The stability of degenerate solitons for derivative nonlinear Schrodinger equations

In this paper, we consider the following nonlinear Schrödinger equation with derivative: \begin{align*} i\partial_tu+\partial_{xx}u+i|u|^{2}\partial_xu+b|u|^4u=0, \quad (t,x) \in $\mathbb{R}$\times$\mathbb{R}$, \quad b\geq 0. \end{align*} For the case $b=0$, the original DNLS, Kwon and Wu \cite{KwonWu2018} proved the conditional orbital stability of degenerate solitons including scaling, phase rotation, and spatial translation with a non-smallness condition, $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{L^6}^6>\sqrt{\delta }$. In this paper, we remove this condition for the non-positive initial energy and momentum, and we extend the stability result for $b\ge 0$.