I am in trouble to understand the "spirit" of the following discussion that my teacher wrote on his notes.
Consider $$\ddot q_k=-\omega_0^2q_k-a(2q_k-q_{k+1}-q_{k-1})\qquad\qquad\qquad (1)$$ with $k\in \mathbb{Z}$. My teacher said that to find the solution of this equation one can proceed to find the normal modes and then write the general solution by superposition.
So he became to consider the function $q_k(t)= e^{i\omega t}e^{isk}$ with $s \in \mathbb{R}$, where $e^{isk}$ is the spatial term and $e^{i\omega t}$ is the oscillatory term.
Then he put the function $q_k(t)$ into the $(1)$ to have a relation that $\omega$ must satisfy (skipping the calculation): $$\omega=\omega(s)=\pm \sqrt{\omega_0^2+4asin^2 \frac{s}{2}}$$ So for any value of $s$ one have two solution $$e^{i\omega(s) t}e^{isk}\qquad\qquad e^{-i\omega(s) t}e^{isk}.$$
Now to build a general solution one can write a superposition of the "waves" found: $$q_k(t) = \int_{-\pi}^{\pi}a(s)e^{i(\omega(s) t+sk)}+b(s)e^{i(-\omega(s)t+sk)}ds.$$
So ( outside all the question under the use of certain type of language like waves, normal mode that I think are linked to the fact that this is some kind of generalization of waves equation - if someone want to say something about this, I will happy - ) my really question is:
What is the spirit under the fact the constructed $q_k(t)$ is a solution of (1)? Why does that integral appear?
The equation is linear: linear combinations of solutions are still solutions.
If you have say finitely many solutions, then you can obtain further solutions by taking a linear combination of these (that is, a finite linear combination).
But in your case you have infinitely many solutions, and so the next step can be to consider the integral of a continuum of solutions since it is a generalization of the notion of linear combination now with the coefficients $a(s)$ and $b(s)$.
Indeed, the integral that you wrote is again a solution of the original equation.