On certain shallow water models, scaling invariance and strict self-adjointness

Abstract

In this work we establish conditions for a class of third order partial differential
equations to be strictly self-adjoint and scale invariant. The obtained family of equations includes the Benjamin-Bona-Mahony, Camassa-Holm and Novikov equations. Using the strict self-adjointness and Ibragimov’s conservation theorem, we establish some local conservation laws for some of the mentioned equations.