Instability of solitary wave solutions for derivative nonlinear Schrödinger equation in endpoint case

We study the stability theory of solitary wave solutions for a type of the derivative nonlinear Schrödinger equationi∂tu+∂x2u+i|u|2∂xu+b|u|4u=0. The equation has a two-parameter family of solitary wave solutions of the formeiω0t+iω12(x−ω1t)−i4∫−∞x−ω1t|φω(η)|2dηφω(x−ω1t). The stability theory in the frequency region of |ω1|<2ω0 was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case ω1=2ω0, in which the elliptic equation of φω is “zero mass”.