# Orbital stability of solitary waves for generalized derivative nonlinear Schr\"odinger equations in the endpoint case

## TL;DR

This paper proves the orbital stability of solitary waves for a generalized derivative nonlinear Schrödinger equation specifically at the endpoint case where the frequency equals the critical value, extending previous stability results.

## Contribution

It establishes the stability of solitary waves at the critical endpoint case, which was previously unresolved in the stability theory for this equation.

## Key findings

- Proves stability of solitary waves at the endpoint case.
- Extends the stability theory to the critical parameter regime.
- Provides rigorous mathematical analysis for the endpoint stability.

## Abstract

We consider the following generalized derivative nonlinear Schr\"odinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2\sigma}\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 $$u_{\omega,c}(t,x)=\Phi_{\omega,c}(x)e^{i\omega t+\frac{ic}2x-\frac i{2\sigma+2}\int_0^x\Phi_{\omega,c}(y)^{2\sigma}dy},$$ 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 $c>0$.