  The rotation modified Kadomtsev Petviashvili equation which is also known as
the Kadomtsev Petviashvili Ostrovsky equation, describes the gradual wave field
diffusion in the transverse direction to the direction of the propagation of
the wave in a rotating frame of reference. This equation is a generalization of
the Ostrovsky equation additionally having weak transverse effects. We
investigate transverse instability and stability of small periodic traveling
waves of the Ostrovsky equation with respect to either periodic or
square-integrable perturbations in the direction of wave propagation and
periodic perturbations in the transverse direction of motion in the
rotation-modified Kadomtsev Petviashvili equation. We also study transverse
stability or instability in generalized rotation modified KP equation by taking
dispersion term as general and quadratic and cubic nonlinearity. As a
consequence, we obtain transverse stability or instability in two dimensional
generalization of RMBO eqution, Ostrovsky Gardner equation, Ostrovsky fKdV
equation, Ostrovsky mKdV equation, Ostrovsky ILW equation, Ostrovsky Whitham
etc.
