Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient $a\left( t\right)$ satisfies the time growth condition

$\epsilon$ is a sufficiently small constant and the nonlinear interaction term $F$ consists of cubic nonlinearities of derivative type

where $\lambda _{1},\lambda _{6}\in \mathbf{R},$ $\lambda _{2},\lambda _{3},\lambda _{4},\lambda _{5}\in \mathbf{C},$ $\lambda _{2}-\lambda _{3}\in \mathbf{R,}$ and $\lambda _{4}-\lambda _{5}\in \mathbf{R}$. We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate $\Vert u(t)\Vert _{\mathbf{L}^{p}}\leq C\epsilon t^{ \frac{1}{p}-\frac{1}{2}}$, for all $t\geq 1$, where $2\leq p\leq \infty$. Furthermore we show that for $\frac{1}{2}<\delta <1$ there exist the usual scattering states, when $b(x)=\lambda _{1}-\left( \lambda _{2}-\lambda _{3}\right) x+\left( \lambda _{4}-\lambda _{5}\right) x^{2}-\lambda _{6}x^{3}=0,$ and the modified scattering states, when $b(x)\neq 0.$