Computation and Visualization of Solutions of Nonlinear Schr - - PowerPoint PPT Presentation

Computation and Visualization of Solutions

• f Nonlinear Schr¨
• dinger Equations in Optics

by OpenFOAM

Goong Chen Department of Mathematics and Institute for Quantum Science and Engineering Texas A&M University, College Station, TX 77843 Joint work with Milivoj Belic (Texas A&M University-Qatar) Alexey Sergeev (Texas A&M University-Qatar)

34th Annual Texas Differential Equations Conference

March 26-27, 2011, University of Texas-Pan American, Edinburg TX

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Abstract

A large number of optical phenomena are modeled as an initial-boundary value problem governed by the nonlinear Schr¨

• dinger equation. In this

talk, we will show numerical computational results for several nonlinear

• ptical dynamic processes containing various forms of nonlinearities. The

numerical scheme is based on a parabolic marching in time and a finite-volume method adapted from heat equation in the fluid-dynamics software OpenFOAM. Supported in part by Texas Norman Hackman Advanced Research Program #010366-0149-2009 from the Texas Higher Education Coordinating Board, and Qatar National Research Fund (QNRF) National Priority Research Program Grant #NPRP09-462-1-074.

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Introduction

Introduction

OpenFOAM (http://www.OpenFOAM.com) is a free, open source computational fluid dynamics (CFD) software package produced by a commercial company, OpenCFD Ltd. It has a large user base across most areas of engineering and science, from both commercial and academic

• rganizations. Here, FOAM means Field Operation and Manipulation.

Running OpenFOAM consists of

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Introduction

Introduction

OpenFOAM (http://www.OpenFOAM.com) is a free, open source computational fluid dynamics (CFD) software package produced by a commercial company, OpenCFD Ltd. It has a large user base across most areas of engineering and science, from both commercial and academic

• rganizations. Here, FOAM means Field Operation and Manipulation.

Running OpenFOAM consists of generating polyhedral mesh for use in Finite Volume Method;

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Introduction

Introduction

OpenFOAM (http://www.OpenFOAM.com) is a free, open source computational fluid dynamics (CFD) software package produced by a commercial company, OpenCFD Ltd. It has a large user base across most areas of engineering and science, from both commercial and academic

• rganizations. Here, FOAM means Field Operation and Manipulation.

Running OpenFOAM consists of generating polyhedral mesh for use in Finite Volume Method; executing a numerical solver for the given differential equation,

Goong Chen Nonlinear Schr¨

• dinger Equations by OpenFOAM

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Introduction

Introduction

OpenFOAM (http://www.OpenFOAM.com) is a free, open source computational fluid dynamics (CFD) software package produced by a commercial company, OpenCFD Ltd. It has a large user base across most areas of engineering and science, from both commercial and academic

• rganizations. Here, FOAM means Field Operation and Manipulation.

Running OpenFOAM consists of generating polyhedral mesh for use in Finite Volume Method; executing a numerical solver for the given differential equation, displaying results.

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Introduction

Many important phenomena in optics are modeled by the nonlinear Schr¨

• dinger equation (NLSE):

i ∂Ψ ∂t + ∆Ψ + F

• |Ψ|2

Ψ = 0 (1) with Dirichlet or Neumann boundary conditions, and initial conditions.

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Mesh generation

Mesh generation

blockMesh is one of the mesh-generating utilities in OpenFOAM. Listing 1 shows a typical file blockMeshDict that is used to construct a mesh on a cylinder consisting of five curvilinear hexagonal blocks.

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Mesh generation

Listing 1. Content of blockMeshDict file containing specifications of hexagonal blocks and their grading in order to construct a mesh on a cylinder. The radius of cylinder is R = 1 and the length L = 4. The OpenFOAM header of the file is not shown to save space. See http://www.openfoam.com/docs/user/blockMesh.php for a tutorial

• n blockMesh utility.

convertToMeters 1 . 0 ; v e r t i c e s ( ( 0 . 5 0 0) (0 −0.5 0) ( −0.5 0 0) (0 0.5 0) (1 0 0) (0 −1 0) (−1 0 0) (0 1 0) ( 0 . 5 0 4) (0 −0.5 4)

