Similarity and difference in the dynamics of plane and 3-dimensional non-linear waves

The method of quasinormal forms is used to demonstrate the mild loss of stability by travelling waves on a circle and a torus. The dynamics in the case of a small elastic-constraint coefficient is studied in a model example. The results obtained and an explicit difference scheme developed expressly for the problem are used in a series of numerical experiments. As a result one distinguishes a subdomain of the parameter space in which the most important features of the dynamics on a torus and a circle are related to self-organisation regimes, where self-organisation regimes on a circle are versions of certain similar regimes on a torus.