For example, the operator $\ 2uu' = (u^2)'$ Can this be approximated by $\ (2u)' = 2u'$ for functions $\ u$ close to zero?
For more complicated nonlinear operators, is there always such a linear approximation expressible as a linear function of $\ u, u', u'', ...$ ?
My hope is to use this to find dispersion relations for nonlinear differential equations of the form $\ u_t=f(u,u_x,u_{xx},...)$.