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Mesh generation

( −0.5 0 4) (0 0.5 4) (1 0 4) (0 −1 4) (−1 0 4) (0 1 4) ) ; b l o c k s ( hex (8 9 10 11 0 1 2 3) (16 16 32) simpleGrading (1 1 1) hex (8 12 13 9 0 4 5 1) (16 16 32) simpleGrading (1 1 1) hex (9 13 14 10 1 5 6 2) (16 16 32) simpleGrading (1 1 1) hex (10 14 15 11 2 6 7 3) (16 16 32) simpleGrading (1 1 1) hex (11 15 12 8 3 7 4 0) (16 16 32) simpleGrading (1 1 1) ) ; edges ( arc 4 5 (0.707 −0.707 0) arc 5 6 ( −0.707 −0.707 0) arc 6 7 ( −0.707 0.707 0) arc 7 4 (0.707 0.707 0)

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Mesh generation

arc 12 13 (0.707 −0.707 4) arc 13 14 ( −0.707 −0.707 4) arc 14 15 ( −0.707 0.707 4) arc 15 12 (0.707 0.707 4) ) ; patches ( w a l l bound ( (0 1 2 3) (0 4 5 1) (1 5 6 2) (2 6 7 3) (3 7 4 0) (8 9 10 11) (8 12 13 9) (9 13 14 10) (10 14 15 11) (11 15 12 8) (4 5 13 12) (5 6 14 13)

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Mesh generation

(6 7 15 14) (7 4 12 15) ) ) ; mergePatchPairs ( ) ;

The result of mesh generation is visualized by a paraFoam utility subroutine as shown on Fig. 1.

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Mesh generation

Figure: A mesh for a cylinder constructed with use of blockMesh utility controlled by blockMeshDict file, see Listing 1. The cylinder is divided in four prism-like blocks and a central block, which is a rectangular parallelepiped with a 16 × 16 × 32 grading. Four side blocks are curvilinear hexahedrons with a similar grading.

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The heat equation

The heat equation

Presently, OpenFOAM itself does not contain subroutines for NLSE. However, a close analogue of NLSE available in OpenFOAM is the heat equation ∂T ∂t − D ∆T = 0 (2) where D is a coefficient of thermal conductivity and T is temperature. Here, we give an example of the use of laplacianFoam solver for this equation. The temperature at initial time t = 0 is given as T(x, y, z)(t=0) = exp

• − 1

16

• (x − 0.4)2 − y2 − z2

. (3) This function is calculated by a separate program. The results are written into the file T, see Listing 2. The file T is placed in the subfolder 0 which is designated for the storage of the initial data.

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The heat equation

Listing 2. Content of the file T with initial conditions data for the tempera-

• ture. 40954 numbers were skipped to save space. boundaryField section

at the end specifies zero gradient (Neumann) boundary conditions.

dimensions [0 0 0 1 0 0 0 ] ; i n t e r n a l F i e l d nonuniform Li s t <s c a l a r > 40960 ( 8.95663 e−22 9.02248 e−22 8.97853 e−22 . . . 0.319731 0.333965 0.340724 ) ; } } boundaryField {

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The heat equation

bound { type z e r o G r a d i e n t ; } }

Plot of the temperature at the initial time is shown in Figure 2.

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The heat equation

Figure: Gaussian distribution of temperature at initial time t = 0 given by equation (3). Blue color correspons to T = 0, and red to T = 1. The cylinder is sliced in order to show the interior.

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The heat equation

The solver laplacianFoam.C of this equation is given in Listing 3. Notice that “fvm::laplacian” means that the Laplacian is calculated implicitly as a matrix, and an implicit Euler method of integration is used. Listing 3. Solving the heat equation

#include ”fvCFD .H” i n t main ( i n t argc , char ∗ argv [ ] ) { # include ” setRootCase .H” # include ” createTime .H” # include ” createMesh .H” # include ” c r e a t e F i e l d s .H” Info < < ”\ n C a l c u l a t i n g temperature d i s t r i b u t i o n \n” << endl ; while ( runTime . loop ( ) ) { Info < < ”Time = ” << runTime . timeName () << n l << endl ; # include ” readSIMPLEControls .H” for ( i n t nonOrth =0; nonOrth<=nNonOrthCorr ; nonOrth++) { s o l v e

