Here is something I've wondered about from time to time: The continental divide in North America is commonly described as the geographic line curve seperating points where a drop of water would drain to the Atlantic from those where it would drain to the Pacific. My question is how to characterize such a curve mathematically given a "reasonable" height function described over a region of the plane. I am not concerned with applied topography but also not interested in exotic pathologies. I'll propose a crude model now but feel free to propose a better one.
MODEL: The domain is the unit disk. A pre-mountain with peak at $(h,k,p)$ is a function $M=M(x,y)=\frac{p}{1+s((x-h)^2+(y-k)^2)}$ where $s>>0$ controls how steep it is and $p>>0$ how high. (note that a sum M_1+M_2 will have local maxima somewhat higher than $p_1$ and $p_2$ and somewhat displaced from $(h_i,k_i)$) The surface will be $b(x,y)(M_1+M_2+\cdots+M_n)$ where the $M_i$ are a large but finite number of pre-mountains and b(x,y) is a function such as $1-x^2$ or $1-x^2-\frac{y^2}{2}$ which is positive except at (-1,0) and (1,0) where it is 0. From each initial point the path of steepest gradient leads somewhere, usually (one might suppose) to $(1,0)$ or $(-1,0).$
Using the crude model as above, or a better one (describe it!) characterize the boundry between the basin of attraction of $(1,0)$ and that of $(-1,0)$
Comments: Of course a ring of mountains could create a pit with a sink in the middle, but that can be ignored or the problem can be changed to "characterize the boundries of the various basins of attraction". At a peak or saddle point the gradient is 0 but usually any direction one goes leads to the same sink. I imagine that there are (useful) applied approximate solutions starting from a grid of sample points with edges joining nearest neighbors. But I'd like some kind of minimax description like the solution of a continuous linear programing problem.
