To apply the algorithm a ranking of derivatives is first specified.
An example of such a ranking is:
. Then any PDE is either linear or nonlinear
in its largest or leading derivative with respect to the ranking.
The *rif* algorithm performs linear eliminations amongst the leading linear
PDEs in the same way as the standard form
algorithm in the case of linear systems.
When any leading nonlinear PDE is differentiated with respect to
an independent variable it becomes linear in its leading derivative.
Integrability conditions are calculated in the usual (linear) way across
leading linear PDEs.
The *rif* algorithm terminates when no *new*
equations were generated relative to the given system.
We defined `new' geometrically:
an equation is new if it lowers the dimension of the existing system
regarded as a submanifold of its jet space.
The *rif*-form of a system is also required to satisfy a
constant rank condition.

An effective realization of the *rif* algorithm was given for
polynomial PDE systems. The resulting *radical_rif*
algorithm realized the constant rank condition by using the
algorithms [29]
for effective construction of the radical of the
polynomial ideal generated by the system (see [64, §5]
for a proof).

During the Gaussian elimination
phase of the algorithm, pivoting can occur, and
in general a finite tree of cases has to be considered.
Independently the team [5] gave a reduction algorithm for polynomially
(but not analytic) nonlinear PDE systems which gives a representation
of the radical of a finitely generated differential ideal.
This algorithm yields cases based on inequations which are
similar to the pivot conditions in our method.
This work underwent considerable theoretical development
in Colin Rust's thesis [74].
He introduced a family of algebraic variations of the *rif* algorithm.
Rust's Relative Riquier bases, only need ideal membership testing,
to obtain formal existence and uniqueness theorems, rather
than radical ideal membership testing.

To test the practical feasibility of our idea we applied a preliminary
implementation of our algorithm to a nonlinear system of 856
PDEs in 8 independent and 8 dependent variables,
arising in the application of the `nonclassical method'
to a coupled nonlinear Schrödinger (CNLS) system.
The *rif*-form of the system was simple enough for its
exact solution to be obtained and new truly nonclassical vector
fields obtained [64, §6].
In joint work with Mansfield and Clarkson
we performed an extensive analysis of the nonclassical
reductions of the CNLS system, which appears
in [22].
There has been considerable interest in explicit solutions that
result from such reductions of
nonlinear Schrödinger equations due to
commercial applications involving the transmission of optical solitons
and quasi-solitons through fibre-optic cables.