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A regularization of the Cross-Newell equation is presented. It is based on asecondary re-modulation along characteristics. This new characteristicCross-Newell equation is not isotropic (has preferred directions), but isuniversal (homogeneous in $\eps$ and equation independent coefficients), hasfourth derivatives, and generates localized solutions that are bi-asymptotic torolls. Re-modulation of rolls in the Ginzburg-Landau equation is used forillustration of the theory.

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