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Orbital stability of solitary waves for generalized derivative nonlinear Schrödinger equations in the endpoint case (1705.04458v2)
Published 12 May 2017 in math.AP
Abstract: We consider the following generalized derivative nonlinear Schr\"odinger equation \begin{equation*} i\partial_tu+\partial2_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 $$u_{\omega,c}(t,x)=\Phi_{\omega,c}(x)e{i\omega t+\frac{ic}2x-\frac i{2\sigma+2}\int_0x\Phi_{\omega,c}(y){2\sigma}dy},$$ with $(\omega,c)$ satisfying $\omega>c2/4$, or $\omega=c2/4$ and $c>0$. The stability theory in the frequency region $\omega>c2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $\omega=c2/4$ and $c>0$.