  In this paper, the nonlinear theory of plasma waves is extended to the
plasmas that their equilibrium state are specified by the non-Maxwellian (here
kappa) distribution. We believe that the extension is very important since most
of the space and some of laboratory plasmas are not in the Maxwellian
equilibrium. Although the linear theory of this issue has been known for the
decades but, to our knowledge, this is the first attempt in opening the gate to
the nonlinear world of plasma waves with the non-Maxwellian equilibrium. As an
example the ion-acoustic solitons are studied in this framework taking into
account the electron trapping in the trough of longitudinal field. It is shown,
as the most important result regarding to the non-Maxwillian equilibrium, that
there is the possibility for the ion-acoustic solitons to move subsonically.
The solitons velocity and their width are monotonically increasing functions of
spectral index ($\kappa$) and approach to the their Maxwellian values as
$\kappa \to \infty$.
