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THORSTEN HOHAGE, FRANK SCHMIDT

On the Numerical Solution of Nonlinear SchrodingerType Equations in Fiber Optics

ZIB-Report 02-04 (Januar 2002)

On the Numerical Solution of Nonlinear SchrodingerType Equations in Fiber Optics

Thorsten Hohage1, Frank Schmidt

Abstract

The aim of this paper is to develop fast methods for the solution of nonlinear Schrodingertype equations in fiber optics. Using the method of lines we have to solve a stiff system ofordinary differential equations where the eigenvalues of the Jacobian are close to the imag-inary axis. This is usually done by a Split Step method. Here we consider the extrapolationof Split Step methods with adaptive order and step size control. For more complicated non-linearities, in particular stimulated Raman scattering, Split Step methods are less efficientsince symmetry is either destroyed or requires much additional effort. In this case we useimplicit Runge Kutta formulas of Gauß type. The key point for the efficient implemen-tation of these methods is that the system of nonlinear algebraic equations can be solvedwithout setting up the Jacobian. The proposed methods are compared to other methods,in particular exponential integrators, the method of Marcuse, and the method of Blow andWood.

1 Introduction

The increasing demand for information transmission has stimulated a large amount of researchin fiber-optic communication systems. A bottle-neck in the simulation of optical networks isthe fast and reliable numerical solution of the pulse-propagation equation in optical fibers. Dueto the use of wavelength division multiplexing (WDM) with more and more channels, nonlineareffects such as self phase modulation (SPM), cross phase modulation (XPM), four wave mixing,and the stimulated Raman effect play an increasing role.

The simplest model for pulse propagation in optical fiber including nonlinear effects is thenonlinear Schrodinger equation:

∂∂z

B � iβ2

2∂2

∂t2 B � iγ�B� 2B (1)

Here B � z � t � is the amplitude modulation function of the rapidly oscillating electric field in acoordinate system moving with the signal (cf. section 2). For γ � 0 eq. (1) is identical to thefree-space linear Schrodinger equation from quantum mechanics, except that the roles of thetime and the space variable are exchanged.

Approximating (1) by the method of lines leads to an ordinary differential equation

B ��� z � � iβ2

2DB � z ��� iγ

�B � z � � 2B � z � (2)

where B j � z � B � t j � z � and where the matrix D approximates ∂2 � ∂t2. Here t j are given collo-cation points in time, and the absolute value function

� ���is applied componentwise. Since the

spectrum of the differential operator is given by σ � 0 � 5iβ2∂2 � ∂t2 � ��� it : t � 0 � , the eigenval-ues of iβ2D will lie on (or close to) the imaginary axis, and their size will grow as the time

1supported by DFG grant number DE293/7-1

1

discretization is refined. This implies that (2) is a stiff differential equation which should besolved by an A-stable method. The second Dahlquist barrier A-stable linear multistep methodsof order � 2 cannot exist. Since we have large eigenvalues on the imaginary axis, A � α � -stablemethods with α � 900 are unstable unless very small space propagation steps are used. Thesame holds true, of course for explicit Runge-Kutta methods.

In this paper we propose implicit Runge-Kutta methods of Gauß-type and show how thenonlinear systems of equations can be solved efficiently.

2 Modelling pulse propagation in optical fibers

We start with the derivation of the equation descreibing pulse propagation in optical fibers.Our presentation follows [1, Chapter 2], but we put a greater emphasis on the Raman effect andpulses with a large spectral bandwidth. Both points are nicely discussed in [2, 8]. Unfortunately,there exist different sign conventions. We follow the IEEE convention. The other convention isobtained by replacing i ��� � 1 by � i and taking the complex conjugate of all quantities.

The electrical field E � t � x � in optical fibers is governed by the equation

curlcurlE � 1c2

∂2E∂t2

� � µ0∂2P∂t2 � (3)

which follows from Maxwell’s equations and the relation D � ε0E � P if it is assumed that thereare no free charges and currents and that the magnetic polarization is zero. As usual, ε0 is thevacuum permittivity, µ0 is the vacuum permeability, and c � � ε0µ0 ��� 1 � 2 is the speed of light.The electric polarization P is a nonlinear function of E, which we decompose into a linear partPL and a nonlinear part PNL. For isotropic materials PL is given by

PL � t � x � � ε0

� t

� ∞χ1 � t � t � � E � t � � x � dt �

with a scalar valued function χ1. We set ε � ω � x � � 1 � 12π ∞

0 eiωtχ1 � t � dt. For optical fibers thenonlinear polarization is given in good approximation by

PNL � t � x � � ε0E � t � x �� t

� ∞χ 3 � � t � t � � �E � t � � x � � 2 dt � (4)

where the kernel χ 3 � is of the form χ 3 � � t � � χ 3 �K δ � t ��� χ 3 �R gR � t � with

gR � t � � τ21 � τ2

2

τ1τ22

e � t � τ2 sin � t � τ1 � �

The parameters χ 3 �K and χ 3 �R determine the strength of the Kerr effect and the stimulated Ramaneffect. Typical parameter values in the function gR are τ1

� 0 � 0122ps and τ2� 0 � 032ps.

We assume that E and PNL are polarized such that E � xE and PNL� xPNL and that ∇

�E 0.

Then the identity curlcurl � � ∆ � graddiv yields the Helmholtz approximation

� ∆E � ω � x � � ω2ε � ω � x �c2 E � ω � x � � µ0ω2PNL � ω � x � (5)

2

for the Fourier transform E � ω � x � : � 12π IR E � t � x � e � iωt dt. Let z be the propagation direction,

i.e. let ε � ω � x � y � z � be independent of z. We make the separation ansatz

E � ω � x � y � z � � ℜF � x � y � V � ω � z � � (6)

For a single-mode fiber the eigenvalue problem

� ∆F � x � y ��� ω2ε � ω � x � y �c2 F � x � y � � �

�β � ω ��� iα � ω �

2 � 2

F � x � y � (7)

has exactly one eigenvalue λ � � � β � ω � � iα � ω � � 2 � 2 with negative real part for all relevantfrequencies ω. β is called the propagation constant of the guided mode, and α is called attenu-ation constant or fiber loss. The frequency dependence of α may be neglected in the following.In the linear case we easily find the separated equation ∂2

zV � � β � iα � 2 � 2V and the solutionE � ω � x � y � z � � e � iβ ω � � α � 2 � zF � x � y � . Hence, the optical power

�E� 2 decays like e � αz which ex-

plains the factor 12 in (7).

