Low-Order Conservation Law Multipliers for a Generalized Fifth-Order KP Family

TL;DR

This paper classifies low-order conservation law multipliers for a generalized fifth-order KP family, revealing structural constraints and identifying regimes where multipliers simplify.

Contribution

It provides a comprehensive classification of low-order multipliers for the generalized fifth-order KP family, including new structural insights.

Findings

All multipliers of order at most two are of order at most one.

In certain regimes, all first-order multipliers reduce to zeroth-order.

A finite list of exceptional cases remains open for further study.

Abstract

We study local conservation law multipliers for a generalized fifth-order Kadomtsev--Petviashvili family whose one-dimensional reductions include the Lax, Sawada--Kotera, and Kaup--Kupershmidt equations. Using the direct multiplier method, we classify zeroth-order multipliers that are independent of the dependent variable within a natural polynomial subclass and construct representative conserved vectors. We then prove that every multiplier of differential order at most two is necessarily of differential order at most one. An unrestricted first-order classification is obtained when the coefficient of the cubic derivative nonlinearity is nonzero, and the same reduction is established on a generic algebraic sub-branch of the complementary case. In these regimes, all first-order multipliers reduce to the zeroth-order family. A finite list of exceptional branches remains open. The results identify the structural sources responsible for the low-order rigidity of the multiplier problem in the generic regimes treated here.