In this paper, we continue the study of the dynamics of the traveling waves for nonlinear Schrödinger equation with derivative (DNLS) in the energy space. Under some technical assumptions on the speed of each traveling wave, the stability of the sum of two traveling waves for DNLS is obtained in the energy space by Martel-Merle-Tsai's analytic approach in [20,21]. As a by-product, we also give an alternative proof of the stability of the single traveling wave in the energy space in [6], where Colin and Ohta made use of the concentration-compactness argument.2000 Mathematics Subject Classification. Primary: 35L70, Secondary: 35Q55.

Abstract. 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 4ω > c 2 and the critical parameters 4ω = 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 4ω = c 2 , c ≤ 0 and the supercritical parameters 4ω < 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 KOn one hand, different with the scattering result for the L 2 -critical NLS in [10], the scattering result of (DNLS) doesn't hold for initial data in K On the other hand, our global result improves the global result in [34,35] (see Corollary 1.6). IntroductionIn this paper, we consider the solitary waves of nonlinear Schrödinger equation with derivative [22,23,28], and has many equivalent forms. For example, it is equivalent to the following equationby the following gauge transformationThe equation (1.1) is L 2 -critical derivative NLS since the scaling transformationleaves both (1.1) and the mass invariant. The mass, momentum and energy of the solution for (1.1) are defined as followingThey are conserved under the flow (1.1) by the local well-posedness theory in H 1 according to the phase rotation, spatial translation and time translation invariances. Since (1.1) or (1.2) doesn't enjoy the Galilean and pseudo-conformal invariance any more, there is no explicit blowup solution for (1.1) and the momentum is not trivial in dealing with the solitary/traveling waves of (1.1) any more. Local well-posedness thery for (1.1) in the energy space was worked out by N. Hayashi and T. Ozawa [16,25]. They combined the fixed point argument with L 4 I W 1 ∞ (R) estimate to construct the local-in-time solution with arbitrary data in the energy space. For other results, we can refer to [14,15]. Since (1.1) isḢ 1 -subcritical case, the maximal lifespan interval of the energy solution only depends on the H 1 norm of initial data.Theorem 1.1. [16, 25] For any u 0 ∈ H 1 (R) and t 0 ∈ R, there exists a unique maximallifespan solution u : I × R → C to (1.1) with u(t 0 ) = u 0 , the map u 0 → u is continuous from H 1 (R) to C(I, H 1 (R)) ∩ L 4 loc (I; W 1,∞ (R)). Moreover the solution has the following properties:(1) I is an open neighborhood of t 0 .(2) The mass, momentum and energy are conserved, that is, for all t ∈ I, M(u)(t) = M(u)(t 0 ), P (u)(t) = P (u)(t 0 ), E(u)(t) = E(u)(t 0 ).

In this paper, we continue the study in [18]. We use the perturbation argument, modulational analysis and the energy argument in [15,16] to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schrödinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case σ > 1 we considered corresponds to the L 2supercritical case.2010 Mathematics Subject Classification. Primary: 35L70, Secondary: 35Q55.

Compared with macroscopic conservation law for the solution of the derivative nonlinear Schrödingger equation (DNLS) with small mass in [21], we show the corresponding microscopic conservation laws for the Schwartz solutions of DNLS with small mass. The new ingredient is to make use of the logarithmic perturbation determinant introduced in [34,35] to show one-parameter family of microscopic conservation laws of the A(κ) flow and the DNLS flow, which is motivated by [11,19,20].

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in [16] to show the orbital instability of the solitary waves Qω,c (x − ct) e i ωt of the generalized derivative nonlinear Schrödinger equation (gDNLS) in the degenerate case c = 2z0√ ω, where z0 = z0 (σ) is the unique zero point of F (z; σ) in (−1, 1). The new ingredients in the proof are the refined modulation decomposition of the solution near Qω,c according to the spectrum property of the linearized operator S ω,c (Qω,c) and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in [7].