Almost global solutions to two classes of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger equations

Consider two kinds of 1-d Hamiltonian Derivative Nonlinear Schr\"odinger (DNLS) equations with respect to different symplectic forms under periodic boundary conditions. The nonlinearities of these equations depend not only on $(x,\psi,\bar{\psi})$ but also on $(\psi_x,\bar{\psi}_x)$, 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.