Can I solve the discreate ODE by considering the continuous case?

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There is a linear version of so-called lattice Schrodinger equation (LSE), it is just a variation form of nonlinear Schrodinger equation. But the LSE is the discrete case on N lattices. I wonder if I can solve the continuous case and then take the solution at specific lattice for the discrete case?
 
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Well first of all I'd like to see the equation.
What is discrete in the equation, space? time? both?
Does it involve an approximation of the derivative or is it a "standart" difference 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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