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NUMERICAL SIMULTIONS for 1+2 DIMENSIONAL COUPLEDNONLINEAR SCHRÖDINGER EQUATION

THIAB R. TAHA and HARINI MEDIKONDURU

University of Georgia, Athens, GA 30602, USA, thiab@cs.uga.edu, harinim@cs.uga.edu

ABSTRACT

The coupled nonlinear Schrödinger equation is of tremendous importance in both theory and applications. Couplednonlinear Schrödinger equation (CNLS) is the vectorial version of the nonlinear Schrödinger equation (NLS). TheNLS equation is the main governing equation in the area of optical solitons. In this paper we introduce sequential andparallel numerical methods for numerical simulations of the 1+ 2 dimensional CNLSE. The parallel methods areimplemented on the zcluster multiprocessor system at the University of Georgia (UGA). Our preliminary numericalresults have shown that these methods give good results and considerable speedup.

Keywords: Split-step method, CNLS, Parallel algorithms, FFTW, MPI

I. INTRODUCTION

In the studies of optical fiber communication systems,the NLS typed equations are highly used and are ofcurrent importance [9]. The CNLS equation is thevectorial version of the NLS equation. Here, we studythe (1+2) dimensional CNLS equations, with x and yderivatives given by [12]

(1.1)where and are two dimensional complex functions.There are many numerical methods to solve the NLSequation like the split-step Fourier (SSF) method whichwas originally proposed by R.H. Hardin andF.D.Tappert [3]. There are different versions of the SSFmethod to solve the CNLS equations. Taha and Xuhave developed SSF methods for CNLS as well as theNLS equations. They have also developed parallelimplementations of these methods [10]. Ismail andTaha have also developed a finite difference approachto numerically simulate the CNLS equation [4].

This paper includes the implementations of split-stepFourier method in the first, second and fourth orderschemes for the two dimensional CNLS equation. Inaddition, the parallel implementations for thesemethods were developed. We used Fastest FourierTransform in the West (FFTW) developed by M.Frigoand S.G. Johnson [2] for the serial and parallel

implementations. Message Passing Interface (MPI)standard was used for the parallel implementations [8].

2. PRELIMINARIES

2.1 THE DISCRETE TWO DIMENSIONAL

FOURIER TRANSFORM

If is a sequence of size , obtained bytaking samples of a continuous function with equalintervals at the direction of and , respectively, thenits discrete Fourier transform (DFT) is given by

(2.1.1)

(2.1.1)where and are the numbers of samples in x and ydirections in both spatial and frequency domains,respectively. And is the two dimensionaldiscrete spectrum ofThe inverse two dimensional DFT flips the sign of theexponent, which is defined as

(2.1.2)

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(2.1.2)It is the “inverse” of the forward two dimensional DFT,in the sense that computing the inverse transfer after theforward transform of a given sequence would yield theoriginal sequence.Both and could be considered aselements of two matrices and ,respectively.

2.2 THE TWO DIMENSIONAL FASTFOURIER TRANSFORM

The Fast Fourier transform (FFT) is an efficientalgorithm to compute the discrete Fourier transform(DFT) and its inverse. A DFT decomposes a sequenceof values into components of different frequencies. It isuseful in many fields. However, calculating DFTdirectly from its definition is often too slow. Instead,FFT is a better way to calculate the same result morequickly.Computing a DFT of N points requires O ( ).However, an FFT can calculate the same result in onlyO ( ) operations [1]. The difference in speedcould be substantial, especially for large data sets where

may be very huge. In this case, FFTs are of greatimportance to a large number of applications, like,digital signal processing and solving partial differentialequations.A two dimensional FFT is achieved by firsttransforming each row, replacing each row with its onedimensional transform FFT and then transforming eachcolumn, replacing each column with its transform. Atwo dimensional FFT of size requires

one dimensional FFT.

2.3 THE FASTEST FOURIER TRANSFORMIN THE WEST

The Fastest Fourier Transform in the West (FFTW) is asoftware library used to calculate DFTs [2]. FFTW is acomprehensive collection of fast C routines forcalculating the DFT in one or more dimensions, of bothreal and complex data, and of arbitrary input size. “Ithas gained a wide acceptance in both academia andindustry, because it provides excellent performance ona variety of machines (even competitive with or fasterthan equivalent libraries supplied by vendors)”.FFTW automatically adapts the DFT algorithm todetails of the underlying hardware (cache size, memory

size, registers, etc.). The inner loop of FFTW isgenerated automatically by a special-purpose compiler[2]. The FFTW begins by generating codelets. Acodelet is a fragment of C code that computes a Fouriertransform of a fixed small size. A composition ofcodelets is called a plan which depends on the size ofthe input and the underline hardware. At runtime, theFFTW’s planner finds the optimal decomposition fortransforms of a specified size on your machine andproduces a plan that contains this information. Theresulting plan can be reused as many times as needed.This makes the FFTW’s relatively expensiveinitialization acceptable. FFTW also includes a shared-memory implementation on top of POSIX threads, anda distributed-memory implementation based on MPI(Message Passing Interface) [8].

