A Complete Classification of Low-Order Conservation Laws for a Generalized Fifth-Order KP Family
A complete classification of local conservation laws with multipliers of differential order at most two is obtained for a generalized fifth-order Kadomtsev--Petviashvili family. The classification is carried out by the direct multiplier method and concerns nonlinear members of the family, with the conservation laws expressed locally in the original dependent variable. It is first shown, uniformly in the parameters, that every multiplier of differential order at most two reduces to first order. The resulting determining equations yield one generic case and two exceptional nonlinear cases. In the generic case, the multipliers involve four arbitrary functions of time. One exceptional case admits an additional multiplier depending on the first longitudinal derivative and involves five arbitrary functions of time, while the other has an enlarged zeroth-order multiplier family involving eight arbitrary functions of time. Representative conserved densities and spatial fluxes are derived for all three cases. The generic nonzero densities represent mass and transverse moment-type quantities. The first exceptional case admits the longitudinal gradient-energy density $\tfrac12 u_x^2$, whereas the second admits higher longitudinal moment densities. No multiplier proportional to $u$ occurs within the classified low-order local family, and hence no $L^2$-type density arises within this classification. The corresponding conserved integrals are obtained under appropriate boundary conditions or sufficient weighted spatial decay.