Strictly speaking, it is not possible to obtain a separated equation for V in the nonlinear casesince F actually depends on ω and is complex valued. Fortunately, both the dependence on ωand the imaginary part of F are very small and may be neglegted in PNL. Hence, multiplying(5) by F � x � y � and integrating over x and y we obtain

∂2V � ω � z �∂z2 �

�β � ω � � iα

2 � 2

V � ω � z � � Aω2

c2 p � ω � z � (8)

with

A : � �F � x � y � � 4 dxdy

�F � x � y � � 2 dxdy

and

p � t � z � � V � t � z �� t

� ∞χ3 � t � t � � �V � t � � z � � 2 dt � �

Now we assume that V � t � z � is of the form

V � t � z � � 12

B � t � z � ei ω0t � β0z � � 12

B � t � z � e � i ω0t � β0z � (9)

where β0� β � ω0 � with a slowly varying envelope function B � t � z � satisfying

B � ω � � z � 0 for�ω � � � ω0

6� (10)

Inserting (9) into (8) and picking out the terms with nonvanishing Fourier transform in theinterval � ω0

2 � 3ω02 � yields�

∂∂z2 � 2iβ0

∂∂z�

�β � ω ��� iα

2 � 2

� β20 � B � ω � � z � � � ω2

c2 q � ω � � z � (11)

3

where ω � ω0 � ω � and

q � t � z � � AB � t � z ��

3χ 3 �K

8

�B � t � z � � 2 � χ 3 �R

4

� t

� ∞gR � t � t � � �B � t � � z � � 2 dt ��� �

The factor 14 comes from neglecting the term

B � t � z � ei ω0t � β0z � � t

� ∞gR � t � t � � e2iω0 t � � t � B � t � � z � 2 dt �

which is almost zero due to our assumption (10). We neglegt the interaction between forwardand backward traveling waves and formally factorize (11) to obtain�

i∂∂z� β0 � ω

c � n2 � q � ω � ��� �i∂∂z� β0 � ω

c � n2 � q � ω � ��� B � ω � � z � � 0

where n � ω � � � β � ω ��� iα � 2 � cω is the (linear) effective refractive index. q can be interpreted as

the square of the nonlinear contribution to the refractive index, and it is usually much smallerthan n2. Hence, we approximate the square root by its first order Taylor approximation andneglegt the first differential operator corresponding backward traveling waves to obtain

i∂∂z

B � ω � � z � ��

β � ω � � iα2� β0 � B � ω � � z � � ω0 � ω �

cn � ω0 � ω � � q � ω � � z � �Now the inverse Fourier transform yields the propagation equation for B

∂∂z

B � t � z � � � i

�β

�ω0 � i

∂∂t � � iα

2� β0 � B � t � z �

� iγ�

1 � τshi∂∂t � � � t

� ∞g � t � t � � �B � t � � z � � 2 dt � � B � t � z �

where

γ : � ω0 � 3χ 3 �K � 2χ 3 �R � A � ω0 �8cn � ω0 � � τsh : � 1

ω0� d

dωln

A � ω �n � ω � �

g � t � � � 1 � fR � δ � t ��� fRgR � t � � fR� 2χ 3 �R

3χ 3 �K � 2χ 3 �R

�

The derivative term in the definition of τsh can usually be neglected. Moreover, it is sufficientto consider the first few terms of the Taylor expansion of β at ω0:

β � ω0 � ω � � β0 � β1ω � � β2

2ω � 2 � β3

6ω � 3

where β j : � β j � � ω0 � . Here β1 can be eliminated from the equation by using a coordinatesystem which moves with the signal at the speed of the group velocity 1 � β1. The functionB � t � z � : � B � t � β1z � z � satisfies

∂∂z

B � t � z � � �D

�� i

∂∂t � � N � B � � � z � � � B � t � z � (12)

4

where

D � ω � � : � � i

�β � ω0 � ω � ��� β0 � β1ω � � iα

2 � � α2� i

�β2

2ω � 2 � β3

6ω � 3 � �

N � B � : � � iγ�

1 � τshi∂∂t � � � t

� ∞g � t � t � � �B � t � � � 2 dt � � �

We will always drop the tilde on B below. Typical values of β2 and γ are β2� � 20ps2 � km and

γ � 2 � Wkm � � 1. If all other parameters are zero, (12) reduces to (1).Let us summarize the most important assumptions we have made in the derivation of equa-

tion (12) and which restrict its validity in practice:

� We assumed that the electric field is polarized in one direction x in the entire fiber.

� We ignored backscattering, in particular the Brilloun effect.

� We only considered single-mode fibers.

3 Time Discretization and Solution of the Linear Schrodingerequation

The easiest way to discretize the time variable in the nonlinear Schrodinger equation is to usethe Fast Fourier Transform (FFT). The linear Schrodinger equation (γ � 0) can be integratedexactly in the Fourier domain. However, since the nonlinear part has to be integrated in thespace domain, at least one FFT and one inverse FFT transform have to be applied in eachspace propagation step. Since large signals with up to several hundred thousands of unknownshave to be propagated in practice, the FFT operations dominate the total cost of the integrationprocess. Therefore, attempts have been made to find alternatives. Some approaches are dis-cussed in subsection 3.2. In this report we mainly focus on space discretization and always useFFT. However, the methods discussed in the following section can also be combined with othertime discretization schemes. Note that FFT introduces an artificial periodization of the model.Therefore, the time interval has to be chosen sufficiently large to obtain reliable results.

3.1 Fast Fourier Transform

We solve the partial differential equation (12) by the method of lines. Given an equidistantmesh of time points t j

� tmin � j∆t, j � 1 � � � � � n, let B � z � � Cn denote the vector given byBi � z � � B � z � ti � . The partial differential equation (12) is approximated by the ordinary differen-tial equation

B ��� z � � � F � 1D � ω ��� F � N � B ��� B � z � (13)

N � B � : � γ � I � τshiF � 1ω � F � diag � � 1 � fR � �B � 2 � fRF � 1 � F � f ��� F � �B � 2 � � �Here F is the Fast Fourier Transform,

ω : � 2πn∆t

�0 � 1 � � � � � n

2� 1 � � n

2� � � � � � 2 � � 1 � T �

5

and � denotes the componentwise multiplication of vectors. Moreover D,� � �

, and� 2 are applied

componentwise.In some cases it is advantagous to work with the equivalent differential equation for B : � F B

given by

B � � z � � � D � ω ��� F N � B � F � 1 � B � z � (14)

3.2 Other methods

One alternative to FFT is to work in the time domain and approximate the differential operators∂ j

∂t j by finite differences or finite elements. In this case one has to cope with the problem ofnumerical dispersion: High frequencies are propagated with wrong speeds in the numericalapproximation. Therefore, either a very fine discretization or high order elements have to beused. The use of wavelets is studied in [16].

