Global asymptotic behavior of solutions to the generalized derivative nonlinear Schr\"odinger equation

This article is concerned with the global asymptotic behavior for the generalized derivative nonlinear Schr\"odinger (gDNLS) equation. When the nonlinear effect is not strong, we show pointwise-in-time dispersive decay for solutions to the gDNLS equation with small initial data in $H^{\frac{1}{2}+}(\mathbb{R})$ utilizing crucially Lorentz-space improvements of the traditional Strichartz inequality. When the nonlinear effect is especially dominant, there exists a sequence of solitary waves that are arbitrary small in the energy space, which means the small data scattering is not true. However, there is evidence that it is not possible for the solitons to be localized in $L^{2}(\mathbb{R})$ and small in $H^{1}(\mathbb{R})$. With small and localized data assumption, we obtain global asymptotic behavior for solutions to the gDNLS equation by using vector field methods combined with the testing by wave packets method.