] T, and

(15)

Y2,) T = [ uT, vT] T, U = [ U O ) , . . . , U(")]T, V = O

E, is the identity n-matrix, and

1

1 + --~-(v(t)) 2

1 + --~-(v('+l)) 2

ah 2

ah 2

2

2

G( u, v) = -~ ~

))

×tan

1 + --~-(v(t+l)) 2 u(t+l)

+ v(t)vU+

1)}. (1~)

Every single transformation in (14) for certain /(1 < l ~< n) is a diffeomorphism, whose inverse is

ah 2 arctan

2 ah 2 p(z) 1 + --~-[ q(0] 2 (17)

U( l) =

1 + -5-[ ~'12 v(~)

)

q( O

Symplectic Methods/or the Ablowitz-Ladik Model

25

with the Jacobian a( u(0 , v(0)

1

a(p('), q(O)

ah2 1 + 2 ([ p(O]2 + [q(,)]2)

= 1 + O(h2).

In the following sections we will apply the symplectic and non-symplectic schemes to integrate the standard Hamiltonian system, and, for different stepsizes It, we will study the solutions and discrete integrals (10) behavior. The later point is very interesting because the symplectic integrators are nonconservative even though they preserve the structure of the phase space. Good conservation properties, which are not ensured a priori are then a test that provides more confidence in the symplectic scheme's behavior. 5.

SYMPLECTIC AND NONSYMPLECTIC SCHEMES

During the recent twelve years, the symplectic difference methods for Hamiltonian systems have been rapidly developed [21-32] (see [19, 20] for an introduction to Hamiltonian dynamics and symplectic geometry). Because we have already changed the A-L model (12) into a standard Hamiltonian system (15), we can use the usual symplectic schemes straightforward. A nonsymplectic scheme will be also used to compare with the symplectic schemes results.

S c h e m e 1 (S1) : T h e m i d p o i n t rule

(18) where T is the step size. The scheme S l is symplectic with a second order accuracy and revertible in r. and it preserves any quadratic invariant of the Hamiltonian H [23, 24]. Moreover, the scheme (18) has the formal energy [32]

/~ = H -

T2/24 H~2(Z['I) 2

+ 774/5760 Hz, (Z[ll) 4 + T4/480 Hz3( Z['])2Z [21 + r4/16OHz~(Z[2I) 2

+ O(~°).

(19)

26

Y.-F. TANG ET AL.

where Z Ill = JVH, Z [21 = (z[ll)zZ[ll = JHz~ JVH, they are 2 p-dim vectors, provided Z ~ R2P; while Hzq (*) stands for the q-linear form for any integer q i> 2. For instance, 2p

Hz2( Z[i])2 = ( J V H ) THzz(JVH)

=

E

yziziZ i[1]Zj

i,j=l

S c h e m e 2 ($2): t h i r d o r d e r s c h e m e

'2= Z+ 2J[VH( K1) + VH(K2)] T J [ 3 V H ( g , ) - ~/-3VH(g2) ] K2= Z + -~

(20)

v J[~/-3VH(K1) + 3VH(K2) ] K2= Z + -6 This Runge-Kutta scheme is of third order but nonsymplectic.

S c h e m e 3 ($3): f o u r t h o r d e r s c h e m e T

2 = Z + -~ J [ V H ( K 1 )

+

VH(K2) ]

T

g 2 = Z + ~ - J[3VH(K1) + (3 - 2 f 3 ) V H ( K 2 ) ]

(21}

T

K 2 = Z + - ~ J[(3 + 2{-3)VH(g,) + 3VH(K2) ] This Runge-Kutta scheme is also symplectic with fourth order, also revertible, and its formal energy (see [32] is T4

T4

T4

/q = H - 432-----~Hz,(z0l) 4 - --H~3(720 Z[11)2Z[2l - --H~2(1440 Z[21)2 (22) Tang et al. have transformed the problem of the convergence of the formal energies series for these symplectic schemes into a problem relating to G r a p h T h e o r y . In [33] they conjectured that I / q - Ht ~< D2~"2 for

Symplectic Methods for the Ablowitz-Ladik Model

27

scheme S l and IH - HI ~< D4T4 for scheme S3, the Ds depending only on the Hamiltonians. If this were true it would imply that the series are convergent. This conjecture will also be tested in the numerical experiments to be presented in the next section. Because of the nonlinearly implicit nature of the schemes displayed above we have to choose a way to approximately solve them. We used simple iteration, which converges without problems when the temporal stepsize T is small enough. 6.

NUMERICAL EXPERIMENTS

In this section we will present the numerical simulation results performed in order to test the accuracy of the symplectic schemes and their conservativity of the invariants in comparison with the non-symplectic scheme, as well as the conjecture mentioned in the preceeding sections. The following initial conditions are used, C10ne-soliton solution

W( x, 0) = 277

e2x~isech[2~( x - xl)],

(23)

C2 T w o - s o l i t o n s o l u t i o n

W( x, 0) = 2nl~-2a e~x'~Zsech[2n,( x - xo)]

+ 2~72~-2a e2x2Xisech[2~?2( x - xb)],

(24)

C3 Three-soliton solution

W( x, 0) = sech[ x - x31

(25)

Unless the contrary is stated the standard value for the nonlinear constant is a = 2.0. In the following, we will call E r r ( A X t ) = A ( t ) - A(O) for any variable A.