A different approach has been suggested by Plura [18]. The idea is to approximate the actionof the matrix exp � hF � 1 diag � D � ω � � F � by an Infinite Impulse Response (IIR) filter.

4 Analysis of Split Step Methods

4.1 Introduction

We consider an ordinary differential equation of the form

B ��� z � � LB � z � � N � B � z � � B � z � (15)

where L and N � B � are matrices.If N � N � X � is constant, then

B � z � h � � ehL�

hNB � z � � (16)

In the simple Split Step method (cf. Table 1) the approximation

ehL�

hN ehLehN �is used. It follows from the Baker-Campbell-Hausdorf formula (cf. e.g. [12]) that the dominanterror contribution in this approximation is due to the term h2 � LN � NL � , and that the simple SplitStep method is locally of order O � h2 � , i.e. globally of order O � h � . For constant N it can be seenthat the accuracy of the Split Step method can be increased by one order if the exponential termare arranged in a symmetric manner. In the nonlinear case this so-called full Split Step method(cf. Table 1) requires about twice as much memory and computation time as the simple SplitStep method since Bh � z � has to be computed and memorized in each step. It is not possible tocombine the factors eh � 2L of two subsequent steps to one factor ehL. This is the case for thereduced Split Step method (cf. Table 1) where N � Bh � z � � is replaced by N � eh � 2LBh � z � � in theexponent. The reduced Split Step method requires about as much memory and computationtime as the simple Split Step method. However, since the true potential N � Bh � z � � is replacedby N � X � , we may expect that it to be less accurate than the full Split Step method. The plotsin Figure 2 show that this is not the case except for the first order soliton. We will explain thisobservation in the next section.

6

An implicit version of the Split Step method has been suggested by Agrawal [19, 1] (cf. Ta-ble 1). The implicit equation can be solved by the fixed point iteration

X0� Bh � z � � Xn

�1� e

h2 Le

h2 N Xn � �

N Bh z � � � e h2 LBh � z � � (17)

The following lemma guarantees the convergence of this iteration for sufficiently small stepsizes h.

Lemma 1 Let N : Cn � Cn � n be continuously differentiable as a mapping from IR2n to IR2n � 2n,and let B � Cn. Then there exists a constant h0

� 0 such that the fixed point equation

X � eh2 Le

h2 N X � �

N B � � e h2 LB (18)

has a unique solution X � Cn for all h � h0. The iteration (17) converges to this solution, and�X � Xn

� � θn

1 � θ� �

X1 � X0� �

If L is skew-Hermitian, then h0 and θ do not depend on L.

Proof. We use the Banach fixed point theorem (cf. [7]). Let f � X � denote the right hand sideof (18), and let R : � 2

�B

�. We have to show that f is a contractive mapping form BR : � � X �

Cn :�X

� � R � into itself for h � h0. The fact that f maps BR into itself for sufficiently small hfollows from the boundedness of N and exp on compact sets. We have

f � X1 ��� f � X2 � � eh2 L

�e

h2 N X1 � �

N B � � � eh2 N X1 � �

N B � � � eh2 LB

� h2

eh2 L� 1

0Dexp

�h2� N � X � t � � � N � B � ��� DN � X � t � � � X1 � X2 � dt e

h2 LB

where X � t � : � tX1 � � 1 � t � X2. The derivative of the matrix exponential function exp : Cn � n �Cn � n is given by

Dexp � X � Y � � 1

0exp � � tX � Y exp � tX � dt

(cf. [12, page 40]). It follows that there exist constants C and h0 such that�Dexp � h � 2 � N � X � t � � �

N � B � � � � � C and �f � X1 � � f � X2 � � � hC

�X1 � X2

�for all X1 � X2

� BR and 0 � h � h0. Hence, f is contractive for h � min � 1 � C � h0 � with contractionfactor θ : � hC. The last statement follows from the fact that eh � 2L is unitary, i.e.

�eh � 2L � � 1 if

L is skew-Hermitian.

4.2 Adjoint and Symmetric Methods

We consider a general ordinary differential equation

B � � z � � f � z � B � z � � (19)

7

simple SS Bh � z � h � � ehLehN Bh z � � Bh � z �full SS Bh � z � h � � e

h2 LehN Bh z � � e h

2 LBh � z �reduced SS Bh � z � h � � e

h2 LehN X � X � X � e

h2 LBh � z �

Agrawal SS Bh � z � h � � eh2 Le

h2� N Bh z �

h � �N Bh z � � � � e

h2 LBh � z �

Table 1: Versions of the Split Step method

with a function f which satisfies a Lipschitz condition with respect to the second variable. (19)induces an evolution Ψ. For z1 � z2

� IR, Ψz2 � z1B0 is defined as B � z2 � where B is the solution tothe initial value problem (19) with the initial condition B � z1 � � B0. Note that the evolution Ψsatisfies

Ψz � zB � Bd

dhΨz

�h � zB � f � z � B � (20)

Ψz3 � z2Ψz2 � z1B � Ψz3 � z1B

and that these properties characterize the evolution uniquely (cf. [6]). For a numerical solu-tion of the differential equation (19) the continuous evolution Ψ is approximated by a discreteevolution Φ. E.g., the discrete evolution of the simple Split Step method is

ΦsimSSz

�h � z B : � ehLehN B � B �

A discrete evolution which satisfies the first two conditions in (20) is called consistent. Ingeneral, a discrete evolution does not satisfy the third condition (otherwise it would be the exactevolution!). However, in some cases a discrete evolution may satisfies the third condition forthe special case z3

� z1.

Definition 2 Let Φ be the discrete evolution of some method. The adjoint evolution Φ�

isdefined by

Φ�

z�

h � zΦz � z�

hB � B � (21)

Φ is called symmetric if Φ� � Φ. The method described by Φ

�

is called adjoint method. Amethod is called symmetric if its discrete evolution is symmetric.

The following theorem on symmetric methods is proved in [9, Theorem 8.10].

