I'm sorry if this is an inappropriate forum to ask this question on, for I fear it is pretty undergraduate-level one :) I was contemplating on the study of non-linear PDEs. Is it possible to reduce a non-linear PDE on $\mathbb{R}^n$ to a distribution or a 'good' PDE on a smooth manifold? It seems to me like a natural step, but I don't know anything about it yet :(



Studying non-linear PDEs with manifolds
The question is vague (Denis is right) but it does make some sense. When looking at explicit solutions of nonlinear PDEs one frequently encounters singularities that remind of projections from some higher dimensional manifold (e.g. in Burger's equation). But I think the OP does not have a clear picture. Locally, working 'on' a manifold instead than on $R^n$ is essentially like changing coordinates in the equation. So usually you do not resolve nonlinearities, you just modify the coefficients of the equation. 


On the other hand, if you consider equations with values into a manifold, then sometimes a very difficult nonlinear term reveals to be a very simple geometric object, 'linear' in some sense. This does not happen by chance; your equation must have some geometric meaning. Best examples are harmonic maps (elliptic) and wave maps (hyperbolic). 


Some years ago, Serge Alinhac tried to develop a whole theory of 'geometric blow up': singularities in the solution which can be resolved by choosing an appropriate coordinate sistem in the target space. But I guess the results where not too satisfying since he abandoned the effort.


