Singular nonlinear partial differential equations.

*(English)*Zbl 0874.35001
Aspects of Mathematics. E28. Wiesbaden: Vieweg. viii, 269 p. (1996).

This book deals with the existence of formal power series solutions, holomorphic solutions, and singular solutions of several classes of singular nonlinear ordinary (ODE) and partial differential equations (PDE).

The book is divided into ten chapters. In Chapter 1 a new proof is given of the classical Maillet’s theorem for algebraic differential equations. Chapter 2 studies the regular singularities in the case of several variables. Interesting applications are proposed to PDEs with a Poincaré vector field, to the PDE of the hypergeometric functions, and others. Chapter 3 deals with the formal and convergent power series solutions of a class of singular PDEs having a linear principal part. As an application, the well known Kaplan’s theorem for local normal forms of singular holomorphic vector fields near the origin is proved. The case of small denominators is considered in more detail. Chapter 4 is devoted to the local theory of ODEs having the form \(xy'= f(x,y)\) in both cases \(f(0,0)=0\) and \(f(0,0)\neq 0\).

The application of the classical method of majorant equations enables the authors to give holomorphic and singular solutions of several classes of nonlinear PDEs in Chapters 5 and 8. In some cases the asymptotic theory is developed and applied to several linear and nonlinear PDEs. In Chapters 6 and 7 Maillet’s type theorems are proved for singular nonlinear PDEs with linear principal part and without linear principal part, respectively. A rather complete study on the existence of holomorphic solutions of the Cauchy problem for nonlinear PDEs is proposed in Chapter 9. The main result here can be formulated as follows: If the Cauchy problem for the nonlinear PDE has a formal power series solution and if some additional algebraic conditions are satisfied then this Cauchy problem has a convergent power series solution.

This book puts together almost all results about the existence of formal, holomorphic, and singular solutions of some classes of singular nonlinear PDEs. It contains many original results of the authors, too. The book can be useful for all specialists working in the domain of asymptotic theory for nonlinear ODEs and PDEs.

The book is divided into ten chapters. In Chapter 1 a new proof is given of the classical Maillet’s theorem for algebraic differential equations. Chapter 2 studies the regular singularities in the case of several variables. Interesting applications are proposed to PDEs with a Poincaré vector field, to the PDE of the hypergeometric functions, and others. Chapter 3 deals with the formal and convergent power series solutions of a class of singular PDEs having a linear principal part. As an application, the well known Kaplan’s theorem for local normal forms of singular holomorphic vector fields near the origin is proved. The case of small denominators is considered in more detail. Chapter 4 is devoted to the local theory of ODEs having the form \(xy'= f(x,y)\) in both cases \(f(0,0)=0\) and \(f(0,0)\neq 0\).

The application of the classical method of majorant equations enables the authors to give holomorphic and singular solutions of several classes of nonlinear PDEs in Chapters 5 and 8. In some cases the asymptotic theory is developed and applied to several linear and nonlinear PDEs. In Chapters 6 and 7 Maillet’s type theorems are proved for singular nonlinear PDEs with linear principal part and without linear principal part, respectively. A rather complete study on the existence of holomorphic solutions of the Cauchy problem for nonlinear PDEs is proposed in Chapter 9. The main result here can be formulated as follows: If the Cauchy problem for the nonlinear PDE has a formal power series solution and if some additional algebraic conditions are satisfied then this Cauchy problem has a convergent power series solution.

This book puts together almost all results about the existence of formal, holomorphic, and singular solutions of some classes of singular nonlinear PDEs. It contains many original results of the authors, too. The book can be useful for all specialists working in the domain of asymptotic theory for nonlinear ODEs and PDEs.

Reviewer: P.Popivanov (Sofia)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35G20 | Nonlinear higher-order PDEs |

35C10 | Series solutions to PDEs |

35C20 | Asymptotic expansions of solutions to PDEs |