It depends on what non-linearity is, i.e. what is its mathematical form. In any case you can calculate a so called fixed point od the equation, for which the dependant variable is a constant, and then you can linearize the equation around this point. The result will be a linear pde equation, which you can solve. The solution would represent vibrational modes and frequencies in close vicinity of a particular solution, i.e. constant solution.  There is no general procedure to solve non-linear wave equation, only a few examples of true non-linear waves are known.

Thank you dear Vladimir Dananic for your thorough response

so, in the nonlinear case, the meaning of nonlinear vibration is the dynamical response of the system, am I right?

Yes, you are right.

Thank you so much