Near-Resonant Steady Mode Interaction
Maria Higuera, Hermann Riecke, Mary Silber
This research was funded by the National Science Foundation
Motivated by the rich variety of complex periodic and quasi-periodic
patterns found in systems such as two-frequency forced Faraday waves,
we study the interaction of two spatially periodic modes that are
nearly resonant. Within
the framework of two coupled one-dimensional Ginzburg-Landau equations
we investigate analytically the stability of the periodic solutions to general
perturbations, including perturbations that do not respect the
periodicity of the pattern, and which may lead to quasi-periodic solutions. We study the
impact of the deviation from exact resonance on the destabilizing
modes and on the final states. In regimes in which the mode
interaction leads to the existence of traveling waves
our numerical simulations reveal
localized waves in which the wavenumbers are resonant and which
drift through a steady background pattern that has an off-resonant wavenumber
ratio.
The weakly nonlinear amplitude equations for two modes with nearly resonating
wave numbers can be written as
Example of a stability diagrams for pure, periodic solution (A1=0)
and for mixed-mode solution (both amplitudes non-zero)
Localized parity-breaking waves collide with localized domains with different wave numbers and eventually are absorbed by them.
The dotted line in b) indicates a wave number ratio of 2:5.
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