Multidimensional partial differential equation systems: Generating new systems via conservation laws, potentials, gauges, subsystems

For many systems of partial differential equations PDEs , including nonlinear ones, one can construct nonlocally related PDE systems. In recent years, such nonlocally related systems have proven to be useful in applications. In particular, they have yielded systematically nonlocal symmetries, nonlocal conservation laws, noninvertible linearizations, and new exact solutions for many different PDE systems of interest. However, the overwhelming majority of new results and theoretical understanding pertain only to PDE systems with two independent variables. The situation for PDE systems with more than two independent variables turns out to be much more complicated due to gauge freedom relating potential variables. The current paper, together with the companion paper A. F. Cheviakov and G. W. Bluman, J. Math. Phys. 51, 103522 2010 , synthesizes and systematically extends known results for nonlocally related systems arising for multidimensional PDE systems, i.e., for PDE systems with three or more independent variables. The presented framework includes potential systems arising from lower-degree conservation laws of a given PDE system. Nonlocally related multidimensional PDE systems are discussed in terms of their construction, properties, and applications. © 2010 American Institute of Physics. doi:10.1063/1.3496380