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

In pattern-forming systems, localized patterns are readily found when stable patterns exist at the same parameter values as the stable unpatterned state. Oscillons are spatially localized, time-periodic structures, which have been found experimentally in systems that are driven by a time-periodic force, for example, in the Faraday wave experiment. This paper examines the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, int...

Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [20] as phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the forced complex Ginzburg-Landau equation in the limit of weak forcing and weak dampin...

We study the excitation of spatial patterns by resonant, multi-frequency forcing in systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. Using weakly nonlinear analysis we show that for small amplitudes only stripe or hexagon patterns are linearly stable, whereas square patterns and patterns involving more than three modes are unstable. In the case of hexagon patterns up- and down-hexagons can be simultaneously stable. The third-order, weakly nonlinear...

Quasipatterns have been found in dissipative systems ranging from Faraday waves in vertically vibrated fluid layers to nonlinear optics. We describe the dynamics of octagonal, decagonal and dodecagonal quasipatterns by means of coupled Ginzburg-Landau equations and study their stability to sideband perturbations analytically using long-wave equations as well as by direct numerical simulation. Of particular interest is the influence of the phason modes, which are associated wi...

We investigate the role weakly damped modes play in the selection of Faraday wave patterns forced with rationally-related frequency components m*omega and n*omega. We use symmetry considerations to argue for the special importance of the weakly damped modes oscillating with twice the frequency of the critical mode, and those oscillating primarily with the "difference frequency" |n-m|*omega and the "sum frequency" (n+m)*omega. We then perform a weakly nonlinear analysis using ...

Many of the interesting patterns seen in recent multi-frequency Faraday experiments can be understood on the basis of three-wave interactions (resonant triads). In this paper we consider two-frequency forcing and focus on a resonant triad that occurs near the bicritical point where two pattern-forming modes with distinct wavenumbers emerge simultaneously. This triad has been observed directly (in the form of rhomboids) and has also been implicated in the formation of quasipat...

Spontaneous pattern formation in a variety of spatially extended nonlinear system always occurs through a modulation instability: homogeneous state of the system becomes unstable with respect to growing modulation modes. Therefore, the manipulation of the modulation instability is of primary importance in controlling and manipulating the character of spatial patterns initiated by that instability. We show that the spatio-temporal periodic modulation of the potential of the sp...

We show how pattern formation in Faraday waves may be manipulated by varying the harmonic content of the periodic forcing function. Our approach relies on the crucial influence of resonant triad interactions coupling pairs of critical standing wave modes with damped, spatio-temporally resonant modes. Under the assumption of weak damping and forcing, we perform a symmetry-based analysis that reveals the damped modes most relevant for pattern selection, and how the strength of ...

In systems that exhibit a bistability between nonlinear traveling waves and the basic state, pairs of fronts connecting these two states can form localized wave pulses whose stability depends on the interaction between the fronts. We investigate the localization of waves within the framework of coupled Ginzburg-Landau equations describing the traveling-wave amplitudes. We find that the introduction of resonant temporal forcing results in a new, tunable mechanism for stabilizi...

Through multiple-scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady-state pattern-forming instabilities, in the case that the wavelengths of the instabilities are nearly in the ratio 1:2. In the case of exact 1:2 resonance the amplitude equations are ODEs; here they are PDEs. We discuss the stability of spatially-periodic solutions to long-wavelength disturbances. By including these modulational effects we are...