Superharmonic instability for regularized long-wave models*

Abstract

Abstract We examine the spectral stability and instability of periodic traveling waves for regularized long-wave models. Examples include the regularized Boussinesq, Benney–Luke, and Benjamin–Bona–Mahony equations. Of particular interest is a striking new instability phenomenon—spectrum off the imaginary axis extending into infinity. The spectrum of the linearized operator of the generalized Korteweg–de Vries equation, for instance, lies along the imaginary axis outside a bounded set. The spectrum for a regularized long-wave model, by contrast, can vary markedly with the parameters of the periodic traveling waves. We perform rigorous spectral asymptotics for short wavelength perturbations to establish conditions under which the spectrum tends to infinity along the imaginary axis or some curve whose real part is nonzero. We conduct numerical experiments which corroborate our analytical findings.