I studied nonlinear PDEs in grad school. My advisor, Ralph Showalter, would remind us occasionally what ‘nonlinear’ means.
“Nonlinear” is not a hypothesis but the lack of a hypothesis.
He meant a couple things by this. First, when people say “nonlinear,” they often mean “not necessarily linear.” That is, they use “nonlinear” as a generalization of linear. If a statement doesn’t hold for linear equations, it can’t hold more generally. So try the linear case first.
Second, and more importantly, you usually have to specify in what way an equation is nonlinear before you can say anything useful. If you’re not assuming linearity, what are you assuming? Maybe you need to assume a function is convex. Or maybe you need to assume an upper or lower bound on a function’s growth. In any case, focus on what you are assuming rather than what you are not assuming, and make your assumptions explicit.
Related post: Three views of differential equations