A theoretical foundation for the use of the parabolic wave equationfourier split. Our main result states that the longtime orbital stability of the exact solution near plane waves transfers to the numerical solution, see subsect. We will call this implementation of the split step method the logarithmic step size method. This method combines a fourier collocation in space with a strang splitting in time. We set up different long bone models for numerical simulation. Stability properties of the wellknown fourier split step method used to simulate a soliton and similar solutions of the nonlinear dirac equations, known as the grossneveu model, are studied numerically and analytically. The ssfm is the technique of choice for solving the nlse due to its easy implementation and speed compared to other methods, notably timedomain finitedifference methods 73. In this paper, we focus on the split step fourier method.
The splitting algorithm is a pseudospectral numerical method which consists in approximate iteratively the solution of the nonlinear evolution equation using exponential operators. Plane wave stability of the splitstep fourier method for. Split step method in the analysis and modeling of optical fiber communication system. Pdf this is a tutorial on how to use a split step method using fourier transform with matlab.
Pdf splitstep fourier method in modeling dual core. Osa fast implementation of the splitstep fourier method. In numerical analysis, the splitstep fourier method is a pseudospectral numerical method used to solve nonlinear partial differential equations like the. We analyze a numerical instability that occurs in the well. Proposed method can be named as compressive split step fourier method cssfm. Using a modified version of the split step fourier method, we analyze the effect of noise on soliton propagation inside erbiumdoped fiber amplifiers. In order to explain the split step fourier method, we use the nonlinear schroedinger equation. We will discuss the fastfouriertransform method, which should be used to e ciently carry out the long series of fourier and inverse fourier transformations needed to propagate the wave function this way for a large number of time steps. Abstract in this paper an approach for decreasing the computational effort required for the split step fourier method ssfm is introduced.
Splitstep fourier method for nonlinear schrodinger. Download pdf download citation view references email request permissions export to collabratec. Osa optimization of the splitstep fourier method in. Plane wave stability of the splitstep fourier method for the. To apply the new method, we considered a complicated system of 16channel wavelengths, where the splitstep fourier method was used to solve the nonlinear schrodinger equations, and the transfer matrix method was applied to describe the coupled mode equations. Assuming i want to propagate a gaussian in free space, im supposed to use. A higherorder splitstep fourier parabolicequation sound. We analyze a numerical instability that occurs in the wellknown splitstep fourier method on the background of a soliton. In this paper an approach for decreasing the computational effort required for the splitstep fourier method ssfm is introduced. Theoretical analysis of a novel wdm optical longhaul network.
The numerical investigation of all optical soliton switching is done by using split step fourier transformation algorithm. Parabolic equation method in underwater acoustics, j. Pdf stability analysis of the splitstep fourier method. This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of most finite. Sorry that this post is long but i am trying to simulate two dimensional schrodinger equation in python using split step method. Oct 27, 2010 we analyze a numerical instability that occurs in the well. There are frequentlyused methods like perturbation method, differential method, splitstep fourier method ssfm et al. Pdf in this paper an approach for decreasing the computational effort required for the splitstep fourier method ssfm is introduced.
The chapter will begin with a list pointing the advantages of the. Pdf an introduction to the split step fourier method using matlab. Theoretical analysis of a novel wdm optical longhaul. Splitstep fourier method in modeling dual core nonlinear fiber coupler james raja. Timedependent schrodinger equation via the splitstep. Pdf stability analysis of the splitstep fourier method on. In this paper we focus on the splitstep fourier method.
Pdf modified splitstep fourier method for the numerical. Proposed method can be named as compressive splitstep fourier method cssfm. A short description of the ssfm method can be found on wikipedia. The numerical analysis of soliton propagation with plit. The linear subequation is still solved by the discrete fourier transform 17. Numerical methods for partial differential equations. Pdf splitstep fourier method in modeling dual core nonlinear. In this talk we employ the well known split step fourier method for the numerical simulation of the nls and cnls equations. We will discuss the fast fourier transform method, which should be used to e ciently carry out the long series of fourier and inverse fourier transformations needed to propagate the wave function this way for a large number of time steps. Notes on the sine gordon equation david gablinger january 31, 2007 abstract in this seminar, we will introduce the sinegordon equation, and solve it using a baecklund transfomation. Splitstep method for generalized nonlinear equations. Fourier splitting method for kawahara type equations. Abstract extensions of the splitstep fourier method ssfm for schrodingertype pulse propagation equations for simulating femtosecond pulses in single and.
