Orbital stability of solitary waves for generalized derivative nonlinear Schrödinger equations in the endpoint case

Abstract

We consider the following generalized derivative nonlinear Schrödinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2σ}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $\sigma \in (0,1)$. The equation has a two-parameter family of solitary waves $$u_{\omega ,c}(t,x)={\Phi }_{\omega ,c}(x)e^{i\omega t+\frac{ic}{2}x-\frac{i}{2\sigma +2}{\int }_0^x{\Phi }_{\omega ,c}(y{)}^{2\sigma }dy},$$ with $(\omega ,c)$ satisfying $\omega >c^2/4$, or $\omega =c^2/4$ and $c>0$. The stability theory in the frequency region $\omega >c^2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $\omega =c^2/4$ and $c>0$.

References in corpus (1)

• Well-posedness for a generalized derivative nonlinear Schrödinger equation

Cited by in corpus (3)

• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
• Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations
• The stability of degenerate solitons for derivative nonlinear Schrodinger equations