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The heat equation

( fvm : : ddt (T) − fvm : : l a p l a c i a n (DT, T) ) ; } # include ” w r i t e .H” Info < < ” ExecutionTime = ” << runTime . elapsedCpuTime () << << ” ClockTime = ” << runTime . elapsedClockTime () << << n l << endl ; } Info < < ”End\n” << endl ; return 0; }

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The heat equation

The time and write control input data are given in Listing 4. Note that time marches from t = 0 to t = 2 with a time step of ∆t = 0.001 and write time interval 0.005. Listing 4. Time and write control data organized in controlDict file. See http://www.openfoam.com/docs/user/controlDict.php for explana- tion of keyword entries.

a p p l i c a t i o n laplacianFoam ; startFrom startTime ; startTime 0; stopAt endTime ; endTime 2; deltaT 0 . 0 0 1 ; w r i t e C o n t r o l runTime ; w r i t e I n t e r v a l 0 . 0 0 5 ; purgeWrite 0; writeFormat a s c i i ; w r i t e P r e c i s i o n 6; writeCompression uncompressed ; timeFormat g e n e r a l ;

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The heat equation

t i m e P r e c i s i o n 6; runTimeModifiable yes ;

The input constant D is given by Listing 5, i.e. in this case D = 1. Numbers in square brackets specify dimensionality of the heat transfer constant, m2/sec. Listing 5. Transport properties organized in transportProperties file.

DT DT [ 0 2 −1 0 0 0 0 ] 1;

Numerical results are shown on Fig. 3.

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The heat equation

Figure: Results of calculations as visualized by paraFoam utility. Heat distribution at initial time t = 0 is shown at the upper left. Subsequent panels show calculated temperature distribution at t = 0.05, 0.1, and 0.25. Total computation time on a laptop is around one minute.

Click to start animation

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Modification for NLSE

Modification for NLSE

The heat equation numerical solver is now modified in order to allow a non-zero imaginary part. Real and imaginary parts of the wave function are stored as a three dimensional vector (Re Ψ, Im Ψ, 0) which is denoted as T, see Listing 6. The identity matrix I corresponds to multiplying by real one, and the antisymmetric matrix II by imaginary i. Notice that “fvc::laplacian” means that the Laplacian is calculated explicitly as a value, i.e., we use an explicit Euler method of integration. The reason of using the explicit method is our so-far inability to easily adjust an implicit method for the case of two coupled equations (for real and imaginary parts).

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Modification for NLSE

Listing 6. Modification of laplacianFoamM.C solver by replacing a scalar variable T by a vector variable T for storing both real and imag- inary parts of the function. The non-linearity is represented by cu- bic and quintic terms, ((CA*II+CA*EPSILON*I)&T)*magSqr(T) and ((CB*II-CB*MU*I)&T)*pow(magSqr(T),2).

#include ”fvCFD .H” i n t main ( i n t argc , char ∗ argv [ ] ) { # include ” setRootCase .H” # include ” createTime .H” # include ” createMesh .H” # include ” c r e a t e F i e l d s .H” Info < < ”\ nSolving non−l i n e a r Schrodinger equation i d p s i / dt − i DELTA p s i + (CA − i EPSILON) | p s i |ˆ2 p s i − (CB + i MU) t e n s o r I (1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 ) ; t e n s o r I I (0 , −1, 0 , 1 , 0 , 0 ,

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Modification for NLSE

0 , 0 , 0 ) ; while ( runTime . loop ( ) ) { Info < < ”Time = ” << runTime . timeName () << n l << endl ; # include ” readSIMPLEControls .H” for ( i n t nonOrth =0; nonOrth<=nNonOrthCorr ; nonOrth++) { s o l v e ( −fvm : : ddt (T) + (( C0∗ I I+C0∗BETA∗ I )& f v c : : l a p l a c i a n (T)) + DELTA∗T + ((CA∗ I I+CA∗EPSILON∗ I )&T)∗ magSqr (T) − ((CB∗ I I −CB∗MU∗ I )&T)∗pow( magSqr (T) ,2) ) ; } # include ” w r i t e .H” Info < < ” ExecutionTime = ” << runTime . elapsedCpuTime () << << ” ClockTime = ” << runTime . elapsedClockTime () << << n l << endl ; } Info < < ”End\n” << endl ;

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Modification for NLSE

return 0; }

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Modification for NLSE

We now introduce seven parameters C0, C1, C2, β, δ, ǫ, and µ in order to accommodate a more general equation i ∂Ψ ∂t +C0(1−iβ)∆Ψ−iδΨ+C1(1−iǫ)|Ψ|2Ψ−C2(1+iµ)|Ψ|4Ψ = 0, (4) If the equation has “Kerr-type” nonlinearity, i.e. F(|u|2) = g|u|2 in (1), then C0 = 1, C1 = g, and C2 = β = δ = ǫ = µ = 0.

