Abstract
Weakly nonlinear spatially periodic patterns coupled to a Goldstone (zero) mode of the phase-field crystal model are investigated. Rotationally invariant equations for the dynamics of the amplitudes of a hexagonal pattern are derived first, which then allows us to determine stability regions for stripes and hexagons. There are parameter regimes in which all periodic patterns become unstable as a result of long-wavelength instabilities generated by the zero mode.
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