We study the global in time existence of small solutions to the generalized derivative nonlinear Schrödinger equations of the form
$\begin{equation} \cases {\mathrm{i}\curpartial_t u + (1/2)\Delta u = \mathcal{N}(u,\nabla u,\overline u,\nabla \overline u), &$(t,x) \in \mathbf{R}\times {\mathbf{R}}^n$,\cr \noalign{\vskip4pt} u(0,x) = u_0 (x),\quad x\in\mathbf{R}^n,\cr} \end{equation}$
where the space dimension
$ n \geqslant 3 $
, the initial data
$ u_0 $
are sufficiently small,
$\bar u$
is the complex conjugate of
$u$
and the nonlinear term
$\mathcal{N}$
is a smooth complex valued function
$\mathbf{C}\times \mathbf{C}^n\times \mathbf{C}\times \mathbf{C}^n\rightarrow \mathbf{C}$
. We assume that
${\mathcal N}$
is a quadratic function in the neighborhood of the origin and always includes at least one derivative, that is,
$|{\mathcal N}(u,w,\bar u,\bar w)| \leqslant C|w|(|u|+|w|)$
, for small
$u$
and
$w$
in the case of space dimensions
$n = 3, 4$
. As a typical example we consider the case of the polynomial type nonlinearity of the form
$ {\mathcal N}(u,w,\bar u,\bar w) = \sum_{\tiny{\matrix{2 \leqslant |\alpha| + |\beta| + |\gamma| \leqslant l\cr \noalign{\vskip2pt} m \leqslant |\beta| + |\gamma| \leqslant l}}} \normalsize\lambda_{\alpha\beta\gamma} u^{\alpha_1}\bar u^{\alpha_2} w^{\beta}\bar w^{\gamma} $
with
$ w = (w_j)_{1\leqslant j\leqslant n}$
,
$ \lambda_{\alpha\beta\gamma} \in \mathbf{C} $
,
$l\geqslant 2,$
$m \geqslant 1$
for
$ n=3,4$
, and
$m \geqslant 0$
for
$ n\geqslant 5 $
.
We prove the global existence of solutions to the Cauchy problem (A) under the condition that the initial data
$u_0 \in \mathbf{H}^{[n/2]+5,0} \cap \mathbf{H}^{[n/2]+3,2}$
, where
$\mathbf{H}^{m,s} = \{ \phi \in \mathbf{L}^2;\ \|\phi\|_{m,s} = \|(1+x^2)^{s/2} (1-\Delta)^{m/2}\phi\|_{\mathbf{L}^2} <\infty\}$
is the weighted Sobolev space. We also show the existence of the usual scattering states. Our result for
$n=3,4$
is an improvement of Hayashi and Hirata, Nonlinear Anal. 31 (1998), 671–685.
