Instability of solitary wave solutions for the nonlinear Schrödinger equation of derivative type in degenerate case. (March 2020) - Ning, Cui
Abstract: We study the stability theory of solitary wave solutions for a type of derivative nonlinear Schrödinger 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.