The Reduced Involutive Form Algorithm for Nonlinear Systems

The rif algorithm does not reduce a system of PDEs to a Differential Gröbner Basis. It reduces systems of nonlinear analytic PDEs to a (reduced involutive) form which is simply related to the involutive form of geometric approaches [75,54].

To apply the algorithm a ranking of derivatives is first specified. An example of such a ranking is: $u \prec u_x$ $\prec u_y$ $\prec u_{xx}$ $\prec u_{xy} \prec ...$. 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.