One of my favorite perspectives on the difficulty of formulating a general theory of PDE in light of the difficulties posed by nonlinearities is Sergiu Klainerman’s “PDE as a unified subject” https://web.math.princeton.edu/~seri/homepage/papers/telaviv.... If I understand correctly, any general theory of PDE would have to incorporate all the subtle behaviors of nonlinear equations such as turbulence (which has thus far evaded a unified description). Indeed, ”solvable” nonlinear systems in physics are so special Wikipedia has a list of them https://en.m.wikipedia.org/wiki/Integrable_system. With this perspective, I’m tempted to say (in a non-precise manner) that solvable systems are the vanishingly small exception to the rule in a frighteningly deep sea of unsolvable equations.