  A direct method for the computation of polynomial conservation laws of
polynomial systems of nonlinear partial differential equations (PDEs) in
multi-dimensions is presented. The method avoids advanced
differential-geometric tools. Instead, it is solely based on calculus,
variational calculus, and linear algebra.
  Densities are constructed as linear combinations of scaling homogeneous terms
with undetermined coefficients. The variational derivative (Euler operator) is
used to compute the undetermined coefficients. The homotopy operator is used to
compute the fluxes.
  The method is illustrated with nonlinear PDEs describing wave phenomena in
fluid dynamics, plasma physics, and quantum physics. For PDEs with parameters,
the method determines the conditions on the parameters so that a sequence of
conserved densities might exist. The existence of a large number of
conservation laws is a predictor for complete integrability. The method is
algorithmic, applicable to a variety of PDEs, and can be implemented in
computer algebra systems such as Mathematica, Maple, and REDUCE.

Exactly Solvable and Integrable Systems