Periodic and solitary waves in systems of coherently coupled nonlinear envelope equations

Exact solutions for two classes of coherently coupled nonlinear envelope equations are derived in terms of products of Jacobi elliptic functions. Physical applications are illustrated in the context of nonlinear optics, namely, polarization of light beams and quadratic (or parametric) solitons. Stabilities of these double-humped solitary pulses are studied by direct numerical simulations. The use of computer is crucial, both in terms of symbolic manipulation in the derivation process and in the implementation of numerical schemes in stability consideration.