28

Y.-F. TANG ET AL.

Initial data (23) is the usual 1-soliton solution which is integrated without problems by most methods. We present here the result of an integration with 7 / = 0.5, X = 0.5, x 1 = 0.0 over the spatial interval x [ - 3 0 0 , 500] and temporal intervals 0 < t ~< 70 for symplectic methods and non-symplectic method, with two different pairs of integration parameters h = 0.3, ~- = 0.02

(26)

h = 0.3, T = 0.01

(27)

Plots in Figure 1 show that the numerical results for the invariants (10) computed using scheme S1 fluctuate in small neighborhoods of the standard values. We have also computed the solution and the m a x i m u m absolute errors with the same h and half ~- and have found that the m a x i m u m errors less than 2 -9 times those in Figure 1 (a)-(f), respectively. These facts are interesting and support the conjecture mentioned in Section 5. Exactly the same results are found when the fourth order symplectic scheme $3 is used as partially shown in Figure 2, the only difference being the better accuracy of this scheme which manifest for example in the smaller error in the numerically computed invariants. On the other hand, it is obvious from Figure 3 that the numerical results for the invariants obtained by using scheme $2 and the same integration parameters decrease with time. The same discrepancies are found when the soliton motion is analyzed (Figure 4): its motions is accurately reproduced by scheme S1 (Figure 5Xa)), but not by scheme S2 (Figure 5(b)) because the soliton amplitude decreases with time, this happening even though the later is a higher order scheme. These results favor the use of the symplectic integrators for our problem, a fact that has been also pointed out in [35] for a repulsive Nonlinear SchrSdinger Equation ((1) with a < 0). The expression in (24) is an initial data for a pair of solitons with different amplitudes and velocities and it is appropriate for the simulation of soliton collisions (assuming that the soliton centers are initially set far away from each other). We have studied the following set of parameters */1 = */z = 0.5, X1 = 0.25, X2 = 0.025, x a = 0.0, x b = 30.0. h = 0.3, ¢ = 0.02

(28)

h = 0.3, "r = 0.01

(29)

Plots in Figure 5 show again the invariant conservation up to a very small error when scheme S l is used to simulate the soliton collision problem.

Symplectic Methods for the Ablowitz-Ladik Model

60t

29

~°~l .o: ot

%

~b~

E

v

W

-400

"

20

40

60

t

75

% ..-L

%_.ot

5(

(d)

2~

w

-2~ -5( -7~

0

20

40

-

60

4

0

0

t

~

20

40

60

t

50]

%

(0

L~J

0

20

40

t

60

t

FIG. 1. Results obtained by using scheme Sl with initial data (26); Evolution of (a)

Err(R1)" 104, (b) Err(Ii)" 106, (c) Err(R2)" 103, (d) Err(I2). 104, (e) Err(R3). 102 and (f) Err( I3)" 102.

30

Y.-F. TANG ET AL.

% E b~

50-

(b) 0-

% -5@ ~-I00-150-200 FIG. 2.

2'0

t

4b

6b

Results obtained by using scheme $2 with initial data (27); Evolution of (a)

Err(R1), (b) Err( I1), (c) Err(R2) , (d) Err( I2), (e) Err( R 3) and (f) Err( I3).

When the temporal stepsize is reduced, the same features concerning the maximum absolute error are found here. Scheme S3 also works with enhanced accuracy as it happended in the simpler one soliton case. With these stepsizes scheme S2 fails again. The comparison of the symplectic 81 solution (Figure 6(a)) and nonsymplectic S2 solution (Fig 6(b)) is presented. Finally, we have used the initial data (25), which is usually considered to be a more difficult "quality" test for numerical schemes because of the appearance of large spatial and temporal gradients in the solution. For a = 2 N 2 ( N = 2, 3 . . . ) Miles [34] has shown that (25) corresponds to a bounded state of N solitons. For the case a = 18, we found that for h < 0.06667, with some proper temporal stepsize which makes simple iteration practicable, the symplectic scheme S3 represents accurately the solu-

Symplectic Methods for the Ablowitz-Ladik Model

31

0.5

(o)

(b) 0.0 -0.5

n. "r" t.. UJ

"I , ~I

1.0. -1.5,

2'0

t

4'0

6'0

(c)

-2.0

20

4'0

1:t

6b

(d)

-I0

"n," ~ -20"

Y UJ

u~ - 2 0 -40" -30 -60

2b

t

6b

4b

100 (

-4q

0

20

t

4b

6b

(e)

(0

I\ ~200-

~-3001 -5o% "

_,ooi 2'o

"

t

4'o

"

60

-600

2b

t

4b

6b

FIG. 3. Results obtained by using scheme 83 with initial data (26); Evolution of (a)

Err(R1)" 106, (b) Err(I1). 10s, (c) Err(R2)" 105. (d) Err(I2)" 106. (e) Err(R3)" 104 and (f) E~(Z3). 104.

