论文标题
圆圈上准线性PDE的非线性Egorov定理和Poincaré-Birkhoff正常形式
A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle
论文作者
论文摘要
在本文中，我们考虑了圆上的一类抽象线性para-differential方程。对于类中的每个方程式，我们证明存在坐标的变化，该坐标将方程式结合到对角线和恒定系数para分化方程中。在汉密尔顿方程式的情况下，我们还将该系统置于Poincaré-Birkhoff正常形式。我们将此转换应用于Schrödinger和Beam方程的准线性扰动，从而获得了很长的存在结果，而无需在初始数据上进行任何对称性。我们还为本杰明·诺克方程的准线性扰动提供了局部时间供应良好。
In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant coefficient para-differential equation. In the case of Hamiltonian equations we also put the system in Poincaré-Birkhoff normal forms. We apply this transformation to quasi-linear perturbations of the Schrödinger and Beam equations, obtaining a long time existence result without requiring any symmetry on the initial data. We also provide the local in time well-posedness for quasi-linear perturbations of the Benjamin-Ono equation.
