Regularizing the Cross-Newell equation by re-modulating along a characteristic angle

A regularization of the Cross-Newell equation is presented. It is based on a secondary re-modulation along characteristics. This new characteristic Cross-Newell equation is not isotropic (has preferred directions), but is universal (homogeneous in $\text{eps}$ and equation independent coefficients), has fourth derivatives, and generates localized solutions that are bi-asymptotic to rolls. Re-modulation of rolls in the Ginzburg-Landau equation is used for illustration of the theory.