The Tanaka Instability of Traveling Waves in Hamiltonian Systems (Mathematical aspects of nonlinear waves and their applications)

Abstract

This paper reviews the linear instability of nonlinear traveling waves in Hamiltonian systems subject to superharmonic perturbations. Tanaka's instability, characterized by a zero eigenvalue with geometric multiplicity of one and algebraic multiplicity equal to or greater than four, occurs in the presence of translationally symmetric traveling wave solutions at the extrema of energy with respect to wave speed. The theory finds application in the study of the superharmonic instability of wave equations with a Hamiltonian structure, such as water waves with constant vorticity.