Linear Asymptotic Stability and Modulation Behavior near Periodic Waves of the Korteweg-de Vries Equation

Luis Miguel Miguel Rodrigues (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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, 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.

Mots clés: oscillatory integrals; modulation systems; periodic traveling waves; cnoidal waves; Korteweg–de Vries equation; asymptotic stability; dispersive estimates

Domaines: Equations aux dérivées partielles [math.AP]

Luis Miguel Miguel Rodrigues. Linear Asymptotic Stability and Modulation Behavior near Periodic Waves of the Korteweg-de Vries Equation. Journal of Functional Analysis, 2018, 274 (9), pp.2553-2605. ⟨10.1016/j.jfa.2018.02.004⟩. ⟨hal-01541239⟩