  In this manuscript, consideration is given to the existence of periodic
traveling-wave solutions to the $abcd$-system. This system was derived by Bona,
Saut, and Chen to describe small amplitude, long wavelength gravity waves on
the surface of water. These exact solutions are formulated in terms of the
Jacobi elliptic function cnoidal. The existence of explicit traveling-wave
solutions is very useful in theoretical investigations such as stability of
solutions, as well as other numerical analysis of the system.
