Cross-Newell Phase Equations

Cross-Newell equations describe the slow modulation of stripe/roll patterns near onset. The phase Φ(x,y,t) satisfies: τ(k)∂_t Φ = −∇·[k B(k²) k̂] where k=∇Φ is the local wavevector. This has two instabilities: Eckhaus (longitudinal compression/dilation) and zigzag (transverse bending). Defects are dislocations — points where Φ is singular and the pattern gains/loses one roll. Grain boundaries separate domains of different orientation. The Newell-Whitehead-Segel equation is the 1D amplitude limit; Cross-Newell captures 2D phase topology.