## Help in Split step fourier method

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 seperate 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)