报告人：金嘉寅（Georgia Institute of Technology）
报告题目：Nonlinear modulational instability of periodic traveling waves for several dispersive PDEs.
报告摘要：We prove nonlinear modulational instability for both periodic and localizedperturbations of periodic traveling waves for several dispersive PDEs,including the KDV type equations (e.g. the Whitham equation, the generalizedKDV equation, the Benjamin-Ono equation), the nonlinear Schr\"{o}dingerequation and the BBM equation. First, the semigroup estimates required for thenonlinear proof are obtained by using the Hamiltonian structures of thelinearized PDEs; Second, for KDV type equations the loss of derivative in thenonlinear term is overcome in two complementary cases: (1) for smoothnonlinear terms and general dispersive operators, we construct higher orderapproximation solutions and then use energy type estimates; (2) for nonlinearterms of low regularity, with some additional assumption on the dispersiveoperator, we use a bootstrap argument to overcome the loss of derivative.This is a joint work with Shasha Liao and Zhiwu Lin.2. Classifications of local dynamics near solitary waves for several Hamiltonian PDEs I &IIIn this series of talks, we classify the dynamics near solitary waves for Hamiltonian PDEs, including the Gross-Pitaevskii equation and the supercritical gKDV equations, by constructing  local invariant manifolds for the solitary waves. There are two main difficulties: (1) How to decompose the energy space along the orbits of solitary waves smoothly ; (2) The nonlinearity does not map energy space to energy space and we need to derive suitable space-time estimates. This is joint work with Zhiwu Lin and Chongchun Zeng.
报告时间：2017年6月12日（星期一）下午16：00-17：00
报告地点：科技楼南楼702学术报告厅