Euclidean invariant phase dynamics for propagating pattern

The Euclidean invariant phase dynamics for propagating spatially modulated structures is considered. The phase dynamics equation of motion is obtained in a general way for sufficiently smooth deformation of a striped pattern. The amplitude equation for a propagating pattern is also derived near a bifurcation threshold where the external parameter can contain a component oscillating in time. The theory is applied to the oscillatory patterns described by the amplitude equation. The new stability criterions for a propagating wave and a nonlinear standing wave are also presented.