Speaker: Zhiwu Lin, Georgia Tech
Date: November 3, 2016
Title: Nonlinear modulational instability of dispersive wave models
Abstract: Modulational instability (also called side band instability, Benjamin-Feir instability) is an important instability mechanism in lots of dispersive wave equations, including 2D water waves and model equations such as KDV, BBM, and Whitham equations. It leads to the breakdown of periodic traveling wave pattern in these modes. In the literature, such instability had been studied a lot by the linearized equation, i.e., the spectra of the linearized operator. With Shasha Liao and Jiayin Jin, we are able to prove nonlinear modulational instability for lots of dispersive models including nonlinear Schrodinger equation, BBM, and KDV type equations (KDV, Benjamin-Ono, Whitham etc). The nonlinear instability is proved for both periodic and localized perturbations. The two main ingredients in the proof are: the semigroup estimates by using the Hamiltonian structures of the PDEs; the construction of higher order approximation solutions with energy estimates to handle the loss of derivative in the nonlinear terms.