Part I
We examine a nonlinear diffusion equation that arises in the study of a number of physical problems, where the equation is nonlinear because the diffusion coefficient is proportional to a power of the concentration. Previous authors have proven using similarity solutions, that this dependence produces fronts (interfaces between regions of zero and nonzero concentration) which propagate with finite speed, as well as waiting-time behavior, where the fronts remain stationary for a finite amount of time before beginning to move. These similarity solutions provide limited information about the solution for general initial conditions, however.
To alleviate this deficiency, we construct approximate solutions for the above nonlinear diffusion equation using singular perturbation theory. We do so by considering the equation in the limit of nearly linear diffusion, but the analysis reveals the basic qualitative behavior outside this limit as well.
The basic behavior follows from the leading-order approximation of a transformed equation, and propagating and waiting fronts are due to the formation (in this approximation) of what we call corner shocks. This enables us to determine for which initial conditions waiting time behavior will occur.
The transformed equation must be solved to first order to find the solution of the original equation to leading order, and when corner shocks occur at a point of nonzero concentration, this first order analysis shows that they become rounded (which we call a corner layer). When a corner shock occurs at a point of zero concentration, this rounding does not take place, and the corner shock remains sharp. This allows us to give a simple procedure for constructing approximate solutions of the nonlinear diffusion equation when corner shocks occur only at points of zero concentration.
Part II
We study a model of reentry roll resonance, a situation encountered when an almost axially symmetric vehicle reenters the earth's atmosphere, using the method of near identity transformations. This method allows us to extend previous results for the case of sustained resonance, when roll buildup occurs.
In particular, we give necessary conditions both for entrainmeat to sustained resonance and for sustained resonance to continue. These conditions imply that it is possible for sustained resonance to last for a finite time and then for unlocking of the resonance to occur. In addition, from the analysis we make a conjecture concerning sufficient conditions for sustained resonance.
