Instability of the solitary wave solutions for the generalized derivative nonlinear schrödinger equation in the critical frequency case

Zihua Guo, Cui Ning, Yifei Wu

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation i∂tu + ∂x2u + i|u|xu = 0. The equation has a two-parameter family of solitary wave solutions of the form ϕω,c(x) = φω,c(x) exp { i 2c x − i+ 2Z−∞x φ2ω,cσ (y)dy }. Here φω,c is some real-valued function. It was proved in [29] that the solitary wave solutions are stable if −2√ω < c < 2z0ω, and unstable if 2z0ω < c < 2√ω for some z0 ∈ (0, 1). We prove the instability at the borderline case c = 2z0ω for 1 < σ < 2, improving the previous results in [7] where 7/6 < σ < 2.

Original languageEnglish
Pages (from-to)339-375
Number of pages37
JournalMathematical Research Letters
Volume27
Issue number2
DOIs
Publication statusPublished - 2020

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