  We give a detailed study of the traveling wave solutions of $(\ell=2)$
Kaup-Boussinesq type of coupled KdV equations. Depending upon the zeros of a
fourth degree polynomial, we have cases where there exist no nontrivial real
solutions, cases where asymptotically decaying to a constant solitary wave
solutions, and cases where there are periodic solutions. All such possible
solutions are given explicitly in the form of Jacobi elliptic functions. Graphs
of some exact solutions in solitary wave and periodic shapes are exhibited.
Extension of our study to the cases $\ell=3$ and $\ell=4$ are also mentioned.
