Multi-bump patterns by a normal form approach

In this paper the behaviour of small solutions in a reaction-diffusion model problem is studied near a co-dimension 2 point. The normal form theory for reversible vector fields is applied on the stationary part of the reaction- diffusion system. This normal form is reduced to a 3-dimensional ODE that is completely integrable. An explicit expression for the solutions to the ODE and therefore for the reaction-diffusion system is given under certain conditions. These solutions have the same multi-bump pattern as the asymptotically stable stationary multi-bump solutions that were found in the numerical simulations of the full reaction-diffusion system.