Instability of solitary wave solutions for the nonlinear Schrödinger equation of derivative type in degenerate case. (March 2020)

• Record Type: Journal Article
• Authors: Ning, Cui
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: Derivative NLS -- Orbital instability -- Solitary wave solutions -- Degenerate case; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111665
• Languages: English
• ISSNs: 0362-546X

Abstract

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.