  Bilinear equation is an important property for integrable nonlinear evolution
equation. Many famous research objects in mathematical physics, such as
Gromov-Witten invariants, can be described in terms of bilinear equations to
show their connections with the integrable systems. Here in this paper, we
mainly discuss the bilinear equations of the transformed tau functions under
the successive applications of the Darboux transformations for the KP
hierarchy, the modified KP hierarchy (Kupershmidt-Kiso version) and the BKP
hierarchy, by the method of the Boson-Fermion correspondence. The Darboux
transformations are considered in the Fermionic picture, by multiplying the
different Fermionic fields on the tau functions. Here the Fermionic fields are
corresponding to the (adjoint) eigenfunctions, whose changes under the Darboux
transformations are showed to be the ones of the squared eigenfunction
potentials in the Bosonic picture, used in the spectral representations of the
(adjoint) eigenfunctions. Then the successive applications of the Darboux
transformations are given in the Fermionic picture. Based upon this, some new
bilinear equations in the Darboux chain are derived, besides the ones of
$(l-l')$ -th modified KP hierarchy. The corresponding examples of these new
bilinear equations are given.
