Stability and Symmetry of Solitary-wave Solutions to Systems Modeling Interactions of Long Waves

We consider systems of equations which arise in modelling strong interactions of weakly nonlinear long waves in dispersive media. For a certain class of such systems, we prove the existence and stability of localized solutions representing coupled solitary waves travelling at a common speed. Our results apply in particular to the systems derived by Gear and Grimshaw and by Liu, Kubota, and Ko as models for interacting gravity waves in a density-stratified fluid. For the latter system, we also prove that any coupled solitary-wave solution must have components which are all symmetric about a common vertical axis.