Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schr\"odinger equation in the critical frequency case

Abstract

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0. $$ The equation has a two-parameter family of solitary wave solutions of the form \begin{align*} \phi_{\omega,c}(x)=\varphi_{\omega,c}(x)\exp{\big\{ i\frac c2 x-\frac{i}{2\sigma+2}\int_{-\infty}^{x}\varphi^{2\sigma}_{\omega,c}(y)dy\big\}}. \end{align*} Here $ \varphi_{\omega,c}$ is some real-valued function. It was proved in \cite{LiSiSu1} that the solitary wave solutions are stable if $-2\sqrt{\omega }<c <2z_0\sqrt{\omega }$, and unstable if $2z_0\sqrt{\omega }<c <2\sqrt{\omega }$ for some $z_0\in(0,1)$. We prove the instability at the borderline case $c =2z_0\sqrt{\omega }$ for $1<\sigma<2$, improving the previous results in \cite{Fu-16-DNLS} where $3/2<\sigma<2$.