Instability of solitary waves for a generalized derivative nonlinear Schrödinger equation in a borderline case

Abstract

We study the orbital instability of solitary waves for a derivative nonlinear Schrödinger equation with a general nonlinearity. We treat a borderline case between stability and instability, which is left as an open problem by Liu, Simpson and Sulem (2013). We give a sufficient condition for instability of a two-parameter family of solitary waves in a degenerate case by extending the results of Ohta (2011), and verify this condition for some cases.