Section: New Results

A new approach to classes of quasilinear PD systems

Participants : Alban Quadrat [correspondent] , Thomas Cluzeau [ENSIL, Univ. Limoges] , Daniel Robertz [Univ. Aachen] .

Many partial differential systems appearing in mathematical physics, engineering sciences and mathematical biology are nonlinear. Unfortunately, algebraic analysis and $D$-module theory, which were successful for the algorithmic study of linear partial differential systems, cannot consider nonlinear PD systems. This project aims at developing a generalization of the algebraic analysis approach to certain classes of quasilinear partial differential systems appearing in mathematical physics and engineering sciences (e.g., Burgers' equation, shallow water, Euler equations for an incompressible fluid, traffic flow, gas flow). In [44] , we have shown how constructive methods of differential algebra and algebraic analysis could be combined to extend results obtained in [45] and [104] for linear PD systems and how they allowed us to compute conservation laws, internal symmetries and decompositions of the solution space of certain classes of quasilinear PD systems. The algorithms have been implemented in the JanetMorphisms package dedicated to this new approach (see Section  5.6 ).