The Existence Of Quasi-Periodic Solutions For Derivative Nonlinear Beam Equation

This paper studies mainly the existence of quasi-periodic solutions for derivative nonlinear beam equation which consists of three chapters. In Chap-ter1, we introduce the historical background, some recent results of KAM theory obtained in the literature and our main work in this paper. Chapter2and Chapter3are the subject of the thesis.In Chapter2, we firstly introduce some definitions and properties and secondly we introduce an abstract KAM theorem, then the measure estimates are given in the last place.In Chapter3, we put the KAM theory applied to the existence of quasi-periodic solutions for derivative nonlinear beam equation. By using the quasi-Toplitz function to study the derivative nonlinear beam equation that are introduced by Berti, Biasco and Procesi[16]. The general KAM theorem is extended by using the properties of the quasi-Toplitz function. The main idea is to translate the equation into Birkhoff normal formal and to introduce the action-angle variables in the Birkhoff normal form, then we can obtain the parameter dependent family of hamiltonians. But we must confirm that the perturbation is a quasi-Toplitz function. Only then can we arrive at a conclusion that the equation admits Cantor families of small-amplitude, ana-lytic, quasi-periodic solutions and the Cantor families have asymptotically full measure at the origin in the set of parameters.