On the development of singularities in linear dispersive systems

Abstract

Benilov, O'Brien & Sazonov (2003) and Benilov (2004) describe “a new type of instability” in a liquid film inside a rotating cylinder. Though their linear systems support only neutrally stable modes, they find explosive disturbances which become singular after a finite time. They suggest that this result casts doubt on the reliability of modal analysis for prediction of instability; and they claim that “Such cases have never been described in the literature, and they are probably extremely rare”. Here, other examples are given, some of which have been known (though not well-known) for many years. A common feature of all these singularities is a local phase synchronization of short-wave modes; but the configuration of Benilov et al. has the additional feature of eigenfunctions that exhibit very large changes in amplitude within the spatial domain. The relevance, or not, of such singularities to real physical systems is discussed.