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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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Are you asking whether you can say $2 u u' \approx 2u'$ if $u\approx 0$? The answer would be then "of course not". For the approximation to stand any chance of being true you may want $u\approx 1$? –  Willie Wong Jul 10 '12 at 11:11
    
There is a common technique of linearization around an equilibrium. But I don't know how linearization can help with dispersion which is a nonlinear phenomenon afaik. –  user31373 Jul 11 '12 at 3:07

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