Modulational instability and variational structure

TL;DR

This paper investigates the modulational instability of periodic traveling waves in Hamiltonian systems, deriving a criterion for instability based on energy and wave parameters, applicable even to nonlocal dispersion operators.

Contribution

It introduces a new instability criterion for Hamiltonian systems with possibly nonlocal dispersion, expanding analysis beyond traditional Evans function methods.

Findings

Derived a bifurcation criterion for modulational instability

Applicable to nonlocal dispersion operators

Validated results with Korteweg-de Vries type equations

Abstract

We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg-de Vries type.