Using a modified version of the splitstep fourier method, we analyze the effect of noise on soliton propagation inside erbiumdoped fiber amplifiers. Theoretical description of the parabolic approximation. Jun 02, 2006 splitstep fourier method for nonlinear schrodinger equation. Optimization of the splitstep fourier method in modeling. Forward propagation above dielectric surfaces is studied using the split step fourier ssf resolution technique. It is shown that using the sparsity property of the simulated signals, the compressive sampling algorithm can be used as a very efficient tool for the split step spectral simulations of various phenomena which can be modeled by using differential equations. Walkoff method in many opticalfiber communications systems, chromatic dispersion is the dominant effect and nonlinearity only plays a. The numerical singularity induced by the reflection coefficient at pseudobrewster incidence is analyzed, and the dmft solution for ssflbc resolution is. Robust splitstep fourier methods for simulating the. Numerical solutions of the schr odinger equation 1 introduction. Because the split step fourier method employs an algorithm of fast fourier transform fft, the analysis speed is faster than most of other analytical methods. Split step fourier method file exchange matlab central. First of all, we deform the nonlinear schroedinger equation into the following expression.
The method is implemented in both cartesian x, y, z and cylindrical r. Numerical solutions of the schr odinger equation 1. Split step fourier propagation beam propagation method. Generalizations i bregman iteration can be used to solve a wide range of nonl1 problems argminju such that au 0 where j and kak2 are convex. In this work we employ a splitstep fourier method to integrate the equation. For testing of the proposed method the nonlinear schrodinger equation and its onesoliton and twosoliton solutions. Furthermore, we also give a numeric solution using a splitstep algorithm, and also present two physical applications of the sinegordon equation. The representation of nlse using splitstep fourier method ssfm 3.
This method combines a fourier collocation in space with a strang splitting in time, see subsect. Pdf an introduction to the split step fourier method. The method is widely used in geophysics to image the earth interior, but to our knowledge has not traditionally been used for bone imaging. Pdf parallel implementations of the splitstep fourier method for. Parallel implementations of the splitstep fourier method for solving. The split bregman method for l1regularized problems. Three distinct types of numerical instability that can occur in this case, are revealed and explained. Furthermore, we also give a numeric solution using a split step algorithm, and also present two physical applications of the sinegordon equation. In 1d, an n element numpy array containing the intial values of \\psi\ at the spatial grid points. In this paper, the ssfm is particularly introduced, and the friendly interface matlab software is. We advance in time in small steps and treat separately the linear and nonlinear parts.
Parallel splitstep fourier methods for the cmkdv equation t. In this paper we focus on the split step fourier method. In this talk we employ the well known splitstep fourier method for the numerical simulation of the nls and cnls equations. We also present a parallelization of the split step fourier method using the fastest fourier. The splitstep fourier method is used in threedimensional parabolicequation pe models to compute underwater sound propagation in one direction i. The introduction of fresnel boundary conditions ssffbc and leontovitch boundary conditions ssflbc is described. Study of instability of the fourier splitstep method for. The principle of the ssfm is simpler among them, it is easy to write program, and employed by many scholars. Ultrafast soliton switching in a twocore fiber coupler is studied by controlling the coupling coefficients of the fiber. Fast fourier transform history twiddle factor ffts noncoprime sublengths 1805 gauss predates even fouriers work on transforms.
If i consider only nonlinear effect in this code, the shape of pusle in time domain does not change righttheory says that nonlinear effects has direct influence on pulse in spectral domain spectral broadening, but inderect influence on pulse shape in time domain. Parallel numerical methods for solving nonlinear evolution. In this method, the equation is discretized in space by a spectral collocation method and in time by a splitting integrator. This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of most finitedifference schemes. The efficiency of the split step method depends on both the timedo. Compressive split step fourier method cihan bay nd r1, y 1department of civil engineering, i. Apr 04, 2016 i am trying to write a very basic matlab code to preform the split step fourier method on the nonlinear schrodinger equation. Sinkin et al optimization of the split step fourier method 63 where and is the number of steps per fiber span.
