  We consider the dynamics and stability of bright soliton stripes in the
two-dimensional nonlinear Schr\"odinger equation with hyperbolic dispersion,
under the action of transverse perturbations. We start by discussing a recently
proposed adiabatic-invariant approximation for transverse instabilities and its
limitations in the bright soliton case. We then focus on a variational
approximation used to reduce the dynamics of the bright-soliton stripe to
effective equations of motion for its transverse shift. The reduction allows us
to address the stripe's snaking instability, which is inherently present in the
system, and follow the ensuing spatiotemporal undulation dynamics. Further,
introducing a channel-shaped potential, we show that the instabilities (not
only flexural, but also those of the necking type) can be attenuated, up to the
point of complete stabilization of the soliton stripe.
