Instability of solitary wave solutions for the nonlinear Schrödinger equation of derivative type in degenerate case

Abstract We study the stability theory of solitary wave solutions for a type of derivative nonlinear Schrodinger equation i ∂ t u + ∂ x 2 u + i | u | 2 ∂ x u + b | u | 4 u = 0 , b > 0 . The equation has a two-parameter family of solitary wave solutions of the form u ω , c ( x , t ) = exp { i ω t + i c 2 ( x − c t ) − i 4 ∫ − ∞ x − c t | φ ω , c ( η ) | 2 d η } φ ω , c ( x − c t ) . Here φ ω , c is some suitable function and − 2 ω c ≤ 2 ω . The stability in the frequency region of − 2 ω c 2 κ ω (for some κ ∈ ( 0 , 1 ) ), and the instability in the frequency region of 2 κ ω c 2 ω were proved by Ohta (2014). Recently, in the endpoint case c = 2 ω , the instability of u ω , c was proved by Ning et al. (2017). Then the stability and instability region has been established except the degenerate case c = 2 κ ω . In this paper, we address the problem and prove its instability in the degenerate case.