thanks for your answer.
1.&3. uu0 and beta are assumed to be of the same order and small.
2. Indeed, the first order derivatives of uu0 and beta are not necessarily small, therefore the must be retained. Higher order derivatives, however, are neglected (with one exception, see below).
The file attached shows the equations and my linearisation procedure, which is ok-ish for this simple problem but more "advanced" equations are much longer and therefore can't be dealt with this way. At the end you can find the desired solution (the mixed derivative of third order as shown here is not small, but this should not matter at this stage).
I guess what is directly associated to my question is clear if you have a look at the attached file at "#manual linearisation", e.g. line 26: subs(-2*d*diff(uu, x,t)*beta*diff(beta,x)=0 : Why does the substitution fail if I omit e.g. the "-2". It should substitute the derivative expression with zero which makes the entire summand zero.
The question of how to get rid of the higher order terms and derivatives is my primary concern.
Thanks again for any help and let me know if anything is unclear.