Mathematical Research Letters

Volume 27 (2020)

Number 2

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

Pages: 339 – 375

DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n2.a2

Authors

Zihua Guo (School of Mathematics, Monash University, Melbourne, Victoria, Australia)

Cui Ning (School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou, Guangdong, China)

Yifei Wu (Center for Applied Mathematics, Tianjin University, Tianjin, China)

Abstract

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation\[i \partial_t u + \partial^2_x u + i {\lvert u \rvert}^{2 \sigma} \partial_x u = 0.\]The equation has a two-parameter family of solitary wave solutions of the form\[\phi_{\omega , c} (x) = \varphi_{\omega ,c} (x) \exp\left \lbrace i \frac{c}{2} x - \frac{i}{2 \sigma + 2} \int^x_{-\infty}\varphi^{2 \sigma}_{\omega , c} (y) dy \right \rbrace \: \textrm{.}\]Here $\varphi_{\omega ,c}$ is some real-valued function. It was proved in [29] that the solitary wave solutions are stable if $-2 \sqrt{\omega} \lt c \lt 2z_0 \sqrt{\omega}$, and unstable if $2z_0 \sqrt{\omega} \lt c \lt 2 \sqrt{\omega}$ for some $z_0 \in (0, 1)$.We prove the instability at the borderline case $c = 2z_0 \sqrt{\omega}$ for $1 \lt \sigma \lt 2$, improving the previous results in [7] where $7/6 \lt \sigma \lt 2$.

Received 25 June 2018

Accepted 31 October 2018

Published 8 June 2020