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.

References in corpus (3)

• The Stability Spectrum for Elliptic Solutions to the Focusing NLS Equation
• Modulational Instability in Equations of KdV Type
• High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs

Cited by in corpus (8)

• Full description of Benjamin-Feir instability of Stokes waves in deep water
• Stability of Traveling wave solutions of Nonlinear Dispersive equations of NLS type
• Global well-posedness for $H^{-1}(\mathbb{R})$ perturbations of KdV with exotic spatial asymptotics
• Spectral instability of small-amplitude periodic waves of the electronic Euler-Poisson system
• Full description of Benjamin-Feir instability for generalized Korteweg-de Vries equations
• Nonlinear modulational instabililty of the Stokes waves in 2d full water waves
• The Benjamin-Feir instability in the infinite depth
• Nonlinear Instability of Periodic Traveling Waves