Near-Resonant, Steady Mode Interaction: Periodic, Quasi-Periodic and Localized Patterns

Stationary periodic patterns are widespread in natural sciences, ranging from nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical and biological media and to macro-scale vegetation and cloud patterns. Their formation is usually due to a primary symmetry breaking of a uniform state to stripes, often followed by secondary instabilities to form zigzag and labyrinthine patterns. These secondary instabilities are well studied under idealized conditions ...

Two-dimensional spatially localized structures in the complex Ginzburg-Landau equation with 1:1 resonance are studied near the simultaneous occurrence of a steady front between two spatially homogeneous equilibria and a supercritical Turing bifurcation on one of them. The bifurcation structures of steady circular fronts and localized target patterns are computed in the Turing-stable and Turing-unstable regimes. In particular, localized target patterns grow along the solution ...

Recent experiments (Kudrolli, Pier and Gollub, 1998) on two-frequency parametrically excited surface waves exhibit an intriguing "superlattice" wave pattern near a codimension-two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wavenumbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Similar patterns have be...

We propose that parametrically excited patterns, also known as Faraday patterns, can be observed in nonlinear fiber resonators, where the coefficient of Kerr nonlinearity is periodically varying along the fiber in resonator. We study the parametric instability analytically on the basis of the Floquet theory, also numerically, by direct integration of the system. Instead of classical Faraday wave excitation scenario, where modulation in time causes formation of patterns in spa...

Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg-Landau equations with nonlinearities of lambda-omega type and establish the existence of on-site and off-site solutions in the case of weak coupling and weak-amplitude dependence of the oscillator periods. Utilizing a core-far-field decomposition, we demonstrate that i...

We study the coupled complex Ginzburg-Landau (CGL) equations for traveling wave systems in the regime where sources and sinks separate patches of left and right-traveling waves. We show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems. We present in detail the framework to analyze these coherent structures. Our counting arguments for the multiplicities of these structures show that independen...

The generalized Swift--Hohenberg equation with a quadratic-cubic nonlinearity is used to study the persistence and decay of localized patterns in the presence of time-periodic parametric forcing. A novel resonance phenomenon between the forcing period and the time required to nucleate one wavelength of the pattern outside the pinning region is identified. The resonance generates distinct regions in parameter space characterized by the net number of wavelengths gained or lost ...

Regular spatial structures emerge in a wide range of different dynamics characterized by local and/or nonlocal coupling terms. In several research fields this has spurred the study of many models, which can explain pattern formation. The modulations of patterns, occurring on long spatial and temporal scales, can not be captured by linear approximation analysis. Here, we show that, starting from a general model with long range couplings displaying patterns, the spatio-temporal...

We analyze pattern-formation scenarios in the two-dimensional (2D) complex Ginzburg-Landau (CGL) equation with the cubic-quintic (CQ) nonlinearity and a cellular potential. The equation models laser cavities with built-in gratings, which are used to stabilize 2D patterns. The pattern-building process is initiated by kicking a localized compound mode, in the form of a dipole, quadrupole, or vortex which is composed of four local peaks. The hopping motion of the kicked mode thr...

We exploit the presence of approximate (broken) symmetries to obtain general scaling laws governing the process of pattern formation in weakly damped Faraday waves. Specifically, we consider a two-frequency forcing function and trace the effects of time translation, time reversal and Hamiltonian structure for three illustrative examples: hexagons, two-mode superlattices, and two-mode rhomboids. By means of explicit parameter symmetries, we show how the size of various three-w...