Theorem 3 Suppose that f and Φ are sufficiently smooth, and that Φ is consistent and sym-metric. Then the global error satisfies

Bh � z ��� B � z � � e2 � z � h2 � e4 � z � h2 � � � � � e2M � 2 � z � h2M � 2 � O � h2M � (22)

for M � IN as h � 0. Here e j are smooth functions.

In particular, a consistent, symmetric method is at least of global order 2.

8

As a first example, let us compute the adjoint evolution ΦsimSS ��

of the simple Split Stepmethod. Let Bh � z � � ΦsimSS

z � z�

h Bh � z � h � � e � hLe � hN Bh z � � Bh � z � h � . The definition (21) implies

ΦsimSS ��

z�

h � z Bh � z � � Bh � z � h � � ehN Bh z �h � � ehLBh � z � �

Note that ΦsimSS ��

is defined implicitly. Even if N � Bh � z � h � � � N � Bh � z � � (e.g. for linearSchroding equations with a constant potential), the simple Split Step method is not symmet-ric in general, since the matrices L and N � Bh � z � � do not commute. An analogous computationshows that

ΦfulSS ��

z�

h � z Bh � z � � Bh � z � h � � eh2 LehN Bh z �

h � � e h2 LBh � z � �

The full Split Step method is symmetric if N � Bh � z � h � � � N � Bh � z � � . For the reduced Split Stepmethod we obtain

ΦredSS ��

z�

h � z Bh � z � � eh2 LX � X � ehN X � e h

2 LBh � z � �In general, the reduced Split Step method if not symmetric. However, if N � X � � � iγdiag � �X � 2 �as in (2), then the diagonal entries of exp � hN � X � � have norm 1, and hence

N � X � � N � ehN X � X � � (23)

Introducing Y � e � hN x � X we find N � Y � � N � e � hN X � X � � N � X � . Therefore,

ΦredSS ��

z�

h � z Bh � z � � eh2 LehN Y � Y � Y � e

h2 LBh � z � �

i.e. the reduced Split Step method is symmetric if N satisfies (23). Finally, it is easy to see thatAgrawal’s method is symmetric.

Let us summarize our results.

Proposition 4 1. The simple Split Step method is not symmetric in general.

2. The full Split Step method is symmetric if N � B � does not depend on B.

3. The reduced Split Step method is symmetric if N satisfies (23).

4. The Split Step method of Agrawal is symmetric.

4.3 Numerical Results

We compared different versions of the Split Step method for the nonlinear Schrodinger equation(1) with γ � 2 � Wkm � � 1 and β2

� � 20ps2 � km. A class of test examples with an explicitelyknown solutions are solitons of order ν � IN. The initial data are given by

B � 0 � t � � ν � Psech

�γPβ2

t � (24)

We tested the Split Step methods for the first and second order soliton with P � 0 � 1W and apropagation distance of 50km.

9

−1000 −800 −600 −400 −200 0 200 400 600 800

−0.2

−0.1

0

0.1

0.2

time [ps]in

tens

ity

amplitude and real part after 0 km

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

10−5

100

frequency [THz]

inte

nsity

frequency

Figure 1: WDM test signal with 2 channels

100

102

104

106

108

101

102

103

104

105

precision=1/(rel. L2−error)

num

ber

of s

teps

1st order soliton

simple Split Stepfull Split Stepreduced Split StepAgrawal

10−2

100

102

104

106

102

103

104

105

precision=1/(rel. L2−error)

num

ber

of s

teps

2nd order soliton

simple Split Stepfull Split Stepreduced Split StepAgrawal

100

102

104

102

103

104

precision=1/(rel. L2−error)

num

ber

of s

teps

1 channel

simple Split Stepfull Split Stepreduced Split StepAgrawal

100

102

104

106

108

101

102

103

104

precision=1/(rel. L2−error)

num

ber

of s

teps

WDM 2 channels

simple Split Stepfull Split Stepreduced Split StepAgrawal

Figure 2: Comparison of different versions of the Split Step method

10

Moreover, we consider a wavelength division multiplexing (WDM) system with 2 channels(cf. Figure 1). The first channel, centered at 0 � 3THz, carries a signal encoding the bit sequence110011101. Initially the signal is located in the left part of the time interval, but due to disper-sion it moves to the right in the given coordinated system. The signal in the second channel,centered at � 0 � 3THz, corresponds to the bit sequence 11100101 and moves to the left. After adistance of 18km the first signal has passed the second signal.

In Figure 2 we plotted the number of steps K � Z � h over the precision p � 1 � � B � Z � �Bh � Z � � with a logarithmic scale on both axes. According to (22) we have asymptoticallyln p � 2ln � Z � 1 �

e2 � Z � � � 1 � 2K � , i.e. that the slope is asymptotically 1 � 2. For first order accu-rate methods the slope is asymptotically 1. In all these test examples the reduced Split Stepmethod and the Agrawal Split Step method are second order accurate. The full Split Stepmethod is only second order accurate for the special case of the first order soliton since hereN � B � z � t � � � � iγ

�B � z � t � � 2 is independent of z. The simple Split Step method is only first order

accurate asymptotically. However, for large step sizes it yields almost the same results as thereduce Split Step methods. The explanation is that the simple Split Step method yields thesame result as the reduced Split Step method applied to the initial data eh � 2LB � 0 � t � if the resultif multiplied by e � h � 2L. However, this is only true for constant step sizes.

4.4 Conclusion

We have discussed criteria for symmetry of Split Step methods and consequences. Symmetryguarantees second order accuracy and is advantageous for exptrapolation methods discussed inthe next section. If only the Kerr effect is considered, the reduced Split Step method yields thebest results. For more complicated nonlinearities Agrawal’s version of the Split Step methodshould be used.

5 Extrapolation of Split Step Methods

5.1 Introduction

Extrapolation of a “basic” method consists in first propagating the signal over a certain distanceseveral times with different step size and then interpolating the results.

Theoretically, extrapolation methods are based on an asymptotic expansion of the followingform

Bh � z � � B � z � � eκ � z � hκ � e2κ � z � h2κ � � � � � eκ M � 1 � hκ M � 1 � � O � hκM � � (25)

It can be shown that such a relation is always satisfied with κ � 1 under certain smoothnessassumptions (cf. [9, Theorem 8.1]). Of course, the first error terms in (25) vanish for higherorder methods. Extrapolation is particularly efficient if κ � 1. We have seen in Theorem 3 that(25) is satisfied with κ � 2 for symmetric methods.

