分野別セミナー

Recent results in the Nikolaevskiy chaos are presented. The Nikolaevskiy chaos is a new type of chaos at onset discovered by one of the authors (MT) in 1996. The chaos is exhibited by solutions of a certain class of 1D nonlinear PDE whose linear parts have the ￣rst order in temporal derivatives and at least sixth order in spatial ones. The simplest example of such a kind is the Nikolaevskiy equation which describes pattern formation in various physical problems (nonlinear acoustics, chemical reactions, laser ablation, etc.). Deep connections between the chaos and the symmetry of the corresponding PDE are revealed. The chaos is associated with a single supercritical bifurcation from the trivial (identically zero) solution to the chaotic state. Its generic feature is interplay of diRerent spatiotemporal scales which gives rise to unusual scaling properties. An identity describing temporal evolution of the net
spectral power density of solutions is obtained. It allows the introduction of new functions which may be regarded as qualitative measures of non-steadiness of the chaos and the rate of its disequilibrium.
Generalized versions of the Nikolaevskiy equation as well as secondary bifurcations of solutions of these equations are discussed too.