I think that many nonlinear PDE have the form (I use d/dt for partial derivative)
du/dt = F(d/dx, u),
where F is a nonlinear operator that can include partial derivatives w.r.t. x and multiplication by function of u. Very often instead of F there is equation
du/dt = A u,
where A is nonlinear differential operator. While there are many possible types of PDE, interesting physical applications concentrate over the first derivative over time. Here we have KdV equation, nonlinear Schrodinger, Sin-Gordon and others.