We introduce a vector pz of polynomials of degree � � M � 1 by

pz � hκ � : � B � z ��� eκ � z � hκ � e2κ � z � h2κ � � � � � eκ M � 1 � hκ M � 1 � �Note that pz � 0 � � B � z � is the exact solution. In order to approximate pz � 0 � by interpolation, weevaluate the polynomial pz at M distinct points � h � n j � κ with an error of order O � hκM � . Here

11

T1 � 1 �

T2 � 1� T2 � 2

.... . .

TM � 1 � 1� � � � � TM � 1 � M � 1� � �

TM � 1� � � � � TM � M � 1

� TM � M

Figure 3: Illustration of the Aitken-Neville Algorithm.

S � � n1 � n2 � � � � � nM � is a given step sequence. The interpolation can be done efficiently by theAitken-Neville algorithm (cf. [7]).

Given some basic method with discrete evolution Φ satisfying (25) and a step sequence S �� n1 � n2 � � � � � nM � , we define an extrapolated evolution ΦS of global order κM by the followingalgorithm:

Extrapolation of a discrete evolution Φ with step sequence S .

1. Compute Tj � 1 : � Φz

�h � z

� n1 � 1n1

h

� � �Φz

� 2n j

h � z� 1

n jhΦz

� 1n j

h � zB for j � 1 � � � � � M. (Note that Tj � 1�

ph � � h � n j � κ ��� O � hκM � !)2. (Aitken-Neville Algorithm) For k � 1 � � � � � M � 1 and j � k � 1 � � � � � M compute

Tj � k�

1 : � Tj � k � Tj � k � Tj � 1 � k�n j

n j � k� κ � 1

(26)

3. Set ΦSz

�h � zB : � TM � M.

The Aitken-Neville Algorithm has the advantage that it provides a complete table of numer-ical results Tj � k which can be used as error estimates for step size and order selection. It is easyto show that Tj � k represents a method of order κk.

Several step sequences S have been suggested:� Romberg sequence: 1 � 2 � 4 � 8 � 16 � 32 � 64 � � � �� Burlisch sequence: 1 � 2 � 3 � 4 � 6 � 8 � 12 � 16 � 24 � 32 � � � �� harmonic sequence: 1 � 2 � 3 � 4 � 5 � 6 � 7 � � � �

We have found the harmonic sequence to be most efficient for the nonlinear Schrodinger equa-tion.

5.2 Step Size Selection

Let us first consider extrapolation of fixed order with variable step size. Since a control of theglobal error is hard to realize, we try to control the local error

�TM � MB � Φz

�h � zB

�. Here we use

the norm �B

�: � 1

n

n

∑j � 1

�B j�2

12

0 2 4 6 8 10 12 14 16 180

0.02

0.04

0.06

0.08

0.1

0.12

position z [km]st

ep s

ize

[km

]

0 2 4 6 8 10 12 14 16 182

2.5

3

3.5

4

4.5

5

position z [km]

orde

r

realizedsuggested

Figure 4: Order and step size selection for the extrapolation of Split Step methods

An estimate of�TM � MB � Φz

�h � zB

�would require a better approximation to Φz

�h � zB than TM � M .

However, if we had such an approximation we should use it for the further computations. There-fore, we try to control the error of the second best approximation TM � M � 1 in the Aitken-Nevillescheme. More precisely, we require that�

TM � 1 � M � TM � M� � Tol (27)

where ε is some given tolerence prescribed by the user. The real error�TM � MB � Φz

�h � zB

�is

usually much smaller. If (27) is violated, we repeat the integration step with a smaller step size.Otherwise, we compute a new “optimal” step size hopt

M for the following step. hoptM is chosen

such that�TM � 1 � M � TM � M

� Tol . Since�TM � 1 � M � TM � M

� c � z � hκ M � 1 � �1

we obtain

hoptM

� ρh

�Tol�

TM � 1 � M � TM � M� � 1

κ�M � 1 ��� 1 � (28)

The safety factor ρ is included to increase the probability that (27) is satisfied in the next step.We use ρ � 0 � 8.

5.3 Order Selection

We want to choose the order of the extrapolation method adaptively such that the amount ofwork per unit step is minimized under the side condition that a given accuracy is guaran-teed. The amount of work of one step of an extrapolation method with step sequence S �

13

� n1 � n2 � � � � � nM � is given approximately by

AM : � M

∑m � 1

sm �

and the work per unit step is measured by

WM : � AM

HM�

Moreover, we introduce the error estimates errM : � �TM � M � 1 � TM � M

� � Tol which must be � 1to satisfy (27).

We proceed with a detailed description of our combined step size and order selection algo-rithm (cf. [9, 3, 4]). Assume for some integration step the step size H and the order M

�3 have

been suggested. Then we compute the first M � 1 lines of the extrapolation scheme in Fig. 3.1) Convergence in line M � 1. If errM � 1 � 1, we accept TM � 1 � M � 1 as numerical solution

and proceed with the new proposed quantities

Mnew�

�M � 1 if M � 3 and WM � 1 � 0 � 9 � WM � 2

M else(29a)

Hnew�

�HMnew if Mnew � M � 1

HM � 1 � AM� AM � 1 � if Mnew

� M �The expression HM � 1 � AM

� AM � 1 is an approximation to WM based on the assumption WM WM � 1 which can be evaluation without computing line M in the extrapolation tableau.

2) First convergence monitor. If errM � 1� 1 we use the heuristic estimate errM � n1

� nM � 2errM � 1

(cf. [9]) to decide if we can expect convergence in line M or M � 1. The condition errM�

1� 1

leads to the criterion

errM � 1� �

nM�

1nM

n21 � 2

� (30)

If (30) is satisfied, we reject the step and restart with Mnew and Hnew given in (29). Otherwisewe compute the Mth line of the extrapolation tableau.

3) Convergence in line M. If errM � 1 we accept TM � M as numerical solution and continuethe integration with the new quantities

Mnew�

��� �� M � 1 if M � 3 and WM � 1 � 0 � 9 � WM

M � 1 if M � Mmax and WM � 0 � 9 � WM � 1

M else

(31a)

Hnew�

�HMnew if Mnew � M

HM � AM�

1� AM � if Mnew

� M � 1 �Here Mmax is the largest admissible order of the extrapolation scheme. We used Mmax � 7, butthis was never realized in our numerical experiments.

