Solutions to the optical cascading equations

Group theoretical methods are used to study the equations describing ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}:{\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ cascading. The equations are shown not to be integrable by inverse scattering techniques. On the other hand, these equations do share some of the nice properties of soliton equations. Large families of explicit analytical solutions are obtained in terms of elliptic functions. In special cases, these periodic solutions reduce to localized ones, i.e., solitary waves. All previously known explicit solutions are recovered, and many additional ones are obtained.