Instability of the solitary wave solutions for the generalized derivative nonlinear Schrodinger equation in the endpoint case

We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrodinger equation i partial derivative(t)u + partial derivative(2)(x)u + i vertical bar u vertical bar(2 sigma)partial derivative(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(t omega t+tc/2(x-ct)-t/2 sigma+2) integral(-infinity x-ci) phi(omega,c2 sigma(y)dy) phi((x-ct)(omega,c)). The stability theory in the frequency region of vertical bar c vertical bar < 2 root omega was thoroughly studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case c = 2 root omega.