Dynamics of Essentially Unstable Nonlinear Waves

Abstract

In this thesis we primarily consider the stability of traveling wave solutions to a modified Kuramoto-Sivashinsky Equation equation modeling nanoscale pattern formation and the St. Venant equations modeling shallow water flow down an inclined plane. Numerical evidence suggests that these equations have no unstable spectrum other than λ =0, however they both have unstable essential spectrum. This unstable essential spectrum manifests as a convecting, oscillating perturbation which grows to a certain size independent on the initial perturbation — precluding stability in the regular L^2(R) space. Exponentially weighted spaces are typically used to handle such instabilities, and in Theorem 5.7 we prove asymptotic orbital linear stability in such an exponentially weighted space. We also discuss difficulties with extending this to a nonlinear stability result. In Section 5.5 we discuss another way of obtaining stability, through ad-hoc periodic wave trains. Chapter 6 concerns the general problem of creating a spectral projection to project away unstable essential spectrum. We consider this problem in the context of spatially periodic-coefficient PDE by proposing a candidate spectral projection defined via the Bloch transform and showing that initial perturbations which activate a sufficiently unstable part of the essential spectrum lead to solutions which are not Lyapunov stable. We also extend these results to dissipative systems of conservation laws. Additional chapters of interest are Chapter 3 where we address finding the spectrum and Chapter 4 where we discuss the numerics which lead to many of the figures in this thesis.