A generalized sideband stability theory via center manifold projection

Abstract

A center manifold theory is used to determine all modes that contribute significantly to the leading-order sideband stability of finite-amplitude monochromatic waves. The classical multiscale theories based on the Ginzburg–Landau equation are extended away from near-critical conditions and are shown to have omitted an important contribution from nonlinear interactions with low wave-number modes. Stability bounds on stable monochromatic waves are reported for dispersive systems that extend the classical Eckhaus bound for nondispersive systems and the Lange–Newell and Benjamin–Feir stability conditions for monochromatic waves with critical wave numbers. These new stability bounds are verified numerically by computing the evolving spectrum of a model equation.