Plasma modes analysis with Laplace and Fourier methods I've seen that in order to investigate plasma's modes it's possible to linearize the system and then Fourier transform it (for longitudinal modes with kinetic description the Laplace transform in time is also used).
After having transformed the system the dispersion relation is derived:
$\omega=\omega(\vec k)$ for Fourier transformed systems
$z=z(\vec k)$ where $z$ is the complex frequency for Laplace-Fourier transformed systems
As far as I understand, if the unknown functions are "monochromatics" $e^{i(\vec k \cdot \vec x-\omega t)}$ where $\vec k$ and $\omega$ satisfies the dispersion relation (and analogous for Laplace-Fourier case)  they are solutions of the system of equations (and for what I've understood these solutions are called the modes of the plasma).
My question is how do we know that those monochromatic are solutions of the system of equations?
I suspect that the answer may be related to the fact the the starting system of equations becomes algebraic after the Fourier transformations but this is just a guess.
It's also possible that I didn't understand the main idea and that they aren't solutions of the system, however, in this case I don't understand the utility of this Fourier analysis.
If you think my question is not clear enough I'll be glad to update it and explain my doubt at better.
 A: 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.
