In this paper, we prove a Birkhoff normal form result for abcd Boussinesq system on the circle with $b = d > 0; a; c < 0$. The result is based on the fact that the nonlinear term has a so-called tame modulus and the frequencies satisfy nonresonant condition.
As a consequence of the Birkhoff normal form, we have that any small amplitude solution remains close to a torus for a long time.
As a consequence of the Birkhoff normal form, we have that any small amplitude solution remains close to a torus for a long time.