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 $c=2\sqrt{ \omega}$.
Categories: math.AP