Domain boundaries in convection patterns

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Abstract
Domain boundaries (DB’s) in convection patterns are studied near the onset of convection within the framework of amplitude equations of the Newell-Whitehead-Segel type. It is demonstrated that DB’s of an arbitrary direction are possible between domains occupied by convection rolls with different orientations. Those DB’s are always immobile, and they perform perfect selection of the rolls’ wave numbers. In certain particular cases, exact solutions for DB’s of this type are found, and their stability against long-wave flexural perturbations is established. DB’s between the rolls and a hexagonal pattern in the slightly overcritical case, and between the hexagons and the conduction (quiescent) state in the slightly subcritical case, are investigated as well. In this case, the DB is, generally speaking, moving with a constant velocity. The exponentially weak effect of pinning of the moving DB by a small-scale underlying structure is studied. Formation of DB’s of the ‘‘quiescent-state–hexagon’’ and ‘‘roll-hexagon’’ types near a sidewall is also investigated. The special case of a DB between two systems of rolls, the angle between which is close to π/3 or 2π/3, is studied in detail. This DB is, as a matter of fact, a bound state of two DB’s of the ‘‘roll-hexagon’’ type. Finally, it is demonstrated that in one-dimensional patterns (rolls) a stationary DB is not possible. In this case, a dynamical problem of decay of a DB-like structure is solved.