Secondary instability of one-dimensional cellular patterns: a gap soliton, black soliton and breather analogy

Abstract

Ten years ago, a normal form theory for the generic secondary instabilities of one-dimensional cellular patterns has been rigorously derived close to the secondary bifurcation threshold. The aim of this paper is to show that some experimental observations of localized patterns, previously unexplained, can be understood if one slightly modifies the former theory in order to take into account a linearly damped eigenmode with particular symmetry property. The model we develop, involving a huge number of unknown parameters, turns out to be quite unsatisfactory from a computational point of view. But we get round this difficulty by developing a strong analogy with the well-known problem of the propagation of a nonlinear wave in a periodic medium. With this analogy in mind, localized solutions of our model, similar to the usual conservative breather, gap or black soliton, are numerically obtained and compared with the experimental observations with quite a good qualitative agreement.