Harmonic oscillator states as oscillating localized structures near Hopf-Turing instability boundary

A set of coupled complex Ginzburg-landau type amplitude equations which operates near a Hopf-Turing instability boundary is analytically investigated to show localized oscillatory patterns. The spatial structure of those patterns are the same as quantum mechanical harmonic oscillator stationary states and can have even or odd symmetry depending on the order of the state. It has been seen that the underlying Turing state plays a major role in the selection of the order of such solutions.