Solitary waves for nonlinear Schr\"odinger equation with derivative

In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesn't enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4\omega>c^2$ and the critical parameters $4\omega=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters $4\omega=c^2, c\leq 0$ and the supercritical parameters $4\omega<c^2$, there is no nontrivial solitary wave for (DNLS). At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for (DNLS) with initial data in the invariant set $\mathcal{K}^+_{\omega,c}\subseteq H^1(\R)$, with $4\omega=c^2, c>0$ or $4\omega>c^2$. On one hand, different with the scattering result for the $L^2$-critical NLS in \cite{Dod:NLS_sct}, the scattering result of (DNLS) doesn't hold for initial data in $\mathcal{K}^+_{\omega,c}$ because of the existence of infinity many small solitary/traveling waves in $\mathcal{K}^+_{\omega,c},$ with $4\omega=c^2, c>0$ or $4\omega>c^2$. On the other hand, our global result improves the global result in \cite{Wu-DNLS, Wu-DNLS2} (see Corollary \ref{cor:gwp}).