zhejiang normal university

Zhejiang Normal University

Split-Step Fourier

Transform Method

李画眉2009 年 11 月

Outline

一、 Algorithms 二、 Stationary solution for 1+1D N

LSE三、 Propagation for 1+1D NLSE四、 Stability Analysis五、 Conclusions

　　　

Algorithms ： Image-time method

By making use of a substitution t-it, then one can employ other appropriate algorithms (FFT, difference scheme, etc.) to obtain stationary solution.

2

2

2

10

2

q qi q q pR x qz x

z iz

2

2

2

10

2

q qq q pR x q

z x

Advantage: independence of initial guess value

Disadvantage: only ground state can be obtained

( ) cos( )R x x

Stationary solution for 1+1D NLSE

22

2

1( )

2

q qpR x q q q

z x

FFT

Linear part2

2

1

2

q q

z x

Nonlinear part2

( )q

pR x q q qz

z2

hz z h

linear linearnonlinear

Programclear allp=1;omega=2;%%-------------------n=2048;hx=0.06;x=(-n/2:n/2-1)*hx;hw=2*pi/(n*hx);w=fftshift((-n/2:n/2-1)*hw);%%-------------------% q=exp(-1*(x).^2/2);q=sech(x);intensity=4.6;u1(:,1)=(abs(q).^2)';%-------------------V=cos(omega*x);

Program%--------------------L=50;nm=5000;h=L/nm;%-------------------for j=1:nm j D=exp(((i*w).^2/2)*h/2); qstep1=ifft(D.*fft(q)); N=exp((p*V+(abs(qstep1)).^2)*h); qstep2=N.*qstep1; q=ifft(D.*fft(qstep2)); q=sqrt(intensity)*q/sqrt(sum(abs(q).^2)*hx); u=abs(q); r=floor(2+(j-1)/50); u1(:,r)=u';end

Program

kin=-sum((q(3:end)-q(1:end-2)).^2)/4/hx;p_i=sum(2*q.^2.*(abs(q).^2+V))*hx;b=(kin+p_i)/2/intensity z=0:h*50:50;figure(1)mesh(x,z,u1');view(0,90)figure(2)plot(x,u1(:,end),'r',x,V,'b')

Energy_b

22 42 ( ) 2

2

dwpR x w w dx

dxb

U

( , ) ( ) exp( )q x z w x ibz2

32

1( )

2

d wbw pR x w w

dx

2U w dx

Energy_b

Programclear allp=1;omega=2;%%-------------------n=2048;hx=0.06;x=(-n/2:n/2-1)*hx;hw=2*pi/(n*hx);w=fftshift((-n/2:n/2-1)*hw);%%-------------------loop=0;

Programfor intensity=1:0.1:5 % q=exp(-1*(x).^2/2); q=sech(x); u1(:,1)=(abs(q).^2)'; %------------------- V=cos(omega*x); %-------------------- L=50; nm=1000; h=L/nm; loop=loop+1 %-------------------

Program

for j=1:nm j; D=exp(((i*w).^2/2)*h/2); qstep1=ifft(D.*fft(q)); N=exp((p*V+(abs(qstep1)).^2)*h); qstep2=N.*qstep1; q=ifft(D.*fft(qstep2)); q=sqrt(intensity)*q/sqrt(sum(abs(q).^2)*hx); end kin=-sum((q(3:end)-q(1:end-2)).^2)/4/hx; p_i=sum(2*q.^2.*(q.^2+p*V))*hx; b(loop)=(kin+p_i)/2/intensityendintensity=1:0.1:5plot(b,intensity)

Numerical results 1, 2p

Iterative process Stationary solution Energy_b

Propagation for 1+1D NLSE NLSE

2

2

2

10

2

q qi q q pR x qz x

Programclear allp=1;omega=2;%%-------------------n=2048;hx=0.06;x=(-n/2:n/2-1)*hx;hw=2*pi/(n*hx);w=fftshift((-n/2:n/2-1)*hw);%%-------------------% q=exp(-1*(x).^2/2);q=sech(x);intensity=1;%-------------------V=cos(omega*x);

