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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 afourth degree polynomial, we have cases where there exist no nontrivial realsolutions, cases where asymptotically decaying to a constant solitary wavesolutions, and cases where there are periodic solutions. All such possiblesolutions are given explicitly in the form of Jacobi elliptic functions. Graphsof 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.

Traveling Wave Solutions of Degenerate Coupled KdV Equation