Instability of the solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation in the endpoint case

We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0, $$ where $1<\sigma<2$. The equation has a two-parameter family of solitary wave solutions of the form $$ u_{\omega,c}(t,x)=e^{i\omega t+i\frac c2(x-ct)-\frac{i}{2\sigma+2}\int_{-\infty}^{x-ct}\varphi^{2\sigma}_{\omega,c}(y)dy}\varphi_{\omega,c}(x-ct). $$ The stability theory in the frequency region of $|c|<2\sqrt{\omega}$ was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case $c=2\sqrt{\omega}$.