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Courant number

Courant number

An explicit method can suffer from numerical instability in case when the Courant number C0∆t/(∆x)2 (where C0 is a coefficient in front of the Laplacian) is larger than one, where ∆x is grid spacing. Fig. 4 shows the

• ccurrence of numerical of instability that starts at spots where the grid

spacing is exceptionally small. Finally, the code gives ”floating point

• verflow” numerical error. Parameter of equation (1) is g = 0. The

cylindrical domain is the same as before, with radius R = 1, length L = 4, with the same mesh as in Section 4.

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Courant number

Figure: Numerical divergence caused by Courant number larger than 1. The subfigures show calculated absolute value of the wave function of equation (1). The time step is ∆t = 1.8 · 10−5. Numerical instability starts to develop at t = 0.002 in a region where the grid spacing is smallest. Then, it spreads to an entire region. Coefficient in front of Laplacian is set to C0 = 10, and the nonlinearity is set to zero.

Click to start animation

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Nonlinear dynamical phenomena

Here, we consider the NLSE with two non-linear, cubic and quintic, terms. The equation to solve is i ∂Ψ ∂t + ∆Ψ + C1|Ψ|2Ψ − C2|Ψ|4Ψ = 0, (5) with the Dirichlet boundary condition. If the constants C1 and C2 are positive, then the equation describes a self-guiding light beam organized in a solitonic structure [2, 3], with two transverse coordinates x, y and the third coordinate t in the direction of the beam propagation. There are four variables, x, y, z, and t. The cubic nonlinearity, |Ψ|2Ψ, induces an effect of self-focusing, while the quintic term |Ψ|4Ψ induces that of de-focusing. In order for a spatial soliton to become stable, the cubic and quintic non-linearities must have opposite signs. We illustrate this behavior by numerical simulations with various parameters C1 and C2.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Initial conditions, mesh, and time step

The equation is computed inside a ball of radius 20 with the center at the

• rigin. The mesh is obtained from a cubic grid 128 × 128 × 128 by

applying the OpenFOAM tool snappyHexMesh, see Figure 5.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Figure: A mesh for the ball obtained from a cubic 128 × 128 × 128 mesh after applying the OpenFOAM tool snappyHexMesh. The ball is cross-sectioned by a plane, to show the interior of the mesh.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

The initial condition is given as Ψ(t=0) = A(x, y, z) e− (x−x0)2+(y−y0)2+(z−z0)2

2R2

, (6) x0 = 0.1, y0 = 0.2, z0 = 0, R = 7, (7) and A(x, y, z) = 1 + 0.1 x + iy (x2 + y2)1/2 (8) where the factor A(x, y, z) is multiplied in order to break rotation symmetry. The time step is chosen as ∆t = 0.0025 that is around 50% of maximal possible time step for which Courant number allows convergence of the

• method. The function is computed up to tmax = 15, with 6000 time steps.

Execution time is about 12 hours.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

A linear Schr¨

• dinger equation

We first treat equation (5) for the case C1 = C2 = 0 (the linear case), i.e., i ∂Ψ

∂t + ∆Ψ = 0. As it is expected, the wave packet slowly disperses with

time, but remains Gaussian, see Figure 6.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Figure: Numerical solution of the linear Schr¨

• dinger equation i ∂Ψ

∂t + ∆Ψ = 0.

Density plot shows absolute value of the function. Blue color corresponds to |Ψ| = 0, and red color to maximal possible absolute value at t = 0. Wave packet slowly disperses with time.

