The generation of instability in inviscid, non-diffusive geophysical flows is generically caused by a resonance between two wave modes. The weakly nonlinear unfolding of this situation is described in the long-wave regime, using a particular two-layer quasi-geostrophic model as an illustrative example. The outcome is a system of two coupled Korteweg-de Vries equations. This system contains a very rich solution set, consisting typically of solitary wave interactions. We will describe some numerical solutions of the coupled Korteweg-de Vries equations, supplemented by perturbation analyses. We also report on some preliminary analogous numerical simulations of the full two-layer quasi-geostrophic system.