Amplitude equations for extended discrete systems: A study of an antiferromagnetic spin chain

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Abstract
We study the effect of the dynamical degrees of freedom resulting from the lattice structure of an antiferromagnet on pattern forming bifurcations. They are examined in a one-dimensional chain of damped and driven classical spin oscillators. In addition to stationary states where all spins are parallel (quasiferromagnetic state), it exhibits states where the spins are lined up on two sublattices (noncollinear state). Besides showing instabilities against large scale perturbations, short wavelengths of the order of the lattice constant become critical even for weak driving fields. A general formalism admitting a dynamical description of discrete oscillator chains in the weakly nonlinear regime beyond the instabilities is developed. The ensuing amplitude equation allows the examination of the formation of patterns in this domain. Special emphasis is laid on instabilities of higher codimension. A codimension-3 bifurcation where all wave numbers become critical simultaneously implies a direct transition to turbulence.