2.4 THE SPLIT-STEP METHOD

The split-step Fourier method is a pseudo-spectralnumerical method used to solve nonlinear partialdifferential equations [9].For example, consider the following equations,

(2.4.1)in which and are linear and nonlinear operators,respectively. Generally, these operators are non-commutative.The two dimensional CNLS equation

(2.4.2)can be rewritten for any real number as,

where,

The solution of (2.4.1) can be advanced from one time-level to the next by using the following formulae [10]

(2.4.3)where denotes the time step. This method is firstorder accurate. It would be exact, if the operators and

are time-independent [5].The procedure for time splitting consists of replacingthe right hand side of the above equations with an

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appropriate combination of products of the exponentialoperators and This produces asplitting error due to the non-commutability of L and N[3]. The best scheme to minimize this error would beusing Baker-Campbell-Hausdorf (BCH) formula [5] forthe two operators and as follows

(2.4.4)where is the coefficient of and , and

the remaining operators are commutators of and, commutators of commutators of and and so on.

The expressions for are rather complicated:

where is the commutator of and, and

From these results, we can obtain the first orderapproximation of the exponential operator in (2.4.3) asfollows [10]

(2.4.5)We usually first solve the nonlinear part

And using this solution, we advance to solving thelinear part

This advancement is carried out in two steps, thusgiving the procedure the name “split-step” method.The second-order approximation of the exponentialoperator in (2.4.2) is given by

, (2.4.6)

The fourth-order approximation of the exponentialoperator in (2.4.3) which preserves the symmetry couldalso be constructed [10],

(2.4.7)

whereω=(2+∛2+1/∛2)/3 (2.4.8)

The operators L and N in (2.4.4) – (2.4.6) may beinterchanged without affecting the order of the method[5].

2.6 MESSAGE PASSING INTERFACEMessage Passing Interface (MPI) is a libraryspecification for message-passing, proposed as astandard by a broadly based committee of vendors,

implementations, and users [8]. Message passing is aparadigm that has been widely used on certain classesof parallel machines, especially those with distributedmemory. Processes running on such machinescommunicate through messages [8].

3.1 1+2 TWO DIMENSIONAL COUPLEDNONLINEAR SCHRÖDINGER EQUATION

The original two dimensional coupled nonlinearSchrödinger Equation (2D CNLS), is represented asfollows:

(3.0.1)where and are complex valued functions withrespect to which represent the amplitudes of twocircularly polarized waves. which is theparameter to distinguish between self focusing or selfdefocusing Kerr nonlinearity. The value of variesover and 7 [12]. We consider (3.0.1) withthe Kerr-type electronic nonlinearity, when and

.The exact one soliton solution of (3.0.l) is given by [12]

Where

where is the time and and are arbitrarycomplex parameters whose values areand = = . Also = = because they arethe complex conjugates of and respectively.

3.1 NUMERICAL METHODSWe start off with studying the two dimensional CNLS

equation (3.0.1) with the solution given in (3.0.2) and(3.0.3) with the values , , , =

= and = = .Here we assume that and satisfythe boundary condition with period . Afternormalizing the spatial period to we have

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(3.1.1)where , which is the half length of the period.

and . Then, theinterval is divided into sub intervals in the

-direction and sub intervals in the - directionrespectively where with grid spacing

and . It is also denoted aswhere and where

.We now advance the solutions of (3.0.1) from time tothe next time-level in the following manner:Firstly, we focus on the nonlinear part of the solution[10]:

(3.1.2)

which can be solved exactly with

(3.1.3)Second, we take the linear part:

(3.1.4)For the two-dimensional DFT , we have

(3.1.5)the inverse DFT is given by

(3.1.6)Combining (3.1.4), (3.1.5) and (3.1.6), we have

which is nothing but

(3.1.7)The split-step Fourier method for the first orderapproximation (2.4.5) is given by (3.1.7), where is thetime step, and are the forward and inverse discreteFourier transforms respectively.To advance in time from to by the split-step Fouriermethod with the second order splitting approximation (2.4.6),we should take the following steps [10]:

(1) Apply (3.1.3) to advance the solution using thenonlinear part

(2) Apply (3.1.7) to advance the solution using thelinear part

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(3) Apply (3.1.3) to advance the solution using thenonlinear part

Advancement in time from to by the split-stepFourier method with the fourth order splittingapproximation (2.4.7) could be obtained with followingsteps [10]:First, advance in time from to using thesecond-order split-step Fourier method, where

Second, advance in time from tousing the second-order split-step

Fourier method.Finally, advance in time from tousing the second-order split-step Fourier method andwe obtain approximations to and

.

