The aim of this book is to put together all the results that are known about the existence of formal, holomorphic and singular solutions of singular non linear partial differential equations. We study the existence of formal power series solutions, holomorphic solutions, and singular solutions of singular non linear partial differential equations. In the first chapter, we introduce operators with regular singularities in the one variable case and we give a new simple proof of the classical Maillet's theorem for algebraic differential equations. In chapter 2, we extend this theory to operators in several variables. The chapter 3 is devoted to the study of formal and convergent power series solutions of a class of singular partial differential equations having a linear part, using the method of iteration and also Newton's method. As an appli­ cation of the former results, we look in chapter 4 at the local theory of differential equations of the form xy' = 1(x,y) and, in particular, we show how easy it is to find the classical results on such an equation when 1(0,0) = 0 and give also the study of such an equation when 1(0,0) #- 0 which was never given before and can be extended to equations of the form Ty = F(x, y) where T is an arbitrary vector field.

Table of Contents:
1 Operators with regular singularities: One variable case.- 1.1 Notations, definitions, examples.- 1.2 The good operators.- 1.3 A class of operators with a regular singularity.- 1.4 Applications to differential equations.- 1.5 The Maillet theorem.- 2 Operators with regular singularities: Several variables case.- A Formal theory.- 2.1 Notations.- 2.2 Linear operators on ?[[x]].- 2.3 Non linear operators on ? f.- 2.4 Solutions of linear equations.- 2.5 Solutions of non linear equations.- B Analytic theory.- 2.6 Notations and definitions.- 2.7 The good operators and the notion of domination.- 2.8 A class of operators having a regular singularity.- 2.9 Applications to partial differential equations.- 3 Formal and convergent solutions of singular partial differential equations.- 3.1 Notations and definitions.- 3.2 Holomorphic solutions of certain equations.- 3.3 Equations with parameters.- 3.4 An application: A theorem of S. Kaplan.- 3.5 The case of small denominators.- 4 Local study of differential equations of the form xy? = f(x,y) near x = 0.- 4.1 Coupling of two differential equations.- 4.2 Behavior of solutions of a differential equation near a regular point.- 4.3 Local study of a differential equation near a singular point of regular type.- 4.4 Study of the Hukuhara equation and of the Hukuhara function.- 5 Holomorphic and singular solutions of non linear singular first order partial differential equations.- 5.1 Notations and definitions.- 5.2 Statement of the main theorem.- 5.3 Holomorphic solutions.- 5.4 Singular solutions.- 5.5 Uniqueness of the solution.- 5.6 Proof of the main theorem 5.2.3.- 5.7 Remarks.- 5.8 Supplementary result.- 6 Maillet ’s type theorems for non linear singular partial differential equations.- 6.1 Implicit function theorem.- 6.2 Nonlinear equations with first order linear part.- 6.3 Non linear equations with higher order linear part.- 6.4 Formal Gevrey index for a particular type of equations — Examples.- 6.5 Supplementary results.- 7 Maillet’s type theorems for non linear singular partial differential equations without linear part.- 7.1 Notations and definitions.- 7.2 Assumptions and results.- 7.3 A basic lemma.- 7.4 Proof of theorem 7.2.5.- 7.5 Complementary results.- 7.6 A remark.- 8 Holomorphic and singular solutions of non linear singular partial differential equations.- 8.1 Holomorphic solutions.- 8.2 Singular solutions: Special case.- 8.3 Singular solutions: General case.- 8.4 Asymptotic study.- 8.5 Completion of the proof of the main theorem.- 9 On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations.- 9.1 Notations and definitions.- 9.2 Results.- 9.3 Proof of theorem 9.2.1.- 9.4 Proof of theorem 9.2.3.- 10 Maillet’s type theorems for non linear singular integro—differential equations.- 10.1 Notations and definitions.- 10.2 The main theorem.- 10.3 Construction of the formal solution.- 10.4 Some discussions.- 10.5 Convergence of the formal solution in the case sl = 1.- 10.6 Convergence of the formal solution in the case sl > 1.- 10.7 Supplementary results and remark.