14

100

102

104

106

108

1010

102

103

104

105

precision=1/(rel. L2−error)

nr o

f FF

T o

pera

tions

2nd order soliton

red. Split StepBlow and Woodadapt Extrap 12adapt Extrap 123Extrap var. order

102

103

104

105

106

107

108

109

102

103

104

105

precision=1/(rel. L2−error)

nr o

f FF

T o

pera

tions

WDM 2 channels

red. Split Stepadapt Extrap 12adapt Extrap 123Extrap var. order

Figure 5: Convergence of extrapolation methods

4) Second convergence monitor. If errM� 1 we again check whether or not we may expect

convergence in line M � 1 of the extrapolation tableau. Since now errM is at our disposal, (30)can be replaced by the more reliable test

errM� �

nM�

1

n1 � 2

� (32)

If (32) is satisfied, the step is rejected and we restart with Mnew and Hnew given by (31). Other-wise we compute line M � 1 of the extrapolation tableau.

5) Convergence in line M � 1. If errM�

1 � 1 we accept TM�

1 � M�

1 as numerical solution andcontinue the integration with

Mnew�

��� �� M � 1 if M � 3 and WM � 1 � 0 � 9 � WM

M � 1 if M � Mmax and WM�

1 � 0 � 9 � WM

M else

(33a)

Hnew� HMnew �

Otherwise we reject the step and restart with Mnew� M � 1 if M � 3 and WM � 1 � 0 � 9 � WM ,

Mnew� M else, and Hnew

� HMnew.If M � 2 is the optimal choice of order, the order tends to flip back and forth between 2 and

3, or it remains at M � 3. This is because in (29b) a too large step size appropriate for M � 3 issuggested. Therefore we estimate H3 and W3 by H3

� ρH � � n1� n3 � 2err2 � � 1 � 5 and W3

� A3� H3

and replace (29b) by

Hnew�

�H2 if W2 � 0 � 9 � W3

H3 else

in the important case M � 3.

5.4 Numerical Experiments

Figure 5 shows the results of the order selection algorithm applied to the 2 channel WDMproblem described in subsection 4.3 (cf. Figure 1). During the overlap of the two signals the

15

order is increased. The convergence of the extrapolation of the reduced Split Step methodwith adaptive step size selection and fixed step sequence S ��� 1 � 2 � and S � � 1 � 2 � 3 � and theconvergence of the extrapolation method with adaptive step size and order selection are plottedin Figure 5.

6 Collocation Methods

6.1 Introduction

Let us consider a general differential equation

B � � z � � f � z � B � z � � � (34)

We make the ansatz

B � z � τh � � s

∑j � 0

γ jτ j (35)

with γ j� Cn and require that B satisfies the differential equation at s collocation points 0 � c1 �

c2 � � � � � cs � 1:

B ��� z � cih � � f � z � cih � B � z � cih � � � i � 1 � � � � � s � (36a)

Moreover, we require

B � z � � B0 (36b)

and set

Φz�

h � zB0� B � z � h � � (36c)

To show that this method is a Runge-Kutta method, we introduce the Lagrange polynomials L j

with respect to � c1 � � � � � cs � . L j is the (unique) polynomial of degree s � 1 satisfying L j � ci � � δi � j

for j � 1 � � � � � s. Since B � is a vector of polynomials of degree s � 1, we have

B ��� z � τh � � s

∑j � 1

k jL j � τ � (37)

with k j� B � � z � c jh � . It follows that

B � z � τh � � B � z ��� h� τ

0B � � z � th � dt (38)

Inserting (38) and (36b) in (36a) and (36c) yields

ki� f � z � cih � B0 � h

s

∑j � 1

k jai � j � � i � 1 � � � � � s (39a)

Φz�

h � zB0� B0 � h

s

∑j � 1

k jb j (39b)

16

with

ai � j : �� ci

0L j � t � dt and b j

� � 1

0L j � t � dt �

Note that a collocation method is uniquely determined by the choice of the points c j. Thehighest possible global order 2s is achieved if the c j are chosen as Gauß quadrature points.The corresponding method, which are called Gauß methods, are known to be A-stable andsymmetric (cf. [10]). In our numerical computations we used the Gauß method of order 6.Other commonly used methods are Radau methods (cs

� 1: order 2s � 1) and Lobatto methods(c1

� 0 and cs� 1: order 2s � 2). The numerical values of the Runge-Kutta coefficients ai � j, b j

and c j of the first Gauß, Radau and Lobatto methods can be found in [10] and [6].

6.2 Numerical solution of the nonlinear equations

For a numerical solution of the system of equations (39) we introduce the new unknowns

yi� h

s

∑j � 1

ai � jk j

and reformulate (39) equivalently as

yi� h

s

∑j � 1

ai � j f � z � c jh � B0 � y j � � i � 1 � � � � � s (40a)

Φz�

h � zB0� B0 �

s

∑j � 1

d jy j (40b)

where dT � bT A � 1 (cf. [6]). Combining y1 � � � � � ys to one large vector Y , we may rewrite thesystem of equations (40a) as

Y � hF � z � Y � � (41)

For the nonlinear Schrodinger equation (14), eq. (40a) has the form

Y � hLY � hN � Y � � (42)

Here L is a block diagonal matrix whose diagonal blocks are D � ω j � A. As opposed to the fixedpoint equation in Agrawal’s method (cf. Lemma 1), eq. (42) should not be solved by a fixedpoint iteration. This is because the imaginary parts of the eigenvalues of L get larger and largeras the time discretization gets finer, and hence the step size h has to be chosen very small toensure convergence. Therefore, we rewrite (42) as Y � � I � hL � � 1hN � Y � and use the fixedpoint iteration

Yn�

1� � I � hL � � 1hN � Yn � � (43)

The iteration (43) can be interpreted as an inexact Newton iteration for (42) where the derivativematrix is approximated by I � hL.

17

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

position z [km]

step

siz

e [k

m]

2nd order soliton

0 5 10 15 20 25 30 35 40 45 503

4

5

6

7

position z [km]

nr o

f New

ton

its

Figure 6: Step size selection and termination of the Newton iteration for collocation methods

6.3 Stopping criterion and starting values for the Newton iteration

We first derive a stopping criterion for the inexact Newton iteration 43 (cf. [5]). If h is suffi-ciently small, the updates ∆Yn : � Yn � Yn � 1, n

�1 satisfy�

∆Yn�

1� � θ

�∆Yn

�with θ � 1. Applying the triangle inequality to

Yn�

1 � Y � � Yn�

1 � Yn�

2 ��� � Yn�

2 � Yn�

3 ��� � � �yields the estimate �

Yn�

1 � Y� � Θ

1 � Θ�∆Yn

� �To obtain a computable error estimate we approximate the convergence rate Θ by the quantities

Θn : � �∆Yn

� � � ∆Yn � 1� � n

�2 �

We have thus derived the stopping criterion

Θn

1 � Θn

�∆Yn

� � κTol (44)

for the iteration (43). Since the error in the Newton iteration should be smaller the the dis-cretization error, which is usually close to Tol , the parameter κ should be � 1. If (44) is notsatisfied after nmax iterations or if θn

� 1 for some n, the step is repeated with step size h � 2. Weused the parameter values κ � 10 � 4 and nmax

� 10.The starting values are computed by extrapolating the polynomials (35) of the previous step.

