  We introduce a method of rigorous analysis of the location and type of
complex singularities for nonlinear higher order PDEs as a function of the
initial data. The method is applied to determine rigorously the asymptotic
structure of singularities of the modified Harry-Dym equation $$ H_t + H_y = -
{1/2} H^3 + H^3 H_{yyy} : H(y, 0) = y^{-1/2} $$ for small time at the
boundaries of the sector of analyticity. Previous work \cite{CPAM},
\cite{invent03} shows existence, uniqueness and Borel summability of solutions
of general PDEs. It is shown that the solution to the above initial value
problem is represented convergently by a series in a fractional power of $t$
down to a small annular neighborhood of a singularity of the leading order
equation. We deduce that the exact solution has a singularity nearby having, to
leading order, the same type.
