  Using Painlev\'e analysis, the Hirota multi-linear method and a direct ansatz
technique, we study analytic solutions of the (1+1)-dimensional complex cubic
and quintic Swift-Hohenberg equations. We consider both standard and
generalized versions of these equations. We have found that a number of exact
solutions exist to each of these equations, provided that the coefficients are
constrained by certain relations. The set of solutions include particular types
of solitary wave solutions, hole (dark soliton) solutions and periodic
solutions in terms of elliptic Jacobi functions and the Weierstrass $\wp$
function. Although these solutions represent only a small subset of the large
variety of possible solutions admitted by the complex cubic and quintic
Swift-Hohenberg equations, those presented here are the first examples of exact
analytic solutions found thus far.
