Help in Split step fourier method

In summary: P and U. The first equation has second order derivatives, which is different from the examples I have seen. I attempt to separate the linear and nonlinear parts of the equation, but face difficulties with the Fourier transform of the linear part due to the second order derivative. I am seeking help from someone who is familiar with the SSFM method to modify the Fourier transform for the linear part in this case. In summary, the conversation discussed the split step Fourier method (SSFM) and its application to a set of 2 equations involving functions P and U. The first equation has second order derivatives, which poses a challenge in applying the SSFM method. The speaker seeks help in modifying the Fourier transform
Hello all,

I have short question at the end , but i wil gve short background.

The subject is regarding the split step Fourier method (SSFM) adn i will be gratefull if someone who know the method can help.

i have the set of 2 equations:

Utt=Uzz-a*U+i*P-(P^2)*U

Pt=-i*P-(P^2)*U

where: P, U are the function need to be find P(z,t) U(z,t)

i = sqrt(-1) a=constant

I try to do the split step Fourier method (SSFM) on this set, the problem is that the first equation derivative are from order 2, and in all the examples i saw it was from order 1.

I try to separate to linear part and nonlinear part:

for the first equation:

U_linear_tt=Uzz-a*U

U_nonlinear_tt= i*P-(P^2)*U

and for the second equation:

P_linear_t=-i*P

P_nonlinear_t=-(P^2)*U

now when i try to do the Fourier transform at the linear part, for the U function, it is problem because the derivative is second order.

if it was from first order the Fourier transform for the linear part becomes:

Fourier_U_linear(t+dt,z)=exp{(-w^2-a)*dt}

how do i modified it on my case when the derivative is second order ?

thanks

I try to do the split step Fourier method (SSFM)

1. What is the Split Step Fourier Method?

The Split Step Fourier Method is a numerical technique used to solve the time-dependent Schrodinger equation in quantum mechanics. It involves splitting the wavefunction into two parts, propagating each part through a potential, and then combining them again to obtain the final solution.

2. Why is the Split Step Fourier Method useful?

This method is useful because it allows for efficient and accurate calculations of quantum systems, particularly those with time-varying potentials. It is also relatively easy to implement and can handle complex systems with multiple dimensions.

3. How does the Split Step Fourier Method work?

The method works by first transforming the wavefunction into the frequency domain using a Fourier transform. Then, the potential is applied to the wavefunction in this frequency domain. The wavefunction is then transformed back into the time domain, and the process is repeated until the desired time evolution is reached.

4. What are the limitations of the Split Step Fourier Method?

One limitation of this method is that it assumes the potential is time-independent during each propagation step. This may not be accurate for systems with rapidly changing potentials. Additionally, the method can become computationally expensive for systems with a large number of dimensions.

5. Are there any applications of the Split Step Fourier Method?

Yes, this method is commonly used in quantum mechanics for studying a variety of systems, such as atoms, molecules, and solids. It has also been applied in other fields, such as signal processing and optics, for solving differential equations with time-varying coefficients.

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