论文标题
在周期性边界条件下大域上的quasilIrIrIrIrIrIrIrInear klein-gordon方程的长时间NLS近似
Long time NLS approximation for the quasilinear Klein-Gordon equation on large domains under periodic boundary conditions
论文作者
论文摘要
我们提供了一类准确的Hamiltonian klein gordon方程的NLS近似值的严格理由，具有大型一维tori $ \ t_l：= \ \ m athbb {r}/（r}/（r}/（2πl\ mathbb {z Z} $ gg，我们证明了这种近似值在\ emph {长时间}刻度上的有效性，这意味着它具有超出立方非线性时间尺度的范围。为了实现这一结果，我们需要执行二阶分析并处理高阶共振波互动。主要困难是由问题的准线性性质以及由准共和度引起的小除数的存在所提供的。该证明基于二分的演算，能量方法，正常形式程序和高低频率分析。
We provide the rigorous justification of the NLS approximation, in Sobolev regularity, for a class of quasilinear Hamiltonian Klein Gordon equations with quadratic nonlinearities on large one-dimensional tori $\T_L:=\mathbb{R}/(2πL \mathbb{Z})$, $L\gg 1$. We prove the validity of this approximation over a \emph{long-time} scale, meaning that it holds beyond the cubic nonlinear time scale. To achieve this result we need to perform a second-order analysis and deal with higher order resonant wave-interactions. The main difficulties are provided by the quasi-linear nature of the problem and the presence of small divisors arising from quasi-resonances. The proof is based on para-differential calculus, energy methods, normal form procedures and a high-low frequencies analysis.
