Abstract
A long-wave model for two-dimensional thermal convection is examined. This model contains both quadratic and cubic nonlinearity unless the problem is symmetrical about the horizontal mid-plane, in which case only the cubic nonlinearity is present. In the asymmetrical case a strong interaction between modes with wavenumbers k and 2 k is responsible for “wavenumber gaps” - regions in parameter space where steady states of a given spatial period cease to exist. In both the symmetrical and asymmetrical cases a 1:3 resonance generates multiple steady solutions of a given spatial period. We calculate steady states and determine their stability. For the asymmetrical case we also solve the initial-value problem and find stable travelling waves, stable modulated travelling waves and stable heteroclinic cycles, as predicted by an analysis of the 1:2 resonance.
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