Consider a class of 1-d Hamiltonian derivative nonlinear Schrödinger equations$$\begin{aligned} \mathbf{i}{\psi }_t=\partial _{xx}\psi + V* \psi + \mathbf{i}\partial _{x}\big ({\partial _{\bar{\psi }} F(x,\psi ,\bar{\psi })}\big ),\quad x\in \mathbb {T}, \end{aligned}$$where \(V\in \Theta _m\) [\(\Theta _m\) is defined in (1.5)]. The nonlinearity of these equations includes \((\psi _x,\bar{\psi }_x)\) and depends on space variable x periodically, which means that the nonlinearity is unbounded (see Definition 1.1) and the momentum set (see Definition 2.2) of the corresponding Hamiltonian function is unbounded. In this paper, we obtain that for any potential V outside a small measure subset of \( {\Theta }_m \), if the initial value is smaller than \(R\ll 1\) in p-Sobolev norm, then the corresponding solution to this equation is also smaller than 2R during a time interval \((-\,cR^{-r_*},cR^{-r_*})\) (for any given positive \(r_*\)). The main methods are constructing Birkhoff normal forms to unbounded Hamiltonian systems which have unbounded momentum sets and using the special symmetry of the Hamiltonian functions to control p-Sobolev norms of the solutions.

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圆上哈密顿量导数非线性Schrödinger方程的几乎整体解

考虑一类一维哈密顿导数非线性Schrödinger方程$$ \ begin {aligned} \ mathbf {i} {\ psi} _t = \ partial _ {xx} \ psi + V * \ psi + \ mathbf {i} \部分_ {x} \ big（{\ partial _ {\\ bar {\ psi}} F（x，\ psi，\ bar {\ psi}）} \ big），\ quad x \ in \ mathbb {T}， \ end {aligned} $$其中\（V \ in \ Theta _m \） [ \（\ Theta _m \）在（1.5）中定义]。这些方程的非线性包括\（（\ psi _x，\ bar {\ psi} _x）\）并周期性地取决于空间变量x，这意味着非线性是无界的（请参见定义1.1）和动量集（请参见定义2.2）相应的哈密顿函数是无界的。在本文中，对于任何潜在的V外的一个小的子集测量\（{\西塔} _m \） ，如果初始值小于\（R \ LL 1 \）在p -Sobolev范数，则相应的解决这个方程也小于2 - [R在时间间隔\（（-\，cR ^ {-r _ *}，cR ^ {-r _ *}）\）（对于任何给定的正\（r _ * \））。主要方法是为具有无界动量集的无界哈密顿系统构造Birkhoff范式，并使用哈密顿函数的特殊对称性来控制解的p -Sobolev范数。