TITLE: How to tell, roughly, which PDE's are interesting to analyse?
QUESTION [7 upvotes]: How can one tell which PDE's are, roughly speaking, perhaps more interesting to analyse? 
Physical motivation is one reason. For example, the KdV $u_t+u_{xxx} - 6uu_x=0$ for a function $u:\mathbb R\times\mathbb R\to\mathbb R$ is a model for many physical systems, such as the propagation of shallow water waves along water bodies with low height. It is also a completely integrable PDE, which makes its analysis more tempting. But putting aside complete integrability (many studied PDE do not have completely integrable variants) why not study the derivative quasilinear equation $u_t + u_{xxx} - 6uu_{xxx}$ or the another derivative semilinear equation $u_t + u_{xxx} - 6uu_{xx}=0$ (I am only putting the KdV here as an example, and am not really asking this particular question for the KdV)?
Aside from physical motivation, how can one pick PDE to analyse?
Edit: Just to be clear, I am asking this from the perspective of PDE research. When I say "analyse," I mean "do PDE research on."

REPLY [6 votes]: My impression is that a PDE, or rather a class of PDEs, is interesting to the analysts when it is relevant for some analytical tool. Let me take a few examples. The list is not exhaustive.

linear constant coefficient PDEs in the whole space ${\mathbb R}^n$ are treated with Fourier analysis.
Elliptic (scalar) PDEs are analyzed with the maximum principle. This is particularly true in the modern treatment.
Linear and semi-linear evolution PDEs are the realm of semi-group theory.
Dispersive linear PDEs obey to Strichartz inequalities.
So-called integrable equations or systems are treated by Lax pairs and spectral theory.
Othere examples in homogenization or in hyperbolic conservation laws are solved by using compensated compactness.