"The coupling of Green-Naghdi and Nonlinear Shallow water equations provides an attractive parameter-free formulation for dispersive water-wave propagation and wave breaking. However, so far, instabilities have been observed in current formulations, as the numerical resolution is increased. In this talk I will present how we can study this issue from the perspective of domain decomposition methods (DDM). For that I will introduce DDM by using the coupling of the BBM and transport equations as an example. We will discuss the importance of using the right boundary conditions for the coupling, most importantly the so called absorbing or transparent boundary conditions, and what we can learn from this ""toy"" model that can be useful for coupling dispersive and nondipersive (hyperbolic) models in more complex settings."