  In this paper we prove the well-posedness issues of the associated initial
value problem, the existence of nontrivial solutions with prescribed
$L^2$-norm, and the stability of associated solitary waves for two classes of
coupled nonlinear dispersive equations. The first problem here describes the
nonlinear interaction between two Schr\"{o}dinger type short waves and a
generalized Korteweg-de Vries type long wave and the second problem describes
the nonlinear interaction of two generalized Korteweg-de Vries type long waves
with a common Schr\"{o}dinger type short wave. The results here extend many of
the previously obtained results for two-component coupled
Schr\"{o}dinger-Korteweg-de Vries systems.