This saves about 1-2 Newton iterations compared to starting the iteration with the zero vector.

18

−1.5 −1 −0.5 0 0.5 1 1.50

2

4

6

8x 10

4

time [ps]

inte

nsity

[W] solution

initial

−300 −200 −100 0 100 200 30010

−40

10−20

100

1020

frequency [THz]

inte

nsity

[W] solution

initial

100

102

104

106

108

103

104

105

106

precision=1/(rel. L2−error)

num

ber

of s

teps

1st order soliton with Raman effect

AgrawalMarcuse 4gauss6

Figure 7: Decay of a first order soliton under the influence of the Raman effect

6.4 Step Size Selection

To implement an adaptive step size selecting algorithm we need an estimate of the error. Inanalogy to the procedure described in [10, p. 133] for the 3-stage Radau IIA method, we discussthe construction of an embedded lower order Runge-Kutta method of the form

Φz�

h � zB0� B0 � h

�b0 f � z � B0 ���

s

∑i � 1

bi f � z � cih � B0 � yi � � bs�

1 f � z � h � Φz�

h � zB0 � � �

with b0�� 0. Given a choice of b0, e.g. b0

� � s � 1 � � 1, we determine the other coefficients bi

such that the embedded evolution Φ has the maximal possible order s � 2. This is the case if bi

satisfy the system of equations ∑s�

1i � 1 bic

ji� 1

j�

1 � b0c j0, j � 0 � � � � � s with c0 : � 0 and cs

�1 : � 1.

The difference err : � Φz�

h � zB0 � Φz�

h � zB0 serves as an error estimate. It satisfies

errsss � hb0 f � z � B0 � � hbs�

1 f � z � h � Φz�

h � zB0 � �s

∑i � 1

eiyi

with � e1 � � � � � es � � � b1 � b1 � � � � � bs � bs � A � 1. We use the following formula to compute a sug-gestion for the new step size:

hopt� 0 � 9nmax � 1

nmax � n

�Tol�err

� � 1 � s �2 �

6.5 Numerical examples

The performance of our step control algorithm and the stopping criterion for the Newton itera-tion is illustrated in Figure 6 for the second order soliton.

The convergence plots in Figure 8 show that the collocation method performs well for sig-nals with a small bandwidth, but not as good for signals with a large bandwidth such as WDMsignals. The reason is that small step sizes are required to approximate the rapid oscillations ofhigh frequencies suffiently well by polynomials.

We now consider the full propagation equation (12) including the Raman effect. In thiscase the propagation of the nonlinearity in Split Step methods can no longer be performed

19

analytically. In principle we could resort to Runge-Kutta methods, but due to rapid oscillationsthis is not an efficient options. Agrawal’s implicit version of the Split Step method discussed insection 4 works significantly better.

As a test example we consider the decay of a first order soliton. Notice the huge red shiftin the spectrum (cf. Figure 7). Due to Raman scattering energy is transfered from higher fre-quencies to lower frequencies. The convergence plot in figure 7 shows that collocation methodsclearly outperforms Agrawal’s method for this example.

6.6 Conclusion

The efficient implementation of collocation methods of Gauß type for nonlinear Schrodinger-type equations including adaptive step size selection and termination of the Newton iteration.These methods can handle complicated nonlinearities effectively. They are not efficient forsignals with a large spectral bandwidth.

7 Other Methods

7.1 Exponential Integrators

Hochbruck and Lubich have studied numerical integrators which are based on the evaluation ofthe exponential of the Jacobian (cf. [13, 14, 15]). These methods are exact for linear differen-tial equations with constant coefficients if the exponential of the coefficient matrix is evaluatedexactly. The idea is to use Krylov subspace approximations to the action of the matrix exponen-tial operator. Although this idea had been considered earlier, Hochbruck and Lubich were thefirst to prove that Krylov approximations to exp � hA � converge much faster than those for thesolution of linear systems � I � hA � x � v ([13]).

Moreover, they developed the first exponential integrator of order 4 ([14]). Figure 8 showsa comparision of their code exp4 and other methods for the nonlinear Schrodinger equation.The poor performance of exp4 can be explained as follows: Let A be a skew-Hermitian matrixwith eigenvalues in an interval on the imaginary axis of length 4ρ. Then the error in the Arnoldiapproximation of exp � hA � is not decay essentially for m � hρ where m is the dimension of theKrylov subspace (cf. [13, Theorem 4 and Figure 3.2]). Since ρ is very large in our application,we either need small step sizes h or a large number m of iterations. In both cases the method isinefficient.

Hochbruck and Lubich also suggested a Gautschi-type method for differential equations ofthe form

B � ��� z � � � AB � z � � g � B � z � � � B � 0 � � B0 � B � � 0 � � B �0 �which is second order accurate ([15]). It is possible to adopt this scheme to the nonlinearSchrodinger equation, but the resulting method is only first order accurate (cf. [11]). We do notknow how to construct higher order methods.

