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These are replies submitted by arashghgood

@Rouben Rostamian  Thank you so much. It is very helpful.

@Scot Gould first of all thank you for your comment. Actually, the whole story is that there is a non-linear PDE in real space. I transformeed it in Fourier space. as a result, I have u(k,t) in Fourier space and the PDE is

$$\hat{\u_{tt}}+i*\gamma\hat{\u_t}-\omega_0^2\hat{\u_}+\frac{3}{4}k \omega_0 \hat{\u_}*\hat{\u_}= 0$$

where \hat{\u_}*\hat{\u_} means convolution. assume that the shape of the u(x,t) has at t=0 is sech(x)^2. I want to solve this equation to in fourier space and then transform the result to real space to see the evolution of u(x,t) over time. above, I just wrote the linear part of the equation. ( Actually I have no idea how to wrote the last term :). i would appreciate if you can help)
in this case the term (i*\hat{\u_t}) in Fourier space is corresponding to u_{x,t} (I mean diff(u(x,t),x,t)