  Full dispersive models of water waves, such as the Whitham equation and the
full dispersion Kadomtsev-Petviashvili (KP) equation, are interesting from both
the physical and mathematical points of view. This paper studies analogous full
dispersive KP models of nonlinear elastic waves propagating in a nonlocal
elastic medium. In particular we consider anti-plane shear elastic waves which
are assumed to be small-amplitude long waves. We propose two different full
dispersive extensions of the KP equation in the case of cubic nonlinearity and
"negative dispersion". One of them is called the Whitham-type full dispersion
KP equation and the other one is called the BBM-type full dispersion KP
equation. Most of the existing KP-type equations in the literature are
particular cases of our full dispersion KP equations. We also introduce the
simplified models of the new proposed full dispersion KP equations by
approximating the operators in the equations. We show that the line solitary
wave solution of a simplified form of the Whitham-type full dispersion KP
equation is linearly unstable to long-wavelength transverse disturbances if the
propagation speed of the line solitary wave is greater than a certain value. A
similar analysis for a simplified form of the BBM-type full dispersion KP
equation does not provide a linear instability assessment.