32

Y.-F. T A N G E T AL.

150,-

(')

100

---"...-"

0~

1

5

150r | 100

--

0

('b)

60

,~,,,,~

0

0

1

5O

t FIG. 4. Propagation of a single soliton, 6(a), computed by using scheme S1 with initial data (26), Z-axis stands for the value of I WI. 6(b) computed by using scheme S2 with initial data (27), Z-axis stands for the value of [ W[.

Symplectic Methods for the Ablowitz-Ladik Model

33

600

(b)

© 400

%

o

:

a3

0-

oE

-2001 -400

2b

t

4b

6b

40

60

6o0t %

.o4°° 1

@

~200~ ,,~

bJ

0-

-200-400

2b

t

(f)

150 o

0

L-W

-50

--150 t FIC. 5.

20

t

40

60

Results obtained by using scheme S l with initial data (28); Evolution of (a)

Err(R1). 103, (b) Err(I1)" 106, (c) Err(R~). 102, (d) Err(Is)" 104, (e) Err(R3)" 101 and (f) Err(/3)' 10s.

34

Y.-F. T A N G E T AL.

lSO-

(0

i

lso I 100J-

(b) 1

t

ok.~.~.~.~..~.Y...d ~ ~<~>.' ~ -.tzj~~ .// o l~__...i~,,

~ X "~

~tm ~ r ~ ~ ~

,

~0

FIG. 6. Propagation of two solitons. 12(a) computed by using scheme Sl with initial data (28), Z-axis stands for the value of I WI. 12(b) computed by using scheme 82 with initial data (29), Z-axis stands for the value of I WI.

Symplectic Methods/or the Ablowitz-Ladik Model

35

~0.0 3.0

0.0

-20.0

FIG. 7. Three soliton bounded state evolution computed by using scheme 83 with spatial stepsize h = 0.06667 and temporal stepsize ~ = 0.002, Z-axis stands for the value of IWI.

tion without problems (see Figure 7 where x 3 = 0.0). This is a very good result and provides convergence to the correct solution with a relatively rough spatial grid (see the simulation results in [7, 14]).

7.

CONCLUDING REMARKS

The Ablowitz-Ladik model of the Nonlinear Schr6dinger Equation is a completely integrable general Hamiltonian system which can be standardized as shown in Section 5. W h e n the Hamiltonian equations for this standard system are considered, the symplectic methods simulate the solu-

36

Y.-F. TANG ET AL.

tions with complete success, while a nonsymplectic scheme studied for comparison fails to do so. As a conclusion, we can say that the symplectic schemes are very accurate and provide some overwhelming superiorities over nonsymplectic ones, mainly presented as long-term tracking ability, perservativity of invariants of the original Hamiltonian up to a very small error and preservativity of global, topological structure of the original systems. In this work we have shown that the spatial Ablowitz-Ladik discretization of the NLS (which enables the Hamiltonian nature and integrability, this being an essential difference with the discretization studied in [16] plus the symplectic time integration is a good combination that performs robustly. On the other hand, the main disadvantage of the symplectic schemes is that due to the nonlinearly implicit nature of the system to be solved, we have to use iterative methods such as the simple iteration method used here or the Newton method. This fact makes the implementation time consuming when compared with spectral schemes [7] or linearly implicit finite difference schemes [14] which are commonly used to integrate the NLS. It seems then that these should be the schemes of choice when noncritical problems are considered, but for more difficult or complicated problems (as it happens for example in delta potentials [35]), the additional guaranties provided by this method (Ablowitz-Ladik + Symplectic integration) may be of high interest, and, in any case, symplectic schemes provide a safe way to check the results of the faster but less accurate methods when simple physical problems are to be studied.

We acknowledge Dr. Fei Zhang and Dr. V. E. Vekslerchik for their valuable suggestions. Y.-F. T also thanks Dr. Song-he Song and Dr. Yu-hong Dai for their help. L. V. acknowledges the support from the DirecciSn General de InvestigaciSn Cient~fiea y T~cnica of Spain under grant PB920226. Y.-F. T. is supported by a fellowship from the ComisiSn Mixta Hispano-China de CooperaciSn Cient~fica y T~cnica. V. M. P.-G. acknowledges support from the DGICT under grant PB92-0798. REFERENCES 1 G.L. Lamb, Elements of Soliton Theory, John Wiley & Sons, New York, 1980. 2 M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM Philadelphia, 1981. 3 R.K. Dodd, J. C. Eibeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equation Academic Press, 1982. 4 A. Hasegawa, Optical Solitons in Fibers, Springer-Verlag, Berlin, 1989.

Symplectic Methods for the Ablowitz-Ladik Model

37

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