Complex Swift-Hohenberg Equation Research Articles

Worm structure in the modified Swift-Hohenberg equation for electroconvection

An anisotropic complex Swift-Hohenberg equation is proposed to study pattern formation in electroconvection. In the subcritical regime, a localized state is found in two dimensions, which resembles the ``worm'' state observed in recent experiment by M. Dennin et al. [Phys. Rev. Lett. 77, 2475 (1996); Science 272, 388 (1996)]. In the corresponding one-dimensional model, a stationary pulse state is discovered, due to a nonadiabatic effect, and it is shown to explain the localization of the ``worm'' state in the two-dimensional model. Based on these results, we believe that the initial bifurcation should be subcritical where the ``worm'' state is observed, and further experiment is suggested to test this scenario.

Standing Wave Patterns for the Complex Swift-Hohenberg Equation

Roll-type standing wave patterns appear as a dissipative structure in the complex Swift-Hohenberg equation. The zigzag-type phase instability and the domain wall motion of standing wave patterns are investigated.

Controlling optical turbulence

A robust global control strategy, implemented as a spatial filter with delayed feedback, is shown to stabilize and steer the weakly turbulent output of a spatially extended system. The latter is described by a generalized complex Swift-Hohenberg equation [J. Lega, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994); Physica D 83, 478 (1995)], which is used as a generic model for pattern formation in the transverse section of semiconductor lasers. Our technique is particularly adapted to optical systems and should provide convenient experimental control of filamentation in wide-aperture lasers.

Swift-Hohenberg equation for optical parametric oscillators.

Pattern formation near threshold in single longitudinal mode and large aspect ratio optical parametric oscillators (OPOs) operating near resonance are shown to be described by a complex Swift-Hohenberg equation. Such an equation is capable of capturing the main wave-vector selection rules for both degenerate and nondegenerate OPOs. \textcopyright{} 1996 The American Physical Society.

Domain walls in wave patterns.

We study the interaction of counterpropagating traveling waves in 2D nonequilibrium media described by the complex Swift-Hohenberg equation (CSHE). Direct numerical integration of CSHE reveals novel features of domain walls separating wave systems: wave-vector selection and transverse instability. Analytical treatment is based on a study of coupled complex Ginzburg-Landau equations for counterpropagating waves. At the threshold we find the stationary (yet unstable) solution corresponding to the selected waves. It is shown that sources of traveling waves exhibit long wavelength instability, whereas sinks remain stable. An analogy with the Kelvin-Helmholtz instability is established. {copyright} {ital 1995} {ital The} {ital American} {ital Physical} {ital Society}.

Universal description of laser dynamics near threshold

Abstract Complex order parameter descriptions of large aspect ratio, single longitudinal mode, two-level lasers with flat end reflectors, valid near onset of lasing and for small detunings of the laser from the peak gain, are given in terms of a complex Swift-Hohenberg equation for Class A and C lasers and by a complex Swift-Hohenberg equation coupled to a mean flow for the case of a Class B laser. The latter coupled system is a physically consistent generalized rate equation model for wide aperture stiff laser systems. These universal order parameter equations provide a connection between spatially homogeneous oscillating states of the complex Ginzburg-Landau equation description of the laser system valid for finite negative detunings, and traveling wave states, described by coupled Newell-Whitehead-Segel equations valid for finite positive detunings. One of the main conclusions of the present paper is that the usual Eckhaus instability boundary associated with a long wavelength phase instability, and which delineates the region of the stable traveling wave solutions for Class A and C lasers, no longer defines the stability boundary for the mathematically stiff Class B laser. Instead a short wavelength phase instability appears causing the stability domain to shrink as a function of increasing stiffness of the system. This prediction is consistent with the strong spatiotemporal filamentation instabilities experimentally observed in a borad area semiconductor laser, a Class B system.

Swift-Hohenberg equation for lasers.

Pattern formation in large aspect ratio, single longitudinal mode, two-level lasers with flat end reflectors, operating near peak gain, is shown to be described by a complex Swift-Hohenberg equation for class A and C lasers and by a complex Swift-Hohenberg equation coupled to a mean flow for the case of a class B laser.