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We study the Swift--Hohenberg equation (SHE) in the presence of an external field. The application of the field leads to a phase diagram with three phases, i.e., stripe, hexagon, and uniform. We focus on coarsening after a quench from the uniform to stripe or hexagon regions. For stripe patterns, we find that the length scale associated with the order-parameter structure factor has the same growth exponent ($\ensuremath{\simeq}1/4$) as for the SHE with zero field. The growth process is slower in the case of hexagonal patterns, with the effective growth exponent varying between $1/6$ and $1/9$, depending on the quench parameters. For deep quenches in the hexagonal phase, the growth process stops at late stages when defect boundaries become pinned.

A theoretical model for studying pattern formation in electroconvection is proposed in the form of a modified Swift-Hohenberg equation. A localized state is found in two dimension, in agreement with the experimentally observed ``worm" state. The corresponding one dimensional model is also studied, and a novel stationary localized state due to nonadiabatic effect is found. The existence of the 1D localized state is shown to be responsible for the formation of the two dimensional ``worm" state in our model.

It is shown that the Swift-Hohenberg equation w\ifmmode \dot{}\else \.{}\fi{}=\ensuremath{\epsilon}w-(${\mathrm{\ensuremath{\nabla}}}^{2}$+${\mathit{k}}_{0}^{2}$${)}^{2}$w-${\mathit{w}}^{3}$ for \ensuremath{\epsilon}g3/2${\mathit{k}}_{0}^{4}$ exhibits a new phase ordering kinetics from the lamellar phase to the uniform state. A domain wall propagates with the velocity c(R)=-(2/R)(\ensuremath{\epsilon}+8${\mathit{k}}_{0}^{4}$)/4${\mathit{k}}_{0}^{2}$ and reduces the domain structure which finally shrinks to a spot. Here ${\mathit{R}}^{\mathrm{\ensuremath{-}}1}$ denotes the curvature of the domain wall. The dynamics of relaxation is investigated by observing how the structure function evolves in time. It is found that the structure function shows the same scaling relationship that is known from the ordering kinetics in the time-dependent Ginzburg-Landau equation. \textcopyright{} 1996 The American Physical Society.

The competition between patterns of different symmetries is studied for the generalized Swift-Hohenberg model in the presence of a large quadratic coupling in one dimension (1D) and 2D. It is shown that hexagons with different phase relations may coexist for some values of the control parameter. Numerical experiments exhibiting the effects of linear spatial ramps of the control parameter on this selection are presented. The analogy with recent patterns obtained experimentally in open chemical reactors is also discussed.

The cubic-quintic Swift-Hohenberg equation (SH35) provides a convenient order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use SH35 with an additional quadratic term to determine the qualitative effects of breaking the midplane reflection symmetry on the properties of spatially localized structures in these systems. Our results describe how the snakes-and-ladders organization of localized structures in SH35 deforms with increasing symmetry breaking and show that the deformation ultimately generates the snakes-and-ladders structure familiar from the quadratic-cubic Swift-Hohenberg equation. Moreover, in nonvariational systems, such as convection, odd-parity convectons necessarily drift when the reflection symmetry is broken, permitting collisions among moving localized structures. Collisions between both identical and nonidentical traveling states are described.

The complex Swift-Hohenberg (CSH) equation is a generic order parameter equation that applies to many physical systems. In the case of class C lasers, it can be obtained from the Maxwell-Bloch equations using the assumptions of slow envelope and small detuning. We show that the resulting CSH equation inevitably contains different asymptotic order terms, associated with the dominance of the effect of dispersion over diffusion. These scale disparities are usually overlooked or simply not mentioned in the literature, assuming that a CSH equation with all terms of the same order still provides qualitative information. In this paper, the asymptotically nonuniform CSH equation is carefully deduced using a simpler scaling-free procedure, and a stability analysis of the simplest solutions together with some numerical simulations are presented, in which the mentioned scale disparities are clearly seen.