7.2 The method of Marcuse

Eq. (14) can be solved explicitly if N � 0. The solution is given by

B � z � � eD ω � z � B � 0 � � j � 1 � � � � � n �20

10−2

100

102

104

106

108

1010

101

102

103

104

105

precision=1/(rel. L2−error)

num

ber

of s

teps

2nd order soliton

exp4GautschiMarcusered. SSExtrapolationgauss 6

100

101

102

103

104

105

106

102

103

104

105

precision=1/(rel. L2−error)

num

ber

of s

teps

1 channel

exp4GautschiMarcusered. SSExtrapolationgauss 6

10−2

100

102

104

106

108

1010

102

103

104

105

precision=1/(rel. L2−error)

num

ber

of s

teps

2 channel WDM, DCF

exp4GautschiMarcusered. SSExtrapolationgauss 6

Figure 8: Comparison various methods for the solution of the nonlinear Schrodinger equation

21

10−2

100

102

104

106

108

1010

102

103

104

105

precision=1/(rel. L2−error)

nr o

f FF

T o

pera

tions

2nd order soliton

ExtrapolationBlow and Wood

101

102

103

104

105

106

102

103

104

105

precision=1/(rel. L2−error)

nr o

f FF

T o

pera

tions

WDM 2 channels, DCF

ExtrapolationBlow and Wood

Figure 9: Comparison of the method of Blow and Wood and second order extrapolation

The high frequency components of this solution oscillate rapidly in space. This is undesirablefor a numerical approximation of these components. Now the idea of Marcuse et.al. [17] is tocompute the function

C � z � : � e � D ω � z � B � z � �Since C � � z � � � B � � z ��� D � ω � B � z � � e � D ω � z, C satisfies the differential equation

C ��� z � � F � N � B � z � � B � z � � � B � z � � F � 1�e � iD ω � z � C � z � � � (45)

This differential equation can be solved by explicit Runge-Kutta methods.The method of Marcuse yields good results for small step sizes. However, for large step

sizes it is often unstable.

7.3 The method of Blow and Wood

We have seen in subsection 4.2 that the leading local error term of symmetric Split Step methodsis given by e � z � t � h3. It may be assumed that e � z � t � varies slowly with z. Therefore, taking 4forward steps of size h followed by 1 backward step of size 2h and 4 forward steps of size heliminates the leading error term as 4h3 � � � 2h � 3 � 4h3 � 0. This is the method of Blow andWood [2]:

ΦBW2h

� Φh� � �

Φh� ��� �

4times

Φ � 2h Φh� � �

Φh� ��� �

4times

(46)

In total we have to take 9 steps to propagate the signal over the distance distance 6h. The sameamount of steps is needed for the second order extrapolation scheme described in section 5.Since both methods eliminate the leading error term, it is not surprising that their performanceis almost the same (cf. Figure 9). However, the extrapolation method has the advantage that italso yields an error estimate, which can be used for adaptive step size selection. Moreover, theconstruction of higher order schemes is straightforward. Therefore, we prefer the extrapolationmethod from section 5 to the method of Blow and Wood.

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7.4 Numerical results and conclusions

We have chosen three test examples to compare the different methods (cf. Figure 8): The secondorder soliton, a one channel system with P � 0 � 1W over a propagation distance of 50km, anda 2 channel system with damping constant α � 0 � 05 including a dispersion compensating fiber(DCF).

The results show that exponential integrators are not efficient for the integration of the non-linear Schrodinger equation since the application of the exponential of the jacobian is too ex-pensive. The method of Marcuse is flexible and often yields good results, but for large step sizesit tends to be unstable. The method of Blow and Wood is similar to second order extrapolation,but it is not clear how to contruct higher order methods and adaptive step size selection in anefficient manner.

In summary, there is no method which performs well in all situations. For high accuraciesand the simple nonlinear Schrodinger equation (1) extrapolation of Split Step methods is mostefficient. For low accuracies it is better to use the reduced Split Step method without extrapo-lation. If the Raman effect is considered and high powers are involved, collocation methods ofGauß type yield the best results.

References

[1] G. Agrawal. Nonlinear Fiber Optics. Academic Press, San Diego, 1989.

[2] K. J. Blow and D. Wood. Theoretical description of transient stimulated Raman scatteringin optical fibers. IEEE J Quantum Electronics, 25(12):2665–2673, 1989.

[3] P. Deuflhard. Order and stepsize control in extrapolation methods. Numer. Math., 41:399–422, 1983.

[4] P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations.SIAM review, 27(4):505–535, 1985.

[5] P. Deuflhard. Newton Methods for Nonlinear Problems. Affine Invariance and AdaptiveAlgorithms. ., In preparation.

[6] P. Deuflhard and F. Bornemann. Numerische Mathematik II. Walter de Gruyter, Berlin,New York, 1994.

[7] P. Deuflhard and A. Hohmann. Numerische Mathematik I. Walter de Gruyter, Berlin, NewYork, 2nd edition, 1993.

[8] J.-P. Elbers. Modellierung und Simulation faseroptischer Ubertragungssysteme. Master’sthesis, Uni. Dortmund, 1996. Diplomarbeit.

[9] E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I. NonstiffProblems, volume 8 of Springer series in Computational Mathematics. Springer-Verlag,Berlin, New York, 2nd edition, 1992.

[10] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, volume 14 of Springer series in Computational Mathematics.Springer-Verlag, Berlin, New York, 1991.

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[11] J. Haring. Entwurf schneller Algorithmen zur Losung der nichtlinearen Schrodinger-Gleichung. Master’s thesis, Uni. Dortmund, 1998. Diplomarbeit.

[12] J. Hilgert and K.-H. Neeb. Lie-Gruppen and Lie-Algebren. Vieweg Verlag, Braunschweig,1991.

[13] M. Hochbruck and C. Lubich. On Krylov subspace approximation of the matrix exponen-tial operator. SIAM J. Numer. Anal., 34:1911–1925, 1997.

[14] M. Hochbruck and C. Lubich. Exponential integrators for large systems of differentialequations. SIAM J. Sci. Comput., 19:1552–1574, 1998.

[15] M. Hochbruck and C. Lubich. A Gautschi-type method for oscillatory second-order dif-ferential equations. Numer. Math., 83:403–426, 1999.

[16] T. Kremp, A. Killi, A. Rieder, and W. Freude. Split-step wavelet collocation methodfor nonlinear pulse propagation. Technical Report 01/14, Institut fur WissenschaftlichesRechnen und Mathematische Modellbildung, Univ. Karlsruhe, Universitat Karlsruhe,76128 Karlsruhe, Germany, 2001.

[17] D. Marcuse, A. R. Chraplyvy, et al. Effect of fiber nonlinearity on long distance transmis-sion. J. Lightwave Tech., 9(1):121–128, 1991.

[18] M. Plura. Makromodellierung und schnelle Simulationsmethoden zur Losung der nicht-linearen Schrodinger-Gleichung. Technical report, Lehrstuhl fur Hochfrequenztechnik,Prof. Voges, Uni. Dortmund, 2001.

[19] M. J. Potasek, G. P. Agrawal, and S. C. Pinault. Analytic and numerical study of pulsebroadening in nonlinear dispersive optical fibers. J.Opt.Soc.Am B, 3(2), 1986.

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