两类一维Hamilton导数非线性薛定谔方程的几乎整体解
Almost global solutions to two classes of 1-d Hamiltonian Derivative Nonlinear Schrödinger equations
摘要：
考虑了在周期边界条件下两类具有不同辛形式的一维Hamilton导数非线性薛定谔（DNLS）方程.这些方程的非线性不仅依赖于$(x,ψ,\barψ)$，而且依赖于$(ψ_x,\barψ_x)$，这意味着这些方程的非线性是无界的。假设非线性周期性地依赖于空间变量$x$。在一定的假设下，对于这两类Hamilton DNLS方程的大多数势，如果初值小于$p$ -Sobolev范数下的$\varepsilon\ll1$，则在一个时间间隔$(-c\varepsilon^{-r_*},c\varepsilon^{-r_*})$内（对于任意给定的正$r_*$），这两类方程的相应解也小于$2\varepsilon$。主要方法是对两类具有无界非线性项的Hamilton系统构造Birkhoff规范形，并利用Hamilton函数无界非线性项的特殊对称性，得到解在$p$ -Sobolev范数下的长时间估计.
英文摘要：
Consider two kinds of 1-d Hamiltonian Derivative Nonlinear Schrödinger (DNLS) equations with respect to different symplectic forms under periodic boundary conditions. The nonlinearities of these equations depend not only on $(x,ψ,\barψ)$ but also on $(ψ_x,\barψ_x)$, which means the nonlinearities of these equations are unbounded. Suppose that the nonlinearities depend on the space-variable $x$ periodically. Under some assumptions, for most potentials of these two kinds of Hamiltonian DNLS equations, if the initial value is smaller than $\varepsilon\ll1$ in $p$-Sobolev norm, then the corresponding solution to these equations is also smaller than $2\varepsilon$ during a time interval $(-c\varepsilon^{-r_*},c\varepsilon^{-r_*})$(for any given positive $r_*$). The main methods are constructing Birkhoff normal forms to two kinds of Hamiltonian systems which have unbounded nonlinearities and using the special symmetry of the unbounded nonlinearities of Hamiltonian functions to obtain a long time estimate of the solution in $p$-Sobolev norm.
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