  A self-gravitating, differentially rotating galactic disc under vertical
hydrostatic equilibrium is supported by the vertical pressure gradient force
against the gravitational collapse. Such discs are known to support various
bending modes e.g., warps, corrugation, or scalloping (typically, higher order
bending modes) of which m=1 bending modes (warps) are the most prevalent ones
in galactic discs. Here, we present a detailed theoretical analysis of the
bending instability in realistic models of disc galaxies in which an
exponential stellar disc is under vertical equilibrium and residing in a cold
rigid dark matter halo. A quadratic eigenvalue equation describing the bending
modes is formulated and solved for the complete eigen spectrum for a set of
model disc galaxies by varying their physical properties such as disc
scale-height, and dark matter halo mass. It is shown that the vertical pressure
gradient force can excite unstable bending modes in such a disc as well as
large scale discrete modes. Further, it is shown that the unstable eigen-modes
in a thinner disc grow faster than those in a thicker disc. The bending
instabilities are found to be suppressed in discs dominated by massive dark
matter halo. We estimate the growth timescales and corresponding wavelength of
the m=1 unstable bending modes in Milky Way like galaxies and discuss its
implication.
