Our goal is to find a macroscopic description of patterns that both un
ifies and simplifies classes of externally stressed, dissipative, patt
ern forming systems, such as convecting fluids, liquid crystals, wideb
and lasers, that are seemingly unrelated at the microscopic level. We
construct an order parameter equation which provides a controlled appr
oximation of the original microscopic field in the limit of large aspe
ct ratios. It is built from, and is a regularization of, the Cross-New
ell phase diffusion equation obtained by averaging over the local peri
odicity of the pattern. Unlike the latter, it is valid for all wavenum
bers and can correctly capture the nucleation, shape and nontrivial pr
operties of the far fields of disclinations, dislocations and grain bo
undaries. It reduces to the Cross-Newell equation away from pattern si
ngularities and to the Newell-Whitehead-Segel equation near onset. As
a consequence, it correctly determines all the long wave instability b
oundaries (zig-zag, Eckhaus-skew-varicose) of the Busse balloon. Far f
rom onset, the order parameter is a real variable but its equation inv
olves a functional corresponding to its local amplitude. The local amp
litude and phase, required for the order parameter equation and the re
construction of the approximation to the original field respectively,
are extracted from the order parameter field by wavelet analysis. Nume
rical comparisons between solutions of the original equation and the r
egularized equation are carried out. We also explore a new class of si
ngular and weak solutions of the Cross-Newell equation which take acco
unt of the energetics of defects as well as their topologies. These so
lutions correspond to convex and concave disclinations and their compo
sites, including saddles, vortices, targets, dislocations and two new
objects, handles and bridges. Finally, we show that phase grain bounda
ries, lines across which the wavevector is discontinuous but the phase
is continuous are captured by shock solutions of the phase diffusion
equation.