Stability of periodic Kuramoto-Sivashinsky waves

Blake Barker (1) , Mathew A. Johnson (2) , Pascal Noble (3) , Luis Miguel Miguel Rodrigues (3) , Kevin Zumbrun (4)

1 Department of mathematics [Bloomington]
2 Department of Mathematics [Lawrence, Kansas]
3 ICJ - Institut Camille Jordan
4 Department of Mathematics IU

In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinski equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of decay. The approach extends readily to other second- and higher-order parabolic equations, for example, the Cahn Hilliard equation or more general thin film models.

Domaines: Equations aux dérivées partielles [math.AP]

Blake Barker, Mathew A. Johnson, Pascal Noble, Luis Miguel Miguel Rodrigues, Kevin Zumbrun. Stability of periodic Kuramoto-Sivashinsky waves. Applied Mathematics Letters, 2012, 25 (5), pp.824-829. ⟨hal-00943583⟩