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# 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

#### 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 $7/6 \lt \sigma \lt 2$.

Z. Guo is supported by ARC DP170101060. C. Ning and Y. Wu are partially supported by NSFC 11901120, 1771325 and 11571118.

Received 25 June 2018

Accepted 31 October 2018

Published 8 June 2020