Click to start animation

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

A self-focusing case

Equation (5) is computed for the case of C1 = 1 and C2 = 0, i.e. i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ = 0. The positive cubic non-linearity causes a

self-focusing effect. Singularity develops, and part of the wave function blows up, see Figure 7.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Figure: Numerical solution of the NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ = 0. There are

two plots combined together, a contour plot of |Ψ| (a ball), and a density plot on the sphere sectioned at x = 8. Wave packet shrinks with time and blows up.

Click to start animation

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Similarly, equation (5) is solved for the case of C1 = 0 and C2 = −1, i.e., i ∂Ψ

∂t + ∆Ψ + |Ψ|4Ψ = 0. Again, the quintic non-linearity causes

self-focusing effects with final blow up, see Figure 8.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Figure: Solution of the NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|4Ψ = 0. Wave packet shrinks

with time and finally blows up.

Click to start animation

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

A defocusing case

Defocusing occurs when the equation takes the form i ∂Ψ

∂t + ∆Ψ − |Ψ|2Ψ = 0. Figure 9 shows the case of C1 = −1 and C2 = 0,

and Figure 10 the case of C1 = 0 and C2 = −1. De-focusing causes the wave function to spread, and finally disperse over the volume.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Figure: Solution of the NLSE i ∂Ψ

∂t + ∆Ψ − |Ψ|2Ψ = 0. Wave packet blows up

and finally disperses over the entire domain of the ball.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Figure: Solution of the NLSE i ∂Ψ

∂t + ∆Ψ − |Ψ|4Ψ = 0. The non-linearity

causes the wave packet to defocus. Finally, it disperses over the entire domain of the ball.

Click to start animation

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Stable case

Stabilization occurs when cubic and quintic non-linearities have opposite

• signs. Figure 11 shows the case of C1 = C2 = 1, i.e., the NLSE

i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ − |Ψ|4Ψ = 0. The wave packet becomes more

compact with time, and finally stabilizes.

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Nonlinear dynamical phenomena Modeling equation Initial conditions, mesh, and time step A lin

Figure: Solution of the NLSE with combined cubic-quintic nonlinearities of

• pposite signs, i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ − |Ψ|4Ψ = 0 corresponding to a stable

• soliton. Wave packet originally excretes small fractions of the wave function,

and finally becomes stable. There are two plots combined together, a contour plot of |Ψ| (a ball), and a density plot on the sphere sec attioned x = 8.

Click to start animation

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Dynamics of the NLSE over a rectangular domain

Dynamics of the NLSE over a rectangular domain

Here, we consider the initial condition Ψ(t=0) = e− x2+y2

16

[1 + 0.01 cos(πz/5)] , (9) We consider (5) with C1 = C2 = 1. The rectangular parallelepiped is square-shaped in (x, y)-plane with size of the side of square a = 48, and has length L = 72 in z-direction. The results are shown in Fig. 12.

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Dynamics of the NLSE over a rectangular domain

Figure: Dynamic snapshots of the numerical solution of the NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ − |Ψ|4Ψ = 0 with a cylindrical-like initial condition given

by equation (9). The panels show calculated absolute value of the wave function with increasing values of time. The plot is a combination of a contour plot with sliced wave function in three planes given by equations x = 6, y = −24, and z = 36. The mesh grading is 96 × 96 × 144. The time step is ∆t = 0.02. Execution time to tmax = 300 (15000 time steps) equals to 55000 seconds.

Click to start animation

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Dynamics of the NLSE over a rectangular domain

The results on a domain with L = 144 are shown on Fig. 13.

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Dynamics of the NLSE over a rectangular domain

Figure: Visualization of solutions of the NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ − |Ψ|4Ψ = 0 with the initial condition given by (9). The

panels show calculated absolute value of the wave function cross sections of the with increasing with? time. The plot is a combination of a contour plot with sliced wave function in three planes given by planes x = 6, y = −24, and z = 72. The mesh grading is 96 × 96 × 288. The time step is ∆t = 0.02. Execution time to tmax = 160 (8000 time steps) equals 58000 second.

Click to start animation

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Dynamics of the NLSE over a rectangular domain

We also tried to solve the equation for smaller z-length of the region, in anticipation that for a sufficiently short cylinder an asymptotic pattern of the solution would emerge. However, in all considered cases we se no asymptotically stable pattern formations. The results for L = 36 are shown

• n Fig. 14, and for L = 18 are shown on Fig. 15.

