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

We study 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. The equation has a two-parameter family of solitary wave solutions of the form phi(omega, c)(x) = phi(omega, c)(x) exp{i c/2x - i/2 sigma + 2 integral(x)(-infinity) phi(2 sigma)(omega,c)(y)dy}. Here phi(omega, c) is some real-valued function. It was proved in [29] that the solitary wave solutions are stable if -2 root omega < c < 2z(0) root omega), and unstable if 2z(0) root omega < c < 2 root omega for some z(0) is an element of (0, 1). We prove the instability at the borderline case c = 2z(0)root omega for 1 < sigma < 2, improving the previous results in [7] where 7/6 < sigma < 2.