On the full dispersion Kadomtsev–Petviashvili equations for dispersive elastic waves

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