Program%%%%%%%%%%%%%%%%L=50;nm=5000;h=L/nm;u1(:,1)=(abs(q).^2)';%-------------------for j=1:nm j D=exp((i*(i*w).^2/2)*h/2); qstep1=ifft(D.*fft(q)); N=exp((i*p*V+i*(abs(qstep1)).^2)*h); qstep2=N.*qstep1; q=ifft(D.*fft(qstep2)); u=abs(q).^2; r=floor(2+(j-1)/50); u1(:,r)=u';end

Program

z=0:50*h:L;mesh(x,z,u1');view(0,90)


Propagation

Stability Analysis for 1+1D NLSE

qqqxpx

q

z

qi 2

2

2

)cos(2

1

Assuming the stationary solution is of the form

exp( )q w x ibz

23

2

1cos( )

2

d wbw p x w w

dx

We consider the stability of the stationary solution w(x) by employing the following algorithms:

Split-step Fourier method

Eigenvalue method

Eigenvalue method

where the perturbation components u, v could grow with a complex rate λ during propagation. Substitute the perturbed solution into equation, we obtained the linear eigen-equations

22

2

22

2

12 cos

2

12 cos

2

d uu w u v p x u bu

dx

d vv w v u p x v bv

dx

The solution is stable if the imaginary parts of λ equal zero.

In eigenvalue method, there are two assumptions for the perturbed stationary solution

*( ) ( )( , ) ( ) ( ) ( )ibz i b z i b zq x z w x e u x e v x e

First assumption: Physica D 237, 3252 (2008)

We begin by discretizing the domain x [-∈ a,a] by placing a grid over the domain. We will use the grid with grid spacing h=2a/N in axis x. We will attempt to approximate the solution at the points on this lattice, and define uk and vk to be functions defined at the point xk≡-a+(k-1)h or the lattice point k, where k=1, 2, ⋯, N+1. Thus we can obtain the difference scheme as follows

1 1

1 1

k k k k k k k

k k k k k k k

u u u v u

v v u v v

with 2 2

2 2

1 1, 2 cos ,

2 k k k k kw p x b wh h

Espically 2 2 2 2 2 3

2 2 2 2 2 3

u u v u

v u v v

and 1

1

N N N N N N

N N N N N N

u u u v

v v u v

due to 1 1 1 10, 0N Nu v u v

namely

The soliton is stable if the imaginary parts λ of equal zero.

2 22 2

1 1, 2 cos ,

2 k k k k kw p x b wh h

where

stability_1D_eigenvalue_1.m

Second assumption: PRL93, 153903 (2004)

( , ) ( ) ( , ) ( , )ibz ibz ibzq x z w x e U x z e iV x z e

with the complex rate δ:

( , ) Re[ ( ) ], ( , ) Re[ ( ) ]z zU x z u x e V x z v x e

After substituting this perturbed solution into equation and linearizing, we obtain a linear eigenvalue problem as follows

22

2

22

2

1cos 3

2

1cos

2

d uv p x u bu w u

dx

d vu p x v bv w v

dx

The soliton is stable if the real parts of δ equal zero.

Discretizing

21 12

21 12

2cos 3

22

cos2

k k kk k k k k k

k k kk k k k k k

u u uv p x u bu w u

hv v v

u p x v bv w vh

Namely

22

1, 2 cos ,

2 k k k kp p x b wh

2 2 2

3 3

1 4 4

1

1 1

2

1

1

0 0 0 0 0 0 3 0 0 0 0

0 0 0 0 0 0 3 0 0 0

0 0 0 0 0 0 0 3 0 0 0

0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 3

0 0 0 0 0 0 0

k

k

k

N N

N

k

k

k

N

v p

p

v p

v

v

p

v

u

u

u

u

u

2

1

2 2

3 3

4 4

1 1

0 0 0 3

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

k

k

k

N N

N N

N N

v

v

v

v

p

p

p

p

p

p

1

2

1

1

N

k

k

k

N

v

u

u

u

u

u

where

stability_1D_eigenvalue_2.m

Split-step Fourier method

qqqxpx

q

z

qi 2

2

2

)cos(2

1

By denoting

2

2 2 *2

1cos 2 0

2

U Ui p x b w U w Uz x

Starting from a white-noise initial condition, we simulate above linear equation for a long distance (hundreds of z units).

