Conservation laws of high-order nonlinear PDEs and the variational conservation laws in the class with mixed derivatives

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 formulas to determine these for higher order flows are somewhat cumbersome but peculiar and become more so as the order increases. We carry out these for a class of high-order partial differential equations from mathematical physics and then consider some specific ones with mixed derivatives. In the latter set of examples, our main focus is that the resultant conserved flows display some previously unknown interesting 'divergence properties' owing to the presence of the mixed derivatives. Overall, we consider a large class of equations of interest and construct some new conservation laws.