Phase dynamics of periodic wavetrains leading to the 5th order KP equation

Using the previous approach outlined in Ratliff and Bridges (2016, 2015), a novel method is presented to derive the fifth order Kadomtsev-Petviashvili(KP) equation from periodic wavetrains. As a result, the coefficients and criterion for the fifth order KP to emerge take a universal form that can be determined a-priori, relating to the system's conservation laws and the termination of a Jordan chain. Moreover, the analysis reveals that generically a mixed dispersive term qXXXY appears within the final phase equation. The theory presented here is complimented by an example from the context of flexural gravity waves in shallow water and a higher order Nonlinear Schrödinger model relevant in plasma physics, demonstrating how the coefficients in this model are determined via elementary calculations.