The dispersion relation for a plasma is often derived from the dielectric tensor of the system (e.g., see https://physics.stackexchange.com/a/264526/59023 for cold plasma example and see https://physics.stackexchange.com/a/138460/59023 for more general derivation of the dielectric tensor), which depends upon the kinetic properties of the plasma (i.e., the details of the velocity distribution function (VDF)). Sometimes one can derive a dispersion relation from the equations of motion, e.g., see https://physics.stackexchange.com/a/350655/59023 for derivation of the MHD fast/magnetosonic wave dispersion relation.
My question is how do we know that those monochromatic are solutions of the system of equations?
This is an over simplification. The dispersion relation defines the wave frequency as a function of the wave vector (or wavenumber, depending on the mode). There is no strict requirement that anything be monochromatic. The solutions are often for a single frequency at a single wave vector, but how these are chosen/found often result from an idealized set of assumptions prior to deriving the dielectric tensor.
For instance, some numerical solvers will often search through the $\omega$-$\mathbf{k}$-$\gamma$ space for the mode with the largest growth rate, $\gamma$, for a given set of particle VDFs. The solution is only for the peak growth rate, but that does not mean that this peak has zero width.
It is a common misconception that waves satisfying a dispersion relation must be monochromatic, but a simple example will illustrate why this is incorrect. Take, for example, acoustic waves. In a time series plot of amplitude the waveform can appear to be a modulated sine wave (e.g., look at the ion acoustic wave example in the following paper https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.99.041101 [it's Open Access so no pay wall]). However, if you look at the frequency power spectrum, the wave is clearly not monochromatic. Yet the mode can also clearly obey the linear acoustic dispersion relation.
A good book to read to learn more about waves is Whitham [1999]. It doesn't focus specifically on plasmas but gives a very generalized discussion of linear and nonlinear waves while also providing physically intuitive explanations of these phenomena.
References
Whitham, G. B. (1999), Linear and Nonlinear Waves, New York, NY: John Wiley & Sons, Inc.; ISBN:0-471-35942-4.
So, is collective motion due to these electromagnetic forces originated from the "small charge imbalance"? Or is it a different mechanism, for example the result of many short-range interactions?
The electrostatic shielding within a Debye sphere results in the plasma behaving like a fluid, i.e., the equivalent of a fluid parcel would be something like a Debye sphere. It is collective because an external electromagnetic force acts on all of the charged particles. Yes, you can apply an electric field externally with a scale length larger than the Debye length. The Debye length is the characteristic scale over which positive and negative charges are balanced. The Debye length does not represent the longest scale length of any electric fields within the plasma. For instance, all electromagnetic waves (of which I am aware) have wavelengths longer than one Debye length (e.g., Langmuir waves have wavelengths on the order of the electron skin depth, which is often much much larger than the Debye length in an over dense plasma).
Best Answer
Long-range is referring to the Coulomb interaction between free-charged particles within a Debye sphere. A Debye sphere may seem small to you, but compared to atomic interactions or neutral-neutral interactions, the distances can be huge.
The collective behavior of a plasma results from the accumulated effects of all these Coulomb potentials. Electric fields do work to get rid of themselves, thus how Debye spheres form. Because of the long-range interactions of the electric fields, you cannot "push" one particle without it affecting all the others within a given Debye sphere. Thus, the system begins to behave somewhat like a fluid.
Note that most plasmas, by volume, in the universe are in the range of weakly collisional to collisionless.
Yes, they are over distances longer than roughly one Debye length. That does not impede collective behavior. Take the water in a glass of water, for example. Any given particle's immediate sphere of influence is defined by its collisional mean free path. In this case we are talking about length scales on the order of sub-microns (it's actually much much smaller than a micron). Unlike a plasma the molecules in water do not care about the collective influence of a surrounding sphere of water molecules, they only care about the next water molecule with which they collide.
Note that when the collide, they don't actually touch. Neutral particle collisions are also mediated by the electric fields from Coulomb potentials but they don't see these fields until the inter-nuclei separation is on the same order of magnitude as the collisional cross-section of interaction. For water-water molecule collisions, the cross-section is on the order of ~10-15 cm2.
In fact, if you are worried about shielding, you can actually think of a monatomic atom as a super simple analogy to a plasma. The nucleus is positively charged while the electron cloud is negatively charged. However, outside the electron cloud, the total electric field from the atom asymptotes to zero rather quickly.