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Dynamics of the NLSE over a rectangular domain

Figure: Visualization of solution of the NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ − |Ψ|4Ψ = 0

with the initial condition given by (9). The length of the cylinder is chosen shorter than on previous examples that are shown on Figures 12 and 13.

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Dynamics of the NLSE over a rectangular domain

Figure: Visualization of solution of the NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ − |Ψ|4Ψ = 0

with the initial condition given by (9). The length of the cylinder is chosen even shorter than the previous example in Figure 14.

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Dynamics of the NLSE over a rectangular domain

In addition, we tried to solve the same equation for a plane-like initial conditions Ψ(t=0) = e− z2

16 [1 + 0.01 sin(πx/5) + 0.01 sin(πy/4)] ,

(10) The results are shown on Fig. 16. For small time, the function almost does not have dependence on x and y coordinates, but finally it becomes essentially three-dimensional because of instability.

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Dynamics of the NLSE over a rectangular domain

Figure: Visualization of solution of the NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ − |Ψ|4Ψ = 0

with the initial condition given by (10). The size of the rectangular parallelepiped is 72 × 72 × 96. Mesh grading is 144 × 144 × 96.

Click to start animation

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Dynamics of the NLSE over a rectangular domain

Finally, we tried to solve a two-dimensional equation i ∂Ψ(x, y) ∂t + ∆Ψ(x, y) +

• |Ψ(x, y)|2

Ψ(x, y) = 0 (11) with the initial condition Ψ(t=0) = e− y2

R2 [1 + 0.001 sin(πx/3)] ,

(12) The results for the square size a = 24 and R = 4 are shown on Fig. 17.

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Dynamics of the NLSE over a rectangular domain

Figure: Solution of a two-dimensional NLSE i ∂Ψ

∂t + ∆Ψ + |Ψ|2Ψ = 0 on a

square 24 × 24 with the initial condition given by (12). Mesh grading is 512 × 512 × 1. Since a one-dimensional soliton does not collapse, the pattern stays stable in y-direction, until instabilities in x-direction break translational

• symmetry. Since in two dimensions solitons exhibit self-focusing and collapse,

the pattern finally diverges.

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A Ginzburg–Landau equation (a 2D case)

A Ginzburg–Landau equation (a 2D case)

The equation takes the form of (13): i ∂Ψ ∂t + (1 − iβ)∆Ψ − iδΨ + (1 − iǫ)|Ψ|2Ψ − (1 + iµ)|Ψ|4Ψ = 0, (13) with zero Neumann (zero flux) boundary conditions. The constants in equation (13) are as follows, β = 0.05, δ = −0.01, ǫ = 0.45, µ = −0.8. (14) The equation is solved in two dimensions inside a square of size 96 with the center at the origin. The mesh is 256 × 256 × 1.

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A Ginzburg–Landau equation (a 2D case)

The initial condition is given as Ψ(t=0) = A(x, y) A0e− (x−x0)2+(y−y0)2

2R2

, (15) A0 = 0.87, x0 = 4, y0 = 2, R = 4.03, (16) and A(x, y) = 1 + 0.1 x + iy (x2 + y2)1/2 (17) where the A(x, y, z) term is multiplied in order to break rotation symmetry. The time step is chosen as ∆t = 0.01 that is around 50% of maximal possible time step for which Courant number allows convergence of the

• method. The function was computed up to tmax = 1000.

Results of calculation are shown on Figure 18. With passing of time, the solution becomes elongated and starts rotating. The radial symmetry of the function breaks at around time t = 500.

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A Ginzburg–Landau equation (a 2D case)

Figure: Solution of Ginzburg–Landau equation given by equation (15) with initial condition given by equation (17) At small times, the solution is a Gaussian-like, with radial symmetry. Then, soliton elongates in one direction and starts rotating.

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Bibliography

• C. Sulem and P.-L. Sulem. The NLSE. Self-focusing and wave
• collapse. Springer, 1999.
• V. Skarka, D. V. Timotijevi´

c, and N. B. Aleksi´

• c. Journal of Optics A:

Pure and Applied Optics, vol. 10, 075102, 2008.

• N. B. Aleksi´

c, V. Skarka, D. V. Timotijevi´ c, and D. Gauthier. Phys.

• Rev. A, vol. 75, 061802(R), 2007.

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