( , ) ( ) ( , )ibzq x z e w x U x z

where w(x) is the fundamental soliton and U(x,z)<<1 is the infinitesimal perturbation, the linearized equation for U(x,z) is

JOSAB21, 973 (2004)

z2

hz z h

FFT FFTRK method

2

2 2 *2

1cos 2 0

2

U Ui p x b w U w Uz x

Note

2

2

1

2

U Ui

z x

2 2 *cos 2U

i p x b w U iw Uz

FFT

RK method

Programclear allticp=1;omega=2;%%-------------------n=1024;hx=0.08;x=(-n/2:n/2-1)*hx;hw=2*pi/(n*hx);w=fftshift((-n/2:n/2-1)*hw);%%-------------------q=exp(-1*(x).^2/2);q=sech(x);intensity=4.6;u1(:,1)=(abs(q).^2)';%-------------------V=cos(omega*x);

Program%--------------------L=50;nm=5000;h=L/nm;%-------------------for j=1:nm j D=exp(((i*w).^2/2)*h/2); qstep1=ifft(D.*fft(q)); N=exp((p*V+(abs(qstep1)).^2)*h); qstep2=N.*qstep1; q=ifft(D.*fft(qstep2)); q=sqrt(intensity)*q/sqrt(sum(abs(q).^2)*hx); u=abs(q); r=floor(2+(j-1)/50); u1(:,r)=u';end

Program

kin=-sum((q(3:end)-q(1:end-2)).^2)/4/hx;p_i=sum(2*q.^2.*(abs(q).^2+V))*hx;b=(kin+p_i)/2/intensity%%--------------------------alpha=1/(2*hx^2);w=u1(:,end)';beta=2*w.^2+p*cos(omega*x)-b-1/hx^2;kai=w.^2;growth_rate1=stability_1D_eigenvalue_1(alpha,beta,kai,w,n)%%-------------------% alpha=1/(2*h^2);% w=phi;p=2*alpha-p*cos(omega*x)+b;% kai=w.^2;growth_rate2=stability_1D_eigenvalue_2(alpha,p,kai,w,n)toc

Subprogram1

function growth_rate=stability_1D_eigenvalue_1(alpha,beta,kai,phi,N)A=[beta(2) kai(2);-kai(2) -beta(2)];B=[alpha 0;0 -alpha];for kk=3:N A=[A [zeros(2*(kk-3),2);B];zeros(2,2*(kk-3)) B [beta(kk) kai(kk);-kai(kk) -beta(kk)]];endgrowth_rate=max(abs(imag(eig(A))));

Subprogram2

function growth_rate=stability_1D_eigenvalue_2(alpha,p,kai,phi,N)for mm=1:N-2 AA2(mm,mm)=-p(mm+1)+3*kai(mm+1); AA2(mm,mm+1)=alpha; AA2(mm+1,mm)=alpha; BB2(mm,mm)=p(mm+1)-kai(mm+1); BB2(mm,mm+1)=-alpha; BB2(mm+1,mm)=-alpha;endAA2(N-1,N-1)=-p(N)+3*kai(N);BB2(N-1,N-1)=p(N)-kai(N);CC2=[zeros(N-1,N-1) AA2;BB2 zeros(N-1,N-1)];delta2=eig(CC2);NN2=length(delta2);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%growth_rate=max(abs(real(delta2)));


stability_eigen_newton_1D_LU_growth.m

Conclusions

As far as ground state is concerned, image-time method is a most effective algorithm to deal with it due to insignificance of initial guess value. It is worth noting that energy conservation law must be met when image-time method is employed.

Split-step Fourior algorithm is a feasible method for stability analysis.

谢谢大家！谢谢大家！

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