  We analyze the dynamics of two-component vector solitons, namely
bright-in-dark solitons, via the variational approximation in Bose-Einstein
condensates. The system is described by a vector nonlinear Schr\"odinger
equation appropriate to multi-component Bose-Einstein condensates (BECs). The
variational approximation is based on hyperbolic secant (hyperbolic tangent)
for the bright (dark) component, which leads to a system of coupled ordinary
differential equations for the evolution of the ansatz parameters. We obtain
the oscillation dynamics of two-component dark-bright vector solitons.
Analytical calculations are performed for same-width components in the vector
soliton and numerical calculations extend the results to arbitrary widths. We
calculate the binding energy of the system and find it proportional to the
intercomponent coupling interaction, and numerically demonstrate the break up
or unbinding of a dark-bright soliton. Our calculations explore observable
eigenmodes, namely the internal oscillation eigenmode and the Goldstone
eigenmode. We find analytically that the density of the bright component is
required to be less than the density of the dark component in order to find the
internal oscillation eigenmode of the vector soliton and support the existence
of the dark-bright soliton. This outcome is confirmed by numerical results.
Numerically, we find that the oscillation frequency is amplitude independent.
For dark-bright vector solitons in $^{87}$Rb we find that the oscillation
frequency range is 90 to 405 Hz, and therefore observable in multi-component
BEC experiments.
