Conservation laws of some 'high-order' nonlinear PDEs using variational and 'partial-variational' principles

Abstract

The construction of conserved vectors using Noether's theorem via a knowledge of a Lagrangian (or via the recently developed concept of partial Lagrangians) is well known. The formulae to determine these for higher-order °ows is some- what cumbersome and becomes more so as the order increases. We carry out these for a class of fourth, ¯fth and sixth order PDEs. In the latter case, we involve the ¯fth-order KdV equation using the concept of `weak' Lagrangians analogous to the third-order KdV case. Then we considered the case of a mixed `high-order' equations working on the Shallow Water Wave and Regularized Long Wave equations. These mixed type equations have not been dealt with thus far using this technique. We ¯nally, analyse the conserved °ows of some multi-variable equations that arises in relativity.