3.2 NUMERICAL EXPERIMENTS

In order for us to test the numerical methods that wereimplemented, we compute the and norms at theterminating time or the infinity norm is themaximum deviation between the numerical and exactsolutions. or the Euclidean norm is the mostcommon norm, calculated by summing the squares ofall the differences between numerical and analyticalsolution and taking the square root.Here we only show the results for the complex function

as the same results reflect for the function . This isdue to the fact that and are two similar functions

involved in the 2D CNLS equations with differentamplitudes which are also the same if the definedparameters’ values are considered. The absolute valueof the numerical solution of i.e. is plotted forevery method implemented in this paper.

Figure 3.1: The exact solution

Table 3.1: Numerical solution of the CNLS using thefirst order SSF method

cpu(s)256 2.812919086E-01 6.187103923E-01 1383.0

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Figure 3.2: Two dimensional CNLSE using the firstorder SSF.

Table 3.3: Numerical solution of the CNLS using thesecond order SSF method

cpu(s)256 2.8129179E-01 6.1871023E-01 1531.45

Figure 3.3: Two dimensional CNLSE using the secondorder SSF.

Table 3.4: Numerical solution of the CNLS using thefourth order SSF method

cpu(s)256 2.8148341E-03 6.1873047E-03 3974.33

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Figure 3.4: Two dimensional CNLSEusing SSF in fourth order.

3.4 PARALLEL IMPLEMENTATION ANDEXPERIMENTSLarge-scale numerical simulations of the 2D CNLS arerequired for many problems in fiber optics. Such asimulation is computationally intensive and timeconsuming using sequential methods described earlier.The parallelization methods for SSF methods arepresented here in this section.Each of the steps from (3.1.5) to (3.1.8) are parallelizedfor the first order SSF method. Similar procedures arefollowed for second and fourth order SSF methods aswell as for those respective steps.

All the parallel algorithms including the ones for thesplit-step Fourier methods are implemented on thezcluster of UGA which is a linux based cluster. Thespeedup factor is defined as

As the number of physical processes (or the number ofprocessors) available increases, the speedup

increases proportionally. In other words, as theproblem size increases, obtained on themultiprocessor computer running the parallel codes inan MPI environment is considerably good given that

is large.For all the numerical experiments, we varied sizes at

.

Table 3.8: Parallel implementation results for first orderSSF method

(sec) 93.74 415.85 1201.23(sec) 60.77 209.16 764.67(sec) 36.21 119.30 465.23(sec) 29.40 88.87 278.54

1.54 1.98 1.572.58 3.48 2.583.19 4.67 4.31

Table 3.9: Parallel implementation results for secondorder SSF method

(sec) 103.63 426.25 1999.38(sec) 65.18 278.16 972.00(sec) 39.70 152.83 791.80(sec) 49.32 112.79 583.42

1.59 1.53 2.052.61 2.78 2.562.08 3.78 3.43

Table 3.10: Parallel implementation results for fourthorder SSF method

(sec) 306.33 1247.14 4790.67(sec) 195.67 817.64 3881.87(sec) 123.34 466.72 1814.51(sec) 163.27 533.48 1157.55

1.56 1.51 1.232.48 2.69 2.641.87 2.33 4.13

CONCLUSION AND FUTURE WORKIn this paper, we have implemented the one solitonsolution of the 2D CNLS developed by Hai-QiangZhang et al. [12]. We have described three different

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split-step methods to solve the 2D CNLS. We have alsoparallelized the methods to compute their speed ups.We have found that these methods give reasonableaccuracy.Also, the conclusion we have arrived upon using theparallel methods is that, for a given problem, the speedup increases as the size of the problem increases whenrunning on a multiprocessor machine.There is prospective future work in this area foraspirants. Hai-Qiang Zhang et al. [12] have also definedthe two soliton solutions for a 2D CNLS on which thesenumerical methods can be applied and tested. Literatureis also available for coupled nonlinear Schrödingerequation in 3D. The application of these numericalmethods on such problems definitely provides scope forgood and challenging future work.

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