Invariance, conservation laws and reductions of some classes of “high” order partial differential equations

Using underlying invariance/symmetry properties and related/associated conservation laws, we investigate some 'high' order nonlinear equations. The multiplier method is mainly used to construct conserved vectors for these equations. When the partial differential equations are reduced to the nonlinear ordinary differential equation (NLODE), exact solutions for the ODEs are constructed and graphical representations of the resulting solutions are provided. In some cases, the solutions obtained are the Jacobi elliptic cosine function and the solitary wave solutions. We study the third-order 'equal width equation' followed by a new fourth-order nonlinear partial differential equation (NLPDE), which was recently established in the literature and, finally, the Korteweg–de Vries (KdV) equation having three dispersion sources.