On Complex Singularity Analysis for Some Linear Partial Differential Equations in

We investigate the existence of local holomorphic solutionsYof linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solutionXof some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutionsYare shown to develop singularities along with estimates of exponential type depending on the growth's rate ofXnear the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders ofXin one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.

The Existence of Analytic Invariant Curves for a Planar Map

In this paper, we study the existence of analytic invariant curves for two-dimensional maps in the complex field C. Employing the method of majorant series, we discuss the eigenvalueof the mapping at a fixed point. We discuss not only thoseat resonance, i.e., at a root of the unity but also thosenear resonance under Brjuno condition.

Majorant series convergence for twistless KAM tori

A self-contained proof of the KAM theorem in the Thirring model is discussed, completely relaxing the ‘strong diophantine property’ hypothesis used previously.