Existence and stability of the traveling waves of the derivative Schrodinger equations in the energy space
主 题: Existence and stability of the traveling waves of the derivative Schrodinger equations in the energy space
报告人: 徐桂香 (北京应用物理与计算数学研究所)
时 间: 2016-11-09 10:30-11:30
地 点: 理科一号楼 1303
On one hand, we consider the existence/nonexistence of traveling waves with two parameters for NLS with derivative (DNLS). For the critical parameters $4\omega=c^2, c\leq 0$ and the supercritical parameters $4\omega
0$ and the subcritical parameters case $4\omega>c^2$, we can show the existence and uniqueness (up to the phase rotation and spatial translation symmetries) of traveling waves by the variational argument. The result shows that the appearance of nontrivial momentum and the structure of traveling waves play important role in the existence of the traveling waves. By the variational characterization of the traveling wave, we can construct two invariant sets of (DNLS), and obtain the global existence of solution for (DNLS) with initial data in one of the invariant sets: $\mathcal{K}^+_{\omega,c}\subseteq H^1(\R)$, for any $(\omega, c) \in \R^2$ with $4\omega=c^2, c>0$ or $4\omega>c^2$. On the other hand, we show the dynamics of the traveling waves for (DNLS) in the energy space. We give an alternative perturbative method to prove the stability of the single traveling wave in the energy space, which was shown by Colin and Ohta and by the concentraction compactness argument. In addtion, under some technical assumptions on the speed of each traveling wave, our perturbative method was successfully show the stability of the sum of the multi-traveling waves for DNLS in the energy space.
