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

We study the stability theory of solitary wave solutions for a type of the derivative nonlinear Schrodinger equation i partial derivative(t)u + partial derivative(2)(x)u + i|u|(2) partial derivative(x)u + b|u|(4) u = 0. The equation has a two-parameter family of solitary wave solutions of the form e(i omega 0t)+i omega(1)/2(x-omega(1)t)-i/4 integral(x-omega 1t)(-infinity) |phi(omega)(eta)(2)d eta(phi omega(x - omega 1t)). The stability theory in the frequency region of |omega(1)| < 2 root omega(0) was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case w(1) = 2 root w(0), in which the elliptic equation of phi(omega) is "zero mass". (C) 2016 Elsevier Inc. All rights reserved.