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."