Linear asymptotic stability and modulation behavior near periodic waves of the Korteweg–de Vries equation

Abstract We provide a detailed study of the dynamics obtained by linearizing the Korteweg–de Vries equation about one of its periodic traveling waves, a cnoidal wave. In a suitable sense, linearly analogous to space-modulated stability [15] , we prove global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. Furthermore, we provide both a leading-order description of the dynamics in terms of slow modulation of local parameters and asymptotic modulation systems and effective initial data for the evolution of those parameters. This requires a global-in-time study of the dynamics generated by a non normal operator with non constant coefficients. On the road we also prove estimates on oscillatory integrals particularly suitable to derive large-time asymptotic systems that could be of some general interest.