Evolution and self formation in some nonlinear systems

The evolution of plasma temperature profiles and the formation of equilibria for such profiles, governed by reaction-diffusion equations, depend on the simultaneous processes of diffusion and creation, e.g. heating, as well as losses, e.g. bremsstrahlung, and furthermore on the effects of boundaries, which play an essential role in the formation of equilibria. A nonlinear diffusion equation including source and loss terms (which are also nonlinear) is studied by means of : (i) a time-dependent central approximation technique, and (ii) direct numerical integration of the nonlinear partial differential equation. The two methods are compared and are found to be in excellent agreement numerically. It is concluded that the methods may be regarded as complementary for the understanding and description of the results. Applications are found, e.g. in the field of fusion reactor plasmas, for which one notices the interesting fact that several different equilibria may occur for the same set of parameter values in the original equation.Wilhelmsson Hans. Evolution and self formation in some nonlinear systems. In: Bulletin de la Classe des sciences, tome 1, 1990. pp. 269-287