Abstract : This thesis is dedicated to the study of nonlinear partial differential equations systems. The chosen approach is using differential algebra. Given a system of differential equations, we seek information about its solutions. To do so, we first compute particular systems (called differential regular chains) such that the union of their solutions coincide with the solutions of the initial system.
This thesis mainly presents new results in symbolic computation. Chapter 2 clarifies the link between regular chains and differential regular chains. Two new algorithms (given in chapters 4 and 5) improve existing algorithms for computing these differential regular chains. These algorithms involve purely algebraic techniques which help reduce expression swell and help avoid unnecessary computations. Previously intractable problems have been solved using these techniques. An algorithm computing the normal form of a differential polynomial modulo a differential regular chain is described in chapter 2.
The last results deal with analysis. The solutions we consider are formal power series. Chapter 3 gives sufficient conditions for a solution to be analytic. The same chapter presents a counter-example to a conjecture dealing with the analyticity of formal solutions.