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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 $\ u_t=f(u,u_x,u_{xx},...)$.

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    There is a common technique of [linearization around an equilibrium](http://www.sosmath.com/diffeq/system/nonlinear/linearization/linearization.html). But I don't know how linearization can help with dispersion which is a nonlinear phenomenon afaik.2012-07-11

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