We study various classes of the nonlinear dynamics of some ‘high’ order parabolic equations (pdes) like the Benjamin-Bona-Mahony-Peregrine-Burger and the Oskolkov-Benjamin-Bona-Mahony-Burgers equations that arise in the study of some wave phenomena. Also, a large class of pdes arising in the modelling of ocean waves are due to Ostrovsky. We determine the invariance properties (through the Lie point symmetry generators) of the nonlinear systems and construct classes of conservation laws for some of the models above and show how the relationship leads to double reductions of the systems. This relationship is determined by a recent result involving ‘multipliers’ that lead to ‘total divergence’.