Title: Nonlinear Modulational Instability of Dispersive PDE Models,
Abstract: We prove nonlinear modulational instability for both periodic and localized
perturbations of periodic traveling waves for several dispersive PDEs,
including the KDV type equations (e.g. the Whitham equation, the generalized
KDV equation, the Benjamin-Ono equation), the nonlinear Schrödinger equation
and the BBM equation. First, the semigroup estimates required for the nonlinear
proof are obtained by using the Hamiltonian structures of the linearized PDEs;
Second, for KDV type equations the loss of derivative in the nonlinear term is
overcome in two complementary cases: (1) for smooth nonlinear terms and general
dispersive operators, we construct higher order approximation solutions and
then use energy type estimates; (2) for nonlinear terms of low regularity, with
some additional assumption on the dispersive operator, we use a bootstrap
argument to overcome the loss of derivative.