  In the present work, we develop an adiabatic invariant approach for the
evolution of quasi-one-dimensional (stripe) solitons embedded in a
two-dimensional Bose-Einstein condensate. The results of the theory are
obtained both for the one-component case of dark soliton stripes, as well as
for the considerably more involved case of the two-component dark-bright (alias
"filled dark") soliton stripes. In both cases, analytical predictions regarding
the stability and dynamics of these structures are obtained. One of our main
findings is the determination of the instability modes of the waves as a
function of the parameters of the system (such as the trap strength and the
chemical potential). Our analytical predictions are favorably compared with
results of direct numerical simulations.
