On a weakly nonlinear basis, we revisit the pattern formation problem in the Boussinesq convection, for which nonlinear terms of the quadratic order are known to vanish from amplitude equations. It is thus necessary to proceed to the quintic-order approximation in order for the amplitude equations to be generic. By deriving the quintic amplitude equations from the governing PDEs, we examined the bifurcation of steady solutions under rigid-free, rigid-rigid and free-free boundary conditions. Right above the criticality, all the axial solutions are obtained including up-and down-hexagons under the asymmetric boundary conditions and hexagons and regular triangles under the symmetric conditions. Hexagons and regular triangles are unstable whereas rolls are stable as has already been predicted by the cubic-order amplitude equations. Irrespective of the boundary conditions, quintic-order terms stabilize hexagons except near the criticality; rolls and hexagons thus coexist stably in an open region. This suggests that amplitude equations of higher order are possible to predict re-entrant hexagons.

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