Understanding Fourier Equations in PDE for Beginners

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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.
 
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Just out of interest which DKS equation does this refer to ?
 
This is the damped kuramoto-sivashinsky equation.
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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