Solitary waves for nonlinear Schrodinger equation with derivative

In this paper, we characterize a family of solitary waves for nonlinear Schrodinger equation (NLS) with derivative (DNLS) by the structure analysis and the variational argument. Since DNLS does not 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 < 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 K-omega,c(+) subset of H-1 (R), with 4 omega = c(2), c > 0 or 4 omega > c(2). On the one hand, different with the scattering result for the L-2-critical NLS in [B. Dodson, Global well-po.sedness and scattering for the mass critical nonlinear Schrodinger equation with mass below the mass of the ground state, Adv. Math. 285(5) (2015) 1589-1618], the scattering result of DNLS does not hold for initial data in K-omega,c(+) because of the existence of infinity many small solitary/traveling waves in 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 [Y. Wu, Global well-po.sedness of the derivative nonlinear Schrodinger equations in energy space, Anal. Partial Differential Equations 6(8) (2013) 1989-2002; Global well-posedness on the derivative nonlinear Schrodinger equation, Anal. Partial Differential Equations 8(5) (2015) 1101-1112] (see Corollary 1.6).