  We study the dynamics of multi-component Bose gas described by the Vector
Nonlinear Schr\"{o}dinger Equation (VNLS), aka the Vector Gross--Pitaevskii
Equation (VGPE) . Through a Madelung transformation, the VNLS can be reduced to
coupled hydrodynamic equations in terms of multiple density and velocity
fields. Using a multi-scaling and a perturbation method along with the Fredholm
alternative, we reduce the problem to a Korteweg de-Vries (KdV) system. This is
of great importance to study more transparently, the obscure features hidden in
VNLS. This ensures that hydrodynamic effects such as dispersion and
nonlinearity are captured at an equal footing. Importantly, before studying the
KdV connection, we provide a rigorous analysis of the linear problem. We write
down a set of theorems along with proofs and associated corollaries that shine
light on the conditions of existence and nature of eigenvalues and eigenvectors
of the linear problem. This rigorous analysis is paramount for understanding
the nonlinear problem and the KdV connection. We provide strong evidence of
agreement between VNLS systems and KdV equations by using soliton solutions as
a platform for comparison. Our results are expected to be relevant not only for
cold atomic gases, but also for nonlinear optics and other branches where VNLS
equations play a defining role.
