# Nonlinear Modulational Instability of Dispersive PDE Models

## TL;DR

This paper establishes nonlinear modulational instability for various dispersive PDEs, including KDV, Schrödinger, and BBM equations, using Hamiltonian structures and advanced energy estimates.

## Contribution

It introduces new methods to prove nonlinear instability for a broad class of dispersive PDEs, handling derivative loss and low regularity nonlinearities.

## Key findings

- Proves nonlinear modulational instability for multiple dispersive PDEs.
- Develops semigroup estimates using Hamiltonian structures.
- Overcomes derivative loss with higher order approximations and bootstrap arguments.

## 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 Schr\"odinger 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.