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

Abstract

We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation $i{\partial }_{t}u+{\partial }_x^{2}u+i{\left|u\right|}^{2\sigma }{\partial }_{x}u=0,$where $1<\sigma <2$. The equation has a two-parameter family of solitary wave solutions of the form $u_{\omega ,c}(t,x)=e^{i\omega t+i\frac{c}{2}(x-ct)-\frac{i}{2\sigma +2}{\int }_{-\infty }^{x-ct}{\phi }_{\omega ,c}^{2\sigma }\left(y\right)dy}{\phi }_{\omega ,c}(x-ct).$The stability theory in the frequency region of $\left|c\right|<2\sqrt{\omega }$ was thoroughly studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case $c=2\sqrt{\omega }$.