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

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 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.