Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations

Abstract

In this work, we consider the following generalized derivative nonlinear Schrödinger equation \begin{align*} i\partial_t u+\partial_{xx} u +i |u|^{2σ}\partial_x u=0, \quad (t,x)\in \mathbb R\times \mathbb R. \end{align*} We prove that when $\sigma \ge 2$, the solution is global and scattering when the initial data is small in $H^s(\mathbb{R})$, $\frac{1}{2}\le s\le 1$. Moreover, we show that when $0<\sigma <2$, there exist a class of solitary wave solutions $\{{\varphi }_c\}$ satisfying $$\Vert {\varphi }_c{\Vert }_{H^1(\mathbb{R})}\to 0,$$ when $c$ tends to some endpoint, which is against the small data scattering statement. Therefore, in this model, the exponent $\sigma \ge 2$ is optimal for small data scattering. We remark that this exponent is larger than the short range exponent and the Strauss exponent.

26 pages