  An asymmetric pair of coupled nonlinear Schr{\"o}dinger (CNLS) equations has
been derived through a multiscale perturbation method applied to a plasma fluid
model, in which two wavepackets of distinct carrier wavenumbers and amplitudes
are allowed to co-propagate and interact. The original fluid model was set up
for a non-magnetized plasma consisting of cold inertial ions evolving against a
$\kappa-$distributed electron background in 1D. The reduction procedure
resulting in the CNLS equations has provided analytical expressions for the
dispersion, self-modulation and cross-coupling coefficients in terms of the
carrier wavenumbers.
  The system admits various types of vector solitons (VSs), physically
representing nonlinear localized electrostatic plasma modes. The possibility
for either bright (B) or dark (D) type excitations for either of the two waves
provides four combinations for the envelope pair (BB, BD, DB, DD). Moreover,
the soliton parameters are also calculated for each type of VS in its
respective area of existence. The dependence of the VS characteristics on the
carrier wavenumbers and the spectral index $\kappa$ has been explored. In
certain cases, the amplitude of one component may exceed its counterpart
(second amplitude) by a factor 2.5 or higher, indicating that extremely
asymmetric waves may be formed due to modulational interactions among the
wavepackets.
  As $\kappa$ decreases from large values, modulational instability (MI) occurs
in larger areas of the parameter plane(s) and with higher growth rates. The
distribution of different types of VSs on the parameter plane(s) also varies
significantly with decreasing $\kappa$, and in fact dramatically for $\kappa$
between $3$ and $2$. Deviation from the Maxwell-Boltzmann picture therefore
seems to favor MI as a precursor to the formation of bright (predominantly)
type envelope excitations and freak waves.
