Stability of rolls and hexagonal patterns in non-potential systems
Pattern formation in Rayleigh-Bénard convection (in a fluid layer heated from below) close to the onset of instability of the flat layer is usually described by a system of three coupled equations of the Newell-Whitehead-Segel (NWS) type for a set of three independent roll systems, the angles between the corresponding wave vectors being 120 degrees. These are coupled PDEs of the first-order in time and of the second-order in two spatial coordinates, containing linear, quadratic, and cubic terms. Within the framework of this system, it is possible to analyze in detail existence and stability conditions for two basic types of patterns which are observed in the convection, viz., rolls and hexagons. A full stability region produced by the analysis is called a Busse balloon. Usually, the governing coupled NWS equations are taken in a form which can be represented as a gradient flow, in terms of a corresponding Lyapunov functional (``potential''). However, in the general case, the system may also contain quadratically nonlinear terms with first-order spatial derivatives, which cannot be derived from a potential. The present paper is devoted to a detailed stability analysis of rolls and hexagons in a system including these non-potential terms. The main result is obtaining a set of the corresponding Busse balloons, taking into account various instabilities. The solutions investigated also include non-equilateral hexagons, which are generated by superpositions of three roll systems with angles between them slightly different from 120 degrees. The authors find new instabilities induced by the non-potential terms, e.g. an oscillatory instability of the rolls. Nonlinear stage of the instability development, characterized by such findings as heteroclinic loops etc., is investigated, too.
scientific article; zbMATH DE number 1398662
scientific article