Propagation modeling using the split step fourier method. Stability properties of the wellknown fourier splitstep method used to simulate a soliton and similar solutions of the nonlinear dirac equations, known as the grossneveu model, are studied numerically and analytically. A reliable splitstep fourier method for the propagation equation of ultrafast pulses in singlemode optical fibers ralf deiterding, roland glowinski, hilde oliver, and stephen poole member, ieee abstractthe extension to the splitstep fourier method ssfm. Various versions of the splitstep fourier method ssfm for the nonlinear. Ultrasound imaging of long bone fractures and healing with. In this paper an approach for decreasing the computational effort required for the split step fourier method ssfm is introduced. One of the most popular numerical methods to solve the perturbed nlse is the splitstep fourier method ssf 2. Pdf an introduction to the split step fourier method using. Then the fourier series of ft is the series cc1a,1 cos iii h b7, sin itt, 18 i i where the fourier. Abstract in this paper an approach for decreasing the computational effort required for the splitstep fourier method ssfm is introduced. A variation on this method is the symmetrized split step fourier method, which takes half a time step using one operator, then takes a fulltime step with only the other, and then takes a second half time step again with only the first. Study of instability of the fourier splitstep method for the. An introduction to the split step fourier method using matlab.
This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of. We will call this implementation of the splitstep method the logarithmic stepsize method. In this work, we used the splitstep fourier imaging method to reconstruct bone models. It is shown that using the sparsity property of the simulated signals, the compressive sampling algorithm can be used as a very efficient tool for the splitstep spectral simulations of various phenomena which can be modeled by using differential equations. Sinkin et al optimization of the splitstep fourier method 63 where and is the number of steps per fiber span. This function performs the splitstep fourier method to solve the 1d timedependent schrodinger equation for a given potential. Hello im having a difficulty using the split step fourier method. We present a parallel version of the wellknown splitstep fourier method ssf for solving the nonlinear schrodinger equation, a mathematical model. Download pdf download citation view references email request permissions export to collabratec alerts metadata. Hanik, fast implementation of the splitstep fourier method using a graphics processing unit, in optical fiber communication conference, osa technical digest cd optical society of america, 2010, paper otud7. Compressive splitstep fourier method cihan bay nd r1, y 1department of civil engineering, i. We analyze a numerical instability that occurs in the wellknown split step fourier method on the background of a soliton.
Numerical method and statement of the main results 2. In ffd method, usually the minimum velocity is used as the reference velocity. We compared the performance of the different implementations for a variety of pulse formats and systems,including higher order solitons, collisions of soliton pulses, a singlechannel periodically. Then the fourier series of ft is the series cc1a,1 cos iii h b7, sin itt, 18 i i where the fourier coefficients a, and b, are defined by means of the. I am trying to write a very basic matlab code to preform the splitstep fourier method on the nonlinear schrodinger equation. A splitstep method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation. Splitstep fourier method for nonlinear schrodinger equation. In fact, noise from forwardpropagating amplified spontaneous emission, associated with a markov. If i consider only nonlinear effect in this code, the shape of pusle in time domain does not change righttheory says that nonlinear effects has direct influence on pulse in spectral domain spectral broadening, but inderect influence on.
Pdf compressive splitstep fourier method researchgate. Sorry that this post is long but i am trying to simulate two dimensional schrodinger equation in python using splitstep method. Splitstep methods for the solution of the nonlinear. Performancewise, gpuwave has been found to be up to three orders of magnitude faster than peec, a comparable cpubased implementation of the splitstep fourier method, and the.
The space variable is discretized by means of a finite difference and a. Plane wave stability of the splitstep fourier method for the nonlinear schr odinger equation erwan faou1,2 ludwig gauckler3 christian lubich4 version of 2 dezember 20 abstract plane wave solutions to the cubic nonlinear schr odinger equation on a torus. Parallel splitstep fourier methods for the cmkdv equation. One dimensional problem of this equation has been explained in this post. Hanik, fast implementation of the split step fourier method using a graphics processing unit, in optical fiber communication conference, osa technical digest cd optical society of america, 2010, paper otud7.
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