Understanding Fourier Equations in PDE for Beginners

[hunter151]

So I suppose my Fourier knowledge is a little bit rusty. Any help would be greatly appreciated.

http://pmgz.net/3259.jpg

How do they get from the original DKS equation to the Fourier space DKS equation (from eq 1 to eq 2)?

Thanks greatly for any help.

 

[masterfox100]

Just out of interest which DKS equation does this refer to ?

 

[hunter151]

This is the damped kuramoto-sivashinsky equation.